Numerical solutions for Helmholtz equations using Bernoulli polynomials

NASA Astrophysics Data System (ADS)

Bicer, Kubra Erdem; Yalcinbas, Salih

2017-07-01

This paper reports a new numerical method based on Bernoulli polynomials for the solution of Helmholtz equations. The method uses matrix forms of Bernoulli polynomials and their derivatives by means of collocation points. Aim of this paper is to solve Helmholtz equations using this matrix relations.

Numerical analysis for distributed-order differential equations

NASA Astrophysics Data System (ADS)

Diethelm, Kai; Ford, Neville J.

2009-03-01

In this paper we present and analyse a numerical method for the solution of a distributed-order differential equation of the general form where m is a positive real number and where the derivative is taken to be a fractional derivative of Caputo type of order r. We give a convergence theory for our method and conclude with some numerical examples.

GLOBAL PROPERTIES OF FULLY CONVECTIVE ACCRETION DISKS FROM LOCAL SIMULATIONS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bodo, G.; Ponzo, F.; Rossi, P.

2015-08-01

We present an approach to deriving global properties of accretion disks from the knowledge of local solutions derived from numerical simulations based on the shearing box approximation. The approach consists of a two-step procedure. First, a local solution valid for all values of the disk height is constructed by piecing together an interior solution obtained numerically with an analytical exterior radiative solution. The matching is obtained by assuming hydrostatic balance and radiative equilibrium. Although in principle the procedure can be carried out in general, it simplifies considerably when the interior solution is fully convective. In these cases, the construction ismore » analogous to the derivation of the Hayashi tracks for protostars. The second step consists of piecing together the local solutions at different radii to obtain a global solution. Here we use the symmetry of the solutions with respect to the defining dimensionless numbers—in a way similar to the use of homology relations in stellar structure theory—to obtain the scaling properties of the various disk quantities with radius.« less

A mathematical solution for the parameters of three interfering resonances

NASA Astrophysics Data System (ADS)

Han, X.; Shen, C. P.

2018-04-01

The multiple-solution problem in determining the parameters of three interfering resonances from a fit to an experimentally measured distribution is considered from a mathematical viewpoint. It is shown that there are four numerical solutions for a fit with three coherent Breit-Wigner functions. Although explicit analytical formulae cannot be derived in this case, we provide some constraint equations between the four solutions. For the cases of nonrelativistic and relativistic Breit-Wigner forms of amplitude functions, a numerical method is provided to derive the other solutions from that already obtained, based on the obtained constraint equations. In real experimental measurements with more complicated amplitude forms similar to Breit-Wigner functions, the same method can be deduced and performed to get numerical solutions. The good agreement between the solutions found using this mathematical method and those directly from the fit verifies the correctness of the constraint equations and mathematical methodology used. Supported by National Natural Science Foundation of China (NSFC) (11575017, 11761141009), the Ministry of Science and Technology of China (2015CB856701) and the CAS Center for Excellence in Particle Physics (CCEPP)

Numerical solution of second order ODE directly by two point block backward differentiation formula

NASA Astrophysics Data System (ADS)

Zainuddin, Nooraini; Ibrahim, Zarina Bibi; Othman, Khairil Iskandar; Suleiman, Mohamed; Jamaludin, Noraini

2015-12-01

Direct Two Point Block Backward Differentiation Formula, (BBDF2) for solving second order ordinary differential equations (ODEs) will be presented throughout this paper. The method is derived by differentiating the interpolating polynomial using three back values. In BBDF2, two approximate solutions are produced simultaneously at each step of integration. The method derived is implemented by using fixed step size and the numerical results that follow demonstrate the advantage of the direct method as compared to the reduction method.

Numerical solution for the velocity-derivative skewness of a low-Reynolds-number decaying Navier-Stokes flow

NASA Technical Reports Server (NTRS)

Deissler, Robert G.

1990-01-01

The variation of the velocity-derivative skewness of a Navier-Stokes flow as the Reynolds number goes toward zero is calculated numerically. The value of the skewness, which has been somewhat controversial, is shown to become small at low Reynolds numbers.

Numerical solution of the exterior oblique derivative BVP using the direct BEM formulation

NASA Astrophysics Data System (ADS)

Čunderlík, Róbert; Špir, Róbert; Mikula, Karol

2016-04-01

The fixed gravimetric boundary value problem (FGBVP) represents an exterior oblique derivative problem for the Laplace equation. A direct formulation of the boundary element method (BEM) for the Laplace equation leads to a boundary integral equation (BIE) where a harmonic function is represented as a superposition of the single-layer and double-layer potential. Such a potential representation is applied to obtain a numerical solution of FGBVP. The oblique derivative problem is treated by a decomposition of the gradient of the unknown disturbing potential into its normal and tangential components. Our numerical scheme uses the collocation with linear basis functions. It involves a triangulated discretization of the Earth's surface as our computational domain considering its complicated topography. To achieve high-resolution numerical solutions, parallel implementations using the MPI subroutines as well as an iterative elimination of far zones' contributions are performed. Numerical experiments present a reconstruction of a harmonic function above the Earth's topography given by the spherical harmonic approach, namely by the EGM2008 geopotential model up to degree 2160. The SRTM30 global topography model is used to approximate the Earth's surface by the triangulated discretization. The obtained BEM solution with the resolution 0.05 deg (12,960,002 nodes) is compared with EGM2008. The standard deviation of residuals 5.6 cm indicates a good agreement. The largest residuals are obviously in high mountainous regions. They are negative reaching up to -0.7 m in Himalayas and about -0.3 m in Andes and Rocky Mountains. A local refinement in the area of Slovakia confirms an improvement of the numerical solution in this mountainous region despite of the fact that the Earth's topography is here considered in more details.

A 1D radiative transfer benchmark with polarization via doubling and adding

NASA Astrophysics Data System (ADS)

Ganapol, B. D.

2017-11-01

Highly precise numerical solutions to the radiative transfer equation with polarization present a special challenge. Here, we establish a precise numerical solution to the radiative transfer equation with combined Rayleigh and isotropic scattering in a 1D-slab medium with simple polarization. The 2-Stokes vector solution for the fully discretized radiative transfer equation in space and direction derives from the method of doubling and adding enhanced through convergence acceleration. Updates to benchmark solutions found in the literature to seven places for reflectance and transmittance as well as for angular flux follow. Finally, we conclude with the numerical solution in a partially randomly absorbing heterogeneous medium.

An explicit closed-form analytical solution for European options under the CGMY model

NASA Astrophysics Data System (ADS)

Chen, Wenting; Du, Meiyu; Xu, Xiang

2017-01-01

In this paper, we consider the analytical pricing of European path-independent options under the CGMY model, which is a particular type of pure jump Le´vy process, and agrees well with many observed properties of the real market data by allowing the diffusions and jumps to have both finite and infinite activity and variation. It is shown that, under this model, the option price is governed by a fractional partial differential equation (FPDE) with both the left-side and right-side spatial-fractional derivatives. In comparison to derivatives of integer order, fractional derivatives at a point not only involve properties of the function at that particular point, but also the information of the function in a certain subset of the entire domain of definition. This ;globalness; of the fractional derivatives has added an additional degree of difficulty when either analytical methods or numerical solutions are attempted. Albeit difficult, we still have managed to derive an explicit closed-form analytical solution for European options under the CGMY model. Based on our solution, the asymptotic behaviors of the option price and the put-call parity under the CGMY model are further discussed. Practically, a reliable numerical evaluation technique for the current formula is proposed. With the numerical results, some analyses of impacts of four key parameters of the CGMY model on European option prices are also provided.

The origin of spurious solutions in computational electromagnetics

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Wu, Jie; Povinelli, L. A.

1995-01-01

The origin of spurious solutions in computational electromagnetics, which violate the divergence equations, is deeply rooted in a misconception about the first-order Maxwell's equations and in an incorrect derivation and use of the curl-curl equations. The divergence equations must be always included in the first-order Maxwell's equations to maintain the ellipticity of the system in the space domain and to guarantee the uniqueness of the solution and/or the accuracy of the numerical solutions. The div-curl method and the least-squares method provide rigorous derivation of the equivalent second-order Maxwell's equations and their boundary conditions. The node-based least-squares finite element method (LSFEM) is recommended for solving the first-order full Maxwell equations directly. Examples of the numerical solutions by LSFEM for time-harmonic problems are given to demonstrate that the LSFEM is free of spurious solutions.

Numerical Solution of the Electron Transport Equation in the Upper Atmosphere

DOE Office of Scientific and Technical Information (OSTI.GOV)

Woods, Mark Christopher; Holmes, Mark; Sailor, William C

A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplifieed problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other to at least two significant digits.

Application of fractional derivative with exponential law to bi-fractional-order wave equation with frictional memory kernel

NASA Astrophysics Data System (ADS)

Cuahutenango-Barro, B.; Taneco-Hernández, M. A.; Gómez-Aguilar, J. F.

2017-12-01

Analytical solutions of the wave equation with bi-fractional-order and frictional memory kernel of Mittag-Leffler type are obtained via Caputo-Fabrizio fractional derivative in the Liouville-Caputo sense. Through the method of separation of variables and Laplace transform method we derive closed-form solutions and establish fundamental solutions. Special cases with homogeneous Dirichlet boundary conditions and nonhomogeneous initial conditions, as well as for the external force are considered. Numerical simulations of the special solutions were done and novel behaviors are obtained.

Approximate Solutions for Ideal Dam-Break Sediment-Laden Flows on Uniform Slopes

NASA Astrophysics Data System (ADS)

Ni, Yufang; Cao, Zhixian; Borthwick, Alistair; Liu, Qingquan

2018-04-01

Shallow water hydro-sediment-morphodynamic (SHSM) models have been applied increasingly widely in hydraulic engineering and geomorphological studies over the past few decades. Analytical and approximate solutions are usually sought to verify such models and therefore confirm their credibility. Dam-break flows are often evoked because such flows normally feature shock waves and contact discontinuities that warrant refined numerical schemes to solve. While analytical and approximate solutions to clear-water dam-break flows have been available for some time, such solutions are rare for sediment transport in dam-break flows. Here we aim to derive approximate solutions for ideal dam-break sediment-laden flows resulting from the sudden release of a finite volume of frictionless, incompressible water-sediment mixture on a uniform slope. The approximate solutions are presented for three typical sediment transport scenarios, i.e., pure advection, pure sedimentation, and concurrent entrainment and deposition. Although the cases considered in this paper are not real, the approximate solutions derived facilitate suitable benchmark tests for evaluating SHSM models, especially presently when shock waves can be numerically resolved accurately with a suite of finite volume methods, while the accuracy of the numerical solutions of contact discontinuities in sediment transport remains generally poorer.

Analytical and numerical solution for wave reflection from a porous wave absorber

NASA Astrophysics Data System (ADS)

Magdalena, Ikha; Roque, Marian P.

2018-03-01

In this paper, wave reflection from a porous wave absorber is investigated theoretically and numerically. The equations that we used are based on shallow water type model. Modification of motion inside the absorber is by including linearized friction term in momentum equation and introducing a filtered velocity. Here, an analytical solution for wave reflection coefficient from a porous wave absorber over a flat bottom is derived. Numerically, we solve the equations using the finite volume method on a staggered grid. To validate our numerical model, comparison of the numerical reflection coefficient is made against the analytical solution. Further, we implement our numerical scheme to study the evolution of surface waves pass through a porous absorber over varied bottom topography.

A new approach to exact optical soliton solutions for the nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Morales-Delgado, V. F.; Gómez-Aguilar, J. F.; Baleanu, Dumitru

2018-05-01

By using the modified homotopy analysis transform method, we construct the analytical solutions of the space-time generalized nonlinear Schrödinger equation involving a new fractional conformable derivative in the Liouville-Caputo sense and the fractional-order derivative with the Mittag-Leffler law. Employing theoretical parameters, we present some numerical simulations and compare the solutions obtained.

Analysis and testing of numerical formulas for the initial value problem

NASA Technical Reports Server (NTRS)

Brown, R. L.; Kovach, K. R.; Popyack, J. L.

1980-01-01

Three computer programs for evaluating and testing numerical integration formulas used with fixed stepsize programs to solve initial value systems of ordinary differential equations are described. A program written in PASCAL SERIES, takes as input the differential equations and produces a FORTRAN subroutine for the derivatives of the system and for computing the actual solution through recursive power series techniques. Both of these are used by STAN, a FORTRAN program that interactively displays a discrete analog of the Liapunov stability region of any two dimensional subspace of the system. The derivatives may be used by CLMP, a FORTRAN program, to test the fixed stepsize formula against a good numerical result and interactively display the solutions.

Expressions of the fundamental equation of gradient elution and a numerical solution of these equations under any gradient profile.

PubMed

Nikitas, P; Pappa-Louisi, A

2005-09-01

The original work carried out by Freiling and Drake in gradient liquid chromatography is rewritten in the current language of reversed-phase liquid chromatography. This allows for the rigorous derivation of the fundamental equation for gradient elution and the development of two alternative expressions of this equation, one of which is free from the constraint that the holdup time must be constant. In addition, the above derivation results in a very simple numerical solution of the various equations of gradient elution under any gradient profile. The theory was tested using eight catechol-related solutes in mobile phases modified with methanol, acetonitrile, or 2-propanol. It was found to be a satisfactory prediction of solute gradient retention behavior even if we used a simple linear description for the isocratic elution of these solutes.

Numerical analysis of the transient response of an axisymmetric ablative char layer considering internal flow effects

NASA Technical Reports Server (NTRS)

Pittman, C. M.; Howser, L. M.

1972-01-01

The differential equations governing the transient response of the char layer of an ablating axisymmetric body, internal pyrolysis gas flow effects being considered, have been derived. These equations have been expanded into finite difference form and programed for numerical solution on a digital computer. Numerical results compare favorably with simplified exact solutions. The complete numerical analysis was used to obtain solutions for two representative body shapes subjected to a typical entry heating environment. Pronounced effects of the lateral flow of pyrolysis gases on the mass flow field within the char layer and the associated surface and pyrolysis interface recession rates are shown.

Structural, thermodynamic, and electrical properties of polar fluids and ionic solutions on a hypersphere: Theoretical aspects

NASA Astrophysics Data System (ADS)

Caillol, J. M.

1992-01-01

We generalize previous work [J. Chem. Phys. 94, 597 (1991)] on an alternative to the Ewald method for the numerical simulations of Coulomb fluids. This new method consists in using as a simulation cell the three-dimensional surface of a four-dimensional sphere, or hypersphere. Here, we consider the case of polar fluids and electrolyte solutions. We derive all the formal expressions which are needed for numerical simulations of such systems. It includes a derivation of the multipolar interactions on a hypersphere, the expansion of the pair-correlation functions on rotational invariants, the expression of the static dielectric constant of a polar liquid, the expressions of the frequency-dependent conductivity and dielectric constant of an ionic solution, and the derivation of the Stillinger-Lovett sum rules for conductive systems.

On mixed derivatives type high dimensional multi-term fractional partial differential equations approximate solutions

NASA Astrophysics Data System (ADS)

Talib, Imran; Belgacem, Fethi Bin Muhammad; Asif, Naseer Ahmad; Khalil, Hammad

2017-01-01

In this research article, we derive and analyze an efficient spectral method based on the operational matrices of three dimensional orthogonal Jacobi polynomials to solve numerically the mixed partial derivatives type multi-terms high dimensions generalized class of fractional order partial differential equations. We transform the considered fractional order problem to an easily solvable algebraic equations with the aid of the operational matrices. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some test problems are considered to confirm the accuracy and validity of the proposed numerical method. The convergence of the method is ensured by comparing our Matlab software simulations based obtained results with the exact solutions in the literature, yielding negligible errors. Moreover, comparative results discussed in the literature are extended and improved in this study.

A Bayesian Hierarchical Model for Glacial Dynamics Based on the Shallow Ice Approximation and its Evaluation Using Analytical Solutions

NASA Astrophysics Data System (ADS)

Gopalan, Giri; Hrafnkelsson, Birgir; Aðalgeirsdóttir, Guðfinna; Jarosch, Alexander H.; Pálsson, Finnur

2018-03-01

Bayesian hierarchical modeling can assist the study of glacial dynamics and ice flow properties. This approach will allow glaciologists to make fully probabilistic predictions for the thickness of a glacier at unobserved spatio-temporal coordinates, and it will also allow for the derivation of posterior probability distributions for key physical parameters such as ice viscosity and basal sliding. The goal of this paper is to develop a proof of concept for a Bayesian hierarchical model constructed, which uses exact analytical solutions for the shallow ice approximation (SIA) introduced by Bueler et al. (2005). A suite of test simulations utilizing these exact solutions suggests that this approach is able to adequately model numerical errors and produce useful physical parameter posterior distributions and predictions. A byproduct of the development of the Bayesian hierarchical model is the derivation of a novel finite difference method for solving the SIA partial differential equation (PDE). An additional novelty of this work is the correction of numerical errors induced through a numerical solution using a statistical model. This error correcting process models numerical errors that accumulate forward in time and spatial variation of numerical errors between the dome, interior, and margin of a glacier.

Analytical and numerical solutions for heat transfer and effective thermal conductivity of cracked media

NASA Astrophysics Data System (ADS)

Tran, A. B.; Vu, M. N.; Nguyen, S. T.; Dong, T. Q.; Le-Nguyen, K.

2018-02-01

This paper presents analytical solutions to heat transfer problems around a crack and derive an adaptive model for effective thermal conductivity of cracked materials based on singular integral equation approach. Potential solution of heat diffusion through two-dimensional cracked media, where crack filled by air behaves as insulator to heat flow, is obtained in a singular integral equation form. It is demonstrated that the temperature field can be described as a function of temperature and rate of heat flow on the boundary and the temperature jump across the cracks. Numerical resolution of this boundary integral equation allows determining heat conduction and effective thermal conductivity of cracked media. Moreover, writing this boundary integral equation for an infinite medium embedding a single crack under a far-field condition allows deriving the closed-form solution of temperature discontinuity on the crack and particularly the closed-form solution of temperature field around the crack. These formulas are then used to establish analytical effective medium estimates. Finally, the comparison between the developed numerical and analytical solutions allows developing an adaptive model for effective thermal conductivity of cracked media. This model takes into account both the interaction between cracks and the percolation threshold.

Asymptotic analysis of dissipative waves with applications to their numerical simulation

NASA Technical Reports Server (NTRS)

Hagstrom, Thomas

1990-01-01

Various problems involving the interplay of asymptotics and numerics in the analysis of wave propagation in dissipative systems are studied. A general approach to the asymptotic analysis of linear, dissipative waves is developed. It was applied to the derivation of asymptotic boundary conditions for numerical solutions on unbounded domains. Applications include the Navier-Stokes equations. Multidimensional traveling wave solutions to reaction-diffusion equations are also considered. A preliminary numerical investigation of a thermo-diffusive model of flame propagation in a channel with heat loss at the walls is presented.

Approximate analytical solutions to the double-stance dynamics of the lossy spring-loaded inverted pendulum.

PubMed

Shahbazi, Mohammad; Saranlı, Uluç; Babuška, Robert; Lopes, Gabriel A D

2016-12-05

This paper introduces approximate time-domain solutions to the otherwise non-integrable double-stance dynamics of the 'bipedal' spring-loaded inverted pendulum (B-SLIP) in the presence of non-negligible damping. We first introduce an auxiliary system whose behavior under certain conditions is approximately equivalent to the B-SLIP in double-stance. Then, we derive approximate solutions to the dynamics of the new system following two different methods: (i) updated-momentum approach that can deal with both the lossy and lossless B-SLIP models, and (ii) perturbation-based approach following which we only derive a solution to the lossless case. The prediction performance of each method is characterized via a comprehensive numerical analysis. The derived representations are computationally very efficient compared to numerical integrations, and, hence, are suitable for online planning, increasing the autonomy of walking robots. Two application examples of walking gait control are presented. The proposed solutions can serve as instrumental tools in various fields such as control in legged robotics and human motion understanding in biomechanics.

Solving fractional optimal control problems within a Chebyshev-Legendre operational technique

NASA Astrophysics Data System (ADS)

Bhrawy, A. H.; Ezz-Eldien, S. S.; Doha, E. H.; Abdelkawy, M. A.; Baleanu, D.

2017-06-01

In this manuscript, we report a new operational technique for approximating the numerical solution of fractional optimal control (FOC) problems. The operational matrix of the Caputo fractional derivative of the orthonormal Chebyshev polynomial and the Legendre-Gauss quadrature formula are used, and then the Lagrange multiplier scheme is employed for reducing such problems into those consisting of systems of easily solvable algebraic equations. We compare the approximate solutions achieved using our approach with the exact solutions and with those presented in other techniques and we show the accuracy and applicability of the new numerical approach, through two numerical examples.

Application of matched asymptotic expansions to lunar and interplanetary trajectories. Volume 1: Technical discussion

NASA Technical Reports Server (NTRS)

Lancaster, J. E.

1973-01-01

Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical solution to the problem on N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The derivation of the second-order solution is summarized by showing the essential steps, some in functional form. The general asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-earth, and interplanetary solutions. The results show that the accuracies of the asymptotic solutions range from an order of magnitude better than conic approximations to that of numerical integration itself. Also, since no iterations are required, the asymptotic boundary value solutions are obtained in a fraction of the time required for comparable numerically integrated solutions. The subject of minimizing the second-order error is discussed, and recommendations made for further work directed toward achieving a uniform accuracy in all applications.

The Bean model in suprconductivity: Variational formulation and numerical solution

DOE Office of Scientific and Technical Information (OSTI.GOV)

Prigozhin, L.

The Bean critical-state model describes the penetration of magnetic field into type-II superconductors. Mathematically, this is a free boundary problem and its solution is of interest in applied superconductivity. We derive a variational formulation for the Bean model and use it to solve two-dimensional and axially symmetric critical-state problems numerically. 25 refs., 9 figs., 1 tab.

Partial differential equations of 3D boundary layer and their numerical solutions in turbomachinery

NASA Astrophysics Data System (ADS)

Zhang, Guoqing; Hua, Yaonan; Wu, Chung-Hua

1991-08-01

This paper studies the 3D boundary layer equations (3DBLE) and their numerical solutions in turbomachinery: (1) the general form of 3DBLE in turbomachines with rotational and curvature effects are derived under the semiorthogonal coordinate system, in which the normal pressure gradient is not equal to zero; (2) the method of solution of the 3DBLE is discussed; (3) the 3D boundary layers on the rotating blade surface, IGV endwall, rotor endwall (with a relatively moving boundary) are numerically solved, and the predicted data correlates well with the measured data; and (4) the comparison is made between the numerical results of 3DBLE with and without normal pressure gradient.

2–stage stochastic Runge–Kutta for stochastic delay differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rosli, Norhayati; Jusoh Awang, Rahimah; Bahar, Arifah

2015-05-15

This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.

Transition and mixing in axisymmetric jets and vortex rings

NASA Technical Reports Server (NTRS)

Allen, G. A., Jr.; Cantwell, B. J.

1986-01-01

A class of impulsively started, axisymmetric, laminar jets produced by a time dependent joint source of momentum are considered. These jets are different flows, each initially at rest in an unbounded fluid. The study is conducted at three levels of detail. First, a generalized set of analytic creeping flow solutions are derived with a method of flow classification. Second, from this set, three specific creeping flow solutions are studied in detail: the vortex ring, the round jet, and the ramp jet. This study involves derivation of vorticity, stream function, entrainment diagrams, and evolution of time lines through computer animation. From entrainment diagrams, critical points are derived and analyzed. The flow geometry is dictated by the properties and location of critical points which undergo bifurcation and topological transformation (a form of transition) with changing Reynolds number. Transition Reynolds numbers were calculated. A state space trajectory was derived describing the topological behavior of these critical points. This state space derivation yielded three states of motion which are universal for all axisymmetric jets. Third, the axisymmetric round jet is solved numerically using the unsteady laminar Navier Stokes equations. These equations were shown to be self similar for the round jet. Numerical calculations were performed up to a Reynolds number of 30 for a 60x60 point mesh. Animations generated from numerical solution showed each of the three states of motion for the round jet, including the Re = 30 case.

Upscaling of Solute Transport in Heterogeneous Media with Non-uniform Flow and Dispersion Fields

DOE Office of Scientific and Technical Information (OSTI.GOV)

Xu, Zhijie; Meakin, Paul

2013-10-01

An analytical and computational model for non-reactive solute transport in periodic heterogeneous media with arbitrary non-uniform flow and dispersion fields within the unit cell of length ε is described. The model lumps the effect of non-uniform flow and dispersion into an effective advection velocity Ve and an effective dispersion coefficient De. It is shown that both Ve and De are scale-dependent (dependent on the length scale of the microscopic heterogeneity, ε), dependent on the Péclet number Pe, and on a dimensionless parameter α that represents the effects of microscopic heterogeneity. The parameter α, confined to the range of [-0.5, 0.5]more » for the numerical example presented, depends on the flow direction and non-uniform flow and dispersion fields. Effective advection velocity Ve and dispersion coefficient De can be derived for any given flow and dispersion fields, and . Homogenized solutions describing the macroscopic variations can be obtained from the effective model. Solutions with sub-unit-cell accuracy can be constructed by homogenized solutions and its spatial derivatives. A numerical implementation of the model compared with direct numerical solutions using a fine grid, demonstrated that the new method was in good agreement with direct solutions, but with significant computational savings.« less

Derivation of phase functions from multiply scattered sunlight transmitted through a hazy atmosphere

NASA Technical Reports Server (NTRS)

Weinman, J. A.; Twitty, J. T.; Browning, S. R.; Herman, B. M.

1975-01-01

The intensity of sunlight multiply scattered in model atmospheres is derived from the equation of radiative transfer by an analytical small-angle approximation. The approximate analytical solutions are compared to rigorous numerical solutions of the same problem. Results obtained from an aerosol-laden model atmosphere are presented. Agreement between the rigorous and the approximate solutions is found to be within a few per cent. The analytical solution to the problem which considers an aerosol-laden atmosphere is then inverted to yield a phase function which describes a single scattering event at small angles. The effect of noisy data on the derived phase function is discussed.

Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography

NASA Astrophysics Data System (ADS)

Medl'a, Matej; Mikula, Karol; Čunderlík, Róbert; Macák, Marek

2018-01-01

The paper presents a numerical solution of the oblique derivative boundary value problem on and above the Earth's topography using the finite volume method (FVM). It introduces a novel method for constructing non-uniform hexahedron 3D grids above the Earth's surface. It is based on an evolution of a surface, which approximates the Earth's topography, by mean curvature. To obtain optimal shapes of non-uniform 3D grid, the proposed evolution is accompanied by a tangential redistribution of grid nodes. Afterwards, the Laplace equation is discretized using FVM developed for such a non-uniform grid. The oblique derivative boundary condition is treated as a stationary advection equation, and we derive a new upwind type discretization suitable for non-uniform 3D grids. The discretization of the Laplace equation together with the discretization of the oblique derivative boundary condition leads to a linear system of equations. The solution of this system gives the disturbing potential in the whole computational domain including the Earth's surface. Numerical experiments aim to show properties and demonstrate efficiency of the developed FVM approach. The first experiments study an experimental order of convergence of the method. Then, a reconstruction of the harmonic function on the Earth's topography, which is generated from the EGM2008 or EIGEN-6C4 global geopotential model, is presented. The obtained FVM solutions show that refining of the computational grid leads to more precise results. The last experiment deals with local gravity field modelling in Slovakia using terrestrial gravity data. The GNSS-levelling test shows accuracy of the obtained local quasigeoid model.

A discontinuous Galerkin method for poroelastic wave propagation: The two-dimensional case

NASA Astrophysics Data System (ADS)

Dudley Ward, N. F.; Lähivaara, T.; Eveson, S.

2017-12-01

In this paper, we consider a high-order discontinuous Galerkin (DG) method for modelling wave propagation in coupled poroelastic-elastic media. The upwind numerical flux is derived as an exact solution for the Riemann problem including the poroelastic-elastic interface. Attenuation mechanisms in both Biot's low- and high-frequency regimes are considered. The current implementation supports non-uniform basis orders which can be used to control the numerical accuracy element by element. In the numerical examples, we study the convergence properties of the proposed DG scheme and provide experiments where the numerical accuracy of the scheme under consideration is compared to analytic and other numerical solutions.

A new numerical treatment based on Lucas polynomials for 1D and 2D sinh-Gordon equation

NASA Astrophysics Data System (ADS)

Oruç, Ömer

2018-04-01

In this paper, a new mixed method based on Lucas and Fibonacci polynomials is developed for numerical solutions of 1D and 2D sinh-Gordon equations. Firstly time variable discretized by central finite difference and then unknown function and its derivatives are expanded to Lucas series. With the help of these series expansion and Fibonacci polynomials, matrices for differentiation are derived. With this approach, finding the solution of sinh-Gordon equation transformed to finding the solution of an algebraic system of equations. Lucas series coefficients are acquired by solving this system of algebraic equations. Then by plugginging these coefficients into Lucas series expansion numerical solutions can be obtained consecutively. The main objective of this paper is to demonstrate that Lucas polynomial based method is convenient for 1D and 2D nonlinear problems. By calculating L2 and L∞ error norms of some 1D and 2D test problems efficiency and performance of the proposed method is monitored. Acquired accurate results confirm the applicability of the method.

Spectral methods in general relativity and large Randall-Sundrum II black holes

NASA Astrophysics Data System (ADS)

Abdolrahimi, Shohreh; Cattoën, Céline; Page, Don N.; \\\\; Yaghoobpour-Tari, Shima

2013-06-01

Using a novel numerical spectral method, we have found solutions for large static Randall-Sundrum II (RSII) black holes by perturbing a numerical AdS5-CFT4 solution to the Einstein equation with a negative cosmological constant Λ that is asymptotically conformal to the Schwarzschild metric. We used a numerical spectral method independent of the Ricci-DeTurck-flow method used by Figueras, Lucietti, and Wiseman for a similar numerical solution. We have compared our black-hole solution to the one Figueras and Wiseman have derived by perturbing their numerical AdS5-CFT4 solution, showing that our solution agrees closely with theirs. We have obtained a closed-form approximation to the metric of the black hole on the brane. We have also deduced the new results that to first order in 1/(-ΛM2), the Hawking temperature and entropy of an RSII static black hole have the same values as the Schwarzschild metric with the same mass, but the horizon area is increased by about 4.7/(-Λ).

Numerical solution of distributed order fractional differential equations

NASA Astrophysics Data System (ADS)

Katsikadelis, John T.

2014-02-01

In this paper a method for the numerical solution of distributed order FDEs (fractional differential equations) of a general form is presented. The method applies to both linear and nonlinear equations. The Caputo type fractional derivative is employed. The distributed order FDE is approximated with a multi-term FDE, which is then solved by adjusting appropriately the numerical method developed for multi-term FDEs by Katsikadelis. Several example equations are solved and the response of mechanical systems described by such equations is studied. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.

Analytic solutions for Long's equation and its generalization

NASA Astrophysics Data System (ADS)

Humi, Mayer

2017-12-01

Two-dimensional, steady-state, stratified, isothermal atmospheric flow over topography is governed by Long's equation. Numerical solutions of this equation were derived and used by several authors. In particular, these solutions were applied extensively to analyze the experimental observations of gravity waves. In the first part of this paper we derive an extension of this equation to non-isothermal flows. Then we devise a transformation that simplifies this equation. We show that this simplified equation admits solitonic-type solutions in addition to regular gravity waves. These new analytical solutions provide new insights into the propagation and amplitude of gravity waves over topography.

Atmospheric guidance law for planar skip trajectories

NASA Technical Reports Server (NTRS)

Mease, K. D.; Mccreary, F. A.

1985-01-01

The applicability of an approximate, closed-form, analytical solution to the equations of motion, as a basis for a deterministic guidance law for controlling the in-plane motion during a skip trajectory, is investigated. The derivation of the solution by the method of matched asymptotic expansions is discussed. Specific issues that arise in the application of the solution to skip trajectories are addressed. Based on the solution, an explicit formula for the approximate energy loss due to an atmospheric pass is derived. A guidance strategy is proposed that illustrates the use of the approximate solution. A numerical example shows encouraging performance.

Comparing the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivative with fractional order: Fractional cubic isothermal auto-catalytic chemical system

NASA Astrophysics Data System (ADS)

Saad, K. M.

2018-03-01

In this work we extend the standard model for a cubic isothermal auto-catalytic chemical system (CIACS) to a new model of a fractional cubic isothermal auto-catalytic chemical system (FCIACS) based on Caputo (C), Caputo-Fabrizio (CF) and Atangana-Baleanu in the Liouville-Caputo sense (ABC) fractional time derivatives, respectively. We present approximate solutions for these extended models using the q -homotopy analysis transform method ( q -HATM). We solve the FCIACS with the C derivative and compare our results with those obtained using the CF and ABC derivatives. The ranges of convergence of the solutions are found and the optimal values of h , the auxiliary parameter, are derived. Finally, these solutions are compared with numerical solutions of the various models obtained using finite differences and excellent agreement is found.

Exact solutions of the Navier-Stokes equations generalized for flow in porous media

NASA Astrophysics Data System (ADS)

Daly, Edoardo; Basser, Hossein; Rudman, Murray

2018-05-01

Flow of Newtonian fluids in porous media is often modelled using a generalized version of the full non-linear Navier-Stokes equations that include additional terms describing the resistance to flow due to the porous matrix. Because this formulation is becoming increasingly popular in numerical models, exact solutions are required as a benchmark of numerical codes. The contribution of this study is to provide a number of non-trivial exact solutions of the generalized form of the Navier-Stokes equations for parallel flow in porous media. Steady-state solutions are derived in the case of flows in a medium with constant permeability along the main direction of flow and a constant cross-stream velocity in the case of both linear and non-linear drag. Solutions are also presented for cases in which the permeability changes in the direction normal to the main flow. An unsteady solution for a flow with velocity driven by a time-periodic pressure gradient is also derived. These solutions form a basis for validating computational models across a wide range of Reynolds and Darcy numbers.

Numerical Modeling of Ablation Heat Transfer

NASA Technical Reports Server (NTRS)

Ewing, Mark E.; Laker, Travis S.; Walker, David T.

2013-01-01

A unique numerical method has been developed for solving one-dimensional ablation heat transfer problems. This paper provides a comprehensive description of the method, along with detailed derivations of the governing equations. This methodology supports solutions for traditional ablation modeling including such effects as heat transfer, material decomposition, pyrolysis gas permeation and heat exchange, and thermochemical surface erosion. The numerical scheme utilizes a control-volume approach with a variable grid to account for surface movement. This method directly supports implementation of nontraditional models such as material swelling and mechanical erosion, extending capabilities for modeling complex ablation phenomena. Verifications of the numerical implementation are provided using analytical solutions, code comparisons, and the method of manufactured solutions. These verifications are used to demonstrate solution accuracy and proper error convergence rates. A simple demonstration of a mechanical erosion (spallation) model is also provided to illustrate the unique capabilities of the method.

Non-Schwarzschild black-hole metric in four dimensional higher derivative gravity: Analytical approximation

NASA Astrophysics Data System (ADS)

Kokkotas, K. D.; Konoplya, R. A.; Zhidenko, A.

2017-09-01

Higher derivative extensions of Einstein gravity are important within the string theory approach to gravity and as alternative and effective theories of gravity. H. Lü, A. Perkins, C. Pope, and K. Stelle [Phys. Rev. Lett. 114, 171601 (2015), 10.1103/PhysRevLett.114.171601] found a numerical solution describing a spherically symmetric non-Schwarzschild asymptotically flat black hole in Einstein gravity with added higher derivative terms. Using the general and quickly convergent parametrization in terms of the continued fractions, we represent this numerical solution in the analytical form, which is accurate not only near the event horizon or far from the black hole, but in the whole space. Thereby, the obtained analytical form of the metric allows one to study easily all the further properties of the black hole, such as thermodynamics, Hawking radiation, particle motion, accretion, perturbations, stability, quasinormal spectrum, etc. Thus, the found analytical approximate representation can serve in the same way as an exact solution.

Computer simulation of solutions of polyharmonic equations in plane domain

NASA Astrophysics Data System (ADS)

Kazakova, A. O.

2018-05-01

A systematic study of plane problems of the theory of polyharmonic functions is presented. A method of reducing boundary problems for polyharmonic functions to the system of integral equations on the boundary of the domain is given and a numerical algorithm for simulation of solutions of this system is suggested. Particular attention is paid to the numerical solution of the main tasks when the values of the function and its derivatives are given. Test examples are considered that confirm the effectiveness and accuracy of the suggested algorithm.

Ineffective higher derivative black hole hair

NASA Astrophysics Data System (ADS)

Goldstein, Kevin; Mashiyane, James Junior

2018-01-01

Inspired by the possibility that the Schwarzschild black hole may not be the unique spherically symmetric vacuum solution to generalizations of general relativity, we consider black holes in pure fourth order higher derivative gravity treated as an effective theory. Such solutions may be of interest in addressing the issue of higher derivative hair or during the later stages of black hole evaporation. Non-Schwarzschild solutions have been studied but we have put earlier results on a firmer footing by finding a systematic asymptotic expansion for the black holes and matching them with known numerical solutions obtained by integrating out from the near-horizon region. These asymptotic expansions can be cast in the form of trans-series expansions which we conjecture will be a generic feature of non-Schwarzschild higher derivative black holes. Excitingly we find a new branch of solutions with lower free energy than the Schwarzschild solution, but as found in earlier work, solutions only seem to exist for black holes with large curvatures, meaning that one should not generically neglect even higher derivative corrections. This suggests that one effectively recovers the nonhair theorems in this context.

A deterministic particle method for one-dimensional reaction-diffusion equations

NASA Technical Reports Server (NTRS)

Mascagni, Michael

1995-01-01

We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of suitably defined particles. We then consider the time explicit and implicit methods for this system of ordinary differential equations and we study a Picard and Newton iteration for the solution of the implicit system. Next we solve numerically this system and study the discretization error both analytically and numerically. Numerical computation shows that this deterministic method is automatically adaptive to large gradients in the solution.

Steering particles by breaking symmetries

NASA Astrophysics Data System (ADS)

Bet, Bram; Samin, Sela; Georgiev, Rumen; Burak Eral, Huseyin; van Roij, René

2018-06-01

We derive general equations of motions for highly-confined particles that perform quasi-two-dimensional motion in Hele-Shaw channels, which we solve analytically, aiming to derive design principles for self-steering particles. Based on symmetry properties of a particle, its equations of motion can be simplified, where we retrieve an earlier-known equation of motion for the orientation of dimer particles consisting of disks (Uspal et al 2013 Nat. Commun. 4), but now in full generality. Subsequently, these solutions are compared with particle trajectories that are obtained numerically. For mirror-symmetric particles, excellent agreement between the analytical and numerical solutions is found. For particles lacking mirror symmetry, the analytic solutions provide means to classify the motion based on particle geometry, while we find that taking the side-wall interactions into account is important to accurately describe the trajectories.

An extension of the Derrida-Lebowitz-Speer-Spohn equation

NASA Astrophysics Data System (ADS)

Bordenave, Charles; Germain, Pierre; Trogdon, Thomas

2015-12-01

We show how the derivation of the Derrida-Lebowitz-Speer-Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy-Widom GOE distribution.

A new Jacobi spectral collocation method for solving 1+1 fractional Schrödinger equations and fractional coupled Schrödinger systems

NASA Astrophysics Data System (ADS)

Bhrawy, A. H.; Doha, E. H.; Ezz-Eldien, S. S.; Van Gorder, Robert A.

2014-12-01

The Jacobi spectral collocation method (JSCM) is constructed and used in combination with the operational matrix of fractional derivatives (described in the Caputo sense) for the numerical solution of the time-fractional Schrödinger equation (T-FSE) and the space-fractional Schrödinger equation (S-FSE). The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, the presented approach is also applied to solve the time-fractional coupled Schrödinger system (T-FCSS). In order to demonstrate the validity and accuracy of the numerical scheme proposed, several numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.

Using a quasi-heat-pulse method to determine heat and moisture transfer properties for porous orthotropic wood products or cellular solid materials

Treesearch

M. A. Dietenberger

2006-01-01

Understanding heat and moisture transfer in a wood specimen as used in the K-tester has led to an unconventional numerical solution arid intriguing protocol to deriving the transfer properties. Laplace transform solutions of Luikov’s differential equations are derived for one-dimensional heat and moisture transfer in porous hygroscopic orthotropic materials and for a...

Variational Methods in Sensitivity Analysis and Optimization for Aerodynamic Applications

NASA Technical Reports Server (NTRS)

Ibrahim, A. H.; Hou, G. J.-W.; Tiwari, S. N. (Principal Investigator)

1996-01-01

Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equation is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Numerical analysis of two-fluid tearing mode instability in a finite aspect ratio cylinder

NASA Astrophysics Data System (ADS)

Ito, Atsushi; Ramos, Jesús J.

2018-01-01

The two-fluid resistive tearing mode instability in a periodic plasma cylinder of finite aspect ratio is investigated numerically for parameters such that the cylindrical aspect ratio and two-fluid effects are of order unity, hence the real and imaginary parts of the mode eigenfunctions and growth rate are comparable. Considering a force-free equilibrium, numerical solutions of the complete eigenmode equations for general aspect ratios and ion skin depths are compared and found to be in very good agreement with the corresponding analytic solutions derived by means of the boundary layer theory [A. Ito and J. J. Ramos, Phys. Plasmas 24, 072102 (2017)]. Scaling laws for the growth rate and the real frequency of the mode are derived from the analytic dispersion relation by using Taylor expansions and Padé approximations. The cylindrical finite aspect ratio effect is inferred from the scaling law for the real frequency of the mode.

Dispersion analysis and measurement of circular cylindrical wedge-like acoustic waveguides.

PubMed

Yu, Tai-Ho

2015-09-01

This study investigated the propagation of flexural waves along the outer edge of a circular cylindrical wedge, the phase velocities, and the corresponding mode displacements. Thus far, only approximate solutions have been derived because the corresponding boundary-value problems are complex. In this study, dispersion curves were determined using the bi-dimensional finite element method and derived through the separation of variables and the Hamilton principle. Modal displacement calculations clarified that the maximal deformations appeared at the outer edge of the wedge tip. Numerical examples indicated how distinct thin-film materials deposited on the outer surface of the circular cylindrical wedge influenced the dispersion curves. Additionally, dispersion curves were measured using a laser-induced guided wave, a knife-edge measurement scheme, and a two-dimensional fast Fourier transform method. Both the numerical and experimental results correlated closely, thus validating the numerical solution. Copyright © 2015 Elsevier B.V. All rights reserved.

Breather management in the derivative nonlinear Schrödinger equation with variable coefficients

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhong, Wei-Ping, E-mail: zhongwp6@126.com; Texas A&M University at Qatar, P.O. Box 23874 Doha; Belić, Milivoj

2015-04-15

We investigate breather solutions of the generalized derivative nonlinear Schrödinger (DNLS) equation with variable coefficients, which is used in the description of femtosecond optical pulses in inhomogeneous media. The solutions are constructed by means of the similarity transformation, which reduces a particular form of the generalized DNLS equation into the standard one, with constant coefficients. Examples of bright and dark breathers of different orders, that ride on finite backgrounds and may be related to rogue waves, are presented. - Highlights: • Exact solutions of a generalized derivative NLS equation are obtained. • The solutions are produced by means of amore » transformation to the usual integrable equation. • The validity of the solutions is verified by comparing them to numerical counterparts. • Stability of the solutions is checked by means of direct simulations. • The model applies to the propagation of ultrashort pulses in optical media.« less

An implicit semianalytic numerical method for the solution of nonequilibrium chemistry problems

NASA Technical Reports Server (NTRS)

Graves, R. A., Jr.; Gnoffo, P. A.; Boughner, R. E.

1974-01-01

The first order differential equation form systems of equations. They are solved by a simple and relatively accurate implicit semianalytic technique which is derived from a quadrature solution of the governing equation. This method is mathematically simpler than most implicit methods and has the exponential nature of the problem embedded in the solution.

Thermodynamics of Inozemtsev's elliptic spin chain

NASA Astrophysics Data System (ADS)

Klabbers, Rob

2016-06-01

We study the thermodynamic behaviour of Inozemtsev's long-range elliptic spin chain using the Bethe ansatz equations describing the spectrum of the model in the infinite-length limit. We classify all solutions of these equations in that limit and argue which of these solutions determine the spectrum in the thermodynamic limit. Interestingly, some of the solutions are not selfconjugate, which puts the model in sharp contrast to one of the model's limiting cases, the Heisenberg XXX spin chain. Invoking the string hypothesis we derive the thermodynamic Bethe ansatz equations (TBA-equations) from which we determine the Helmholtz free energy in thermodynamic equilibrium and derive the associated Y-system. We corroborate our results by comparing numerical solutions of the TBA-equations to a direct computation of the free energy for the finite-length hamiltonian. In addition we confirm numerically the interesting conjecture put forward by Finkel and González-López that the original and supersymmetric versions of Inozemtsev's elliptic spin chain are equivalent in the thermodynamic limit.

Asymptotic-induced numerical methods for conservation laws

NASA Technical Reports Server (NTRS)

Garbey, Marc; Scroggs, Jeffrey S.

1990-01-01

Asymptotic-induced methods are presented for the numerical solution of hyperbolic conservation laws with or without viscosity. The methods consist of multiple stages. The first stage is to obtain a first approximation by using a first-order method, such as the Godunov scheme. Subsequent stages of the method involve solving internal-layer problems identified by using techniques derived via asymptotics. Finally, a residual correction increases the accuracy of the scheme. The method is derived and justified with singular perturbation techniques.

Two-Dimensional Model for Reactive-Sorption Columns of Cylindrical Geometry: Analytical Solutions and Moment Analysis.

PubMed

Khan, Farman U; Qamar, Shamsul

2017-05-01

A set of analytical solutions are presented for a model describing the transport of a solute in a fixed-bed reactor of cylindrical geometry subjected to the first (Dirichlet) and third (Danckwerts) type inlet boundary conditions. Linear sorption kinetic process and first-order decay are considered. Cylindrical geometry allows the use of large columns to investigate dispersion, adsorption/desorption and reaction kinetic mechanisms. The finite Hankel and Laplace transform techniques are adopted to solve the model equations. For further analysis, statistical temporal moments are derived from the Laplace-transformed solutions. The developed analytical solutions are compared with the numerical solutions of high-resolution finite volume scheme. Different case studies are presented and discussed for a series of numerical values corresponding to a wide range of mass transfer and reaction kinetics. A good agreement was observed in the analytical and numerical concentration profiles and moments. The developed solutions are efficient tools for analyzing numerical algorithms, sensitivity analysis and simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment. © The Author 2017. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com.

Application of matched asymptotic expansions to lunar and interplanetary trajectories. Volume 2: Derivations of second-order asymptotic boundary value solutions

NASA Technical Reports Server (NTRS)

Lancaster, J. E.

1973-01-01

Previously published asymptotic solutions for lunar and interplanetery trajectories have been modified and combined to formulate a general analytical solution to the problem of N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The complete derivation of the second-order solution, including the application of a regorous matching principle, is given. It is shown that the outer and inner expansions can be matched in a region of order mu to the alpha power, where 2/5 alpha 1/2, and mu (the moon/earth or planet/sun mass ratio) is much less than one. The second-order asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-Earth, and interplanetary solutions. Each is presented as an explicit analytical solution which does not require iterative steps to satisfy the boundary conditions. The complete derivation of each solution is shown, as well as instructions for numerical evaluation. For Vol. 1, see N73-27738.

Complexiton and solitary wave solutions of the coupled nonlinear Maccari’s system using two integration schemes

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Aliyu, Aliyu Isa; Yusuf, Abdullahi; Baleanu, Dumitru; Nuray, Elif

2018-01-01

In this paper, we consider a coupled nonlinear Maccari’s system (CNMS) which describes the motion of isolated waves localized in a small part of space. There are some integration tools that are adopted to retrieve the solitary wave solutions. They are the modified F-Expansion and the generalized projective Riccati equation methods. Topological, non-topological, complexiton, singular and trigonometric function solutions are derived. A comparison between the results in this paper and the well-known results in the literature is also given. The derived structures of the obtained solutions offer a rich platform to study the nonlinear CNMS. Numerical simulation of the obtained solutions are presented with interesting figures showing the physical meaning of the solutions.

Couette flow through a porous medium with heat and mass transfer in the presence of tranverse magnetic field

NASA Astrophysics Data System (ADS)

Lawanya, T.; Vidhya, M.; Govindarajan, A.

2018-04-01

This present paper deals with the investigation of couette flow of a viscous electrically conducting incompressible fluid three dimensionally through a porous medium in presence of transverse magnetic field. Approximate Solution of equations of motion and energy equations are derived using series solution method. Hartmann number, Schmidt number and Grashoff number (or) modified Grashoff number for mass transfer on the velocity and temperature distribution are numerically discussed and shown graphically. The Nusselt number and skin friction coefficients atthe plate are derived and their numerical values are shown graphically. It is seen that in the main flow direction the velocity profiles decreases due to either an increase in Schmidt number (Or) Hartmann number.

Electro-magneto interaction in fractional Green-Naghdi thermoelastic solid with a cylindrical cavity

NASA Astrophysics Data System (ADS)

Ezzat, M. A.; El-Bary, A. A.

2018-01-01

A unified mathematical model of Green-Naghdi's thermoelasticty theories (GN), based on fractional time-derivative of heat transfer is constructed. The model is applied to solve a one-dimensional problem of a perfect conducting unbounded body with a cylindrical cavity subjected to sinusoidal pulse heating in the presence of an axial uniform magnetic field. Laplace transform techniques are used to get the general analytical solutions in Laplace domain, and the inverse Laplace transforms based on Fourier expansion techniques are numerically implemented to obtain the numerical solutions in time domain. Comparisons are made with the results predicted by the two theories. The effects of the fractional derivative parameter on thermoelastic fields for different theories are discussed.

Fractional calculus in hydrologic modeling: A numerical perspective

PubMed Central

Benson, David A.; Meerschaert, Mark M.; Revielle, Jordan

2013-01-01

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus. PMID:23524449

Wind laws for shockless initialization. [numerical forecasting model

NASA Technical Reports Server (NTRS)

Ghil, M.; Shkoller, B.

1976-01-01

A system of diagnostic equations for the velocity field, or wind laws, was derived for each of a number of models of large-scale atmospheric flow. The derivation in each case is mathematically exact and does not involve any physical assumptions not already present in the prognostic equations, such as nondivergence or vanishing of derivatives of the divergence. Therefore, initial states computed by solving these diagnostic equations should be compatible with the type of motion described by the prognostic equations of the model and should not generate initialization shocks when inserted into the model. Numerical solutions of the diagnostic system corresponding to a barotropic model are exhibited. Some problems concerning the possibility of implementing such a system in operational numerical weather prediction are discussed.

Optimality conditions for the numerical solution of optimization problems with PDE constraints :

DOE Office of Scientific and Technical Information (OSTI.GOV)

Aguilo Valentin, Miguel Alejandro; Ridzal, Denis

2014-03-01

A theoretical framework for the numerical solution of partial di erential equation (PDE) constrained optimization problems is presented in this report. This theoretical framework embodies the fundamental infrastructure required to e ciently implement and solve this class of problems. Detail derivations of the optimality conditions required to accurately solve several parameter identi cation and optimal control problems are also provided in this report. This will allow the reader to further understand how the theoretical abstraction presented in this report translates to the application.

Properties of finite difference models of non-linear conservative oscillators

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1988-01-01

Finite-difference (FD) approaches to the numerical solution of the differential equations describing the motion of a nonlinear conservative oscillator are investigated analytically. A generalized formulation of the Duffing and modified Duffing equations is derived and analyzed using several FD techniques, and it is concluded that, although it is always possible to contstruct FD models of conservative oscillators which are themselves conservative, caution is required to avoid numerical solutions which do not accurately reflect the properties of the original equation.

A higher order numerical method for time fractional partial differential equations with nonsmooth data

NASA Astrophysics Data System (ADS)

Xing, Yanyuan; Yan, Yubin

2018-03-01

Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu [20] (2016), where k is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 as in Gao et al. [11] (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O (k 3 - α), 0 < α < 1 for any fixed tn > 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Second-order numerical solution of time-dependent, first-order hyperbolic equations

NASA Technical Reports Server (NTRS)

Shah, Patricia L.; Hardin, Jay

1995-01-01

A finite difference scheme is developed to find an approximate solution of two similar hyperbolic equations, namely a first-order plane wave and spherical wave problem. Finite difference approximations are made for both the space and time derivatives. The result is a conditionally stable equation yielding an exact solution when the Courant number is set to one.

Limit Theorems and Their Relation to Solute Transport in Simulated Fractured Media

NASA Astrophysics Data System (ADS)

Reeves, D. M.; Benson, D. A.; Meerschaert, M. M.

2003-12-01

Solute particles that travel through fracture networks are subject to wide velocity variations along a restricted set of directions. This may result in super-Fickian dispersion along a few primary scaling directions. The fractional advection-dispersion equation (FADE), a modification of the original advection-dispersion equation in which a fractional derivative replaces the integer-order dispersion term, has the ability to model rapid, non-Gaussian solute transport. The FADE assumes that solute particle motions converge to either α -stable or operator stable densities, which are modeled by spatial fractional derivatives. In multiple dimensions, the multi-fractional dispersion derivative dictates the order and weight of differentiation in all directions, which correspond to the statistics of large particle motions in all directions. This study numerically investigates the presence of super- Fickian solute transport through simulated two-dimensional fracture networks. An ensemble of networks is gen

Black holes in higher derivative gravity.

PubMed

Lü, H; Perkins, A; Pope, C N; Stelle, K S

2015-05-01

Extensions of Einstein gravity with higher-order derivative terms arise in string theory and other effective theories, as well as being of interest in their own right. In this Letter we study static black-hole solutions in the example of Einstein gravity with additional quadratic curvature terms. A Lichnerowicz-type theorem simplifies the analysis by establishing that they must have vanishing Ricci scalar curvature. By numerical methods we then demonstrate the existence of further black-hole solutions over and above the Schwarzschild solution. We discuss some of their thermodynamic properties, and show that they obey the first law of thermodynamics.

Approximate method of variational Bayesian matrix factorization/completion with sparse prior

NASA Astrophysics Data System (ADS)

Kawasumi, Ryota; Takeda, Koujin

2018-05-01

We derive the analytical expression of a matrix factorization/completion solution by the variational Bayes method, under the assumption that the observed matrix is originally the product of low-rank, dense and sparse matrices with additive noise. We assume the prior of a sparse matrix is a Laplace distribution by taking matrix sparsity into consideration. Then we use several approximations for the derivation of a matrix factorization/completion solution. By our solution, we also numerically evaluate the performance of a sparse matrix reconstruction in matrix factorization, and completion of a missing matrix element in matrix completion.

Orientation of doubly rotated quartz plates.

PubMed

Sherman, J R

1989-01-01

A derivation from classical spherical trigonometry of equations to compute the orientation of doubly-rotated quartz blanks from Bragg X-ray data is discussed. These are usually derived by compact and efficient vector methods, which are reviewed briefly. They are solved by generating a quadratic equation with numerical coefficients. Two methods exist for performing the computation from measurements against two planes: a direct solution by a quadratic equation and a process of convergent iteration. Both have a spurious solution. Measurement against three lattice planes yields a set of three linear equations the solution of which is an unambiguous result.

Boundary particle method for Laplace transformed time fractional diffusion equations

NASA Astrophysics Data System (ADS)

Fu, Zhuo-Jia; Chen, Wen; Yang, Hai-Tian

2013-02-01

This paper develops a novel boundary meshless approach, Laplace transformed boundary particle method (LTBPM), for numerical modeling of time fractional diffusion equations. It implements Laplace transform technique to obtain the corresponding time-independent inhomogeneous equation in Laplace space and then employs a truly boundary-only meshless boundary particle method (BPM) to solve this Laplace-transformed problem. Unlike the other boundary discretization methods, the BPM does not require any inner nodes, since the recursive composite multiple reciprocity technique (RC-MRM) is used to convert the inhomogeneous problem into the higher-order homogeneous problem. Finally, the Stehfest numerical inverse Laplace transform (NILT) is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding BPM solutions. In comparison with finite difference discretization, the LTBPM introduces Laplace transform and Stehfest NILT algorithm to deal with time fractional derivative term, which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability. Consequently, it can effectively simulate long time-history fractional diffusion systems. Error analysis and numerical experiments demonstrate that the present LTBPM is highly accurate and computationally efficient for 2D and 3D time fractional diffusion equations.

Efficient numerical method for solving Cauchy problem for the Gamma equation

NASA Astrophysics Data System (ADS)

Koleva, Miglena N.

2011-12-01

In this work we consider Cauchy problem for the so called Gamma equation, derived by transforming the fully nonlinear Black-Scholes equation for option price into a quasilinear parabolic equation for the second derivative (Greek) Γ = VSS of the option price V. We develop an efficient numerical method for solving the model problem concerning different volatility terms. Using suitable change of variables the problem is transformed on finite interval, keeping original behavior of the solution at the infinity. Then we construct Picard-Newton algorithm with adaptive mesh step in time, which can be applied also in the case of non-differentiable functions. Results of numerical simulations are given.

Penalty methods for the numerical solution of American multi-asset option problems

NASA Astrophysics Data System (ADS)

Nielsen, Bjørn Fredrik; Skavhaug, Ola; Tveito, Aslak

2008-12-01

We derive and analyze a penalty method for solving American multi-asset option problems. A small, non-linear penalty term is added to the Black-Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semi-implicit finite difference schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, the experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case.

A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel

NASA Astrophysics Data System (ADS)

Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Torres, L.; Escobar-Jiménez, R. F.

2018-02-01

A reaction-diffusion system can be represented by the Gray-Scott model. The reaction-diffusion dynamic is described by a pair of time and space dependent Partial Differential Equations (PDEs). In this paper, a generalization of the Gray-Scott model by using variable-order fractional differential equations is proposed. The variable-orders were set as smooth functions bounded in (0 , 1 ] and, specifically, the Liouville-Caputo and the Atangana-Baleanu-Caputo fractional derivatives were used to express the time differentiation. In order to find a numerical solution of the proposed model, the finite difference method together with the Adams method were applied. The simulations results showed the chaotic behavior of the proposed model when different variable-orders are applied.

Fully Nonlinear Modeling and Analysis of Precision Membranes

NASA Technical Reports Server (NTRS)

Pai, P. Frank; Young, Leyland G.

2003-01-01

High precision membranes are used in many current space applications. This paper presents a fully nonlinear membrane theory with forward and inverse analyses of high precision membrane structures. The fully nonlinear membrane theory is derived from Jaumann strains and stresses, exact coordinate transformations, the concept of local relative displacements, and orthogonal virtual rotations. In this theory, energy and Newtonian formulations are fully correlated, and every structural term can be interpreted in terms of vectors. Fully nonlinear ordinary differential equations (ODES) governing the large static deformations of known axisymmetric membranes under known axisymmetric loading (i.e., forward problems) are presented as first-order ODES, and a method for obtaining numerically exact solutions using the multiple shooting procedure is shown. A method for obtaining the undeformed geometry of any axisymmetric membrane with a known inflated geometry and a known internal pressure (i.e., inverse problems) is also derived. Numerical results from forward analysis are verified using results in the literature, and results from inverse analysis are verified using known exact solutions and solutions from the forward analysis. Results show that the membrane theory and the proposed numerical methods for solving nonlinear forward and inverse membrane problems are accurate.

Torsional vibration of a cracked rod by variational formulation and numerical analysis

NASA Astrophysics Data System (ADS)

Chondros, T. G.; Labeas, G. N.

2007-04-01

The torsional vibration of a circumferentially cracked cylindrical shaft is studied through an "exact" analytical solution and a numerical finite element (FE) analysis. The Hu-Washizu-Barr variational formulation is used to develop the differential equation and the boundary conditions of the cracked rod. The equations of motion for a uniform cracked rod in torsional vibration are derived and solved, and the Rayleigh quotient is used to further approximate the natural frequencies of the cracked rod. Results for the problem of the torsional vibration of a cylindrical shaft with a peripheral crack are provided through an analytical solution based on variational formulation to derive the equation of motion and a numerical analysis utilizing a parametric three-dimensional (3D) solid FE model of the cracked rod. The crack is modelled as a continuous flexibility based on fracture mechanics principles. The variational formulation results are compared with the FE alternative. The sensitivity of the FE discretization with respect to the analytical results is assessed.

Algorithms for Computing the Magnetic Field, Vector Potential, and Field Derivatives for Circular Current Loops in Cylindrical Coordinates

DOE Office of Scientific and Technical Information (OSTI.GOV)

Walstrom, Peter Lowell

A numerical algorithm for computing the field components B r and B z and their r and z derivatives with open boundaries in cylindrical coordinates for circular current loops is described. An algorithm for computing the vector potential is also described. For the convenience of the reader, derivations of the final expressions from their defining integrals are given in detail, since their derivations (especially for the field derivatives) are not all easily found in textbooks. Numerical calculations are based on evaluation of complete elliptic integrals using the Bulirsch algorithm cel. Since cel can evaluate complete elliptic integrals of a fairlymore » general type, in some cases the elliptic integrals can be evaluated without first reducing them to forms containing standard Legendre forms. The algorithms avoid the numerical difficulties that many of the textbook solutions have for points near the axis because of explicit factors of 1=r or 1=r 2 in the some of the expressions.« less

Non-Linear Spring Equations and Stability

ERIC Educational Resources Information Center

Fay, Temple H.; Joubert, Stephan V.

2009-01-01

We discuss the boundary in the Poincare phase plane for boundedness of solutions to spring model equations of the form [second derivative of]x + x + epsilonx[superscript 2] = Fcoswt and the [second derivative of]x + x + epsilonx[superscript 3] = Fcoswt and report the results of a systematic numerical investigation on the global stability of…

Automatic numerical evaluation of vacancy-mediated transport for arbitrary crystals: Onsager coefficients in the dilute limit using a Green function approach

NASA Astrophysics Data System (ADS)

Trinkle, Dallas R.

2017-10-01

A general solution for vacancy-mediated diffusion in the dilute-vacancy/dilute-solute limit for arbitrary crystal structures is derived from the master equation. A general numerical approach to the vacancy lattice Green function reduces to the sum of a few analytic functions and numerical integration of a smooth function over the Brillouin zone for arbitrary crystals. The Dyson equation solves for the Green function in the presence of a solute with arbitrary but finite interaction range to compute the transport coefficients accurately, efficiently and automatically, including cases with very large differences in solute-vacancy exchange rates. The methodology takes advantage of the space group symmetry of a crystal to reduce the complexity of the matrix inversion in the Dyson equation. An open-source implementation of the algorithm is available, and numerical results are presented for the convergence of the integration error of the bare vacancy Green function, and tracer correlation factors for a variety of crystals including wurtzite (hexagonal diamond) and garnet.

Features in simulation of crystal growth using the hyperbolic PFC equation and the dependence of the numerical solution on the parameters of the computational grid

DOE Office of Scientific and Technical Information (OSTI.GOV)

Starodumov, Ilya; Kropotin, Nikolai

2016-08-10

We investigate the three-dimensional mathematical model of crystal growth called PFC (Phase Field Crystal) in a hyperbolic modification. This model is also called the modified model PFC (originally PFC model is formulated in parabolic form) and allows to describe both slow and rapid crystallization processes on atomic length scales and on diffusive time scales. Modified PFC model is described by the differential equation in partial derivatives of the sixth order in space and second order in time. The solution of this equation is possible only by numerical methods. Previously, authors created the software package for the solution of the Phasemore » Field Crystal problem, based on the method of isogeometric analysis (IGA) and PetIGA program library. During further investigation it was found that the quality of the solution can strongly depends on the discretization parameters of a numerical method. In this report, we show the features that should be taken into account during constructing the computational grid for the numerical simulation.« less

Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation

NASA Technical Reports Server (NTRS)

Kouatchou, Jules

1999-01-01

In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.

Three-wave resonant interactions: Multi-dark-dark-dark solitons, breathers, rogue waves, and their interactions and dynamics

NASA Astrophysics Data System (ADS)

Zhang, Guoqiang; Yan, Zhenya; Wen, Xiao-Yong

2018-03-01

We investigate three-wave resonant interactions through both the generalized Darboux transformation method and numerical simulations. Firstly, we derive a simple multi-dark-dark-dark-soliton formula through the generalized Darboux transformation. Secondly, we use the matrix analysis method to avoid the singularity of transformed potential functions and to find the general nonsingular breather solutions. Moreover, through a limit process, we deduce the general rogue wave solutions and give a classification by their dynamics including bright, dark, four-petals, and two-peaks rogue waves. Ever since the coexistence of dark soliton and rogue wave in non-zero background, their interactions naturally become a quite appealing topic. Based on the N-fold Darboux transformation, we can derive the explicit solutions to depict their interactions. Finally, by performing extensive numerical simulations we can predict whether these dark solitons and rogue waves are stable enough to propagate. These results can be available for several physical subjects such as fluid dynamics, nonlinear optics, solid state physics, and plasma physics.

Analysis of social optimum for staggered shifts in a single-entry traffic corridor with no late arrivals

NASA Astrophysics Data System (ADS)

Li, Chuan-Yao; Huang, Hai-Jun; Tang, Tie-Qiao

2017-03-01

This paper investigates the traffic flow dynamics under the social optimum (SO) principle in a single-entry traffic corridor with staggered shifts from the analytical and numerical perspectives. The LWR (Lighthill-Whitham and Richards) model and the Greenshield's velocity-density function are utilized to describe the dynamic properties of traffic flow. The closed-form SO solution is analytically derived and some numerical examples are used to further testify the analytical solution. The optimum proportion of the numbers of commuters with different desired arrival times is further discussed, where the analytical and numerical results both indicate that the cumulative outflow curve under the SO principle is piecewise smooth.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chen, Zheng; Huang, Hongying; Yan, Jue

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8], [9], [19] and [21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β 0,β 1) in the numerical flux formula, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. As a result, a sequence of numerical examples are carried outmore » to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.« less

The Space-Time Conservation Element and Solution Element Method: A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws. 1; The Two Dimensional Time Marching Schemes

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung; Wang, Xiao-Yen; Chow, Chuen-Yen

1998-01-01

A new high resolution and genuinely multidimensional numerical method for solving conservation laws is being, developed. It was designed to avoid the limitations of the traditional methods. and was built from round zero with extensive physics considerations. Nevertheless, its foundation is mathmatically simple enough that one can build from it a coherent, robust. efficient and accurate numerical framework. Two basic beliefs that set the new method apart from the established methods are at the core of its development. The first belief is that, in order to capture physics more efficiently and realistically, the modeling, focus should be placed on the original integral form of the physical conservation laws, rather than the differential form. The latter form follows from the integral form under the additional assumption that the physical solution is smooth, an assumption that is difficult to realize numerically in a region of rapid chance. such as a boundary layer or a shock. The second belief is that, with proper modeling of the integral and differential forms themselves, the resulting, numerical solution should automatically be consistent with the properties derived front the integral and differential forms, e.g., the jump conditions across a shock and the properties of characteristics. Therefore a much simpler and more robust method can be developed by not using the above derived properties explicitly.

Response of a Rotating Propeller to Aerodynamic Excitation

NASA Technical Reports Server (NTRS)

Arnoldi, Walter E.

1949-01-01

The flexural vibration of a rotating propeller blade with clamped shank is analyzed with the object of presenting, in matrix form, equations for the elastic bending moments in forced vibration resulting from aerodynamic forces applied at a fixed multiple of rotational speed. Matrix equations are also derived which define the critical speeds end mode shapes for any excitation order and the relation between critical speed and blade angle. Reference is given to standard works on the numerical solution of matrix equations of the forms derived. The use of a segmented blade as an approximation to a continuous blade provides a simple means for obtaining the matrix solution from the integral equation of equilibrium, so that, in the numerical application of the method presented, the several matrix arrays of the basic physical characteristics of the propeller blade are of simple form, end their simplicity is preserved until, with the solution in sight, numerical manipulations well-known in matrix algebra yield the desired critical speeds and mode shapes frame which the vibration at any operating condition may be synthesized. A close correspondence between the familiar Stodola method and the matrix method is pointed out, indicating that any features of novelty are characteristic not of the analytical procedure but only of the abbreviation, condensation, and efficient organization of the numerical procedure made possible by the use of classical matrix theory.

A Galleria Boundary Element Method for two-dimensional nonlinear magnetostatics

NASA Astrophysics Data System (ADS)

Brovont, Aaron D.

The Boundary Element Method (BEM) is a numerical technique for solving partial differential equations that is used broadly among the engineering disciplines. The main advantage of this method is that one needs only to mesh the boundary of a solution domain. A key drawback is the myriad of integrals that must be evaluated to populate the full system matrix. To this day these integrals have been evaluated using numerical quadrature. In this research, a Galerkin formulation of the BEM is derived and implemented to solve two-dimensional magnetostatic problems with a focus on accurate, rapid computation. To this end, exact, closed-form solutions have been derived for all the integrals comprising the system matrix as well as those required to compute fields in post-processing; the need for numerical integration has been eliminated. It is shown that calculation of the system matrix elements using analytical solutions is 15-20 times faster than with numerical integration of similar accuracy. Furthermore, through the example analysis of a c-core inductor, it is demonstrated that the present BEM formulation is a competitive alternative to the Finite Element Method (FEM) for linear magnetostatic analysis. Finally, the BEM formulation is extended to analyze nonlinear magnetostatic problems via the Dual Reciprocity Method (DRBEM). It is shown that a coarse, meshless analysis using the DRBEM is able to achieve RMS error of 3-6% compared to a commercial FEM package in lightly saturated conditions.

Numerical Solutions of the Mean-Value Theorem: New Methods for Downward Continuation of Potential Fields

NASA Astrophysics Data System (ADS)

Zhang, Chong; Lü, Qingtian; Yan, Jiayong; Qi, Guang

2018-04-01

Downward continuation can enhance small-scale sources and improve resolution. Nevertheless, the common methods have disadvantages in obtaining optimal results because of divergence and instability. We derive the mean-value theorem for potential fields, which could be the theoretical basis of some data processing and interpretation. Based on numerical solutions of the mean-value theorem, we present the convergent and stable downward continuation methods by using the first-order vertical derivatives and their upward continuation. By applying one of our methods to both the synthetic and real cases, we show that our method is stable, convergent and accurate. Meanwhile, compared with the fast Fourier transform Taylor series method and the integrated second vertical derivative Taylor series method, our process has very little boundary effect and is still stable in noise. We find that the characters of the fading anomalies emerge properly in our downward continuation with respect to the original fields at the lower heights.

Analytical solutions for benchmarking cold regions subsurface water flow and energy transport models: one-dimensional soil thaw with conduction and advection

USGS Publications Warehouse

Kurylyk, Barret L.; McKenzie, Jeffrey M; MacQuarrie, Kerry T. B.; Voss, Clifford I.

2014-01-01

Numerous cold regions water flow and energy transport models have emerged in recent years. Dissimilarities often exist in their mathematical formulations and/or numerical solution techniques, but few analytical solutions exist for benchmarking flow and energy transport models that include pore water phase change. This paper presents a detailed derivation of the Lunardini solution, an approximate analytical solution for predicting soil thawing subject to conduction, advection, and phase change. Fifteen thawing scenarios are examined by considering differences in porosity, surface temperature, Darcy velocity, and initial temperature. The accuracy of the Lunardini solution is shown to be proportional to the Stefan number. The analytical solution results obtained for soil thawing scenarios with water flow and advection are compared to those obtained from the finite element model SUTRA. Three problems, two involving the Lunardini solution and one involving the classic Neumann solution, are recommended as standard benchmarks for future model development and testing.

Analytic computation of energy derivatives - Relationships among partial derivatives of a variationally determined function

NASA Technical Reports Server (NTRS)

King, H. F.; Komornicki, A.

1986-01-01

Formulas are presented relating Taylor series expansion coefficients of three functions of several variables, the energy of the trial wave function (W), the energy computed using the optimized variational wave function (E), and the response function (lambda), under certain conditions. Partial derivatives of lambda are obtained through solution of a recursive system of linear equations, and solution through order n yields derivatives of E through order 2n + 1, extending Puley's application of Wigner's 2n + 1 rule to partial derivatives in couple perturbation theory. An examination of numerical accuracy shows that the usual two-term second derivative formula is less stable than an alternative four-term formula, and that previous claims that energy derivatives are stationary properties of the wave function are fallacious. The results have application to quantum theoretical methods for the computation of derivative properties such as infrared frequencies and intensities.

A split finite element algorithm for the compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Baker, A. J.

1979-01-01

An accurate and efficient numerical solution algorithm is established for solution of the high Reynolds number limit of the Navier-Stokes equations governing the multidimensional flow of a compressible essentially inviscid fluid. Finite element interpolation theory is used within a dissipative formulation established using Galerkin criteria within the Method of Weighted Residuals. An implicit iterative solution algorithm is developed, employing tensor product bases within a fractional steps integration procedure, that significantly enhances solution economy concurrent with sharply reduced computer hardware demands. The algorithm is evaluated for resolution of steep field gradients and coarse grid accuracy using both linear and quadratic tensor product interpolation bases. Numerical solutions for linear and nonlinear, one, two and three dimensional examples confirm and extend the linearized theoretical analyses, and results are compared to competitive finite difference derived algorithms.

A family of four stages embedded explicit six-step methods with eliminated phase-lag and its derivatives for the numerical solution of the second order problems

NASA Astrophysics Data System (ADS)

Simos, T. E.

2017-11-01

A family of four stages high algebraic order embedded explicit six-step methods, for the numerical solution of second order initial or boundary-value problems with periodical and/or oscillating solutions, are studied in this paper. The free parameters of the new proposed methods are calculated solving the linear system of equations which is produced by requesting the vanishing of the phase-lag of the methods and the vanishing of the phase-lag's derivatives of the schemes. For the new obtained methods we investigate: • Its local truncation error (LTE) of the methods.• The asymptotic form of the LTE obtained using as model problem the radial Schrödinger equation.• The comparison of the asymptotic forms of LTEs for several methods of the same family. This comparison leads to conclusions on the efficiency of each method of the family.• The stability and the interval of periodicity of the obtained methods of the new family of embedded finite difference pairs.• The applications of the new obtained family of embedded finite difference pairs to the numerical solution of several second order problems like the radial Schrödinger equation, astronomical problems etc. The above applications lead to conclusion on the efficiency of the methods of the new family of embedded finite difference pairs.

Numerical Manifold Method for the Forced Vibration of Thin Plates during Bending

PubMed Central

Jun, Ding; Song, Chen; Wei-Bin, Wen; Shao-Ming, Luo; Xia, Huang

2014-01-01

A novel numerical manifold method was derived from the cubic B-spline basis function. The new interpolation function is characterized by high-order coordination at the boundary of a manifold element. The linear elastic-dynamic equation used to solve the bending vibration of thin plates was derived according to the principle of minimum instantaneous potential energy. The method for the initialization of the dynamic equation and its solution process were provided. Moreover, the analysis showed that the calculated stiffness matrix exhibited favorable performance. Numerical results showed that the generalized degrees of freedom were significantly fewer and that the calculation accuracy was higher for the manifold method than for the conventional finite element method. PMID:24883403

Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage

NASA Astrophysics Data System (ADS)

Han, Weimin; Shillor, Meir; Sofonea, Mircea

2001-12-01

We consider a model for quasistatic frictional contact between a viscoelastic body and a foundation. The material constitutive relation is assumed to be nonlinear. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, the evolution of which is determined by a parabolic inclusion. The contact is modeled with the normal compliance condition and the associated version of Coulomb's law of dry friction. We derive a variational formulation for the problem and prove the existence of its unique weak solution. We then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions.

Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.; Youssri, Y. H.

2013-10-01

In this paper, we present a new second kind Chebyshev (S2KC) operational matrix of derivatives. With the aid of S2KC, an algorithm is described to obtain numerical solutions of a class of linear and nonlinear Lane-Emden type singular initial value problems (IVPs). The idea of obtaining such solutions is essentially based on reducing the differential equation with its initial conditions to a system of algebraic equations. Two illustrative examples concern relevant physical problems (the Lane-Emden equations of the first and second kind) are discussed to demonstrate the validity and applicability of the suggested algorithm. Numerical results obtained are comparing favorably with the analytical known solutions.

Computing generalized Langevin equations and generalized Fokker-Planck equations.

PubMed

Darve, Eric; Solomon, Jose; Kia, Amirali

2009-07-07

The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.

Direct Linearization and Adjoint Approaches to Evaluation of Atmospheric Weighting Functions and Surface Partial Derivatives: General Principles, Synergy and Areas of Application

NASA Technical Reports Server (NTRS)

Ustino, Eugene A.

2006-01-01

This slide presentation reviews the observable radiances as functions of atmospheric parameters and of surface parameters; the mathematics of atmospheric weighting functions (WFs) and surface partial derivatives (PDs) are presented; and the equation of the forward radiative transfer (RT) problem is presented. For non-scattering atmospheres this can be done analytically, and all WFs and PDs can be computed analytically using the direct linearization approach. For scattering atmospheres, in general case, the solution of the forward RT problem can be obtained only numerically, but we need only two numerical solutions: one of the forward RT problem and one of the adjoint RT problem to compute all WFs and PDs we can think of. In this presentation we discuss applications of both the linearization and adjoint approaches

A second-order cell-centered Lagrangian ADER-MOOD finite volume scheme on multidimensional unstructured meshes for hydrodynamics

NASA Astrophysics Data System (ADS)

Boscheri, Walter; Dumbser, Michael; Loubère, Raphaël; Maire, Pierre-Henri

2018-04-01

In this paper we develop a conservative cell-centered Lagrangian finite volume scheme for the solution of the hydrodynamics equations on unstructured multidimensional grids. The method is derived from the Eucclhyd scheme discussed in [47,43,45]. It is second-order accurate in space and is combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves. Second-order of accuracy in time is achieved via the ADER (Arbitrary high order schemes using DERivatives) approach. A large set of numerical test cases is proposed to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior on discontinuous profiles, general robustness ensuring physical admissibility of the numerical solution, and precision where appropriate.

Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. III: nonspherical Schwarzschild waves and singularities at null infinity

NASA Astrophysics Data System (ADS)

Frauendiener, Jörg; Hennig, Jörg

2018-03-01

We extend earlier numerical and analytical considerations of the conformally invariant wave equation on a Schwarzschild background from the case of spherically symmetric solutions, discussed in Frauendiener and Hennig (2017 Class. Quantum Grav. 34 045005), to the case of general, nonsymmetric solutions. A key element of our approach is the modern standard representation of spacelike infinity as a cylinder. With a decomposition into spherical harmonics, we reduce the four-dimensional wave equation to a family of two-dimensional equations. These equations can be used to study the behaviour at the cylinder, where the solutions turn out to have, in general, logarithmic singularities at infinitely many orders. We derive regularity conditions that may be imposed on the initial data, in order to avoid the first singular terms. We then demonstrate that the fully pseudospectral time evolution scheme can be applied to this problem leading to a highly accurate numerical reconstruction of the nonsymmetric solutions. We are particularly interested in the behaviour of the solutions at future null infinity, and we numerically show that the singularities spread to null infinity from the critical set, where the cylinder approaches null infinity. The observed numerical behaviour is consistent with similar logarithmic singularities found analytically on the critical set. Finally, we demonstrate that even solutions with singularities at low orders can be obtained with high accuracy by virtue of a coordinate transformation that converts solutions with logarithmic singularities into smooth solutions.

Advanced numerical methods for three dimensional two-phase flow calculations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Toumi, I.; Caruge, D.

1997-07-01

This paper is devoted to new numerical methods developed for both one and three dimensional two-phase flow calculations. These methods are finite volume numerical methods and are based on the use of Approximate Riemann Solvers concepts to define convective fluxes versus mean cell quantities. The first part of the paper presents the numerical method for a one dimensional hyperbolic two-fluid model including differential terms as added mass and interface pressure. This numerical solution scheme makes use of the Riemann problem solution to define backward and forward differencing to approximate spatial derivatives. The construction of this approximate Riemann solver uses anmore » extension of Roe`s method that has been successfully used to solve gas dynamic equations. As far as the two-fluid model is hyperbolic, this numerical method seems very efficient for the numerical solution of two-phase flow problems. The scheme was applied both to shock tube problems and to standard tests for two-fluid computer codes. The second part describes the numerical method in the three dimensional case. The authors discuss also some improvements performed to obtain a fully implicit solution method that provides fast running steady state calculations. Such a scheme is not implemented in a thermal-hydraulic computer code devoted to 3-D steady-state and transient computations. Some results obtained for Pressurised Water Reactors concerning upper plenum calculations and a steady state flow in the core with rod bow effect evaluation are presented. In practice these new numerical methods have proved to be stable on non staggered grids and capable of generating accurate non oscillating solutions for two-phase flow calculations.« less

Simulation of a Single-Element Lean-Direct Injection Combustor Using a Polyhedral Mesh Derived from Hanging-Node Elements

NASA Technical Reports Server (NTRS)

Wey, Thomas; Liu, Nan-Suey

2013-01-01

This paper summarizes the procedures of generating a polyhedral mesh derived from hanging-node elements as well as presents sample results from its application to the numerical solution of a single element lean direct injection (LDI) combustor using an open-source version of the National Combustion Code (NCC).

Numerical simulation of wave-induced fluid flow seismic attenuation based on the Cole-Cole model.

PubMed

Picotti, Stefano; Carcione, José M

2017-07-01

The acoustic behavior of porous media can be simulated more realistically using a stress-strain relation based on the Cole-Cole model. In particular, seismic velocity dispersion and attenuation in porous rocks is well described by mesoscopic-loss models. Using the Zener model to simulate wave propagation is a rough approximation, while the Cole-Cole model provides an optimal description of the physics. Here, a time-domain algorithm is proposed based on the Grünwald-Letnikov numerical approximation of the fractional derivative involved in the time-domain representation of the Cole-Cole model, while the spatial derivatives are computed with the Fourier pseudospectral method. The numerical solution is successfully tested against an analytical solution. The methodology is applied to a model of saline aquifer, where carbon dioxide (CO 2 ) is injected. To follow the migration of the gas and detect possible leakages, seismic monitoring surveys should be carried out periodically. To this aim, the sensitivity of the seismic method must be carefully assessed for the specific case. The simulated test considers a possible leakage in the overburden, above the caprock, where the sandstone is partially saturated with gas and brine. The numerical examples illustrate the implementation of the theory.

A new flux-conserving numerical scheme for the steady, incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Scott, James R.

1994-01-01

This paper is concerned with the continued development of a new numerical method, the space-time solution element (STS) method, for solving conservation laws. The present work focuses on the two-dimensional, steady, incompressible Navier-Stokes equations. Using first an integral approach, and then a differential approach, the discrete flux conservation equations presented in a recent paper are rederived. Here a simpler method for determining the flux expressions at cell interfaces is given; a systematic and rigorous derivation of the conditions used to simulate the differential form of the governing conservation law(s) is provided; necessary and sufficient conditions for a discrete approximation to satisfy a conservation law in E2 are derived; and an estimate of the local truncation error is given. A specific scheme is then constructed for the solution of the thin airfoil boundary layer problem. Numerical results are presented which demonstrate the ability of the scheme to accurately resolve the developing boundary layer and wake regions using grids which are much coarser than those employed by other numerical methods. It is shown that ten cells in the cross-stream direction are sufficient to accurately resolve the developing airfoil boundary layer.

Self-similar solutions to isothermal shock problems

NASA Astrophysics Data System (ADS)

Deschner, Stephan C.; Illenseer, Tobias F.; Duschl, Wolfgang J.

We investigate exact solutions for isothermal shock problems in different one-dimensional geometries. These solutions are given as analytical expressions if possible, or are computed using standard numerical methods for solving ordinary differential equations. We test the numerical solutions against the analytical expressions to verify the correctness of all numerical algorithms. We use similarity methods to derive a system of ordinary differential equations (ODE) yielding exact solutions for power law density distributions as initial conditions. Further, the system of ODEs accounts for implosion problems (IP) as well as explosion problems (EP) by changing the initial or boundary conditions, respectively. Taking genuinely isothermal approximations into account leads to additional insights of EPs in contrast to earlier models. We neglect a constant initial energy contribution but introduce a parameter to adjust the initial mass distribution of the system. Moreover, we show that due to this parameter a constant initial density is not allowed for isothermal EPs. Reasonable restrictions for this parameter are given. Both, the (genuinely) isothermal implosion as well as the explosion problem are solved for the first time.

A Numerical Simulation of Scattering from One-Dimensional Inhomogeneous Dielectric Random Surfaces

NASA Technical Reports Server (NTRS)

Sarabandi, Kamal; Oh, Yisok; Ulaby, Fawwaz T.

1996-01-01

In this paper, an efficient numerical solution for the scattering problem of inhomogeneous dielectric rough surfaces is presented. The inhomogeneous dielectric random surface represents a bare soil surface and is considered to be comprised of a large number of randomly positioned dielectric humps of different sizes, shapes, and dielectric constants above an impedance surface. Clods with nonuniform moisture content and rocks are modeled by inhomogeneous dielectric humps and the underlying smooth wet soil surface is modeled by an impedance surface. In this technique, an efficient numerical solution for the constituent dielectric humps over an impedance surface is obtained using Green's function derived by the exact image theory in conjunction with the method of moments. The scattered field from a sample of the rough surface is obtained by summing the scattered fields from all the individual humps of the surface coherently ignoring the effect of multiple scattering between the humps. The statistical behavior of the scattering coefficient sigma(sup 0) is obtained from the calculation of scattered fields of many different realizations of the surface. Numerical results are presented for several different roughnesses and dielectric constants of the random surfaces. The numerical technique is verified by comparing the numerical solution with the solution based on the small perturbation method and the physical optics model for homogeneous rough surfaces. This technique can be used to study the behavior of scattering coefficient and phase difference statistics of rough soil surfaces for which no analytical solution exists.

Fluid Flow and Solidification Under Combined Action of Magnetic Fields and Microgravity

NASA Technical Reports Server (NTRS)

Li, B. Q.; Shu, Y.; Li, K.; deGroh, H. C.

2002-01-01

Mathematical models, both 2-D and 3-D, are developed to represent g-jitter induced fluid flows and their effects on solidification under combined action of magnetic fields and microgravity. The numerical model development is based on the finite element solution of governing equations describing the transient g-jitter driven fluid flows, heat transfer and solutal transport during crystal growth with and without an applied magnetic field in space vehicles. To validate the model predictions, a ground-based g-jitter simulator is developed using the oscillating wall temperatures where timely oscillating fluid flows are measured using a laser PIV system. The measurements are compared well with numerical results obtained from the numerical models. Results show that a combined action derived from magnetic damping and microgravity can be an effective means to control the melt flow and solutal transport in space single crystal growth systems.

An upwind space-time conservation element and solution element scheme for solving dusty gas flow model

NASA Astrophysics Data System (ADS)

Rehman, Asad; Ali, Ishtiaq; Qamar, Shamsul

An upwind space-time conservation element and solution element (CE/SE) scheme is extended to numerically approximate the dusty gas flow model. Unlike central CE/SE schemes, the current method uses the upwind procedure to derive the numerical fluxes through the inner boundary of conservation elements. These upwind fluxes are utilized to calculate the gradients of flow variables. For comparison and validation, the central upwind scheme is also applied to solve the same dusty gas flow model. The suggested upwind CE/SE scheme resolves the contact discontinuities more effectively and preserves the positivity of flow variables in low density flows. Several case studies are considered and the results of upwind CE/SE are compared with the solutions of central upwind scheme. The numerical results show better performance of the upwind CE/SE method as compared to the central upwind scheme.

Numerical Analysis of Incipient Separation on 53 Deg Swept Diamond Wing

NASA Technical Reports Server (NTRS)

Frink, Neal T.

2015-01-01

A systematic analysis of incipient separation and subsequent vortex formation from moderately swept blunt leading edges is presented for a 53 deg swept diamond wing. This work contributes to a collective body of knowledge generated within the NATO/STO AVT-183 Task Group titled 'Reliable Prediction of Separated Flow Onset and Progression for Air and Sea Vehicles'. The objective is to extract insights from the experimentally measured and numerically computed flow fields that might enable turbulence experts to further improve their models for predicting swept blunt leading-edge flow separation. Details of vortex formation are inferred from numerical solutions after establishing a good correlation of the global flow field and surface pressure distributions between wind tunnel measurements and computed flow solutions. From this, significant and sometimes surprising insights into the nature of incipient separation and part-span vortex formation are derived from the wealth of information available in the computational solutions.

An adaptive gridless methodology in one dimension

DOE Office of Scientific and Technical Information (OSTI.GOV)

Snyder, N.T.; Hailey, C.E.

1996-09-01

Gridless numerical analysis offers great potential for accurately solving for flow about complex geometries or moving boundary problems. Because gridless methods do not require point connection, the mesh cannot twist or distort. The gridless method utilizes a Taylor series about each point to obtain the unknown derivative terms from the current field variable estimates. The governing equation is then numerically integrated to determine the field variables for the next iteration. Effects of point spacing and Taylor series order on accuracy are studied, and they follow similar trends of traditional numerical techniques. Introducing adaption by point movement using a spring analogymore » allows the solution method to track a moving boundary. The adaptive gridless method models linear, nonlinear, steady, and transient problems. Comparison with known analytic solutions is given for these examples. Although point movement adaption does not provide a significant increase in accuracy, it helps capture important features and provides an improved solution.« less

Challenges to Applying a Metamodel for Groundwater Flow Beyond Underlying Numerical Model Boundaries

NASA Astrophysics Data System (ADS)

Reeves, H. W.; Fienen, M. N.; Feinstein, D.

2015-12-01

Metamodels of environmental behavior offer opportunities for decision support, adaptive management, and increased stakeholder engagement through participatory modeling and model exploration. Metamodels are derived from calibrated, computationally demanding, numerical models. They may potentially be applied to non-modeled areas to provide screening or preliminary analysis tools for areas that do not yet have the benefit of more comprehensive study. In this decision-support mode, they may be fulfilling a role often accomplished by application of analytical solutions. The major challenge to transferring a metamodel to a non-modeled area is how to quantify the spatial data in the new area of interest in such a way that it is consistent with the data used to derive the metamodel. Tests based on transferring a metamodel derived from a numerical groundwater-flow model of the Lake Michigan Basin to other glacial settings across the northern U.S. show that the spatial scale of the numerical model must be appropriately scaled to adequately represent different settings. Careful GIS analysis of the numerical model, metamodel, and new area of interest is required for successful transfer of results.

QCD triple Pomeron coupling from string amplitudes

NASA Astrophysics Data System (ADS)

Bialas, A.; Navelet, H.; Peschanski, R.

1998-06-01

Using the recent solution of the triple Pomeron coupling in the QCD dipole picture as a closed string amplitude with six legs, its analytical form in terms of hypergeometric functions and numerical value are derived.

Accurate boundary conditions for exterior problems in gas dynamics

NASA Technical Reports Server (NTRS)

Hagstrom, Thomas; Hariharan, S. I.

1988-01-01

The numerical solution of exterior problems is typically accomplished by introducing an artificial, far field boundary and solving the equations on a truncated domain. For hyperbolic systems, boundary conditions at this boundary are often derived by imposing a principle of no reflection. However, waves with spherical symmetry in gas dynamics satisfy equations where incoming and outgoing Riemann variables are coupled. This suggests that natural reflections may be important. A reflecting boundary condition is proposed based on an asymptotic solution of the far field equations. Nonlinear energy estimates are obtained for the truncated problem and numerical experiments presented to validate the theory.

Accurate boundary conditions for exterior problems in gas dynamics

NASA Technical Reports Server (NTRS)

Hagstrom, Thomas; Hariharan, S. I.

1988-01-01

The numerical solution of exterior problems is typically accomplished by introducing an artificial, far-field boundary and solving the equations on a truncated domain. For hyperbolic systems, boundary conditions at this boundary are often derived by imposing a principle of no reflection. However, waves with spherical symmetry in gas dynamics satisfy equations where incoming and outgoing Riemann variables are coupled. This suggests that natural reflections may be important. A reflecting boundary condition is proposed based on an asymptotic solution of the far-field equations. Nonlinear energy estimates are obtained for the truncated problem and numerical experiments presented to validate the theory.

Numeric Solutions of Dirac-Gursey Spinor Field Equation Under External Gaussian White Noise

NASA Astrophysics Data System (ADS)

Aydogmus, Fatma

2016-06-01

In this paper, we consider the Dirac-Gursey spinor field equation that has particle-like solutions derived classical field equations so-called instantons, formed by using Heisenberg ansatz, under the effect of an additional Gaussian white noise term. Our purpose is to understand how the behavior of spinor-type excited instantons in four dimensions can be affected by noise. Thus, we simulate the phase portraits and Poincaré sections of the obtained system numerically both with and without noise. Recurrence plots are also given for more detailed information regarding the system.

Modeling electro-magneto-hydrodynamic thermo-fluidic transport of biofluids with new trend of fractional derivative without singular kernel

NASA Astrophysics Data System (ADS)

Abdulhameed, M.; Vieru, D.; Roslan, R.

2017-10-01

This paper investigates the electro-magneto-hydrodynamic flow of the non-Newtonian behavior of biofluids, with heat transfer, through a cylindrical microchannel. The fluid is acted by an arbitrary time-dependent pressure gradient, an external electric field and an external magnetic field. The governing equations are considered as fractional partial differential equations based on the Caputo-Fabrizio time-fractional derivatives without singular kernel. The usefulness of fractional calculus to study fluid flows or heat and mass transfer phenomena was proven. Several experimental measurements led to conclusion that, in such problems, the models described by fractional differential equations are more suitable. The most common time-fractional derivative used in Continuum Mechanics is Caputo derivative. However, two disadvantages appear when this derivative is used. First, the definition kernel is a singular function and, secondly, the analytical expressions of the problem solutions are expressed by generalized functions (Mittag-Leffler, Lorenzo-Hartley, Robotnov, etc.) which, generally, are not adequate to numerical calculations. The new time-fractional derivative Caputo-Fabrizio, without singular kernel, is more suitable to solve various theoretical and practical problems which involve fractional differential equations. Using the Caputo-Fabrizio derivative, calculations are simpler and, the obtained solutions are expressed by elementary functions. Analytical solutions of the biofluid velocity and thermal transport are obtained by means of the Laplace and finite Hankel transforms. The influence of the fractional parameter, Eckert number and Joule heating parameter on the biofluid velocity and thermal transport are numerically analyzed and graphic presented. This fact can be an important in Biochip technology, thus making it possible to use this analysis technique extremely effective to control bioliquid samples of nanovolumes in microfluidic devices used for biological analysis and medical diagnosis.

Generalized analytical solutions to sequentially coupled multi-species advective-dispersive transport equations in a finite domain subject to an arbitrary time-dependent source boundary condition

NASA Astrophysics Data System (ADS)

Chen, Jui-Sheng; Liu, Chen-Wuing; Liang, Ching-Ping; Lai, Keng-Hsin

2012-08-01

SummaryMulti-species advective-dispersive transport equations sequentially coupled with first-order decay reactions are widely used to describe the transport and fate of the decay chain contaminants such as radionuclide, chlorinated solvents, and nitrogen. Although researchers attempted to present various types of methods for analytically solving this transport equation system, the currently available solutions are mostly limited to an infinite or a semi-infinite domain. A generalized analytical solution for the coupled multi-species transport problem in a finite domain associated with an arbitrary time-dependent source boundary is not available in the published literature. In this study, we first derive generalized analytical solutions for this transport problem in a finite domain involving arbitrary number of species subject to an arbitrary time-dependent source boundary. Subsequently, we adopt these derived generalized analytical solutions to obtain explicit analytical solutions for a special-case transport scenario involving an exponentially decaying Bateman type time-dependent source boundary. We test the derived special-case solutions against the previously published coupled 4-species transport solution and the corresponding numerical solution with coupled 10-species transport to conduct the solution verification. Finally, we compare the new analytical solutions derived for a finite domain against the published analytical solutions derived for a semi-infinite domain to illustrate the effect of the exit boundary condition on coupled multi-species transport with an exponential decaying source boundary. The results show noticeable discrepancies between the breakthrough curves of all the species in the immediate vicinity of the exit boundary obtained from the analytical solutions for a finite domain and a semi-infinite domain for the dispersion-dominated condition.

Mass-conservative reconstruction of Galerkin velocity fields for transport simulations

NASA Astrophysics Data System (ADS)

Scudeler, C.; Putti, M.; Paniconi, C.

2016-08-01

Accurate calculation of mass-conservative velocity fields from numerical solutions of Richards' equation is central to reliable surface-subsurface flow and transport modeling, for example in long-term tracer simulations to determine catchment residence time distributions. In this study we assess the performance of a local Larson-Niklasson (LN) post-processing procedure for reconstructing mass-conservative velocities from a linear (P1) Galerkin finite element solution of Richards' equation. This approach, originally proposed for a-posteriori error estimation, modifies the standard finite element velocities by imposing local conservation on element patches. The resulting reconstructed flow field is characterized by continuous fluxes on element edges that can be efficiently used to drive a second order finite volume advective transport model. Through a series of tests of increasing complexity that compare results from the LN scheme to those using velocity fields derived directly from the P1 Galerkin solution, we show that a locally mass-conservative velocity field is necessary to obtain accurate transport results. We also show that the accuracy of the LN reconstruction procedure is comparable to that of the inherently conservative mixed finite element approach, taken as a reference solution, but that the LN scheme has much lower computational costs. The numerical tests examine steady and unsteady, saturated and variably saturated, and homogeneous and heterogeneous cases along with initial and boundary conditions that include dry soil infiltration, alternating solute and water injection, and seepage face outflow. Typical problems that arise with velocities derived from P1 Galerkin solutions include outgoing solute flux from no-flow boundaries, solute entrapment in zones of low hydraulic conductivity, and occurrences of anomalous sources and sinks. In addition to inducing significant mass balance errors, such manifestations often lead to oscillations in concentration values that can moreover cause the numerical solution to explode. These problems do not occur when using LN post-processed velocities.

Pinching solutions of slender cylindrical jets

NASA Technical Reports Server (NTRS)

Papageorgiou, Demetrios T.; Orellana, Oscar

1993-01-01

Simplified equations for slender jets are derived for a circular jet of one fluid flowing into an ambient second fluid, the flow being confined in a circular tank. Inviscid flows are studied which include both surface tension effects and Kelvin-Helmholtz instability. For slender jets a coupled nonlinear system of equations is found for the jet shape and the axial velocity jump across it. The equations can break down after a finite time and similarity solutions are constructed, and studied analytically and numerically. The break-ups found pertain to the jet pinching after a finite time, without violation of the slender jet ansatz. The system is conservative and admissible singular solutions are those which conserve the total energy, mass, and momentum. Such solutions are constructed analytically and numerically, and in the case of vortex sheets with no surface tension certain solutions are given in closed form.

The impact of the form of the Euler equations for radial flow in cylindrical and spherical coordinates on numerical conservation and accuracy

NASA Astrophysics Data System (ADS)

Crittenden, P. E.; Balachandar, S.

2018-07-01

The radial one-dimensional Euler equations are often rewritten in what is known as the geometric source form. The differential operator is identical to the Cartesian case, but source terms result. Since the theory and numerical methods for the Cartesian case are well-developed, they are often applied without modification to cylindrical and spherical geometries. However, numerical conservation is lost. In this article, AUSM^+-up is applied to a numerically conservative (discrete) form of the Euler equations labeled the geometric form, a nearly conservative variation termed the geometric flux form, and the geometric source form. The resulting numerical methods are compared analytically and numerically through three types of test problems: subsonic, smooth, steady-state solutions, Sedov's similarity solution for point or line-source explosions, and shock tube problems. Numerical conservation is analyzed for all three forms in both spherical and cylindrical coordinates. All three forms result in constant enthalpy for steady flows. The spatial truncation errors have essentially the same order of convergence, but the rate constants are superior for the geometric and geometric flux forms for the steady-state solutions. Only the geometric form produces the correct shock location for Sedov's solution, and a direct connection between the errors in the shock locations and energy conservation is found. The shock tube problems are evaluated with respect to feature location using an approximation with a very fine discretization as the benchmark. Extensions to second order appropriate for cylindrical and spherical coordinates are also presented and analyzed numerically. Conclusions are drawn, and recommendations are made. A derivation of the steady-state solution is given in the Appendix.

The impact of the form of the Euler equations for radial flow in cylindrical and spherical coordinates on numerical conservation and accuracy

NASA Astrophysics Data System (ADS)

Crittenden, P. E.; Balachandar, S.

2018-03-01

The radial one-dimensional Euler equations are often rewritten in what is known as the geometric source form. The differential operator is identical to the Cartesian case, but source terms result. Since the theory and numerical methods for the Cartesian case are well-developed, they are often applied without modification to cylindrical and spherical geometries. However, numerical conservation is lost. In this article, AUSM^+ -up is applied to a numerically conservative (discrete) form of the Euler equations labeled the geometric form, a nearly conservative variation termed the geometric flux form, and the geometric source form. The resulting numerical methods are compared analytically and numerically through three types of test problems: subsonic, smooth, steady-state solutions, Sedov's similarity solution for point or line-source explosions, and shock tube problems. Numerical conservation is analyzed for all three forms in both spherical and cylindrical coordinates. All three forms result in constant enthalpy for steady flows. The spatial truncation errors have essentially the same order of convergence, but the rate constants are superior for the geometric and geometric flux forms for the steady-state solutions. Only the geometric form produces the correct shock location for Sedov's solution, and a direct connection between the errors in the shock locations and energy conservation is found. The shock tube problems are evaluated with respect to feature location using an approximation with a very fine discretization as the benchmark. Extensions to second order appropriate for cylindrical and spherical coordinates are also presented and analyzed numerically. Conclusions are drawn, and recommendations are made. A derivation of the steady-state solution is given in the Appendix.

A unified convergence theory of a numerical method, and applications to the replenishment policies.

PubMed

Mi, Xiang-jiang; Wang, Xing-hua

2004-01-01

In determining the replenishment policy for an inventory system, some researchers advocated that the iterative method of Newton could be applied to the derivative of the total cost function in order to get the optimal solution. But this approach requires calculation of the second derivative of the function. Avoiding this complex computation we use another iterative method presented by the second author. One of the goals of this paper is to present a unified convergence theory of this method. Then we give a numerical example to show the application of our theory.

The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

PubMed

Khader, M M

2013-10-01

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

Evolution of castalagin and vescalagin in ethanol solutions. Identification of new derivatives.

PubMed

Puech, J L; Mertz, C; Michon, V; Le Guernevé, C; Doco, T; Hervé Du Penhoat, C

1999-05-01

Brandies, cognac, armagnac, whiskeys, and rums are aged in oak barrels to improve their organoleptic properties. During this period, numerous compounds such as ellagitannins are extracted from the wood and can subsequently be transformed into new derivatives by chemical reactions. Model solutions of castalagin and vescalagin have been studied to determine the behavior of polyphenols in ethanol-water. Upon prolonged exposure to 40 and 70% (v/v) ethanol at room temperature, hemiketal derivatives containing ethoxy groups have been characterized by LC/MS and NMR. These compounds further evolve to afford the corresponding ketals. They have also been detected in the extracts of oak wood stored under similar conditions.

Dynamics with infinitely many time derivatives in Friedmann-Robertson-Walker background and rolling tachyons

NASA Astrophysics Data System (ADS)

Joukovskaya, Liudmila

2009-02-01

Dynamics with infinitely many time derivatives has place in string field theory and have been profoundly investigated there. Recently there has been considerable interest in theories with infinitely many derivatives in the cosmological context in view of new features which these theories might accommodate owing to nonlocal interaction. In present work we continue investigation of such models, as a concrete example we study the dynamics of unstable D-brane in the open string theory in the Friedmann-Robertson-Walker background. We construct numerical solutions describing dynamical interpolation between the perturbative and non-perturbative vacua. The obtained solutions have several interesting properties and might be of interest from the cosmological points of view.

Efficient computation of the Grünwald-Letnikov fractional diffusion derivative using adaptive time step memory

NASA Astrophysics Data System (ADS)

MacDonald, Christopher L.; Bhattacharya, Nirupama; Sprouse, Brian P.; Silva, Gabriel A.

2015-09-01

Computing numerical solutions to fractional differential equations can be computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In general, numerical approaches that depend on truncating part of the system history while efficient, can suffer from high degrees of error and inaccuracy. Here we present an adaptive time step memory method for smooth functions applied to the Grünwald-Letnikov fractional diffusion derivative. This method is computationally efficient and results in smaller errors during numerical simulations. Sampled points along the system's history at progressively longer intervals are assumed to reflect the values of neighboring time points. By including progressively fewer points backward in time, a temporally 'weighted' history is computed that includes contributions from the entire past of the system, maintaining accuracy, but with fewer points actually calculated, greatly improving computational efficiency.

Rapid computation of directional wellbore drawdown in a confined aquifer via Poisson resummation

NASA Astrophysics Data System (ADS)

Blumenthal, Benjamin J.; Zhan, Hongbin

2016-08-01

We have derived a rapidly computed analytical solution for drawdown caused by a partially or fully penetrating directional wellbore (vertical, horizontal, or slant) via Green's function method. The mathematical model assumes an anisotropic, homogeneous, confined, box-shaped aquifer. Any dimension of the box can have one of six possible boundary conditions: 1) both sides no-flux; 2) one side no-flux - one side constant-head; 3) both sides constant-head; 4) one side no-flux; 5) one side constant-head; 6) free boundary conditions. The solution has been optimized for rapid computation via Poisson Resummation, derivation of convergence rates, and numerical optimization of integration techniques. Upon application of the Poisson Resummation method, we were able to derive two sets of solutions with inverse convergence rates, namely an early-time rapidly convergent series (solution-A) and a late-time rapidly convergent series (solution-B). From this work we were able to link Green's function method (solution-B) back to image well theory (solution-A). We then derived an equation defining when the convergence rate between solution-A and solution-B is the same, which we termed the switch time. Utilizing the more rapidly convergent solution at the appropriate time, we obtained rapid convergence at all times. We have also shown that one may simplify each of the three infinite series for the three-dimensional solution to 11 terms and still maintain a maximum relative error of less than 10-14.

Stochastic responses of Van der Pol vibro-impact system with fractional derivative damping excited by Gaussian white noise.

PubMed

Xiao, Yanwen; Xu, Wei; Wang, Liang

2016-03-01

This paper focuses on the study of the stochastic Van der Pol vibro-impact system with fractional derivative damping under Gaussian white noise excitation. The equations of the original system are simplified by non-smooth transformation. For the simplified equation, the stochastic averaging approach is applied to solve it. Then, the fractional derivative damping term is facilitated by a numerical scheme, therewith the fourth-order Runge-Kutta method is used to obtain the numerical results. And the numerical simulation results fit the analytical solutions. Therefore, the proposed analytical means to study this system are proved to be feasible. In this context, the effects on the response stationary probability density functions (PDFs) caused by noise excitation, restitution condition, and fractional derivative damping are considered, in addition the stochastic P-bifurcation is also explored in this paper through varying the value of the coefficient of fractional derivative damping and the restitution coefficient. These system parameters not only influence the response PDFs of this system but also can cause the stochastic P-bifurcation.

Stochastic responses of Van der Pol vibro-impact system with fractional derivative damping excited by Gaussian white noise

DOE Office of Scientific and Technical Information (OSTI.GOV)

Xiao, Yanwen; Xu, Wei, E-mail: weixu@nwpu.edu.cn; Wang, Liang

2016-03-15

This paper focuses on the study of the stochastic Van der Pol vibro-impact system with fractional derivative damping under Gaussian white noise excitation. The equations of the original system are simplified by non-smooth transformation. For the simplified equation, the stochastic averaging approach is applied to solve it. Then, the fractional derivative damping term is facilitated by a numerical scheme, therewith the fourth-order Runge-Kutta method is used to obtain the numerical results. And the numerical simulation results fit the analytical solutions. Therefore, the proposed analytical means to study this system are proved to be feasible. In this context, the effects onmore » the response stationary probability density functions (PDFs) caused by noise excitation, restitution condition, and fractional derivative damping are considered, in addition the stochastic P-bifurcation is also explored in this paper through varying the value of the coefficient of fractional derivative damping and the restitution coefficient. These system parameters not only influence the response PDFs of this system but also can cause the stochastic P-bifurcation.« less

Rotation relaxation splitting for optimizing parallel RF excitation pulses with T1 - and T2 -relaxations in MRI

NASA Astrophysics Data System (ADS)

Majewski, Kurt

2018-03-01

Exact solutions of the Bloch equations with T1 - and T2 -relaxation terms for piecewise constant magnetic fields are numerically challenging. We therefore investigate an approximation for the achieved magnetization in which rotations and relaxations are split into separate operations. We develop an estimate for its accuracy and explicit first and second order derivatives with respect to the complex excitation radio frequency voltages. In practice, the deviation between an exact solution of the Bloch equations and this rotation relaxation splitting approximation seems negligible. Its computation times are similar to exact solutions without relaxation terms. We apply the developed theory to numerically optimize radio frequency excitation waveforms with T1 - and T2 -relaxations in several examples.

Numerical uncertainty in computational engineering and physics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hemez, Francois M

2009-01-01

Obtaining a solution that approximates ordinary or partial differential equations on a computational mesh or grid does not necessarily mean that the solution is accurate or even 'correct'. Unfortunately assessing the quality of discrete solutions by questioning the role played by spatial and temporal discretizations generally comes as a distant third to test-analysis comparison and model calibration. This publication is contributed to raise awareness of the fact that discrete solutions introduce numerical uncertainty. This uncertainty may, in some cases, overwhelm in complexity and magnitude other sources of uncertainty that include experimental variability, parametric uncertainty and modeling assumptions. The concepts ofmore » consistency, convergence and truncation error are overviewed to explain the articulation between the exact solution of continuous equations, the solution of modified equations and discrete solutions computed by a code. The current state-of-the-practice of code and solution verification activities is discussed. An example in the discipline of hydro-dynamics illustrates the significant effect that meshing can have on the quality of code predictions. A simple method is proposed to derive bounds of solution uncertainty in cases where the exact solution of the continuous equations, or its modified equations, is unknown. It is argued that numerical uncertainty originating from mesh discretization should always be quantified and accounted for in the overall uncertainty 'budget' that supports decision-making for applications in computational physics and engineering.« less

Sound Emission of Rotor Induced Deformations of Generator Casings

NASA Technical Reports Server (NTRS)

Polifke, W.; Mueller, B.; Yee, H. C.; Mansour, Nagi (Technical Monitor)

2001-01-01

The casing of large electrical generators can be deformed slightly by the rotor's magnetic field. The sound emission produced by these periodic deformations, which could possibly exceed guaranteed noise emission limits, is analysed analytically and numerically. From the deformation of the casing, the normal velocity of the generator's surface is computed. Taking into account the corresponding symmetry, an analytical solution for the acoustic pressure outside the generator is round in terms of the Hankel function of second order. The normal velocity or the generator surface provides the required boundary condition for the acoustic pressure and determines the magnitude of pressure oscillations. For the numerical simulation, the nonlinear 2D Euler equations are formulated In a perturbation form for low Mach number Computational Aeroacoustics (CAA). The spatial derivatives are discretized by the classical sixth-order central interior scheme and a third-order boundary scheme. Spurious high frequency oscillations are damped by a characteristic-based artificial compression method (ACM) filter. The time derivatives are approximated by the classical 4th-order Runge-Kutta method. The numerical results are In excellent agreement with the analytical solution.

The RKGL method for the numerical solution of initial-value problems

NASA Astrophysics Data System (ADS)

Prentice, J. S. C.

2008-04-01

We introduce the RKGL method for the numerical solution of initial-value problems of the form y'=f(x,y), y(a)=[alpha]. The method is a straightforward modification of a classical explicit Runge-Kutta (RK) method, into which Gauss-Legendre (GL) quadrature has been incorporated. The idea is to enhance the efficiency of the method by reducing the number of times the derivative f(x,y) needs to be computed. The incorporation of GL quadrature serves to enhance the global order of the method by, relative to the underlying RK method. Indeed, the RKGL method has a global error of the form Ahr+1+Bh2m, where r is the order of the RK method and m is the number of nodes used in the GL component. In this paper we derive this error expression and show that RKGL is consistent, convergent and strongly stable.

Kinematic validation of a quasi-geostrophic model for the fast dynamics in the Earth's outer core

NASA Astrophysics Data System (ADS)

Maffei, S.; Jackson, A.

2017-09-01

We derive a quasi-geostrophic (QG) system of equations suitable for the description of the Earth's core dynamics on interannual to decadal timescales. Over these timescales, rotation is assumed to be the dominant force and fluid motions are strongly invariant along the direction parallel to the rotation axis. The diffusion-free, QG system derived here is similar to the one derived in Canet et al. but the projection of the governing equations on the equatorial disc is handled via vertical integration and mass conservation is applied to the velocity field. Here we carefully analyse the properties of the resulting equations and we validate them neglecting the action of the Lorentz force in the momentum equation. We derive a novel analytical solution describing the evolution of the magnetic field under these assumptions in the presence of a purely azimuthal flow and an alternative formulation that allows us to numerically solve the evolution equations with a finite element method. The excellent agreement we found with the analytical solution proves that numerical integration of the QG system is possible and that it preserves important physical properties of the magnetic field. Implementation of magnetic diffusion is also briefly considered.

Effect of Numerical Error on Gravity Field Estimation for GRACE and Future Gravity Missions

NASA Astrophysics Data System (ADS)

McCullough, Christopher; Bettadpur, Srinivas

2015-04-01

In recent decades, gravity field determination from low Earth orbiting satellites, such as the Gravity Recovery and Climate Experiment (GRACE), has become increasingly more effective due to the incorporation of high accuracy measurement devices. Since instrumentation quality will only increase in the near future and the gravity field determination process is computationally and numerically intensive, numerical error from the use of double precision arithmetic will eventually become a prominent error source. While using double-extended or quadruple precision arithmetic will reduce these errors, the numerical limitations of current orbit determination algorithms and processes must be accurately identified and quantified in order to adequately inform the science data processing techniques of future gravity missions. The most obvious numerical limitation in the orbit determination process is evident in the comparison of measured observables with computed values, derived from mathematical models relating the satellites' numerically integrated state to the observable. Significant error in the computed trajectory will corrupt this comparison and induce error in the least squares solution of the gravitational field. In addition, errors in the numerically computed trajectory propagate into the evaluation of the mathematical measurement model's partial derivatives. These errors amalgamate in turn with numerical error from the computation of the state transition matrix, computed using the variational equations of motion, in the least squares mapping matrix. Finally, the solution of the linearized least squares system, computed using a QR factorization, is also susceptible to numerical error. Certain interesting combinations of each of these numerical errors are examined in the framework of GRACE gravity field determination to analyze and quantify their effects on gravity field recovery.

Functional entropy variables: A new methodology for deriving thermodynamically consistent algorithms for complex fluids, with particular reference to the isothermal Navier–Stokes–Korteweg equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liu, Ju, E-mail: jliu@ices.utexas.edu; Gomez, Hector; Evans, John A.

2013-09-01

We propose a new methodology for the numerical solution of the isothermal Navier–Stokes–Korteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionallymore » stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws.« less

Time-dependent flow model of a generalized Burgers' fluid with fractional derivatives through a cylindrical domain: An exact and numerical approach

NASA Astrophysics Data System (ADS)

Safdar, Rabia; Imran, M.; Khalique, Chaudry Masood

2018-06-01

Exact solutions for velocity field and tangential stress for rotational flow of a generalized Burgers' fluid within an infinite circular pipe are derived by using the methods of Laplace and finite Hankel transformations. Firstly we take the position of fluid at rest and then the fluid flow due to the rotation of the pipe around the axis of flow having time dependant angular velocity. The exact solutions are presented in terms of the generalized Ga,b,c (., t) -functions. The corresponding results can be freely specified for the same results of Burgers', Oldroyd B, Maxwell, second grade and Newtonian fluids (performing the same motion) as particular cases of the results obtained earlier. The impact of the different parameters, individually and in comparison, are represented by graphical demonstrations. Secondly the numerical solutions for velocity and stress are also obtained with the help of Laplace transformation, Gaver Stehfest's algorithm and MATHCAD. Finally a comparison of both methods for the same problem is done and shows the consistency of results.

The unified acoustic and aerodynamic prediction theory of advanced propellers in the time domain

NASA Technical Reports Server (NTRS)

Farassat, F.

1984-01-01

This paper presents some numerical results for the noise of an advanced supersonic propeller based on a formulation published last year. This formulation was derived to overcome some of the practical numerical difficulties associated with other acoustic formulations. The approach is based on the Ffowcs Williams-Hawkings equation and time domain analysis is used. To illustrate the method of solution, a model problem in three dimensions and based on the Laplace equation is solved. A brief sketch of derivation of the acoustic formula is then given. Another model problem is used to verify validity of the acoustic formulation. A recent singular integral equation for aerodynamic applications derived from the acoustic formula is also presented here.

Phantom behavior bounce with tachyon and non-minimal derivative coupling

DOE Office of Scientific and Technical Information (OSTI.GOV)

Banijamali, A.; Fazlpour, B., E-mail: a.banijamali@nit.ac.ir, E-mail: b.fazlpour@umz.ac.ir

2012-01-01

The bouncing cosmology provides a successful solution of the cosmological singularity problem. In this paper, we study the bouncing behavior of a single scalar field model with tachyon field non-minimally coupled to itself, its derivative and to the curvature. By utilizing the numerical calculations we will show that the bouncing solution can appear in the universe dominated by such a quintom matter with equation of state crossing the phantom divide line. We also investigate the classical stability of our model using the phase velocity of the homogeneous perturbations of the tachyon scalar field.

A Posteriori Finite Element Bounds for Sensitivity Derivatives of Partial-Differential-Equation Outputs. Revised

NASA Technical Reports Server (NTRS)

Lewis, Robert Michael; Patera, Anthony T.; Peraire, Jaume

1998-01-01

We present a Neumann-subproblem a posteriori finite element procedure for the efficient and accurate calculation of rigorous, 'constant-free' upper and lower bounds for sensitivity derivatives of functionals of the solutions of partial differential equations. The design motivation for sensitivity derivative error control is discussed; the a posteriori finite element procedure is described; the asymptotic bounding properties and computational complexity of the method are summarized; and illustrative numerical results are presented.

Anisotropic cosmological solutions in R + R^2 gravity

NASA Astrophysics Data System (ADS)

Müller, Daniel; Ricciardone, Angelo; Starobinsky, Alexei A.; Toporensky, Aleksey

2018-04-01

In this paper we investigate the past evolution of an anisotropic Bianchi I universe in R+R^2 gravity. Using the dynamical system approach we show that there exists a new two-parameter set of solutions that includes both an isotropic "false radiation" solution and an anisotropic generalized Kasner solution, which is stable. We derive the analytic behavior of the shear from a specific property of f( R) gravity and the analytic asymptotic form of the Ricci scalar when approaching the initial singularity. Finally, we numerically check our results.

Solutions of the benchmark problems by the dispersion-relation-preserving scheme

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.; Shen, H.; Kurbatskii, K. A.; Auriault, L.

1995-01-01

The 7-point stencil Dispersion-Relation-Preserving scheme of Tam and Webb is used to solve all the six categories of the CAA benchmark problems. The purpose is to show that the scheme is capable of solving linear, as well as nonlinear aeroacoustics problems accurately. Nonlinearities, inevitably, lead to the generation of spurious short wave length numerical waves. Often, these spurious waves would overwhelm the entire numerical solution. In this work, the spurious waves are removed by the addition of artificial selective damping terms to the discretized equations. Category 3 problems are for testing radiation and outflow boundary conditions. In solving these problems, the radiation and outflow boundary conditions of Tam and Webb are used. These conditions are derived from the asymptotic solutions of the linearized Euler equations. Category 4 problems involved solid walls. Here, the wall boundary conditions for high-order schemes of Tam and Dong are employed. These conditions require the use of one ghost value per boundary point per physical boundary condition. In the second problem of this category, the governing equations, when written in cylindrical coordinates, are singular along the axis of the radial coordinate. The proper boundary conditions at the axis are derived by applying the limiting process of r approaches 0 to the governing equations. The Category 5 problem deals with the numerical noise issue. In the present approach, the time-independent mean flow solution is computed first. Once the residual drops to the machine noise level, the incident sound wave is turned on gradually. The solution is marched in time until a time-periodic state is reached. No exact solution is known for the Category 6 problem. Because of this, the problem is formulated in two totally different ways, first as a scattering problem then as a direct simulation problem. There is good agreement between the two numerical solutions. This offers confidence in the computed results. Both formulations are solved as initial value problems. As such, no Kutta condition is required at the trailing edge of the airfoil.

Elastic constants and pressure derivative of elastic constants of Si1-xGex solid solution

NASA Astrophysics Data System (ADS)

Jivani, A. R.; Baria, J. K.; Vyas, P. S.; Jani, A. R.

2013-02-01

Elastic properties of Si1-xGex solid solution with arbitrary (atomic) concentration (x) are studied using the pseudo-alloy atom model based on the pseudopotential theory and on the higher-order perturbation scheme with the application of our own proposed model potential. We have used local-field correction function proposed by Sarkar et al to study Si-Ge system. The Elastic constants and pressure derivatives of elastic constants of the solid solution is investigated with different concentration x of Ge. It is found in the present study that the calculated numerical values of the aforesaid physical properties of Si-Ge system are function of x. The elastic constants (C11, C12 and C44) decrease linearly with increase in concentration x and pressure derivative of elastic constants (C11, C12 and C44) increase with the concentration x of Ge. This study provides better set of theoretical results for such solid solution for further comparison either with theoretical or experimental results.

Exact analytical solutions of continuity equation for electron beams precipitating in Coulomb collisions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dobranskis, R. R.; Zharkova, V. V., E-mail: valentina.zharkova@northumbria.ac.uk

2014-06-10

The original continuity equation (CE) used for the interpretation of the power law energy spectra of beam electrons in flares was written and solved for an electron beam flux while ignoring an additional free term with an electron density. In order to remedy this omission, the original CE for electron flux, considering beam's energy losses in Coulomb collisions, was first differentiated by the two independent variables: depth and energy leading to partial differential equation for an electron beam density instead of flux with the additional free term. The analytical solution of this partial differential continuity equation (PDCE) is obtained bymore » using the method of characteristics. This solution is further used to derive analytical expressions for mean electron spectra for Coulomb collisions and to carry out numeric calculations of hard X-ray (HXR) photon spectra for beams with different parameters. The solutions revealed a significant departure of electron densities at lower energies from the original results derived from the CE for the flux obtained for Coulomb collisions. This departure is caused by the additional exponential term that appeared in the updated solutions for electron differential density leading to its faster decrease at lower energies (below 100 keV) with every precipitation depth similar to the results obtained with numerical Fokker-Planck solutions. The effects of these updated solutions for electron densities on mean electron spectra and HXR photon spectra are also discussed.« less

Peridynamic thermal diffusion

DOE Office of Scientific and Technical Information (OSTI.GOV)

Oterkus, Selda; Madenci, Erdogan, E-mail: madenci@email.arizona.edu; Agwai, Abigail

This study presents the derivation of ordinary state-based peridynamic heat conduction equation based on the Lagrangian formalism. The peridynamic heat conduction parameters are related to those of the classical theory. An explicit time stepping scheme is adopted for numerical solution of various benchmark problems with known solutions. It paves the way for applying the peridynamic theory to other physical fields such as neutronic diffusion and electrical potential distribution.

The interaction of Dirac particles with non-abelian gauge fields and gravity - bound states

NASA Astrophysics Data System (ADS)

Finster, Felix; Smoller, Joel; Yau, Shing-Tung

2000-09-01

We consider a spherically symmetric, static system of a Dirac particle interacting with classical gravity and an SU(2) Yang-Mills field. The corresponding Einstein-Dirac-Yang-Mills equations are derived. Using numerical methods, we find different types of soliton-like solutions of these equations and discuss their properties. Some of these solutions are stable even for arbitrarily weak gravitational coupling.

Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method

NASA Astrophysics Data System (ADS)

Jain, Sonal

2018-01-01

In this paper, we aim to use the alternative numerical scheme given by Gnitchogna and Atangana for solving partial differential equations with integer and non-integer differential operators. We applied this method to fractional diffusion model and fractional Buckmaster models with non-local fading memory. The method yields a powerful numerical algorithm for fractional order derivative to implement. Also we present in detail the stability analysis of the numerical method for solving the diffusion equation. This proof shows that this method is very stable and also converges very quickly to exact solution and finally some numerical simulation is presented.

Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation

PubMed Central

Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui

2014-01-01

Computations of rarefied gas dynamical flows governed by the semiclassical Boltzmann ellipsoidal-statistical (ES) kinetic model equation using an accurate numerical method are presented. The semiclassical ES model was derived through the maximum entropy principle and conserves not only the mass, momentum and energy, but also contains additional higher order moments that differ from the standard quantum distributions. A different decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. The numerical method in phase space combines the discrete-ordinate method in momentum space and the high-resolution shock capturing method in physical space. Numerical solutions of two-dimensional Riemann problems for two configurations covering various degrees of rarefaction are presented and various contours of the quantities unique to this new model are illustrated. When the relaxation time becomes very small, the main flow features a display similar to that of ideal quantum gas dynamics, and the present solutions are found to be consistent with existing calculations for classical gas. The effect of a parameter that permits an adjustable Prandtl number in the flow is also studied. PMID:25104904

Analytic theory of orbit contraction

NASA Technical Reports Server (NTRS)

Vinh, N. X.; Longuski, J. M.; Busemann, A.; Culp, R. D.

1977-01-01

The motion of a satellite in orbit, subject to atmospheric force and the motion of a reentry vehicle are governed by gravitational and aerodynamic forces. This suggests the derivation of a uniform set of equations applicable to both cases. For the case of satellite motion, by a proper transformation and by the method of averaging, a technique appropriate for long duration flight, the classical nonlinear differential equation describing the contraction of the major axis is derived. A rigorous analytic solution is used to integrate this equation with a high degree of accuracy, using Poincare's method of small parameters and Lagrange's expansion to explicitly express the major axis as a function of the eccentricity. The solution is uniformly valid for moderate and small eccentricities. For highly eccentric orbits, the asymptotic equation is derived directly from the general equation. Numerical solutions were generated to display the accuracy of the analytic theory.

Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

DOE PAGES

Chen, Zheng; Huang, Hongying; Yan, Jue

2015-12-21

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8], [9], [19] and [21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β 0,β 1) in the numerical flux formula, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. As a result, a sequence of numerical examples are carried outmore » to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.« less

Radiative flow of Carreau liquid in presence of Newtonian heating and chemical reaction

NASA Astrophysics Data System (ADS)

Hayat, T.; Ullah, Ikram; Ahmad, B.; Alsaedi, A.

Objective of this article is to investigate the magnetohydrodynamic (MHD) boundary layer stretched flow of Carreau fluid in the presence of Newtonian heating. Sheet is presumed permeable. Analysis is studied in the presence of chemical reaction and thermal radiation. Mathematical formulation is established by using the boundary layer approximations. The resultant nonlinear flow analysis is computed for the convergent solutions. Interval of convergence via numerical data and plots are obtained and verified. Impact of numerous pertinent variables on the velocity, temperature and concentration is outlined. Numerical data for surface drag coefficient, surface heat transfer (local Nusselt number) and mass transfer (local Sherwood number) is executed and inspected. Comparison of skin friction coefficient in limiting case is made for the verification of current derived solutions.

Burton-Miller-type singular boundary method for acoustic radiation and scattering

NASA Astrophysics Data System (ADS)

Fu, Zhuo-Jia; Chen, Wen; Gu, Yan

2014-08-01

This paper proposes the singular boundary method (SBM) in conjunction with Burton and Miller's formulation for acoustic radiation and scattering. The SBM is a strong-form collocation boundary discretization technique using the singular fundamental solutions, which is mathematically simple, easy-to-program, meshless and introduces the concept of source intensity factors (SIFs) to eliminate the singularities of the fundamental solutions. Therefore, it avoids singular numerical integrals in the boundary element method (BEM) and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions (MFS). In the present method, we derive the SIFs of exterior Helmholtz equation by means of the SIFs of exterior Laplace equation owing to the same order of singularities between the Laplace and Helmholtz fundamental solutions. In conjunction with the Burton-Miller formulation, the SBM enhances the quality of the solution, particularly in the vicinity of the corresponding interior eigenfrequencies. Numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2D and 3D unbounded domains in comparison with the analytical solutions, the boundary element solutions and Dirichlet-to-Neumann finite element solutions.

Incremental analysis of large elastic deformation of a rotating cylinder

NASA Technical Reports Server (NTRS)

Buchanan, G. R.

1976-01-01

The effect of finite deformation upon a rotating, orthotropic cylinder was investigated using a general incremental theory. The incremental equations of motion are developed using the variational principle. The governing equations are derived using the principle of virtual work for a body with initial stress. The governing equations are reduced to those for the title problem and a numerical solution is obtained using finite difference approximations. Since the problem is defined in terms of one independent space coordinate, the finite difference grid can be modified as the incremental deformation occurs without serious numerical difficulties. The nonlinear problem is solved incrementally by totaling a series of linear solutions.

Analytic solution and pulse area theorem for three-level atoms

NASA Astrophysics Data System (ADS)

Shchedrin, Gavriil; O'Brien, Chris; Rostovtsev, Yuri; Scully, Marlan O.

2015-12-01

We report an analytic solution for a three-level atom driven by arbitrary time-dependent electromagnetic pulses. In particular, we consider far-detuned driving pulses and show an excellent match between our analytic result and the numerical simulations. We use our solution to derive a pulse area theorem for three-level V and Λ systems without making the rotating wave approximation. Formulated as an energy conservation law, this pulse area theorem can be used to understand pulse propagation through three-level media.

Analytic theory of photoacoustic wave generation from a spheroidal droplet.

PubMed

Li, Yong; Fang, Hui; Min, Changjun; Yuan, Xiaocong

2014-08-25

In this paper, we develop an analytic theory for describing the photoacoustic wave generation from a spheroidal droplet and derive the first complete analytic solution. Our derivation is based on solving the photoacoustic Helmholtz equation in spheroidal coordinates with the separation-of-variables method. As the verification, besides carrying out the asymptotic analyses which recover the standard solutions for a sphere, an infinite cylinder and an infinite layer, we also confirm that the partial transmission and reflection model previously demonstrated for these three geometries still stands. We expect that this analytic solution will find broad practical uses in interpreting experiment results, considering that its building blocks, the spheroidal wave functions (SWFs), can be numerically calculated by the existing computer programs.

The radiated noise from isotropic turbulence revisited

NASA Technical Reports Server (NTRS)

Lilley, Geoffrey M.

1993-01-01

The noise radiated from isotropic turbulence at low Mach numbers and high Reynolds numbers, as derived by Proudman (1952), was the first application of Lighthill's Theory of Aerodynamic Noise to a complete flow field. The theory presented by Proudman involves the assumption of the neglect of retarded time differences and so replaces the second-order retarded-time and space covariance of Lighthill's stress tensor, Tij, and in particular its second time derivative, by the equivalent simultaneous covariance. This assumption is a valid approximation in the derivation of the second partial derivative of Tij/derivative of t exp 2 covariance at low Mach numbers, but is not justified when that covariance is reduced to the sum of products of the time derivatives of equivalent second-order velocity covariances as required when Gaussian statistics are assumed. The present paper removes these assumptions and finds that although the changes in the analysis are substantial, the change in the numerical result for the total acoustic power is small. The present paper also considers an alternative analysis which does not neglect retarded times. It makes use of the Lighthill relationship, whereby the fourth-order Tij retarded-time covariance is evaluated from the square of similar second order covariance, which is assumed known. In this derivation, no statistical assumptions are involved. This result, using distributions for the second-order space-time velocity squared covariance based on the Direct Numerical Simulation (DNS) results of both Sarkar and Hussaini(1993) and Dubois(1993), is compared with the re-evaluation of Proudman's original model. These results are then compared with the sound power derived from a phenomenological model based on simple approximations to the retarded-time/space covariance of Txx. Finally, the recent numerical solutions of Sarkar and Hussaini(1993) for the acoustic power are compared with the results obtained from the analytic solutions.

Common aero vehicle autonomous reentry trajectory optimization satisfying waypoint and no-fly zone constraints

NASA Astrophysics Data System (ADS)

Jorris, Timothy R.

2007-12-01

To support the Air Force's Global Reach concept, a Common Aero Vehicle is being designed to support the Global Strike mission. "Waypoints" are specified for reconnaissance or multiple payload deployments and "no-fly zones" are specified for geopolitical restrictions or threat avoidance. Due to time critical targets and multiple scenario analysis, an autonomous solution is preferred over a time-intensive, manually iterative one. Thus, a real-time or near real-time autonomous trajectory optimization technique is presented to minimize the flight time, satisfy terminal and intermediate constraints, and remain within the specified vehicle heating and control limitations. This research uses the Hypersonic Cruise Vehicle (HCV) as a simplified two-dimensional platform to compare multiple solution techniques. The solution techniques include a unique geometric approach developed herein, a derived analytical dynamic optimization technique, and a rapidly emerging collocation numerical approach. This up-and-coming numerical technique is a direct solution method involving discretization then dualization, with pseudospectral methods and nonlinear programming used to converge to the optimal solution. This numerical approach is applied to the Common Aero Vehicle (CAV) as the test platform for the full three-dimensional reentry trajectory optimization problem. The culmination of this research is the verification of the optimality of this proposed numerical technique, as shown for both the two-dimensional and three-dimensional models. Additionally, user implementation strategies are presented to improve accuracy and enhance solution convergence. Thus, the contributions of this research are the geometric approach, the user implementation strategies, and the determination and verification of a numerical solution technique for the optimal reentry trajectory problem that minimizes time to target while satisfying vehicle dynamics and control limitation, and heating, waypoint, and no-fly zone constraints.

Comment on "Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs", [Commun Nonlinear Sci Numer Simulat 47 (2017) 253-266

NASA Astrophysics Data System (ADS)

Li, Xiangzheng

2018-06-01

A counterexample is given to show that the product rule of the Caputo fractional derivatives does not hold except on a special point. The function-expansion method of separation variable proposed by Rui[Commun Nonlinear Sci Numer Simulat 47 (2017) 253-266] based on the product rule must be modified.

Algorithms for the Fractional Calculus: A Selection of Numerical Methods

NASA Technical Reports Server (NTRS)

Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Yu.

2003-01-01

Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.

Numerical approach to optimal portfolio in a power utility regime-switching model

NASA Astrophysics Data System (ADS)

Gyulov, Tihomir B.; Koleva, Miglena N.; Vulkov, Lubin G.

2017-12-01

We consider a system of weakly coupled degenerate semi-linear parabolic equations of optimal portfolio in a regime-switching with power utility function, derived by A.R. Valdez and T. Vargiolu [14]. First, we discuss some basic properties of the solution of this system. Then, we develop and analyze implicit-explicit, flux limited finite difference schemes for the differential problem. Numerical experiments are discussed.

Unsteady Flow Simulation: A Numerical Challenge

DTIC Science & Technology

2003-03-01

drive to convergence the numerical unsteady term. The time marching procedure is based on the approximate implicit Newton method for systems of non...computed through analytical derivatives of S. The linear system stemming from equation (3) is solved at each integration step by the same iterative method...significant reduction of memory usage, thanks to the reduced dimensions of the linear system matrix during the implicit marching of the solution. The

Valuation of financial models with non-linear state spaces

NASA Astrophysics Data System (ADS)

Webber, Nick

2001-02-01

A common assumption in valuation models for derivative securities is that the underlying state variables take values in a linear state space. We discuss numerical implementation issues in an interest rate model with a simple non-linear state space, formulating and comparing Monte Carlo, finite difference and lattice numerical solution methods. We conclude that, at least in low dimensional spaces, non-linear interest rate models may be viable.

Physiology driven adaptivity for the numerical solution of the bidomain equations.

PubMed

Whiteley, Jonathan P

2007-09-01

Previous work [Whiteley, J. P. IEEE Trans. Biomed. Eng. 53:2139-2147, 2006] derived a stable, semi-implicit numerical scheme for solving the bidomain equations. This scheme allows the timestep used when solving the bidomain equations numerically to be chosen by accuracy considerations rather than stability considerations. In this study we modify this scheme to allow an adaptive numerical solution in both time and space. The spatial mesh size is determined by the gradient of the transmembrane and extracellular potentials while the timestep is determined by the values of: (i) the fast sodium current; and (ii) the calcium release from junctional sarcoplasmic reticulum to myoplasm current. For two-dimensional simulations presented here, combining the numerical algorithm in the paper cited above with the adaptive algorithm presented here leads to an increase in computational efficiency by a factor of around 250 over previous work, together with significantly less computational memory being required. The speedup for three-dimensional simulations is likely to be more impressive.

Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative

NASA Astrophysics Data System (ADS)

Owolabi, Kolade M.

2018-01-01

In this paper, we model an ecological system consisting of a predator and two preys with the newly derived two-step fractional Adams-Bashforth method via the Atangana-Baleanu derivative in the Caputo sense. We analyze the dynamical system for correct choice of parameter values that are biologically meaningful. The local analysis of the main model is based on the application of qualitative theory for ordinary differential equations. By using the fixed point theorem idea, we establish the existence and uniqueness of the solutions. Convergence results of the new scheme are verified in both space and time. Dynamical wave phenomena of solutions are verified via some numerical results obtained for different values of the fractional index, which have some interesting ecological implications.

Numerical methods for the weakly compressible Generalized Langevin Model in Eulerian reference frame

DOE Office of Scientific and Technical Information (OSTI.GOV)

Azarnykh, Dmitrii, E-mail: d.azarnykh@tum.de; Litvinov, Sergey; Adams, Nikolaus A.

2016-06-01

A well established approach for the computation of turbulent flow without resolving all turbulent flow scales is to solve a filtered or averaged set of equations, and to model non-resolved scales by closures derived from transported probability density functions (PDF) for velocity fluctuations. Effective numerical methods for PDF transport employ the equivalence between the Fokker–Planck equation for the PDF and a Generalized Langevin Model (GLM), and compute the PDF by transporting a set of sampling particles by GLM (Pope (1985) [1]). The natural representation of GLM is a system of stochastic differential equations in a Lagrangian reference frame, typically solvedmore » by particle methods. A representation in a Eulerian reference frame, however, has the potential to significantly reduce computational effort and to allow for the seamless integration into a Eulerian-frame numerical flow solver. GLM in a Eulerian frame (GLMEF) formally corresponds to the nonlinear fluctuating hydrodynamic equations derived by Nakamura and Yoshimori (2009) [12]. Unlike the more common Landau–Lifshitz Navier–Stokes (LLNS) equations these equations are derived from the underdamped Langevin equation and are not based on a local equilibrium assumption. Similarly to LLNS equations the numerical solution of GLMEF requires special considerations. In this paper we investigate different numerical approaches to solving GLMEF with respect to the correct representation of stochastic properties of the solution. We find that a discretely conservative staggered finite-difference scheme, adapted from a scheme originally proposed for turbulent incompressible flow, in conjunction with a strongly stable (for non-stochastic PDE) Runge–Kutta method performs better for GLMEF than schemes adopted from those proposed previously for the LLNS. We show that equilibrium stochastic fluctuations are correctly reproduced.« less

Optimal economic order quantity for buyer-distributor-vendor supply chain with backlogging derived without derivatives

NASA Astrophysics Data System (ADS)

Teng, Jinn-Tsair; Cárdenas-Barrón, Leopoldo Eduardo; Lou, Kuo-Ren; Wee, Hui Ming

2013-05-01

In this article, we first complement an inappropriate mathematical error on the total cost in the previously published paper by Chung and Wee [2007, 'Optimal the Economic Lot Size of a Three-stage Supply Chain With Backlogging Derived Without Derivatives', European Journal of Operational Research, 183, 933-943] related to buyer-distributor-vendor three-stage supply chain with backlogging derived without derivatives. Then, an arithmetic-geometric inequality method is proposed not only to simplify the algebraic method of completing prefect squares, but also to complement their shortcomings. In addition, we provide a closed-form solution to integral number of deliveries for the distributor and the vendor without using complex derivatives. Furthermore, our method can solve many cases in which their method cannot, because they did not consider that a squared root of a negative number does not exist. Finally, we use some numerical examples to show that our proposed optimal solution is cheaper to operate than theirs.

Asymptotically and exactly energy balanced augmented flux-ADER schemes with application to hyperbolic conservation laws with geometric source terms

NASA Astrophysics Data System (ADS)

Navas-Montilla, A.; Murillo, J.

2016-07-01

In this work, an arbitrary order HLL-type numerical scheme is constructed using the flux-ADER methodology. The proposed scheme is based on an augmented Derivative Riemann solver that was used for the first time in Navas-Montilla and Murillo (2015) [1]. Such solver, hereafter referred to as Flux-Source (FS) solver, was conceived as a high order extension of the augmented Roe solver and led to the generation of a novel numerical scheme called AR-ADER scheme. Here, we provide a general definition of the FS solver independently of the Riemann solver used in it. Moreover, a simplified version of the solver, referred to as Linearized-Flux-Source (LFS) solver, is presented. This novel version of the FS solver allows to compute the solution without requiring reconstruction of derivatives of the fluxes, nevertheless some drawbacks are evidenced. In contrast to other previously defined Derivative Riemann solvers, the proposed FS and LFS solvers take into account the presence of the source term in the resolution of the Derivative Riemann Problem (DRP), which is of particular interest when dealing with geometric source terms. When applied to the shallow water equations, the proposed HLLS-ADER and AR-ADER schemes can be constructed to fulfill the exactly well-balanced property, showing that an arbitrary quadrature of the integral of the source inside the cell does not ensure energy balanced solutions. As a result of this work, energy balanced flux-ADER schemes that provide the exact solution for steady cases and that converge to the exact solution with arbitrary order for transient cases are constructed.

High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields

NASA Astrophysics Data System (ADS)

He, Yang; Sun, Yajuan; Zhang, Ruili; Wang, Yulei; Liu, Jian; Qin, Hong

2016-09-01

We construct high order symmetric volume-preserving methods for the relativistic dynamics of a charged particle by the splitting technique with processing. By expanding the phase space to include the time t, we give a more general construction of volume-preserving methods that can be applied to systems with time-dependent electromagnetic fields. The newly derived methods provide numerical solutions with good accuracy and conservative properties over long time of simulation. Furthermore, because of the use of an accuracy-enhancing processing technique, the explicit methods obtain high-order accuracy and are more efficient than the methods derived from standard compositions. The results are verified by the numerical experiments. Linear stability analysis of the methods shows that the high order processed method allows larger time step size in numerical integrations.

Asymptotic analysis of numerical wave propagation in finite difference equations

NASA Technical Reports Server (NTRS)

Giles, M.; Thompkins, W. T., Jr.

1983-01-01

An asymptotic technique is developed for analyzing the propagation and dissipation of wave-like solutions to finite difference equations. It is shown that for each fixed complex frequency there are usually several wave solutions with different wavenumbers and the slowly varying amplitude of each satisfies an asymptotic amplitude equation which includes the effects of smoothly varying coefficients in the finite difference equations. The local group velocity appears in this equation as the velocity of convection of the amplitude. Asymptotic boundary conditions coupling the amplitudes of the different wave solutions are also derived. A wavepacket theory is developed which predicts the motion, and interaction at boundaries, of wavepackets, wave-like disturbances of finite length. Comparison with numerical experiments demonstrates the success and limitations of the theory. Finally an asymptotic global stability analysis is developed.

Solute drag in polycrystalline materials: Derivation and numerical analysis of a variational model for the effect of solute on the motion of boundaries and junctions during coarsening

NASA Astrophysics Data System (ADS)

Wilson, Seth Robert

A mathematical model that results in an expression for the local acceleration of a network of sharp interfaces interacting with an ambient solute field is proposed. This expression comprises a first-order differential equation for the local velocity that, given the appropriate initial conditions, may be used to predict the subsequent time evolution of the system, including non-steady state absorption and desorption of solute. Evolution equations for both interfaces and the junction of interfaces are derived by maximizing a functional approximating the rate at which the local Gibbs free energy density decreases, as a function of the local solute content and the instantaneous velocity. The model has been formulated in three dimensions, and non-equilibrium effects such as grain boundary diffusion, solute gradients, and time-dependant segregation are taken into account. As a consequence of this model, it is shown that both interfaces and the junctions between interfaces obey evolution equations that closely resemble Newton's second law. In particular, the concept of "thrust" in variable-mass systems is shown to have a direct analog in solute-interface interaction. Numerical analysis of the equations that result reveals that a double cusp catastrophe governs the behavior of the solute-interface system, for which trajectories that include hysteresis, slip-stick motion, and jerky motion are all conceivable. The geometry of the cusp catastrophe is quantified, and a number of relations between physical parameters and system behavior are consequently predicted.

Three-dimensional solutions for the thermal buckling and sensitivity derivatives of temperature-sensitive multilayered angle-ply plates

NASA Technical Reports Server (NTRS)

Noor, A. K.; Burton, W. S.

1992-01-01

Analytic three-dimensional thermoelasticity solutions are presented for the thermal buckling of multilayered angle-ply composite plates with temperature-dependent thermoelastic properties. Both the critical temperatures and the sensitivity derivatives are computed. The sensitivity derivatives measure the sensitivity of the buckling response to variations in the different lamination and material parameters of the plate. The plates are assumed to have rectangular geometry and an antisymmetric lamination with respect to the middle plane. The temperature is assumed to be independent of the surface coordinates, but has an arbitrary symmetric variation through the thickness of the plate. The prebuckling deformations are accounted for. Numerical results are presented, for plates subjected to uniform temperature increase, showing the effects of temperature-dependent material properties on the prebuckling stresses, critical temperatures, and their sensitivity derivatives.

Time-periodic solutions of the Benjamin-Ono equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ambrose , D.M.; Wilkening, Jon

2008-04-01

We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one ofmore » the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.« less

Three-dimensional time-dependent computer modeling of the electrothermal atomizers for analytical spectrometry

NASA Astrophysics Data System (ADS)

Tsivilskiy, I. V.; Nagulin, K. Yu.; Gilmutdinov, A. Kh.

2016-02-01

A full three-dimensional nonstationary numerical model of graphite electrothermal atomizers of various types is developed. The model is based on solution of a heat equation within solid walls of the atomizer with a radiative heat transfer and numerical solution of a full set of Navier-Stokes equations with an energy equation for a gas. Governing equations for the behavior of a discrete phase, i.e., atomic particles suspended in a gas (including gas-phase processes of evaporation and condensation), are derived from the formal equations molecular kinetics by numerical solution of the Hertz-Langmuir equation. The following atomizers test the model: a Varian standard heated electrothermal vaporizer (ETV), a Perkin Elmer standard THGA transversely heated graphite tube with integrated platform (THGA), and the original double-stage tube-helix atomizer (DSTHA). The experimental verification of computer calculations is carried out by a method of shadow spectral visualization of the spatial distributions of atomic and molecular vapors in an analytical space of an atomizer.

Development of numerical techniques for simulation of magnetogasdynamics and hypersonic chemistry

NASA Astrophysics Data System (ADS)

Damevin, Henri-Marie

Magnetogasdynamics, the science concerned with the mutual interaction between electromagnetic field and flow of electrically conducting gas, offers promising advances in flow control and propulsion of future hypersonic vehicles. Numerical simulations are essential for understanding phenomena, and for research and development. The current dissertation is devoted to the development and validation of numerical algorithms for the solution of multidimensional magnetogasdynamic equations and the simulation of hypersonic high-temperature effects. Governing equations are derived, based on classical magnetogasdynamic assumptions. Two sets of equations are considered, namely the full equations and equations in the low magnetic Reynolds number approximation. Equations are expressed in a suitable formulation for discretization by finite differences in a computational space. For the full equations, Gauss law for magnetism is enforced using Powell's methodology. The time integration method is a four-stage modified Runge-Kutta scheme, amended with a Total Variation Diminishing model in a postprocessing stage. The eigensystem, required for the Total Variation Diminishing scheme, is derived in generalized three-dimensional coordinate system. For the simulation of hypersonic high-temperature effects, two chemical models are utilized, namely a nonequilibrium model and an equilibrium model. A loosely coupled approach is implemented to communicate between the magnetogasdynamic equations and the chemical models. The nonequilibrium model is a one-temperature, five-species, seventeen-reaction model solved by an implicit flux-vector splitting scheme. The chemical equilibrium model computes thermodynamics properties using curve fit procedures. Selected results are provided, which explore the different features of the numerical algorithms. The shock-capturing properties are validated for shock-tube simulations using numerical solutions reported in the literature. The computations of superfast flows over corners and in convergent channels demonstrate the performances of the algorithm in multiple dimensions. The implementation of diffusion terms is validated by solving the magnetic Rayleigh problem and Hartmann problem, for which analytical solutions are available. Prediction of blunt-body type flow are investigated and compared with numerical solutions reported in the literature. The effectiveness of the chemical models for hypersonic flow over blunt body is examined in various flow conditions. It is shown that the proposed schemes perform well in a variety of test cases, though some limitations have been identified.

An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - III. Viscoelastic attenuation

NASA Astrophysics Data System (ADS)

Käser, Martin; Dumbser, Michael; de la Puente, Josep; Igel, Heiner

2007-01-01

We present a new numerical method to solve the heterogeneous anelastic, seismic wave equations with arbitrary high order accuracy in space and time on 3-D unstructured tetrahedral meshes. Using the velocity-stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method combines the Discontinuous Galerkin (DG) finite element (FE) method with the ADER approach using Arbitrary high order DERivatives for flux calculations. The DG approach, in contrast to classical FE methods, uses a piecewise polynomial approximation of the numerical solution which allows for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann problems can be applied as in the finite volume framework. The main idea of the ADER time integration approach is a Taylor expansion in time in which all time derivatives are replaced by space derivatives using the so-called Cauchy-Kovalewski procedure which makes extensive use of the governing PDE. Due to the ADER time integration technique the same approximation order in space and time is achieved automatically and the method is a one-step scheme advancing the solution for one time step without intermediate stages. To this end, we introduce a new unrolled recursive algorithm for efficiently computing the Cauchy-Kovalewski procedure by making use of the sparsity of the system matrices. The numerical convergence analysis demonstrates that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes while computational cost and storage space for a desired accuracy can be reduced when applying higher degree approximation polynomials. In addition, we investigate the increase in computing time, when the number of relaxation mechanisms due to the generalized Maxwell body are increased. An application to a well-acknowledged test case and comparisons with analytic and reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-DG approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

Controllability of semi-infinite rod heating by a point source

NASA Astrophysics Data System (ADS)

Khurshudyan, A.

2018-04-01

The possibility of control over heating of a semi-infinite thin rod by a point source concentrated at an inner point of the rod, is studied. Quadratic and piecewise constant solutions of the problem are derived, and the possibilities of solving appropriate problems of optimal control are indicated. Determining of the parameters of the piecewise constant solution is reduced to a problem of nonlinear programming. Numerical examples are considered.

On a perturbed Sparre Andersen risk model with multi-layer dividend strategy

NASA Astrophysics Data System (ADS)

Yang, Hu; Zhang, Zhimin

2009-10-01

In this paper, we consider a perturbed Sparre Andersen risk model, in which the inter-claim times are generalized Erlang(n) distributed. Under the multi-layer dividend strategy, piece-wise integro-differential equations for the discounted penalty functions are derived, and a recursive approach is applied to express the solutions. A numerical example to calculate the ruin probabilities is given to illustrate the solution procedure.

A transient laboratory method for determining the hydraulic properties of 'tight' rocks-I. Theory

USGS Publications Warehouse

Hsieh, P.A.; Tracy, J.V.; Neuzil, C.E.; Bredehoeft, J.D.; Silliman, Stephen E.

1981-01-01

Transient pulse testing has been employed increasingly in the laboratory to measure the hydraulic properties of rock samples with low permeability. Several investigators have proposed a mathematical model in terms of an initial-boundary value problem to describe fluid flow in a transient pulse test. However, the solution of this problem has not been available. In analyzing data from the transient pulse test, previous investigators have either employed analytical solutions that are derived with the use of additional, restrictive assumptions, or have resorted to numerical methods. In Part I of this paper, a general, analytical solution for the transient pulse test is presented. This solution is graphically illustrated by plots of dimensionless variables for several cases of interest. The solution is shown to contain, as limiting cases, the more restrictive analytical solutions that the previous investigators have derived. A method of computing both the permeability and specific storage of the test sample from experimental data will be presented in Part II. ?? 1981.

Unsteady free convection flow of viscous fluids with analytical results by employing time-fractional Caputo-Fabrizio derivative (without singular kernel)

NASA Astrophysics Data System (ADS)

Ali Shah, Nehad; Mahsud, Yasir; Ali Zafar, Azhar

2017-10-01

This article introduces a theoretical study for unsteady free convection flow of an incompressible viscous fluid. The fluid flows near an isothermal vertical plate. The plate has a translational motion with time-dependent velocity. The equations governing the fluid flow are expressed in fractional differential equations by using a newly defined time-fractional Caputo-Fabrizio derivative without singular kernel. Explicit solutions for velocity, temperature and solute concentration are obtained by applying the Laplace transform technique. As the fractional parameter approaches to one, solutions for the ordinary fluid model are extracted from the general solutions of the fractional model. The results showed that, for the fractional model, the obtained solutions for velocity, temperature and concentration exhibit stationary jumps discontinuity across the plane at t=0 , while the solutions are continuous functions in the case of the ordinary model. Finally, numerical results for flow features at small-time are illustrated through graphs for various pertinent parameters.

Efficient modeling of phase jitter in dispersion-managed soliton systems.

PubMed

McKinstrie, C J; Xie, C; Lakoba, T I

2002-11-01

The variational method is used to derive correlation equations that model phase jitter in dispersion-managed soliton systems. The predictions of these correlation equations are consistent with numerical solutions of the nonlinear Schrödinger equation on which they are based.

On multiple solutions of non-Newtonian Carreau fluid flow over an inclined shrinking sheet

NASA Astrophysics Data System (ADS)

Khan, Masood; Sardar, Humara; Gulzar, M. Mudassar; Alshomrani, Ali Saleh

2018-03-01

This paper presents the multiple solutions of a non-Newtonian Carreau fluid flow over a nonlinear inclined shrinking surface in presence of infinite shear rate viscosity. The governing boundary layer equations are derived for the Carreau fluid with infinite shear rate viscosity. The suitable transformations are employed to alter the leading partial differential equations to a set of ordinary differential equations. The consequential non-linear ODEs are solved numerically by an active numerical approach namely Runge-Kutta Fehlberg fourth-fifth order method accompanied by shooting technique. Multiple solutions are presented graphically and results are shown for various physical parameters. It is important to state that the velocity and momentum boundary layer thickness reduce with increasing viscosity ratio parameter in shear thickening fluid while opposite trend is observed for shear thinning fluid. Another important observation is that the wall shear stress is significantly decreased by the viscosity ratio parameter β∗ for the first solution and opposite trend is observed for the second solution.

The L sub 1 finite element method for pure convection problems

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan

1991-01-01

The least squares (L sub 2) finite element method is introduced for 2-D steady state pure convection problems with smooth solutions. It is proven that the L sub 2 method has the same stability estimate as the original equation, i.e., the L sub 2 method has better control of the streamline derivative. Numerical convergence rates are given to show that the L sub 2 method is almost optimal. This L sub 2 method was then used as a framework to develop an iteratively reweighted L sub 2 finite element method to obtain a least absolute residual (L sub 1) solution for problems with discontinuous solutions. This L sub 1 finite element method produces a nonoscillatory, nondiffusive and highly accurate numerical solution that has a sharp discontinuity in one element on both coarse and fine meshes. A robust reweighting strategy was also devised to obtain the L sub 1 solution in a few iterations. A number of examples solved by using triangle and bilinear elements are presented.

Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations

NASA Astrophysics Data System (ADS)

Ohkitani, Koji

2012-09-01

We study the generalised 2D surface quasi-geostrophic (SQG) equation, where the active scalar is given by a fractional power α of Laplacian applied to the stream function. This includes the 2D SQG and Euler equations as special cases. Using Poincaré's successive approximation to higher α-derivatives of the active scalar, we derive a variational equation for describing perturbations in the generalized SQG equation. In particular, in the limit α → 0, an asymptotic equation is derived on a stretched time variable τ = αt, which unifies equations in the family near α = 0. The successive approximation is also discussed at the other extreme of the 2D Euler limit α = 2-0. Numerical experiments are presented for both limits. We consider whether the solution behaves in a more singular fashion, with more effective nonlinearity, when α is increased. Two competing effects are identified: the regularizing effect of a fractional inverse Laplacian (control by conservation) and cancellation by symmetry (nonlinearity depletion). Near α = 0 (complete depletion), the solution behaves in a more singular fashion as α increases. Near α = 2 (maximal control by conservation), the solution behave in a more singular fashion, as α decreases, suggesting that there may be some α in [0, 2] at which the solution behaves in the most singular manner. We also present some numerical results of the family for α = 0.5, 1, and 1.5. On the original time t, the H1 norm of θ generally grows more rapidly with increasing α. However, on the new time τ, this order is reversed. On the other hand, contour patterns for different α appear to be similar at fixed τ, even though the norms are markedly different in magnitude. Finally, point-vortex systems for the generalized SQG family are discussed to shed light on the above problems of time scale.

A {3,2}-Order Bending Theory for Laminated Composite and Sandwich Beams

NASA Technical Reports Server (NTRS)

Cook, Geoffrey M.; Tessler, Alexander

1998-01-01

A higher-order bending theory is derived for laminated composite and sandwich beams thus extending the recent {1,2}-order theory to include third-order axial effect without introducing additional kinematic variables. The present theory is of order {3,2} and includes both transverse shear and transverse normal deformations. A closed-form solution to the cylindrical bending problem is derived and compared with the corresponding exact elasticity solution. The numerical comparisons are focused on the most challenging material systems and beam aspect ratios which include moderate-to-thick unsymmetric composite and sandwich laminates. Advantages and limitations of the theory are discussed.

The Mimetic Finite Element Method and the Virtual Element Method for elliptic problems with arbitrary regularity.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Manzini, Gianmarco

2012-07-13

We develop and analyze a new family of virtual element methods on unstructured polygonal meshes for the diffusion problem in primal form, that use arbitrarily regular discrete spaces V{sub h} {contained_in} C{sup {alpha}} {element_of} N. The degrees of freedom are (a) solution and derivative values of various degree at suitable nodes and (b) solution moments inside polygons. The convergence of the method is proven theoretically and an optimal error estimate is derived. The connection with the Mimetic Finite Difference method is also discussed. Numerical experiments confirm the convergence rate that is expected from the theory.

Energy loss in spark gap switches

DOE Office of Scientific and Technical Information (OSTI.GOV)

Oreshkin, V. I., E-mail: oreshkin@ovpe.hcei.tsc.ru; Lavrinovich, I. V.; National Research Tomsk Polytechnic University, Lenin Avenue 30, 634050 Tomsk

2014-04-15

The paper reports on numerical study of the energy loss in spark gap switches. The operation of the switches is analyzed using the Braginsky model which allows calculation of the time dependence of the spark channel resistance. The Braginsky equation is solved simultaneously with generator circuit equations for different load types. Based on the numerical solutions, expressions which determine both the energy released in a spark gap switch and the switching time are derived.

Series expansions of rotating two and three dimensional sound fields.

PubMed

Poletti, M A

2010-12-01

The cylindrical and spherical harmonic expansions of oscillating sound fields rotating at a constant rate are derived. These expansions are a generalized form of the stationary sound field expansions. The derivations are based on the representation of interior and exterior sound fields using the simple source approach and determination of the simple source solutions with uniform rotation. Numerical simulations of rotating sound fields are presented to verify the theory.

Parametric study of minimum reactor mass in energy-storage dc-to-dc converters

NASA Technical Reports Server (NTRS)

Wong, R. C.; Owen, H. A., Jr.; Wilson, T. G.

1981-01-01

Closed-form analytical solutions for the design equations of a minimum-mass reactor for a two-winding voltage-or-current step-up converter are derived. A quantitative relationship between the three parameters - minimum total reactor mass, maximum output power, and switching frequency - is extracted from these analytical solutions. The validity of the closed-form solution is verified by a numerical minimization procedure. A computer-aided design procedure using commercially available toroidal cores and magnet wires is also used to examine how the results from practical designs follow the predictions of the analytical solutions.

Implementation of Preconditioned Dual-Time Procedures in OVERFLOW

NASA Technical Reports Server (NTRS)

Pandya, Shishir A.; Venkateswaran, Sankaran; Pulliam, Thomas H.; Kwak, Dochan (Technical Monitor)

2003-01-01

Preconditioning methods have become the method of choice for the solution of flowfields involving the simultaneous presence of low Mach and transonic regions. It is well known that these methods are important for insuring accurate numerical discretization as well as convergence efficiency over various operating conditions such as low Mach number, low Reynolds number and high Strouhal numbers. For unsteady problems, the preconditioning is introduced within a dual-time framework wherein the physical time-derivatives are used to march the unsteady equations and the preconditioned time-derivatives are used for purposes of numerical discretization and iterative solution. In this paper, we describe the implementation of the preconditioned dual-time methodology in the OVERFLOW code. To demonstrate the performance of the method, we employ both simple and practical unsteady flowfields, including vortex propagation in a low Mach number flow, flowfield of an impulsively started plate (Stokes' first problem) arid a cylindrical jet in a low Mach number crossflow with ground effect. All the results demonstrate that the preconditioning algorithm is responsible for improvements to both numerical accuracy and convergence efficiency and, thereby, enables low Mach number unsteady computations to be performed at a fraction of the cost of traditional time-marching methods.

Numerical analysis of fractional MHD Maxwell fluid with the effects of convection heat transfer condition and viscous dissipation

NASA Astrophysics Data System (ADS)

Bai, Yu; Jiang, Yuehua; Liu, Fawang; Zhang, Yan

2017-12-01

This paper investigates the incompressible fractional MHD Maxwell fluid due to a power function accelerating plate with the first order slip, and the numerical analysis on the flow and heat transfer of fractional Maxwell fluid has been done. Moreover the deformation motion of fluid micelle is simply analyzed. Nonlinear velocity equation are formulated with multi-term time fractional derivatives in the boundary layer governing equations, and convective heat transfer boundary condition and viscous dissipation are both taken into consideration. A newly finite difference scheme with L1-algorithm of governing equations are constructed, whose convergence is confirmed by the comparison with analytical solution. Numerical solutions for velocity and temperature show the effects of pertinent parameters on flow and heat transfer of fractional Maxwell fluid. It reveals that the fractional derivative weakens the effects of motion and heat conduction. The larger the Nusselt number is, the greater the heat transfer capacity of fluid becomes, and the temperature gradient at the wall becomes more significantly. The lower Reynolds number enhances the viscosity of the fluid because it is the ratio of the viscous force and the inertia force, which resists the flow and heat transfer.

Particle flows to shape and voltage surface discontinuities in the electron sheath surrounding a high voltage solar array in LEO

NASA Technical Reports Server (NTRS)

Metz, Roger N.

1991-01-01

This paper discusses the numerical modeling of electron flows from the sheath surrounding high positively biased objects in LEO (Low Earth Orbit) to regions of voltage or shape discontinuity on the biased surfaces. The sheath equations are derived from the Two-fluid, Warm Plasma Model. An equipotential corner and a plane containing strips of alternating voltage bias are treated in two dimensions. A self-consistent field solution of the sheath equations is outlined and is pursued through one cycle. The electron density field is determined by numerical solution of Poisson's equation for the electrostatic potential in the sheath using the NASCAP-LEO relation between electrostatic potential and charge density. Electron flows are calculated numerically from the electron continuity equation. Magnetic field effects are not treated.

Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves

NASA Astrophysics Data System (ADS)

Barker, Blake; Jung, Soyeun; Zumbrun, Kevin

2018-03-01

Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from subcritical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram - specifically, determination of rigorous Eckhaus-type stability conditions - remains an interesting open problem.

Approximate solution of space and time fractional higher order phase field equation

NASA Astrophysics Data System (ADS)

Shamseldeen, S.

2018-03-01

This paper is concerned with a class of space and time fractional partial differential equation (STFDE) with Riesz derivative in space and Caputo in time. The proposed STFDE is considered as a generalization of a sixth-order partial phase field equation. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine the optimal convergence control parameter. Two numerical examples are studied, considering periodic and non-periodic initial conditions, to justify the efficiency and the accuracy of the adopted iterative approach. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated.

Spherically symmetric vacuum solutions arising from trace dynamics modifications to gravitation

NASA Astrophysics Data System (ADS)

Adler, Stephen L.; Ramazanoğlu, Fethi M.

2015-12-01

We derive the equations governing static, spherically symmetric vacuum solutions to the Einstein equations, as modified by the frame-dependent effective action (derived from trace dynamics) that gives an alternative explanation of the origin of "dark energy". We give analytic and numerical results for the solutions of these equations, first in polar coordinates, and then in isotropic coordinates. General features of the static case are that: (i) there is no horizon, since g00 is nonvanishing for finite values of the polar radius, and only vanishes (in isotropic coordinates) at the internal singularity, (ii) the Ricci scalar R vanishes identically, and (iii) there is a physical singularity at cosmological distances. The large distance singularity may be an artifact of the static restriction, since we find that the behavior at large distances is altered in a time-dependent solution using the McVittie Ansatz.

Using a derivative-free optimization method for multiple solutions of inverse transport problems

DOE PAGES

Armstrong, Jerawan C.; Favorite, Jeffrey A.

2016-01-14

Identifying unknown components of an object that emits radiation is an important problem for national and global security. Radiation signatures measured from an object of interest can be used to infer object parameter values that are not known. This problem is called an inverse transport problem. An inverse transport problem may have multiple solutions and the most widely used approach for its solution is an iterative optimization method. This paper proposes a stochastic derivative-free global optimization algorithm to find multiple solutions of inverse transport problems. The algorithm is an extension of a multilevel single linkage (MLSL) method where a meshmore » adaptive direct search (MADS) algorithm is incorporated into the local phase. Furthermore, numerical test cases using uncollided fluxes of discrete gamma-ray lines are presented to show the performance of this new algorithm.« less

Revisited Fisher's equation in a new outlook: A fractional derivative approach

NASA Astrophysics Data System (ADS)

Alquran, Marwan; Al-Khaled, Kamel; Sardar, Tridip; Chattopadhyay, Joydev

2015-11-01

The well-known Fisher equation with fractional derivative is considered to provide some characteristics of memory embedded into the system. The modified model is analyzed both analytically and numerically. A comparatively new technique residual power series method is used for finding approximate solutions of the modified Fisher model. A new technique combining Sinc-collocation and finite difference method is used for numerical study. The abundance of the bird species Phalacrocorax carbois considered as a test bed to validate the model outcome using estimated parameters. We conjecture non-diffusive and diffusive fractional Fisher equation represents the same dynamics in the interval (memory index, α ∈(0.8384 , 0.9986)). We also observe that when the value of memory index is close to zero, the solutions bifurcate and produce a wave-like pattern. We conclude that the survivability of the species increases for long range memory index. These findings are similar to Fisher observation and act in a similar fashion that advantageous genes do.

Cortical geometry as a determinant of brain activity eigenmodes: Neural field analysis

NASA Astrophysics Data System (ADS)

Gabay, Natasha C.; Robinson, P. A.

2017-09-01

Perturbation analysis of neural field theory is used to derive eigenmodes of neural activity on a cortical hemisphere, which have previously been calculated numerically and found to be close analogs of spherical harmonics, despite heavy cortical folding. The present perturbation method treats cortical folding as a first-order perturbation from a spherical geometry. The first nine spatial eigenmodes on a population-averaged cortical hemisphere are derived and compared with previous numerical solutions. These eigenmodes contribute most to brain activity patterns such as those seen in electroencephalography and functional magnetic resonance imaging. The eigenvalues of these eigenmodes are found to agree with the previous numerical solutions to within their uncertainties. Also in agreement with the previous numerics, all eigenmodes are found to closely resemble spherical harmonics. The first seven eigenmodes exhibit a one-to-one correspondence with their numerical counterparts, with overlaps that are close to unity. The next two eigenmodes overlap the corresponding pair of numerical eigenmodes, having been rotated within the subspace spanned by that pair, likely due to second-order effects. The spatial orientations of the eigenmodes are found to be fixed by gross cortical shape rather than finer-scale cortical properties, which is consistent with the observed intersubject consistency of functional connectivity patterns. However, the eigenvalues depend more sensitively on finer-scale cortical structure, implying that the eigenfrequencies and consequent dynamical properties of functional connectivity depend more strongly on details of individual cortical folding. Overall, these results imply that well-established tools from perturbation theory and spherical harmonic analysis can be used to calculate the main properties and dynamics of low-order brain eigenmodes.

Lagrangian averaging, nonlinear waves, and shock regularization

NASA Astrophysics Data System (ADS)

Bhat, Harish S.

In this thesis, we explore various models for the flow of a compressible fluid as well as model equations for shock formation, one of the main features of compressible fluid flows. We begin by reviewing the variational structure of compressible fluid mechanics. We derive the barotropic compressible Euler equations from a variational principle in both material and spatial frames. Writing the resulting equations of motion requires certain Lie-algebraic calculations that we carry out in detail for expository purposes. Next, we extend the derivation of the Lagrangian averaged Euler (LAE-alpha) equations to the case of barotropic compressible flows. The derivation in this thesis involves averaging over a tube of trajectories etaepsilon centered around a given Lagrangian flow eta. With this tube framework, the LAE-alpha equations are derived by following a simple procedure: start with a given action, expand via Taylor series in terms of small-scale fluid fluctuations xi, truncate, average, and then model those terms that are nonlinear functions of xi. We then analyze a one-dimensional subcase of the general models derived above. We prove the existence of a large family of traveling wave solutions. Computing the dispersion relation for this model, we find it is nonlinear, implying that the equation is dispersive. We carry out numerical experiments that show that the model possesses smooth, bounded solutions that display interesting pattern formation. Finally, we examine a Hamiltonian partial differential equation (PDE) that regularizes the inviscid Burgers equation without the addition of standard viscosity. Here alpha is a small parameter that controls a nonlinear smoothing term that we have added to the inviscid Burgers equation. We show the existence of a large family of traveling front solutions. We analyze the initial-value problem and prove well-posedness for a certain class of initial data. We prove that in the zero-alpha limit, without any standard viscosity, solutions of the PDE converge strongly to weak solutions of the inviscid Burgers equation. We provide numerical evidence that this limit satisfies an entropy inequality for the inviscid Burgers equation. We demonstrate a Hamiltonian structure for the PDE.

A highly accurate analytical solution for the surface fields of a short vertical wire antenna lying on a multilayer ground

NASA Astrophysics Data System (ADS)

Parise, M.

2018-01-01

A highly accurate analytical solution is derived to the electromagnetic problem of a short vertical wire antenna located on a stratified ground. The derivation consists of three steps. First, the integration path of the integrals describing the fields of the dipole is deformed and wrapped around the pole singularities and the two vertical branch cuts of the integrands located in the upper half of the complex plane. This allows to decompose the radiated field into its three contributions, namely the above-surface ground wave, the lateral wave, and the trapped surface waves. Next, the square root terms responsible for the branch cuts are extracted from the integrands of the branch-cut integrals. Finally, the extracted square roots are replaced with their rational representations according to Newton's square root algorithm, and residue theorem is applied to give explicit expressions, in series form, for the fields. The rigorous integration procedure and the convergence of square root algorithm ensure that the obtained formulas converge to the exact solution. Numerical simulations are performed to show the validity and robustness of the developed formulation, as well as its advantages in terms of time cost over standard numerical integration procedures.

Quasi-static responses and variational principles in gradient plasticity

NASA Astrophysics Data System (ADS)

Nguyen, Quoc-Son

2016-12-01

Gradient models have been much discussed in the literature for the study of time-dependent or time-independent processes such as visco-plasticity, plasticity and damage. This paper is devoted to the theory of Standard Gradient Plasticity at small strain. A general and consistent mathematical description available for common time-independent behaviours is presented. Our attention is focussed on the derivation of general results such as the description of the governing equations for the global response and the derivation of related variational principles in terms of the energy and the dissipation potentials. It is shown that the quasi-static response under a loading path is a solution of an evolution variational inequality as in classical plasticity. The rate problem and the rate minimum principle are revisited. A time-discretization by the implicit scheme of the evolution equation leads to the increment problem. An increment of the response associated with a load increment is a solution of a variational inequality and satisfies also a minimum principle if the energy potential is convex. The increment minimum principle deals with stables solutions of the variational inequality. Some numerical methods are discussed in view of the numerical simulation of the quasi-static response.

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations - Part 1: Nonhydrostatic inertia-gravity modes

NASA Astrophysics Data System (ADS)

Konor, Celal S.; Randall, David A.

2018-05-01

We have used a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the nonhydrostatic anelastic inertia-gravity modes on a midlatitude f plane. The dispersion equations are derived from the linearized anelastic equations that are discretized on the Z, C, D, CD, (DC), A, E and B horizontal grids, and on the L and CP vertical grids. The effects of both horizontal grid spacing and vertical wavenumber are analyzed, and the role of nonhydrostatic effects is discussed. We also compare the results of the normal-mode analyses with numerical solutions obtained by running linearized numerical models based on the various horizontal grids. The sources and behaviors of the computational modes in the numerical simulations are also examined.Our normal-mode analyses with the Z, C, D, A, E and B grids generally confirm the conclusions of previous shallow-water studies for the cyclone-resolving scales (with low horizontal wavenumbers). We conclude that, aided by nonhydrostatic effects, the Z and C grids become overall more accurate for cloud-resolving resolutions (with high horizontal wavenumbers) than for the cyclone-resolving scales.A companion paper, Part 2, discusses the impacts of the discretization on the Rossby modes on a midlatitude β plane.

On Dynamics of Spinning Structures

NASA Technical Reports Server (NTRS)

Gupta, K. K.; Ibrahim, A.

2012-01-01

This paper provides details of developments pertaining to vibration analysis of gyroscopic systems, that involves a finite element structural discretization followed by the solution of the resulting matrix eigenvalue problem by a progressive, accelerated simultaneous iteration technique. Thus Coriolis, centrifugal and geometrical stiffness matrices are derived for shell and line elements, followed by the eigensolution details as well as solution of representative problems that demonstrates the efficacy of the currently developed numerical procedures and tools.

Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations

NASA Astrophysics Data System (ADS)

Ford, Neville J.; Connolly, Joseph A.

2009-07-01

We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.

Radiation-driven winds of hot stars. VI - Analytical solutions for wind models including the finite cone angle effect

NASA Technical Reports Server (NTRS)

Kudritzki, R. P.; Pauldrach, A.; Puls, J.; Abbott, D. C.

1989-01-01

Analytical solutions for radiation-driven winds of hot stars including the important finite cone angle effect (see Pauldrach et al., 1986; Friend and Abbott, 1986) are derived which approximate the detailed numerical solutions of the exact wind equation of motion very well. They allow a detailed discussion of the finite cone angle effect and provide for given line force parameters k, alpha, delta definite formulas for mass-loss rate M and terminal velocity v-alpha as function of stellar parameters.

Analytical solution for wave propagation through a graded index interface between a right-handed and a left-handed material.

PubMed

Dalarsson, Mariana; Tassin, Philippe

2009-04-13

We have investigated the transmission and reflection properties of structures incorporating left-handed materials with graded index of refraction. We present an exact analytical solution to Helmholtz' equation for a graded index profile changing according to a hyperbolic tangent function along the propagation direction. We derive expressions for the field intensity along the graded index structure, and we show excellent agreement between the analytical solution and the corresponding results obtained by accurate numerical simulations. Our model straightforwardly allows for arbitrary spectral dispersion.

Spikes and matter inhomogeneities in massless scalar field models

NASA Astrophysics Data System (ADS)

Coley, A. A.; Lim, W. C.

2016-01-01

We shall discuss the general relativistic generation of spikes in a massless scalar field or stiff perfect fluid model. We first investigate orthogonally transitive (OT) G 2 stiff fluid spike models both heuristically and numerically, and give a new exact OT G 2 stiff fluid spike solution. We then present a new two-parameter family of non-OT G 2 stiff fluid spike solutions, obtained by the generalization of non-OT G 2 vacuum spike solutions to the stiff fluid case by applying Geroch's transformation on a Jacobs seed. The dynamics of these new stiff fluid spike solutions is qualitatively different from that of the vacuum spike solutions in that the matter (stiff fluid) feels the spike directly and the stiff fluid spike solution can end up with a permanent spike. We then derive the evolution equations of non-OT G 2 stiff fluid models, including a second perfect fluid, in full generality, and briefly discuss some of their qualitative properties and their potential numerical analysis. Finally, we discuss how a fluid, and especially a stiff fluid or massless scalar field, affects the physics of the generation of spikes.

A comparison of capillary and rotational viscometry of aqueous solutions of hypromellose.

PubMed

Sklubalová, Z; Zatloukal, Z

2007-10-01

A comparison of capillary and rotational viscometry of gentle pseudoplastic solutions of hypromellose (HPMC 4000) by using only single-point value of viscosity is difficult. Single-point comparison becomes topical in consequence to the pharmacopoeial requirement that the apparent viscosity of 2% hypromellose solution should be read at the shear rate of approximately 10 s(-1). This communication is focused on the estimation of the suitable shear rate, D eta, at which the apparent viscosity read using the rotational viscometer is numerically equal to the dynamic viscosity read using a capillary viscometer. For the solutions of HPMC in concentrations up to 2% w/v, the non-linear regression equations generated showed the influencing of the D eta value by the dynamic viscosity and/or by the originally derived linear velocity of the solution flowing through the capillary viscometer tube. To compare the apparent viscosity read using the rotational viscometer with the dynamic viscosity read using capillary viscometer, the exact estimation of the shear rate D eta at which both viscosities are numerically equal is essential since it is markedly affected by the concentration of HPMC solution.

Analysis of groundwater flow and stream depletion in L-shaped fluvial aquifers

NASA Astrophysics Data System (ADS)

Lin, Chao-Chih; Chang, Ya-Chi; Yeh, Hund-Der

2018-04-01

Understanding the head distribution in aquifers is crucial for the evaluation of groundwater resources. This article develops a model for describing flow induced by pumping in an L-shaped fluvial aquifer bounded by impermeable bedrocks and two nearly fully penetrating streams. A similar scenario for numerical studies was reported in Kihm et al. (2007). The water level of the streams is assumed to be linearly varying with distance. The aquifer is divided into two subregions and the continuity conditions of the hydraulic head and flux are imposed at the interface of the subregions. The steady-state solution describing the head distribution for the model without pumping is first developed by the method of separation of variables. The transient solution for the head distribution induced by pumping is then derived based on the steady-state solution as initial condition and the methods of finite Fourier transform and Laplace transform. Moreover, the solution for stream depletion rate (SDR) from each of the two streams is also developed based on the head solution and Darcy's law. Both head and SDR solutions in the real time domain are obtained by a numerical inversion scheme called the Stehfest algorithm. The software MODFLOW is chosen to compare with the proposed head solution for the L-shaped aquifer. The steady-state and transient head distributions within the L-shaped aquifer predicted by the present solution are compared with the numerical simulations and measurement data presented in Kihm et al. (2007).

Backscattering and Nonparaxiality Arrest Collapse of Damped Nonlinear Waves

NASA Technical Reports Server (NTRS)

Fibich, G.; Ilan, B.; Tsynkov, S.

2002-01-01

The critical nonlinear Schrodinger equation (NLS) models the propagation of intense laser light in Kerr media. This equation is derived from the more comprehensive nonlinear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. It is known that if the input power of the laser beam (i.e., L(sub 2) norm of the initial solution) is sufficiently high, then the NLS model predicts that the beam will self-focus to a point (i.e.. collapse) at a finite propagation distance. Mathematically, this behavior corresponds to the formation of a singularity in the solution of the NLS. A key question which has been open for many years is whether the solution to the NLH, i.e., the 'parent' equation, may nonetheless exist and remain regular everywhere, in particular for those initial conditions (input powers) that lead to blowup in the NLS. In the current study, we address this question by introducing linear damping into both models and subsequently comparing the numerical solutions of the damped NLH (boundary-value problem) with the corresponding solutions of the damped NLS (initial-value problem). Linear damping is introduced in much the same way as done when analyzing the classical constant-coefficient Helmholtz equation using the limiting absorption principle. Numerically, we have found that it provides a very efficient tool for controlling the solutions of both the NLH and NHS. In particular, we have been able to identify initial conditions for which the NLS solution does become singular. whereas the NLH solution still remains regular everywhere. We believe that our finding of a larger domain of existence for the NLH than that for the NLS is accounted for by precisely those mechanisms, that have been neglected when deriving the NLS from the NLH, i.e., nonparaxiality and backscattering.

Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function

NASA Astrophysics Data System (ADS)

Conway, John T.; Cohl, Howard S.

2010-06-01

A new method is presented for Fourier decomposition of the Helmholtz Green function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Green function are split into their half advanced + half retarded and half advanced-half retarded components, and closed form solutions for these components are then obtained in terms of a Horn function and a Kampé de Fériet function respectively. Series solutions for the Fourier coefficients are given in terms of associated Legendre functions, Bessel and Hankel functions and a hypergeometric function. These series are derived either from the closed form 2-dimensional hypergeometric solutions or from an integral representation, or from both. A simple closed form far-field solution for the general Fourier coefficient is derived from the Hankel series. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented. Fourth order ordinary differential equations for the Fourier coefficients are also given and discussed briefly.

Group invariant solution for a pre-existing fracture driven by a power-law fluid in impermeable rock

NASA Astrophysics Data System (ADS)

Fareo, A. G.; Mason, D. P.

2013-12-01

The effect of power-law rheology on hydraulic fracturing is investigated. The evolution of a two-dimensional fracture with non-zero initial length and driven by a power-law fluid is analyzed. Only fluid injection into the fracture is considered. The surrounding rock mass is impermeable. With the aid of lubrication theory and the PKN approximation a partial differential equation for the fracture half-width is derived. Using a linear combination of the Lie-point symmetry generators of the partial differential equation, the group invariant solution is obtained and the problem is reduced to a boundary value problem for an ordinary differential equation. Exact analytical solutions are derived for hydraulic fractures with constant volume and with constant propagation speed. The asymptotic solution near the fracture tip is found. The numerical solution for general working conditions is obtained by transforming the boundary value problem to a pair of initial value problems. Throughout the paper, hydraulic fracturing with shear thinning, Newtonian and shear thickening fluids are compared.

Transient Dupuit Interface Flow with partially penetrating features

NASA Astrophysics Data System (ADS)

Bakker, Mark

1998-11-01

A comprehensive potential is presented for Dupuit interface flow in coastal aquifers where both the fresh water and salt water are moving. The resulting potential flow problem may be solved, for incompressible confined aquifers, using analytic functions. The vertical velocity of the interface may then be computed analytically and the change of the position of the interface may be simulated by numerical integration through time, starting with a known (or estimated) initial position. The upconing of the interface below a partially penetrating ditch or well may be studied if Dupuit solutions for such features are available. A new Dupuit solution is derived for a ditch that penetrates the aquifer partially from above; a Dupuit solution for a partially penetrating well may be obtained following a similar derivation. The new Dupuit solution is combined with the interface solution to simulate the upconing of an initially horizontal interface below a series of partially penetrating ditches; the interface converges to the known steady state position.

Combined structures-controls optimization of lattice trusses

NASA Technical Reports Server (NTRS)

Balakrishnan, A. V.

1991-01-01

The role that distributed parameter model can play in CSI is demonstrated, in particular in combined structures controls optimization problems of importance in preliminary design. Closed form solutions can be obtained for performance criteria such as rms attitude error, making possible analytical solutions of the optimization problem. This is in contrast to the need for numerical computer solution involving the inversion of large matrices in traditional finite element model (FEM) use. Another advantage of the analytic solution is that it can provide much needed insight into phenomena that can otherwise be obscured or difficult to discern from numerical computer results. As a compromise in level of complexity between a toy lab model and a real space structure, the lattice truss used in the EPS (Earth Pointing Satellite) was chosen. The optimization problem chosen is a generic one: of minimizing the structure mass subject to a specified stability margin and to a specified upper bond on the rms attitude error, using a co-located controller and sensors. Standard FEM treating each bar as a truss element is used, while the continuum model is anisotropic Timoshenko beam model. Performance criteria are derived for each model, except that for the distributed parameter model, explicit closed form solutions was obtained. Numerical results obtained by the two model show complete agreement.

Special solutions to Chazy equation

NASA Astrophysics Data System (ADS)

Varin, V. P.

2017-02-01

We consider the classical Chazy equation, which is known to be integrable in hypergeometric functions. But this solution has remained purely existential and was never used numerically. We give explicit formulas for hypergeometric solutions in terms of initial data. A special solution was found in the upper half plane H with the same tessellation of H as that of the modular group. This allowed us to derive some new identities for the Eisenstein series. We constructed a special solution in the unit disk and gave an explicit description of singularities on its natural boundary. A global solution to Chazy equation in elliptic and theta functions was found that allows parametrization of an arbitrary solution to Chazy equation. The results have applications to analytic number theory.

Computation of rapidly varied unsteady, free-surface flow

USGS Publications Warehouse

Basco, D.R.

1987-01-01

Many unsteady flows in hydraulics occur with relatively large gradients in free surface profiles. The assumption of hydrostatic pressure distribution with depth is no longer valid. These are rapidly-varied unsteady flows (RVF) of classical hydraulics and also encompass short wave propagation of coastal hydraulics. The purpose of this report is to present an introductory review of the Boussinnesq-type differential equations that describe these flows and to discuss methods for their numerical integration. On variable slopes and for large scale (finite-amplitude) disturbances, three independent derivational methods all gave differences in the motion equation for higher order terms. The importance of these higher-order terms for riverine applications must be determined by numerical experiments. Care must be taken in selection of the appropriate finite-difference scheme to minimize truncation error effects and the possibility of diverging (double mode) numerical solutions. It is recommended that practical hydraulics cases be established and tested numerically to demonstrate the order of differences in solution with those obtained from the long wave equations of St. Venant. (USGS)

Numerical simulation of the hydrodynamical combustion to strange quark matter

NASA Astrophysics Data System (ADS)

Niebergal, Brian; Ouyed, Rachid; Jaikumar, Prashanth

2010-12-01

We present results from a numerical solution to the burning of neutron matter inside a cold neutron star into stable u,d,s quark matter. Our method solves hydrodynamical flow equations in one dimension with neutrino emission from weak equilibrating reactions, and strange quark diffusion across the burning front. We also include entropy change from heat released in forming the stable quark phase. Our numerical results suggest burning front laminar speeds of 0.002-0.04 times the speed of light, much faster than previous estimates derived using only a reactive-diffusive description. Analytic solutions to hydrodynamical jump conditions with a temperature-dependent equation of state agree very well with our numerical findings for fluid velocities. The most important effect of neutrino cooling is that the conversion front stalls at lower density (below ≈2 times saturation density). In a two-dimensional setting, such rapid speeds and neutrino cooling may allow for a flame wrinkle instability to develop, possibly leading to detonation.

Simple Elasticity Modeling and Failure Prediction for Composite Flexbeams

NASA Technical Reports Server (NTRS)

Makeev, Andrew; Armanios, Erian; OBrien, T. Kevin (Technical Monitor)

2001-01-01

A simple 2D boundary element analysis, suitable for developing cost effective models for tapered composite laminates, is presented. Constant stress and displacement elements are used. Closed-form fundamental solutions are derived. Numerical results are provided for several configurations to illustrate the accuracy of the model.

Numerical Differentiation of Noisy, Nonsmooth Data

DOE PAGES

Chartrand, Rick

2011-01-01

We consider the problem of differentiating a function specified by noisy data. Regularizing the differentiation process avoids the noise amplification of finite-difference methods. We use total-variation regularization, which allows for discontinuous solutions. The resulting simple algorithm accurately differentiates noisy functions, including those which have a discontinuous derivative.

Numerical solution of the time fractional reaction-diffusion equation with a moving boundary

NASA Astrophysics Data System (ADS)

Zheng, Minling; Liu, Fawang; Liu, Qingxia; Burrage, Kevin; Simpson, Matthew J.

2017-06-01

A fractional reaction-diffusion model with a moving boundary is presented in this paper. An efficient numerical method is constructed to solve this moving boundary problem. Our method makes use of a finite difference approximation for the temporal discretization, and spectral approximation for the spatial discretization. The stability and convergence of the method is studied, and the errors of both the semi-discrete and fully-discrete schemes are derived. Numerical examples, motivated by problems from developmental biology, show a good agreement with the theoretical analysis and illustrate the efficiency of our method.

On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach

NASA Astrophysics Data System (ADS)

Gerstmayr, Johannes; Irschik, Hans

2008-12-01

In finite element methods that are based on position and slope coordinates, a representation of axial and bending deformation by means of an elastic line approach has become popular. Such beam and plate formulations based on the so-called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories. Examining the existing planar absolute nodal coordinate element, which uses a curvature proportional bending strain expression, it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and, even more serious, the normal force is incorrect. A correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived. The strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the St. Venant-Kirchhoff material: therefore, the strain energy is derived for the latter case and compared to classical nonlinear rod theories. The error in the original absolute nodal coordinate formulation is documented by numerical examples. The numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach, while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications. The numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre-curved static and dynamic problems, including buckling and eigenvalue analysis. The resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy-momentum conservation.

Fiber-reinforced materials: finite elements for the treatment of the inextensibility constraint

NASA Astrophysics Data System (ADS)

Auricchio, Ferdinando; Scalet, Giulia; Wriggers, Peter

2017-12-01

The present paper proposes a numerical framework for the analysis of problems involving fiber-reinforced anisotropic materials. Specifically, isotropic linear elastic solids, reinforced by a single family of inextensible fibers, are considered. The kinematic constraint equation of inextensibility in the fiber direction leads to the presence of an undetermined fiber stress in the constitutive equations. To avoid locking-phenomena in the numerical solution due to the presence of the constraint, mixed finite elements based on the Lagrange multiplier, perturbed Lagrangian, and penalty method are proposed. Several boundary-value problems under plane strain conditions are solved and numerical results are compared to analytical solutions, whenever the derivation is possible. The performed simulations allow to assess the performance of the proposed finite elements and to discuss several features of the developed formulations concerning the effective approximation for the displacement and fiber stress fields, mesh convergence, and sensitivity to penalty parameters.

A stochastic delay model for pricing debt and equity: Numerical techniques and applications

NASA Astrophysics Data System (ADS)

Tambue, Antoine; Kemajou Brown, Elisabeth; Mohammed, Salah

2015-01-01

Delayed nonlinear models for pricing corporate liabilities and European options were recently developed. Using self-financed strategy and duplication we were able to derive a Random Partial Differential Equation (RPDE) whose solutions describe the evolution of debt and equity values of a corporate in the last delay period interval in the accompanied paper (Kemajou et al., 2012) [14]. In this paper, we provide robust numerical techniques to solve the delayed nonlinear model for the corporate value, along with the corresponding RPDEs modeling the debt and equity values of the corporate. Using financial data from some firms, we forecast and compare numerical solutions from both the nonlinear delayed model and classical Merton model with the real corporate data. From this comparison, it comes up that in corporate finance the past dependence of the firm value process may be an important feature and therefore should not be ignored.

Discretizing singular point sources in hyperbolic wave propagation problems

DOE PAGES

Petersson, N. Anders; O'Reilly, Ossian; Sjogreen, Bjorn; ...

2016-06-01

Here, we develop high order accurate source discretizations for hyperbolic wave propagation problems in first order formulation that are discretized by finite difference schemes. By studying the Fourier series expansions of the source discretization and the finite difference operator, we derive sufficient conditions for achieving design accuracy in the numerical solution. Only half of the conditions in Fourier space can be satisfied through moment conditions on the source discretization, and we develop smoothness conditions for satisfying the remaining accuracy conditions. The resulting source discretization has compact support in physical space, and is spread over as many grid points as themore » number of moment and smoothness conditions. In numerical experiments we demonstrate high order of accuracy in the numerical solution of the 1-D advection equation (both in the interior and near a boundary), the 3-D elastic wave equation, and the 3-D linearized Euler equations.« less

Recursive analytical solution describing artificial satellite motion perturbed by an arbitrary number of zonal terms

NASA Technical Reports Server (NTRS)

Mueller, A. C.

1977-01-01

An analytical first order solution has been developed which describes the motion of an artificial satellite perturbed by an arbitrary number of zonal harmonics of the geopotential. A set of recursive relations for the solution, which was deduced from recursive relations of the geopotential, was derived. The method of solution is based on Von-Zeipel's technique applied to a canonical set of two-body elements in the extended phase space which incorporates the true anomaly as a canonical element. The elements are of Poincare type, that is, they are regular for vanishing eccentricities and inclinations. Numerical results show that this solution is accurate to within a few meters after 500 revolutions.

Multicritical points of the O(N) scalar theory in 2 < d < 4 for large N

NASA Astrophysics Data System (ADS)

Katsis, A.; Tetradis, N.

2018-05-01

We solve analytically the renormalization-group equation for the potential of the O (N)-symmetric scalar theory in the large-N limit and in dimensions 2 < d < 4, in order to look for nonperturbative fixed points that were found numerically in a recent study. We find new real solutions with singularities in the higher derivatives of the potential at its minimum, and complex solutions with branch cuts along the negative real axis.

Almost periodic cellular neural networks with neutral-type proportional delays

NASA Astrophysics Data System (ADS)

Xiao, Songlin

2018-03-01

This paper presents a new result on the existence, uniqueness and generalised exponential stability of almost periodic solutions for cellular neural networks with neutral-type proportional delays and D operator. Based on some novel differential inequality techniques, a testable condition is derived to ensure that all the state trajectories of the system converge to an almost periodic solution with a positive exponential convergence rate. The effectiveness of the obtained result is illustrated by a numerical example.

Exact Riemann solutions of the Ripa model for flat and non-flat bottom topographies

NASA Astrophysics Data System (ADS)

Rehman, Asad; Ali, Ishtiaq; Qamar, Shamsul

2018-03-01

This article is concerned with the derivation of exact Riemann solutions for Ripa model considering flat and non-flat bottom topographies. The Ripa model is a system of shallow water equations accounting for horizontal temperature gradients. In the case of non-flat bottom topography, the mass, momentum and energy conservation principles are utilized to relate the left and right states across the step-type bottom topography. The resulting system of algebraic equations is solved iteratively. Different numerical case studies of physical interest are considered. The solutions obtained from developed exact Riemann solvers are compared with the approximate solutions of central upwind scheme.

Stability analysis and wave dynamics of an extended hybrid traffic flow model

NASA Astrophysics Data System (ADS)

Wang, Yu-Qing; Zhou, Chao-Fan; Li, Wei-Kang; Yan, Bo-Wen; Jia, Bin; Wang, Ji-Xin

2018-02-01

The stability analysis and wave dynamic properties of an extended hybrid traffic flow model, WZY model, are intensively studied in this paper. The linear stable condition obtained by the linear stability analysis is presented. Besides, by means of analyzing Korteweg-de Vries equation, we present soliton waves in the metastable region. Moreover, the multiscale perturbation technique is applied to derive the traveling wave solution of the model. Furthermore, by means of performing Darboux transformation, the first-order and second-order doubly-periodic solutions and rational solutions are presented. It can be found that analytical solutions match well with numerical simulations.

A MATLAB-Aided Method for Teaching Calculus-Based Business Mathematics

ERIC Educational Resources Information Center

Liang, Jiajuan; Pan, William S. Y.

2009-01-01

MATLAB is a powerful package for numerical computation. MATLAB contains a rich pool of mathematical functions and provides flexible plotting functions for illustrating mathematical solutions. The course of calculus-based business mathematics consists of two major topics: 1) derivative and its applications in business; and 2) integration and its…

Light diffusion in N-layered turbid media: steady-state domain.

PubMed

Liemert, André; Kienle, Alwin

2010-01-01

We deal with light diffusion in N-layered turbid media. The steady-state diffusion equation is solved for N-layered turbid media having a finite or an infinitely thick N'th layer. Different refractive indices are considered in the layers. The Fourier transform formalism is applied to derive analytical solutions of the fluence rate in Fourier space. The inverse Fourier transform is calculated using four different methods to test their performance and accuracy. Further, to avoid numerical errors, approximate formulas in Fourier space are derived. Fast solutions for calculation of the spatially resolved reflectance and transmittance from the N-layered turbid media ( approximately 10 ms) with small relative differences (<10(-7)) are found. Additionally, the solutions of the diffusion equation are compared to Monte Carlo simulations for turbid media having up to 20 layers.

Transient well flow in leaky multiple-aquifer systems

NASA Astrophysics Data System (ADS)

Hemker, C. J.

1985-10-01

A previously developed eigenvalue analysis approach to groundwater flow in leaky multiple aquifers is used to derive exact solutions for transient well flow problems in leaky and confined systems comprising any number of aquifers. Equations are presented for the drawdown distribution in systems of infinite extent, caused by wells penetrating one or more of the aquifers completely and discharging each layer at a constant rate. Since the solution obtained may be regarded as a combined analytical-numerical technique, a type of one-dimensional modelling can be applied to find approximate solutions for several complicating conditions. Numerical evaluations are presented as time-drawdown curves and include effects of storage in the aquitard, unconfined conditions, partially penetrating wells and stratified aquifers. The outcome of calculations for relatively simple systems compares very well with published corresponding results. The proposed multilayer solution can be a valuable tool in aquifer test evaluation, as it provides the analytical expression required to enable the application of existing computer methods to the determination of aquifer characteristics.

Swimming in a two-dimensional Brinkman fluid: Computational modeling and regularized solutions

NASA Astrophysics Data System (ADS)

Leiderman, Karin; Olson, Sarah D.

2016-02-01

The incompressible Brinkman equation represents the homogenized fluid flow past obstacles that comprise a small volume fraction. In nondimensional form, the Brinkman equation can be characterized by a single parameter that represents the friction or resistance due to the obstacles. In this work, we derive an exact fundamental solution for 2D Brinkman flow driven by a regularized point force and describe the numerical method to use it in practice. To test our solution and method, we compare numerical results with an analytic solution of a stationary cylinder in a uniform Brinkman flow. Our method is also compared to asymptotic theory; for an infinite-length, undulating sheet of small amplitude, we recover an increasing swimming speed as the resistance is increased. With this computational framework, we study a model swimmer of finite length and observe an enhancement in propulsion and efficiency for small to moderate resistance. Finally, we study the interaction of two swimmers where attraction does not occur when the initial separation distance is larger than the screening length.

Effect of solute immobilization on the stability problem within the fractional model in the solute analog of the Horton-Rogers-Lapwood problem.

PubMed

Klimenko, Lyudmila S; Maryshev, Boris S

2017-11-24

The paper is devoted to the linear stability analysis within the solute analogue of the Horton-Rogers-Lapwood (HRL) problem. The solid nanoparticles are treated as solute within the continuous approach. Therefore, we consider the infinite horizontal porous layer saturated with a mixture (carrier fluid and solute). Solute transport in porous media is very often complicated by solute immobilization on a solid matrix of porous media. Solute immobilization (solute sorption) is taken into account within the fractal model of the MIM approach. According to this model a solute in porous media immobilizes within random time intervals and the distribution of such random variable does not have a finite mean value, which has a good agreement with some experiments. The solute concentration difference between the layer boundaries is assumed as constant. We consider two cases of horizontal external filtration flux: constant and time-modulated. For the constant flux the system of equations that determines the frequency of neutral oscillations and the critical value of the Rayleigh-Darcy number is derived. Neutral curves of the critical parameters on the governing parameters are plotted. Stability maps are obtained numerically in a wide range of parameters of the system. We have found that taking immobilization into account leads to an increase in the critical value of the Rayleigh-Darcy number with an increase in the intensity of the external filtration flux. The case of weak time-dependent external flux is investigated analytically. We have shown that the modulated external flux leads to an increase in the critical value of the Rayleigh-Darcy number and a decrease in the critical wave number. For moderate time-dependent filtration flux the differential equation with Caputo fractional derivatives has been obtained for the description of the behavior near the convection instability threshold. This equation is analyzed numerically by the Floquet method; the parametric excitation of convection is observed.

Efficient Numerical Methods for Nonlinear-Facilitated Transport and Exchange in a Blood-Tissue Exchange Unit

PubMed Central

Poulain, Christophe A.; Finlayson, Bruce A.; Bassingthwaighte, James B.

2010-01-01

The analysis of experimental data obtained by the multiple-indicator method requires complex mathematical models for which capillary blood-tissue exchange (BTEX) units are the building blocks. This study presents a new, nonlinear, two-region, axially distributed, single capillary, BTEX model. A facilitated transporter model is used to describe mass transfer between plasma and intracellular spaces. To provide fast and accurate solutions, numerical techniques suited to nonlinear convection-dominated problems are implemented. These techniques are the random choice method, an explicit Euler-Lagrange scheme, and the MacCormack method with and without flux correction. The accuracy of the numerical techniques is demonstrated, and their efficiencies are compared. The random choice, Euler-Lagrange and plain MacCormack method are the best numerical techniques for BTEX modeling. However, the random choice and Euler-Lagrange methods are preferred over the MacCormack method because they allow for the derivation of a heuristic criterion that makes the numerical methods stable without degrading their efficiency. Numerical solutions are also used to illustrate some nonlinear behaviors of the model and to show how the new BTEX model can be used to estimate parameters from experimental data. PMID:9146808

Efficient control schemes with limited computation complexity for Tomographic AO systems on VLTs and ELTs

NASA Astrophysics Data System (ADS)

Petit, C.; Le Louarn, M.; Fusco, T.; Madec, P.-Y.

2011-09-01

Various tomographic control solutions have been proposed during the last decades to ensure efficient or even optimal closed-loop correction to tomographic Adaptive Optics (AO) concepts such as Laser Tomographic AO (LTAO), Multi-Conjugate AO (MCAO). The optimal solution, based on Linear Quadratic Gaussian (LQG) approach, as well as suboptimal but efficient solutions such as Pseudo-Open Loop Control (POLC) require multiple Matrix Vector Multiplications (MVM). Disregarding their respective performance, these efficient control solutions thus exhibit strong increase of on-line complexity and their implementation may become difficult in demanding cases. Among them, two cases are of particular interest. First, the system Real-Time Computer architecture and implementation is derived from past or present solutions and does not support multiple MVM. This is the case of the AO Facility which RTC architecture is derived from the SPARTA platform and inherits its simple MVM architecture, which does not fit with LTAO control solutions for instance. Second, considering future systems such as Extremely Large Telescopes, the number of degrees of freedom is twenty to one hundred times bigger than present systems. In these conditions, tomographic control solutions can hardly be used in their standard form and optimized implementation shall be considered. Single MVM tomographic control solutions represent a potential solution, and straightforward solutions such as Virtual Deformable Mirrors have been already proposed for LTAO but with tuning issues. We investigate in this paper the possibility to derive from tomographic control solutions, such as POLC or LQG, simplified control solutions ensuring simple MVM architecture and that could be thus implemented on nowadays systems or future complex systems. We theoretically derive various solutions and analyze their respective performance on various systems thanks to numerical simulation. We discuss the optimization of their performance and stability issues with respect to classic control solutions. We finally discuss off-line computation and implementation constraints.

Boundary-layer effects in composite laminates. I - Free-edge stress singularities. II - Free-edge stress solutions and basic characteristics

NASA Technical Reports Server (NTRS)

Wang, S. S.; Choi, I.

1982-01-01

The fundamental nature of the boundary-layer effect in fiber-reinforced composite laminates is formulated in terms of the theory of anisotropic elasticity. The basic structure of the boundary-layer field solution is obtained by using Lekhnitskii's stress potentials (1963). The boundary-layer stress field is found to be singular at composite laminate edges, and the exact order or strength of the boundary layer stress singularity is determined using an eigenfunction expansion method. A complete solution to the boundary-layer problem is then derived, and the convergence and accuracy of the solution are analyzed, comparing results with existing approximate numerical solutions. The solution method is demonstrated for a symmetric graphite-epoxy composite.

Particlelike solutions of the Einstein-Dirac equations

NASA Astrophysics Data System (ADS)

Finster, Felix; Smoller, Joel; Yau, Shing-Tung

1999-05-01

The coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state are derived. Using numerical methods, we construct an infinite number of solitonlike solutions of these equations. The stability of the solutions is analyzed. For weak coupling (i.e., small rest mass of the fermions), all the solutions are linearly stable (with respect to spherically symmetric perturbations), whereas for stronger coupling, both stable and unstable solutions exist. For the physical interpretation, we discuss how the energy of the fermions and the (ADM) mass behave as functions of the rest mass of the fermions. Although gravitation is not renormalizable, our solutions of the Einstein-Dirac equations are regular and well behaved even for strong coupling.

Application of ANNs approach for wave-like and heat-like equations

NASA Astrophysics Data System (ADS)

Jafarian, Ahmad; Baleanu, Dumitru

2017-12-01

Artificial neural networks are data processing systems which originate from human brain tissue studies. The remarkable abilities of these networks help us to derive desired results from complicated raw data. In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems. It should be noted that many physical phenomena such as coupling currents in a flat multi-strand two-layer super conducting cable, non-homogeneous elastic waves in soils and earthquake stresses, are described by initial-boundary value wave and heat partial differential equations with variable coefficients. To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned initial-boundary value problem. Finally, several computer simulations confirmed the theoretical results and demonstrating applicability of the method.

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations – Part 2: Quasi-geostrophic Rossby modes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Konor, Celal S.; Randall, David A.

We use a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the quasi-geostrophic anelastic baroclinic and barotropic Rossby modes on a midlatitude β plane. The dispersion equations are derived for the linearized anelastic system, discretized on the Z, C, D, CD, (DC), A, E and B horizontal grids, and on the L and CP vertical grids. The effects of various horizontal grid spacings and vertical wavenumbers are discussed. A companion paper, Part 1, discusses the impacts of the discretization on the inertia–gravity modes on a midlatitude f plane.The results of our normal-modemore » analyses for the Rossby waves overall support the conclusions of the previous studies obtained with the shallow-water equations. We identify an area of disagreement with the E-grid solution.« less

Explosive magnetorotational instability in Keplerian disks

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shtemler, Yu., E-mail: shtemler@bgu.ac.il; Liverts, E., E-mail: eliverts@bgu.ac.il; Mond, M., E-mail: mond@bgu.ac.il

Differentially rotating disks under the effect of axial magnetic field are prone to a nonlinear explosive magnetorotational instability (EMRI). The dynamic equations that govern the temporal evolution of the amplitudes of three weakly detuned resonantly interacting modes are derived. As distinct from exponential growth in the strict resonance triads, EMRI occurs due to the resonant interactions of an MRI mode with stable Alfvén–Coriolis and magnetosonic modes. Numerical solutions of the dynamic equations for amplitudes of a triad indicate that two types of perturbations behavior can be excited for resonance conditions: (i) EMRI which leads to infinite values of the threemore » amplitudes within a finite time, and (ii) bounded irregular oscillations of all three amplitudes. Asymptotic explicit solutions of the dynamic equations are obtained for EMRI regimes and are shown to match the numerical solutions near the explosion time.« less

On the Huygens absorbing boundary conditions for electromagnetics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Berenger, Jean-Pierre

A new absorbing boundary condition (ABC) is presented for the solution of Maxwell equations in unbounded spaces. Called the Huygens ABC, this condition is a generalization of two previously published ABCs, namely the multiple absorbing surfaces (MAS) and the re-radiating boundary condition (rRBC). The properties of the Huygens ABC are derived theoretically in continuous spaces and in the finite-difference (FDTD) discretized space. A solution is proposed to render the Huygens ABC effective for the absorption of evanescent waves. Numerical experiments with the FDTD method show that the effectiveness of the Huygens ABC is close to that of the PML ABCmore » in some realistic problems of numerical electromagnetics. It is also shown in the paper that a combination of the Huygens ABC with the PML ABC is very well suited to the solution of some particular problems.« less

Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction

NASA Astrophysics Data System (ADS)

Deswal, Sunita; Kalkal, Kapil Kumar; Sheoran, Sandeep Singh

2016-09-01

A mathematical model of fractional order two-temperature generalized thermoelasticity with diffusion and initial stress is proposed to analyze the transient wave phenomenon in an infinite thermoelastic half-space. The governing equations are derived in cylindrical coordinates for a two dimensional axi-symmetric problem. The analytical solution is procured by employing the Laplace and Hankel transforms for time and space variables respectively. The solutions are investigated in detail for a time dependent heat source. By using numerical inversion method of integral transforms, we obtain the solutions for displacement, stress, temperature and diffusion fields in physical domain. Computations are carried out for copper material and displayed graphically. The effect of fractional order parameter, two-temperature parameter, diffusion, initial stress and time on the different thermoelastic and diffusion fields is analyzed on the basis of analytical and numerical results. Some special cases have also been deduced from the present investigation.

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

NASA Astrophysics Data System (ADS)

Shirvany, Yazdan; Hayati, Mohsen; Moradian, Rostam

2008-12-01

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations – Part 2: Quasi-geostrophic Rossby modes

DOE PAGES

Konor, Celal S.; Randall, David A.

2018-05-08

We use a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the quasi-geostrophic anelastic baroclinic and barotropic Rossby modes on a midlatitude β plane. The dispersion equations are derived for the linearized anelastic system, discretized on the Z, C, D, CD, (DC), A, E and B horizontal grids, and on the L and CP vertical grids. The effects of various horizontal grid spacings and vertical wavenumbers are discussed. A companion paper, Part 1, discusses the impacts of the discretization on the inertia–gravity modes on a midlatitude f plane.The results of our normal-modemore » analyses for the Rossby waves overall support the conclusions of the previous studies obtained with the shallow-water equations. We identify an area of disagreement with the E-grid solution.« less

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations - Part 2: Quasi-geostrophic Rossby modes

NASA Astrophysics Data System (ADS)

Konor, Celal S.; Randall, David A.

2018-05-01

We use a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the quasi-geostrophic anelastic baroclinic and barotropic Rossby modes on a midlatitude β plane. The dispersion equations are derived for the linearized anelastic system, discretized on the Z, C, D, CD, (DC), A, E and B horizontal grids, and on the L and CP vertical grids. The effects of various horizontal grid spacings and vertical wavenumbers are discussed. A companion paper, Part 1, discusses the impacts of the discretization on the inertia-gravity modes on a midlatitude f plane.The results of our normal-mode analyses for the Rossby waves overall support the conclusions of the previous studies obtained with the shallow-water equations. We identify an area of disagreement with the E-grid solution.

A Numerical Method for Predicting Rayleigh Surface Wave Velocity in Anisotropic Crystals (Postprint)

DTIC Science & Technology

2017-09-05

generalized version of the equations are very difficult to derive, even in symbolic math languages such as Mathematica. As a result, the equations are...formalism, Math . Mech. Solids 9 (1) (2004) 5–15. [8] M. Destrade, The explicit secular equation for surface acoustic waves in monoclinic elastic crystals...Q. J. Mech. Appl. Math . 55 (2) (2002) 297–311. [10] D. Taylor, Surface waves in anisotropic media: the secular equation and its numerical solution

Formulation of a General Technique for Predicting Pneumatic Attenuation Errors in Airborne Pressure Sensing Devices

NASA Technical Reports Server (NTRS)

Whitmore, Stephen A.

1988-01-01

Presented is a mathematical model derived from the Navier-Stokes equations of momentum and continuity, which may be accurately used to predict the behavior of conventionally mounted pneumatic sensing systems subject to arbitrary pressure inputs. Numerical techniques for solving the general model are developed. Both step and frequency response lab tests were performed. These data are compared with solutions of the mathematical model and show excellent agreement. The procedures used to obtain the lab data are described. In-flight step and frequency response data were obtained. Comparisons with numerical solutions of the math model show good agreement. Procedures used to obtain the flight data are described. Difficulties encountered with obtaining the flight data are discussed.

An adaptive time-stepping strategy for solving the phase field crystal model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhang, Zhengru, E-mail: zrzhang@bnu.edu.cn; Ma, Yuan, E-mail: yuner1022@gmail.com; Qiao, Zhonghua, E-mail: zqiao@polyu.edu.hk

2013-09-15

In this work, we will propose an adaptive time step method for simulating the dynamics of the phase field crystal (PFC) model. The numerical simulation of the PFC model needs long time to reach steady state, and then large time-stepping method is necessary. Unconditionally energy stable schemes are used to solve the PFC model. The time steps are adaptively determined based on the time derivative of the corresponding energy. It is found that the use of the proposed time step adaptivity cannot only resolve the steady state solution, but also the dynamical development of the solution efficiently and accurately. Themore » numerical experiments demonstrate that the CPU time is significantly saved for long time simulations.« less

On approximating guided waves in plates with thin anisotropic coatings by means of effective boundary conditions

PubMed

Niklasson; Datta; Dunn

2000-09-01

In this paper, effective boundary conditions for elastic wave propagation in plates with thin coatings are derived. These effective boundary conditions are used to obtain an approximate dispersion relation for guided waves in an isotropic plate with thin anisotropic coating layers. The accuracy of the effective boundary conditions is investigated numerically by comparison with exact solutions for two different material systems. The systems considered consist of a metallic core with thin superconducting coatings. It is shown that for wavelengths long compared to the coating thickness there is excellent agreement between the approximate and exact solutions for both systems. Furthermore, numerical results presented might be used to characterize coating properties by ultrasonic techniques.

Soliton and kink jams in traffic flow with open boundaries.

PubMed

Muramatsu, M; Nagatani, T

1999-07-01

Soliton density wave is investigated numerically and analytically in the optimal velocity model (a car-following model) of a one-dimensional traffic flow with open boundaries. Soliton density wave is distinguished from the kink density wave. It is shown that the soliton density wave appears only at the threshold of occurrence of traffic jams. The Korteweg-de Vries (KdV) equation is derived from the optimal velocity model by the use of the nonlinear analysis. It is found that the traffic soliton appears only near the neutral stability line. The soliton solution is analytically obtained from the perturbed KdV equation. It is shown that the soliton solution obtained from the nonlinear analysis is consistent with that of the numerical simulation.

Exact quantum numbers of collapsed and non-collapsed two-string solutions in the spin-1/2 Heisenberg spin chain

NASA Astrophysics Data System (ADS)

Deguchi, Tetsuo; Ranjan Giri, Pulak

2016-04-01

Every solution of the Bethe-ansatz equations (BAEs) is characterized by a set of quantum numbers, by which we can evaluate it numerically. However, no general rule is known how to give quantum numbers for the physical solutions of BAE. For the spin-1/2 XXX chain we rigorously derive all the quantum numbers for the complete set of the Bethe-ansatz eigenvectors in the two down-spin sector with any chain length N. Here we obtain them both for real and complex solutions. We also show that all the solutions associated with them are distinct. Consequently, we prove the completeness of the Bethe ansatz and give an exact expression for the number of real solutions which correspond to collapsed bound-state solutions (i.e., two-string solutions) in the sector: 2[(N-1)/2-(N/π ){{tan}}-1(\\sqrt{N-1})] in terms of Gauss’ symbol. Moreover, we prove in the sector the scheme conjectured by Takahashi for solving BAE systematically. We also suggest that by applying the present method we can derive the quantum numbers for the spin-1/2 XXZ chain.

Applications of Analytical Self-Similar Solutions of Reynolds-Averaged Models for Instability-Induced Turbulent Mixing

NASA Astrophysics Data System (ADS)

Hartland, Tucker; Schilling, Oleg

2017-11-01

Analytical self-similar solutions to several families of single- and two-scale, eddy viscosity and Reynolds stress turbulence models are presented for Rayleigh-Taylor, Richtmyer-Meshkov, and Kelvin-Helmholtz instability-induced turbulent mixing. The use of algebraic relationships between model coefficients and physical observables (e.g., experimental growth rates) following from the self-similar solutions to calibrate a member of a given family of turbulence models is shown. It is demonstrated numerically that the algebraic relations accurately predict the value and variation of physical outputs of a Reynolds-averaged simulation in flow regimes that are consistent with the simplifying assumptions used to derive the solutions. The use of experimental and numerical simulation data on Reynolds stress anisotropy ratios to calibrate a Reynolds stress model is briefly illustrated. The implications of the analytical solutions for future Reynolds-averaged modeling of hydrodynamic instability-induced mixing are briefly discussed. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

A three-dimensional semi-analytical solution for predicting drug release through the orifice of a spherical device.

PubMed

Simon, Laurent; Ospina, Juan

2016-07-25

Three-dimensional solute transport was investigated for a spherical device with a release hole. The governing equation was derived using the Fick's second law. A mixed Neumann-Dirichlet condition was imposed at the boundary to represent diffusion through a small region on the surface of the device. The cumulative percentage of drug released was calculated in the Laplace domain and represented by the first term of an infinite series of Legendre and modified Bessel functions of the first kind. Application of the Zakian algorithm yielded the time-domain closed-form expression. The first-order solution closely matched a numerical solution generated by Mathematica(®). The proposed method allowed computation of the characteristic time. A larger surface pore resulted in a smaller effective time constant. The agreement between the numerical solution and the semi-analytical method improved noticeably as the size of the orifice increased. It took four time constants for the device to release approximately ninety-eight of its drug content. Copyright © 2016 Elsevier B.V. All rights reserved.

Quantification of mixing in vesicle suspensions using numerical simulations in two dimensions.

PubMed

Kabacaoğlu, G; Quaife, B; Biros, G

2017-02-01

We study mixing in Stokesian vesicle suspensions in two dimensions on a cylindrical Couette apparatus using numerical simulations. The vesicle flow simulation is done using a boundary integral method, and the advection-diffusion equation for the mixing of the solute is solved using a pseudo-spectral scheme. We study the effect of the area fraction, the viscosity contrast between the inside (the vesicles) and the outside (the bulk) fluid, the initial condition of the solute, and the mixing metric. We compare mixing in the suspension with mixing in the Couette apparatus without vesicles. On the one hand, the presence of vesicles in most cases slightly suppresses mixing. This is because the solute can be only diffused across the vesicle interface and not advected. On the other hand, there exist spatial distributions of the solute for which the unperturbed Couette flow completely fails to mix whereas the presence of vesicles enables mixing. We derive a simple condition that relates the velocity and solute and can be used to characterize the cases in which the presence of vesicles promotes mixing.

Quantification of mixing in vesicle suspensions using numerical simulations in two dimensions

PubMed Central

Quaife, B.; Biros, G.

2017-01-01

We study mixing in Stokesian vesicle suspensions in two dimensions on a cylindrical Couette apparatus using numerical simulations. The vesicle flow simulation is done using a boundary integral method, and the advection-diffusion equation for the mixing of the solute is solved using a pseudo-spectral scheme. We study the effect of the area fraction, the viscosity contrast between the inside (the vesicles) and the outside (the bulk) fluid, the initial condition of the solute, and the mixing metric. We compare mixing in the suspension with mixing in the Couette apparatus without vesicles. On the one hand, the presence of vesicles in most cases slightly suppresses mixing. This is because the solute can be only diffused across the vesicle interface and not advected. On the other hand, there exist spatial distributions of the solute for which the unperturbed Couette flow completely fails to mix whereas the presence of vesicles enables mixing. We derive a simple condition that relates the velocity and solute and can be used to characterize the cases in which the presence of vesicles promotes mixing. PMID:28344432

An irregular lattice method for elastic wave propagation

NASA Astrophysics Data System (ADS)

O'Brien, Gareth S.; Bean, Christopher J.

2011-12-01

Lattice methods are a class of numerical scheme which represent a medium as a connection of interacting nodes or particles. In the case of modelling seismic wave propagation, the interaction term is determined from Hooke's Law including a bond-bending term. This approach has been shown to model isotropic seismic wave propagation in an elastic or viscoelastic medium by selecting the appropriate underlying lattice structure. To predetermine the material constants, this methodology has been restricted to regular grids, hexagonal or square in 2-D or cubic in 3-D. Here, we present a method for isotropic elastic wave propagation where we can remove this lattice restriction. The methodology is outlined and a relationship between the elastic material properties and an irregular lattice geometry are derived. The numerical method is compared with an analytical solution for wave propagation in an infinite homogeneous body along with comparing the method with a numerical solution for a layered elastic medium. The dispersion properties of this method are derived from a plane wave analysis showing the scheme is more dispersive than a regular lattice method. Therefore, the computational costs of using an irregular lattice are higher. However, by removing the regular lattice structure the anisotropic nature of fracture propagation in such methods can be removed.

Computational reacting gas dynamics

NASA Technical Reports Server (NTRS)

Lam, S. H.

1993-01-01

In the study of high speed flows at high altitudes, such as that encountered by re-entry spacecrafts, the interaction of chemical reactions and other non-equilibrium processes in the flow field with the gas dynamics is crucial. Generally speaking, problems of this level of complexity must resort to numerical methods for solutions, using sophisticated computational fluid dynamics (CFD) codes. The difficulties introduced by reacting gas dynamics can be classified into three distinct headings: (1) the usually inadequate knowledge of the reaction rate coefficients in the non-equilibrium reaction system; (2) the vastly larger number of unknowns involved in the computation and the expected stiffness of the equations; and (3) the interpretation of the detailed reacting CFD numerical results. The research performed accepts the premise that reacting flows of practical interest in the future will in general be too complex or 'untractable' for traditional analytical developments. The power of modern computers must be exploited. However, instead of focusing solely on the construction of numerical solutions of full-model equations, attention is also directed to the 'derivation' of the simplified model from the given full-model. In other words, the present research aims to utilize computations to do tasks which have traditionally been done by skilled theoreticians: to reduce an originally complex full-model system into an approximate but otherwise equivalent simplified model system. The tacit assumption is that once the appropriate simplified model is derived, the interpretation of the detailed numerical reacting CFD numerical results will become much easier. The approach of the research is called computational singular perturbation (CSP).

Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions and a numerical result.

PubMed

Wu, Yang; Kelly, Damien P

2014-12-12

The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf's treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of [Formula: see text] and [Formula: see text] type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of [Formula: see text] and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by [Formula: see text], where [Formula: see text] is the replica order. These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.

Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions and a numerical result

NASA Astrophysics Data System (ADS)

Wu, Yang; Kelly, Damien P.

2014-12-01

The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf's treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of ? and ? type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of ? and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by ?, where ? is the replica order. These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.

A Taylor weak-statement algorithm for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Baker, A. J.; Kim, J. W.

1987-01-01

Finite element analysis, applied to computational fluid dynamics (CFD) problem classes, presents a formal procedure for establishing the ingredients of a discrete approximation numerical solution algorithm. A classical Galerkin weak-statement formulation, formed on a Taylor series extension of the conservation law system, is developed herein that embeds a set of parameters eligible for constraint according to specification of suitable norms. The derived family of Taylor weak statements is shown to contain, as special cases, over one dozen independently derived CFD algorithms published over the past several decades for the high speed flow problem class. A theoretical analysis is completed that facilitates direct qualitative comparisons. Numerical results for definitive linear and nonlinear test problems permit direct quantitative performance comparisons.

Numerical Analysis of an H 1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

PubMed Central

Wang, Jinfeng; Zhao, Meng; Zhang, Min; Liu, Yang; Li, Hong

2014-01-01

We discuss and analyze an H 1-Galerkin mixed finite element (H 1-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H 1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H 1-GMFE method. Based on the discussion on the theoretical error analysis in L 2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H 1-norm. Moreover, we derive and analyze the stability of H 1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure. PMID:25184148

High-order asynchrony-tolerant finite difference schemes for partial differential equations

NASA Astrophysics Data System (ADS)

Aditya, Konduri; Donzis, Diego A.

2017-12-01

Synchronizations of processing elements (PEs) in massively parallel simulations, which arise due to communication or load imbalances between PEs, significantly affect the scalability of scientific applications. We have recently proposed a method based on finite-difference schemes to solve partial differential equations in an asynchronous fashion - synchronization between PEs is relaxed at a mathematical level. While standard schemes can maintain their stability in the presence of asynchrony, their accuracy is drastically affected. In this work, we present a general methodology to derive asynchrony-tolerant (AT) finite difference schemes of arbitrary order of accuracy, which can maintain their accuracy when synchronizations are relaxed. We show that there are several choices available in selecting a stencil to derive these schemes and discuss their effect on numerical and computational performance. We provide a simple classification of schemes based on the stencil and derive schemes that are representative of different classes. Their numerical error is rigorously analyzed within a statistical framework to obtain the overall accuracy of the solution. Results from numerical experiments are used to validate the performance of the schemes.

Landau-Zener extension of the Tavis-Cummings model: Structure of the solution

DOE PAGES

Sun, Chen; Sinitsyn, Nikolai A.

2016-09-07

We explore the recently discovered solution of the driven Tavis-Cummings model (DTCM). It describes interaction of an arbitrary number of two-level systems with a bosonic mode that has linearly time-dependent frequency. We derive compact and tractable expressions for transition probabilities in terms of the well-known special functions. In this form, our formulas are suitable for fast numerical calculations and analytical approximations. As an application, we obtain the semiclassical limit of the exact solution and compare it to prior approximations. Furthermore, we also reveal connection between DTCM and q-deformed binomial statistics.

Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus

NASA Astrophysics Data System (ADS)

Chen, Shanzhen; Jiang, Xiaoyun

2012-08-01

In this paper, analytical solutions to time-fractional partial differential equations in a multi-layer annulus are presented. The final solutions are obtained in terms of Mittag-Leffler function by using the finite integral transform technique and Laplace transform technique. In addition, the classical diffusion equation (α=1), the Helmholtz equation (α→0) and the wave equation (α=2) are discussed as special cases. Finally, an illustrative example problem for the three-layer semi-circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative.

The exact fundamental solution for the Benes tracking problem

NASA Astrophysics Data System (ADS)

Balaji, Bhashyam

2009-05-01

The universal continuous-discrete tracking problem requires the solution of a Fokker-Planck-Kolmogorov forward equation (FPKfe) for an arbitrary initial condition. Using results from quantum mechanics, the exact fundamental solution for the FPKfe is derived for the state model of arbitrary dimension with Benes drift that requires only the computation of elementary transcendental functions and standard linear algebra techniques- no ordinary or partial differential equations need to be solved. The measurement process may be an arbitrary, discrete-time nonlinear stochastic process, and the time step size can be arbitrary. Numerical examples are included, demonstrating its utility in practical implementation.

Numerical simulation of freshwater/seawater interaction in a dual-permeability karst system with conduits: the development of discrete-continuum VDFST-CFP model

NASA Astrophysics Data System (ADS)

Xu, Zexuan; Hu, Bill

2016-04-01

Dual-permeability karst aquifers of porous media and conduit networks with significant different hydrological characteristics are widely distributed in the world. Discrete-continuum numerical models, such as MODFLOW-CFP and CFPv2, have been verified as appropriate approaches to simulate groundwater flow and solute transport in numerical modeling of karst hydrogeology. On the other hand, seawater intrusion associated with fresh groundwater resources contamination has been observed and investigated in numbers of coastal aquifers, especially under conditions of sea level rise. Density-dependent numerical models including SEAWAT are able to quantitatively evaluate the seawater/freshwater interaction processes. A numerical model of variable-density flow and solute transport - conduit flow process (VDFST-CFP) is developed to provide a better description of seawater intrusion and submarine groundwater discharge in a coastal karst aquifer with conduits. The coupling discrete-continuum VDFST-CFP model applies Darcy-Weisbach equation to simulate non-laminar groundwater flow in the conduit system in which is conceptualized and discretized as pipes, while Darcy equation is still used in continuum porous media. Density-dependent groundwater flow and solute transport equations with appropriate density terms in both conduit and porous media systems are derived and numerically solved using standard finite difference method with an implicit iteration procedure. Synthetic horizontal and vertical benchmarks are created to validate the newly developed VDFST-CFP model by comparing with other numerical models such as variable density SEAWAT, couplings of constant density groundwater flow and solute transport MODFLOW/MT3DMS and discrete-continuum CFPv2/UMT3D models. VDFST-CFP model improves the simulation of density dependent seawater/freshwater mixing processes and exchanges between conduit and matrix. Continuum numerical models greatly overestimated the flow rate under turbulent flow condition but discrete-continuum models provide more accurate results. Parameters sensitivities analysis indicates that conduit diameter and friction factor, matrix hydraulic conductivity and porosity are important parameters that significantly affect variable-density flow and solute transport simulation. The pros and cons of model assumptions, conceptual simplifications and numerical techniques in VDFST-CFP are discussed. In general, the development of VDFST-CFP model is an innovation in numerical modeling methodology and could be applied to quantitatively evaluate the seawater/freshwater interaction in coastal karst aquifers. Keywords: Discrete-continuum numerical model; Variable density flow and transport; Coastal karst aquifer; Non-laminar flow

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations – Part 1: Nonhydrostatic inertia–gravity modes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Konor, Celal S.; Randall, David A.

We have used a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the nonhydrostatic anelastic inertia–gravity modes on a midlatitude f plane. The dispersion equations are derived from the linearized anelastic equations that are discretized on the Z, C, D, CD, (DC), A, E and B horizontal grids, and on the L and CP vertical grids. The effects of both horizontal grid spacing and vertical wavenumber are analyzed, and the role of nonhydrostatic effects is discussed. We also compare the results of the normal-mode analyses with numerical solutions obtained by runningmore » linearized numerical models based on the various horizontal grids. The sources and behaviors of the computational modes in the numerical simulations are also examined.Our normal-mode analyses with the Z, C, D, A, E and B grids generally confirm the conclusions of previous shallow-water studies for the cyclone-resolving scales (with low horizontal wavenumbers). We conclude that, aided by nonhydrostatic effects, the Z and C grids become overall more accurate for cloud-resolving resolutions (with high horizontal wavenumbers) than for the cyclone-resolving scales.A companion paper, Part 2, discusses the impacts of the discretization on the Rossby modes on a midlatitude β plane.« less

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations – Part 1: Nonhydrostatic inertia–gravity modes

DOE PAGES

Konor, Celal S.; Randall, David A.

2018-05-08

We have used a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the nonhydrostatic anelastic inertia–gravity modes on a midlatitude f plane. The dispersion equations are derived from the linearized anelastic equations that are discretized on the Z, C, D, CD, (DC), A, E and B horizontal grids, and on the L and CP vertical grids. The effects of both horizontal grid spacing and vertical wavenumber are analyzed, and the role of nonhydrostatic effects is discussed. We also compare the results of the normal-mode analyses with numerical solutions obtained by runningmore » linearized numerical models based on the various horizontal grids. The sources and behaviors of the computational modes in the numerical simulations are also examined.Our normal-mode analyses with the Z, C, D, A, E and B grids generally confirm the conclusions of previous shallow-water studies for the cyclone-resolving scales (with low horizontal wavenumbers). We conclude that, aided by nonhydrostatic effects, the Z and C grids become overall more accurate for cloud-resolving resolutions (with high horizontal wavenumbers) than for the cyclone-resolving scales.A companion paper, Part 2, discusses the impacts of the discretization on the Rossby modes on a midlatitude β plane.« less

Electrostatic forces in the Poisson-Boltzmann systems

NASA Astrophysics Data System (ADS)

Xiao, Li; Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray

2013-09-01

Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely used in structural and functional analyses of biomolecules. A limitation of the numerical strategies is that it is conceptually difficult to incorporate these types of models into molecular mechanics simulations, mainly because of the issue in assigning atomic forces. In this theoretical study, we first derived the Maxwell stress tensor for molecular systems obeying the full nonlinear Poisson-Boltzmann equation. We further derived formulations of analytical electrostatic forces given the Maxwell stress tensor and discussed the relations of the formulations with those published in the literature. We showed that the formulations derived from the Maxwell stress tensor require a weaker condition for its validity, applicable to nonlinear Poisson-Boltzmann systems with a finite number of singularities such as atomic point charges and the existence of discontinuous dielectric as in the widely used classical piece-wise constant dielectric models.

Analysis of cavitation bubble dynamics in a liquid

NASA Technical Reports Server (NTRS)

Fontenot, L. L.; Lee, Y. C.

1971-01-01

General differential equations governing the dynamics of the cavitation bubbles in a liquid were derived. With the assumption of spherical symmetry the governing equations were simplified. Closed form solutions were obtained for simple cases, and numerical solutions were calculated for complicated ones. The growth and the collapse of the bubble were analyzed, oscillations of the bubbles were studied, and the stability of the cavitation bubbles were investigated. The results show that the cavitation bubbles are unstable, and the oscillation is not sinusoidal.

Implementation of a block Lanczos algorithm for Eigenproblem solution of gyroscopic systems

NASA Technical Reports Server (NTRS)

Gupta, Kajal K.; Lawson, Charles L.

1987-01-01

The details of implementation of a general numerical procedure developed for the accurate and economical computation of natural frequencies and associated modes of any elastic structure rotating along an arbitrary axis are described. A block version of the Lanczos algorithm is derived for the solution that fully exploits associated matrix sparsity and employs only real numbers in all relevant computations. It is also capable of determining multiple roots and proves to be most efficient when compared to other, similar, exisiting techniques.

Numerical solution of linear and nonlinear Fredholm integral equations by using weighted mean-value theorem.

PubMed

Altürk, Ahmet

2016-01-01

Mean value theorems for both derivatives and integrals are very useful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. In this article, a semi-analytical method that is based on weighted mean-value theorem for obtaining solutions for a wide class of Fredholm integral equations of the second kind is introduced. Illustrative examples are provided to show the significant advantage of the proposed method over some existing techniques.

A positive and entropy-satisfying finite volume scheme for the Baer-Nunziato model

NASA Astrophysics Data System (ADS)

Coquel, Frédéric; Hérard, Jean-Marc; Saleh, Khaled

2017-02-01

We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer-Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in [16] for the isentropic Baer-Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer-Nunziato model, namely Schwendeman-Wahle-Kapila's Godunov-type scheme [39] and Tokareva-Toro's HLLC scheme [44]. The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.

A positive and entropy-satisfying finite volume scheme for the Baer–Nunziato model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Coquel, Frédéric, E-mail: frederic.coquel@cmap.polytechnique.fr; Hérard, Jean-Marc, E-mail: jean-marc.herard@edf.fr; Saleh, Khaled, E-mail: saleh@math.univ-lyon1.fr

We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer–Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in for the isentropic Baer–Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound aremore » also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer–Nunziato model, namely Schwendeman–Wahle–Kapila's Godunov-type scheme and Tokareva–Toro's HLLC scheme . The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.« less

Analytical Solutions for an Escape Problem in a Disc with an Arbitrary Distribution of Exit Holes Along Its Boundary

NASA Astrophysics Data System (ADS)

Marshall, J. S.

2016-12-01

We analytically construct solutions for the mean first-passage time and splitting probabilities for the escape problem of a particle moving with continuous Brownian motion in a confining planar disc with an arbitrary distribution (i.e., of any number, size and spacing) of exit holes/absorbing sections along its boundary. The governing equations for these quantities are Poisson's equation with a (non-zero) constant forcing term and Laplace's equation, respectively, and both are subject to a mixture of homogeneous Neumann and Dirichlet boundary conditions. Our solutions are expressed as explicit closed formulae written in terms of a parameterising variable via a conformal map, using special transcendental functions that are defined in terms of an associated Schottky group. They are derived by exploiting recent results for a related problem of fluid mechanics that describes a unidirectional flow over "no-slip/no-shear" surfaces, as well as results from potential theory, all of which were themselves derived using the same theory of Schottky groups. They are exact up to the determination of a finite set of mapping parameters, which is performed numerically. Their evaluation also requires the numerical inversion of the parameterising conformal map. Computations for a series of illustrative examples are also presented.

DRBEM solution of the acid-mediated tumour invasion model with time-dependent carrying capacities

NASA Astrophysics Data System (ADS)

Meral, Gülnihal

2017-07-01

It is known that the pH level of the extracellular tumour environment directly effects the progression of the tumour. In this study, the mathematical model for the acid-mediated tumour cell invasion consisting of a system of nonlinear reaction diffusion equations describing the interaction between the density of the tumour cells, normal cells and the concentration of ? protons produced by the tumour cells is solved numerically using the combined application of dual reciprocity boundary element method (DRBEM) and finite difference method. The space derivatives in the model are discretised by DRBEM using the fundamental solution of Laplace equation considering the time derivative and the nonlinearities as the nonhomogenity. The resulting systems of ordinary differential equations after the application of DRBEM are then discretised using forward difference. Because of the highly nonlinear character of the model, there arises difficulties in solving the model especially for two-dimensions and the boundary-only nature of DRBEM discretisation gives the advantage of having solutions with a lower computational cost. The proposed method is tested with different kinds of carrying capacities which also depend on time. The results of the numerical simulations are compared among each case and seen to confirm the expected behaviour of the model.

Mechanics of additively manufactured porous biomaterials based on the rhombicuboctahedron unit cell.

PubMed

Hedayati, R; Sadighi, M; Mohammadi-Aghdam, M; Zadpoor, A A

2016-01-01

Thanks to recent developments in additive manufacturing techniques, it is now possible to fabricate porous biomaterials with arbitrarily complex micro-architectures. Micro-architectures of such biomaterials determine their physical and biological properties, meaning that one could potentially improve the performance of such biomaterials through rational design of micro-architecture. The relationship between the micro-architecture of porous biomaterials and their physical and biological properties has therefore received increasing attention recently. In this paper, we studied the mechanical properties of porous biomaterials made from a relatively unexplored unit cell, namely rhombicuboctahedron. We derived analytical relationships that relate the micro-architecture of such porous biomaterials, i.e. the dimensions of the rhombicuboctahedron unit cell, to their elastic modulus, Poisson's ratio, and yield stress. Finite element models were also developed to validate the analytical solutions. Analytical and numerical results were compared with experimental data from one of our recent studies. It was found that analytical solutions and numerical results show a very good agreement particularly for smaller values of apparent density. The elastic moduli predicted by analytical and numerical models were in very good agreement with experimental observations too. While in excellent agreement with each other, analytical and numerical models somewhat over-predicted the yield stress of the porous structures as compared to experimental data. As the ratio of the vertical struts to the inclined struts, α, approaches zero and infinity, the rhombicuboctahedron unit cell respectively approaches the octahedron (or truncated cube) and cube unit cells. For those limits, the analytical solutions presented here were found to approach the analytic solutions obtained for the octahedron, truncated cube, and cube unit cells, meaning that the presented solutions are generalizations of the analytical solutions obtained for several other types of porous biomaterials. Copyright © 2015 Elsevier Ltd. All rights reserved.

Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface

NASA Astrophysics Data System (ADS)

Coco, Armando; Russo, Giovanni

2018-05-01

In this paper we propose a second-order accurate numerical method to solve elliptic problems with discontinuous coefficients (with general non-homogeneous jumps in the solution and its gradient) in 2D and 3D. The method consists of a finite-difference method on a Cartesian grid in which complex geometries (boundaries and interfaces) are embedded, and is second order accurate in the solution and the gradient itself. In order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface, two numerical values are assigned on grid points that are close to the interface: a real value, that represents the numerical solution on that grid point, and a ghost value, that represents the numerical solution extrapolated from the other side of the interface, obtained by enforcing the assigned non-homogeneous jump conditions on the solution and its flux. The method is also extended to the case of matrix coefficient. The linear system arising from the discretization is solved by an efficient multigrid approach. Unlike the 1D case, grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient. A proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor, comparable with the one obtained by Local Fourier Analysis for rectangular domains. The method is robust enough to handle large jump in the coefficients: order of accuracy, monotonicity of the errors and good convergence factor are maintained by the scheme.

The sound field of a rotating dipole in a plug flow.

PubMed

Wang, Zhao-Huan; Belyaev, Ivan V; Zhang, Xiao-Zheng; Bi, Chuan-Xing; Faranosov, Georgy A; Dowell, Earl H

2018-04-01

An analytical far field solution for a rotating point dipole source in a plug flow is derived. The shear layer of the jet is modelled as an infinitely thin cylindrical vortex sheet and the far field integral is calculated by the stationary phase method. Four numerical tests are performed to validate the derived solution as well as to assess the effects of sound refraction from the shear layer. First, the calculated results using the derived formulations are compared with the known solution for a rotating dipole in a uniform flow to validate the present model in this fundamental test case. After that, the effects of sound refraction for different rotating dipole sources in the plug flow are assessed. Then the refraction effects on different frequency components of the signal at the observer position, as well as the effects of the motion of the source and of the type of source are considered. Finally, the effect of different sound speeds and densities outside and inside the plug flow is investigated. The solution obtained may be of particular interest for propeller and rotor noise measurements in open jet anechoic wind tunnels.

Soliton-cnoidal interactional wave solutions for the reduced Maxwell-Bloch equations

NASA Astrophysics Data System (ADS)

Huang, Li-Li; Qiao, Zhi-Jun; Chen, Yong

2018-02-01

Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell-Bloch equations. The nonlocal symmetries of the reduced Maxwell-Bloch equations are obtained by the truncated Painleve expansion approach and the Mobious invariant property. The nonlocal symmetries are localized to a prolonged system by introducing suitable auxiliary dependent variables. The extended system can be closed and a novel Lie point symmetry system is constructed. By solving the initial value problems, a new type of finite symmetry transformations is obtained to derive periodic waves, Ma breathers and breathers travelling on the background of periodic line waves. Then rich exact interactional solutions are derived between solitary waves and other waves including cnoidal waves, rational waves, Painleve waves, and periodic waves through similarity reductions. In particular, several new types of localized excitations including rogue waves are found, which stem from the arbitrary function generated in the process of similarity reduction. By computer numerical simulation, the dynamics of these localized excitations and interactional solutions are discussed, which exhibit meaningful structures.

Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation.

PubMed

Van Gorder, Robert A

2013-04-01

We provide a formulation of the local induction approximation (LIA) for the motion of a vortex filament in the Cartesian reference frame (the extrinsic coordinate system) which allows for scaling of the reference coordinate. For general monotone scalings of the reference coordinate, we derive an equation for the planar solution to the derivative nonlinear Schrödinger equation governing the LIA. We proceed to solve this equation perturbatively in small amplitude through an application of multiple-scales analysis, which allows for accurate computation of the period of the planar vortex filament. The perturbation result is shown to agree strongly with numerical simulations, and we also relate this solution back to the solution obtained in the arclength reference frame (the intrinsic coordinate system). Finally, we discuss nonmonotone coordinate scalings and their application for finding self-intersections of vortex filaments. These self-intersecting vortex filaments are likely unstable and collapse into other structures or dissipate completely.

Dynamics from a mathematical model of a two-state gas laser

NASA Astrophysics Data System (ADS)

Kleanthous, Antigoni; Hua, Tianshu; Manai, Alexandre; Yawar, Kamran; Van Gorder, Robert A.

2018-05-01

Motivated by recent work in the area, we consider the behavior of solutions to a nonlinear PDE model of a two-state gas laser. We first review the derivation of the two-state gas laser model, before deriving a non-dimensional model given in terms of coupled nonlinear partial differential equations. We then classify the steady states of this system, in order to determine the possible long-time asymptotic solutions to this model, as well as corresponding stability results, showing that the only uniform steady state (the zero motion state) is unstable, while a linear profile in space is stable. We then provide numerical simulations for the full unsteady model. We show for a wide variety of initial conditions that the solutions tend toward the stable linear steady state profiles. We also consider traveling wave solutions, and determine the unique wave speed (in terms of the other model parameters) which allows wave-like solutions to exist. Despite some similarities between the model and the inviscid Burger's equation, the solutions we obtain are much more regular than the solutions to the inviscid Burger's equation, with no evidence of shock formation or loss of regularity.

Analytical Solution for Optimum Design of Furrow Irrigation Systems

NASA Astrophysics Data System (ADS)

Kiwan, M. E.

1996-05-01

An analytical solution for the optimum design of furrow irrigation systems is derived. The non-linear calculus optimization method is used to formulate a general form for designing the optimum system elements under circumstances of maximizing the water application efficiency of the system during irrigation. Different system bases and constraints are considered in the solution. A full irrigation water depth is considered to be achieved at the tail of the furrow line. The solution is based on neglecting the recession and depletion times after off-irrigation. This assumption is valid in the case of open-end (free gradient) furrow systems rather than closed-end (closed dike) systems. Illustrative examples for different systems are presented and the results are compared with the output obtained using an iterative numerical solution method. The final derived solution is expressed as a function of the furrow length ratio (the furrow length to the water travelling distance). The function of water travelling developed by Reddy et al. is considered for reaching the optimum solution. As practical results from the study, the optimum furrow elements for free gradient systems can be estimated to achieve the maximum application efficiency, i.e. furrow length, water inflow rate and cutoff irrigation time.

A 2D nonlinear multiring model for blood flow in large elastic arteries

NASA Astrophysics Data System (ADS)

Ghigo, Arthur R.; Fullana, Jose-Maria; Lagrée, Pierre-Yves

2017-12-01

In this paper, we propose a two-dimensional nonlinear ;multiring; model to compute blood flow in axisymmetric elastic arteries. This model is designed to overcome the numerical difficulties of three-dimensional fluid-structure interaction simulations of blood flow without using the over-simplifications necessary to obtain one-dimensional blood flow models. This multiring model is derived by integrating over concentric rings of fluid the simplified long-wave Navier-Stokes equations coupled to an elastic model of the arterial wall. The resulting system of balance laws provides a unified framework in which both the motion of the fluid and the displacement of the wall are dealt with simultaneously. The mathematical structure of the multiring model allows us to use a finite volume method that guarantees the conservation of mass and the positivity of the numerical solution and can deal with nonlinear flows and large deformations of the arterial wall. We show that the finite volume numerical solution of the multiring model provides at a reasonable computational cost an asymptotically valid description of blood flow velocity profiles and other averaged quantities (wall shear stress, flow rate, ...) in large elastic and quasi-rigid arteries. In particular, we validate the multiring model against well-known solutions such as the Womersley or the Poiseuille solutions as well as against steady boundary layer solutions in quasi-rigid constricted and expanded tubes.

Monotonic Derivative Correction for Calculation of Supersonic Flows

ERIC Educational Resources Information Center

Bulat, Pavel V.; Volkov, Konstantin N.

2016-01-01

Aim of the study: This study examines numerical methods for solving the problems in gas dynamics, which are based on an exact or approximate solution to the problem of breakdown of an arbitrary discontinuity (the Riemann problem). Results: Comparative analysis of finite difference schemes for the Euler equations integration is conducted on the…

Improving BeiDou real-time precise point positioning with numerical weather models

NASA Astrophysics Data System (ADS)

Lu, Cuixian; Li, Xingxing; Zus, Florian; Heinkelmann, Robert; Dick, Galina; Ge, Maorong; Wickert, Jens; Schuh, Harald

2017-09-01

Precise positioning with the current Chinese BeiDou Navigation Satellite System is proven to be of comparable accuracy to the Global Positioning System, which is at centimeter level for the horizontal components and sub-decimeter level for the vertical component. But the BeiDou precise point positioning (PPP) shows its limitation in requiring a relatively long convergence time. In this study, we develop a numerical weather model (NWM) augmented PPP processing algorithm to improve BeiDou precise positioning. Tropospheric delay parameters, i.e., zenith delays, mapping functions, and horizontal delay gradients, derived from short-range forecasts from the Global Forecast System of the National Centers for Environmental Prediction (NCEP) are applied into BeiDou real-time PPP. Observational data from stations that are capable of tracking the BeiDou constellation from the International GNSS Service (IGS) Multi-GNSS Experiments network are processed, with the introduced NWM-augmented PPP and the standard PPP processing. The accuracy of tropospheric delays derived from NCEP is assessed against with the IGS final tropospheric delay products. The positioning results show that an improvement in convergence time up to 60.0 and 66.7% for the east and vertical components, respectively, can be achieved with the NWM-augmented PPP solution compared to the standard PPP solutions, while only slight improvement in the solution convergence can be found for the north component. A positioning accuracy of 5.7 and 5.9 cm for the east component is achieved with the standard PPP that estimates gradients and the one that estimates no gradients, respectively, in comparison to 3.5 cm of the NWM-augmented PPP, showing an improvement of 38.6 and 40.1%. Compared to the accuracy of 3.7 and 4.1 cm for the north component derived from the two standard PPP solutions, the one of the NWM-augmented PPP solution is improved to 2.0 cm, by about 45.9 and 51.2%. The positioning accuracy for the up component improves from 11.4 and 13.2 cm with the two standard PPP solutions to 8.0 cm with the NWM-augmented PPP solution, an improvement of 29.8 and 39.4%, respectively.

Constrained orbital intercept-evasion

NASA Astrophysics Data System (ADS)

Zatezalo, Aleksandar; Stipanovic, Dusan M.; Mehra, Raman K.; Pham, Khanh

2014-06-01

An effective characterization of intercept-evasion confrontations in various space environments and a derivation of corresponding solutions considering a variety of real-world constraints are daunting theoretical and practical challenges. Current and future space-based platforms have to simultaneously operate as components of satellite formations and/or systems and at the same time, have a capability to evade potential collisions with other maneuver constrained space objects. In this article, we formulate and numerically approximate solutions of a Low Earth Orbit (LEO) intercept-maneuver problem in terms of game-theoretic capture-evasion guaranteed strategies. The space intercept-evasion approach is based on Liapunov methodology that has been successfully implemented in a number of air and ground based multi-player multi-goal game/control applications. The corresponding numerical algorithms are derived using computationally efficient and orbital propagator independent methods that are previously developed for Space Situational Awareness (SSA). This game theoretical but at the same time robust and practical approach is demonstrated on a realistic LEO scenario using existing Two Line Element (TLE) sets and Simplified General Perturbation-4 (SGP-4) propagator.

Analysis of the discontinuous Galerkin method applied to the European option pricing problem

NASA Astrophysics Data System (ADS)

Hozman, J.

2013-12-01

In this paper we deal with a numerical solution of a one-dimensional Black-Scholes partial differential equation, an important scalar nonstationary linear convection-diffusion-reaction equation describing the pricing of European vanilla options. We present a derivation of the numerical scheme based on the space semidiscretization of the model problem by the discontinuous Galerkin method with nonsymmetric stabilization of diffusion terms and with the interior and boundary penalty. The main attention is paid to the investigation of a priori error estimates for the proposed scheme. The appended numerical experiments illustrate the theoretical results and the potency of the method, consequently.

Computation of forces arising from the polarizable continuum model within the domain-decomposition paradigm

NASA Astrophysics Data System (ADS)

Gatto, Paolo; Lipparini, Filippo; Stamm, Benjamin

2017-12-01

The domain-decomposition (dd) paradigm, originally introduced for the conductor-like screening model, has been recently extended to the dielectric Polarizable Continuum Model (PCM), resulting in the ddPCM method. We present here a complete derivation of the analytical derivatives of the ddPCM energy with respect to the positions of the solute's atoms and discuss their efficient implementation. As it is the case for the energy, we observe a quadratic scaling, which is discussed and demonstrated with numerical tests.

Calculation of Sensitivity Derivatives in an MDAO Framework

NASA Technical Reports Server (NTRS)

Moore, Kenneth T.

2012-01-01

During gradient-based optimization of a system, it is necessary to generate the derivatives of each objective and constraint with respect to each design parameter. If the system is multidisciplinary, it may consist of a set of smaller "components" with some arbitrary data interconnection and process work ow. Analytical derivatives in these components can be used to improve the speed and accuracy of the derivative calculation over a purely numerical calculation; however, a multidisciplinary system may include both components for which derivatives are available and components for which they are not. Three methods to calculate the sensitivity of a mixed multidisciplinary system are presented: the finite difference method, where the derivatives are calculated numerically; the chain rule method, where the derivatives are successively cascaded along the system's network graph; and the analytic method, where the derivatives come from the solution of a linear system of equations. Some improvements to these methods, to accommodate mixed multidisciplinary systems, are also presented; in particular, a new method is introduced to allow existing derivatives to be used inside of finite difference. All three methods are implemented and demonstrated in the open-source MDAO framework OpenMDAO. It was found that there are advantages to each of them depending on the system being solved.

Series solution for two-frequency Bragg interaction using the Korpel-Poon multiple-scattering model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Appel, R.K.; Somekh, M.G.

1993-03-01

The two-frequency acousto-optic interaction is analytically solved in the Bragg regime by use of a multiple-scattering model that was previously described by Korpel and Poon [J. Opt. Soc. Am. 70, 817-820 (1980)]. The method uses Feynman diagrams to conceptualize the problem and demonstrates the applicability of such a method to model a relatively complex system. The solution presented is compared with that derived by Hecht [IEEE Trans. Sonics Ultrason. SU-24, 7-18 (1977)], who used a coupled-mode approach. The derivation of the authors' solution is relatively simple and leads to a formulation that appears to be more compact. Numerical evaluations havemore » demonstrated their equivalence. The authors present results that illustrate the dependence of the diffracted beam intensities on the amplitude of the two acoustic waves. 21 refs., 8 figs.« less

Statistical mechanics of an ideal active fluid confined in a channel

NASA Astrophysics Data System (ADS)

Wagner, Caleb; Baskaran, Aparna; Hagan, Michael

The statistical mechanics of ideal active Brownian particles (ABPs) confined in a channel is studied by obtaining the exact solution of the steady-state Smoluchowski equation for the 1-particle distribution function. The solution is derived using results from the theory of two-way diffusion equations, combined with an iterative procedure that is justified by numerical results. Using this solution, we quantify the effects of confinement on the spatial and orientational order of the ensemble. Moreover, we rigorously show that both the bulk density and the fraction of particles on the channel walls obey simple scaling relations as a function of channel width. By considering a constant-flux steady state, an effective diffusivity for ABPs is derived which shows signatures of the persistent motion that characterizes ABP trajectories. Finally, we discuss how our techniques generalize to other active models, including systems whose activity is modeled in terms of an Ornstein-Uhlenbeck process.

Front and pulse solutions for the complex Ginzburg-Landau equation with higher-order terms.

PubMed

Tian, Huiping; Li, Zhonghao; Tian, Jinping; Zhou, Guosheng

2002-12-01

We investigate one-dimensional complex Ginzburg-Landau equation with higher-order terms and discuss their influences on the multiplicity of solutions. An exact analytic front solution is presented. By stability analysis for the original partial differential equation, we derive its necessary stability condition for amplitude perturbations. This condition together with the exact front solution determine the region of parameter space where the uniformly translating front solution can exist. In addition, stable pulses, chaotic pulses, and attenuation pulses appear generally if the parameters are out of the range. Finally, applying these analysis into the optical transmission system numerically we find that the stable transmission of optical pulses can be achieved if the parameters are appropriately chosen.

Iterative discrete ordinates solution of the equation for surface-reflected radiance

NASA Astrophysics Data System (ADS)

Radkevich, Alexander

2017-11-01

This paper presents a new method of numerical solution of the integral equation for the radiance reflected from an anisotropic surface. The equation relates the radiance at the surface level with BRDF and solutions of the standard radiative transfer problems for a slab with no reflection on its surfaces. It is also shown that the kernel of the equation satisfies the condition of the existence of a unique solution and the convergence of the successive approximations to that solution. The developed method features two basic steps: discretization on a 2D quadrature, and solving the resulting system of algebraic equations with successive over-relaxation method based on the Gauss-Seidel iterative process. Presented numerical examples show good coincidence between the surface-reflected radiance obtained with DISORT and the proposed method. Analysis of contributions of the direct and diffuse (but not yet reflected) parts of the downward radiance to the total solution is performed. Together, they represent a very good initial guess for the iterative process. This fact ensures fast convergence. The numerical evidence is given that the fastest convergence occurs with the relaxation parameter of 1 (no relaxation). An integral equation for BRDF is derived as inversion of the original equation. The potential of this new equation for BRDF retrievals is analyzed. The approach is found not viable as the BRDF equation appears to be an ill-posed problem, and it requires knowledge the surface-reflected radiance on the entire domain of both Sun and viewing zenith angles.

Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with Mexican hat connectivity.

PubMed

Heitmann, Stewart; Ermentrout, G Bard

2015-06-01

Spatiotemporal waves of synchronized activity are known to arise in oscillatory neural networks with lateral inhibitory coupling. How such patterns respond to dynamic changes in coupling strength is largely unexplored. The present study uses analysis and simulation to investigate the evolution of wave patterns when the strength of lateral inhibition is varied dynamically. Neural synchronization was modeled by a spatial ring of Kuramoto oscillators with Mexican hat lateral coupling. Broad bands of coexisting stable wave solutions were observed at all levels of inhibition. The stability of these waves was formally analyzed in both the infinite ring and the finite ring. The broad range of multi-stability predicted hysteresis in transitions between neighboring wave solutions when inhibition is slowly varied. Numerical simulation confirmed the predicted transitions when inhibition was ramped down from a high initial value. However, non-wave solutions emerged from the uniform solution when inhibition was ramped upward from zero. These solutions correspond to spatially periodic deviations of phase that we call ripple states. Numerical continuation showed that stable ripple states emerge from synchrony via a supercritical pitchfork bifurcation. The normal form of this bifurcation was derived analytically, and its predictions compared against the numerical results. Ripple states were also found to bifurcate from wave solutions, but these were locally unstable. Simulation also confirmed the existence of hysteresis and ripple states in two spatial dimensions. Our findings show that spatial synchronization patterns can remain structurally stable despite substantial changes in network connectivity.

Self-Similar Compressible Free Vortices

NASA Technical Reports Server (NTRS)

vonEllenrieder, Karl

1998-01-01

Lie group methods are used to find both exact and numerical similarity solutions for compressible perturbations to all incompressible, two-dimensional, axisymmetric vortex reference flow. The reference flow vorticity satisfies an eigenvalue problem for which the solutions are a set of two-dimensional, self-similar, incompressible vortices. These solutions are augmented by deriving a conserved quantity for each eigenvalue, and identifying a Lie group which leaves the reference flow equations invariant. The partial differential equations governing the compressible perturbations to these reference flows are also invariant under the action of the same group. The similarity variables found with this group are used to determine the decay rates of the velocities and thermodynamic variables in the self-similar flows, and to reduce the governing partial differential equations to a set of ordinary differential equations. The ODE's are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are used to examine the dependencies of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Also, experimental data on compressible free vortex flow are compared to the analytical results, the evolution of vortices from initial states which are not self-similar is discussed, and the energy transfer in a slightly-compressible vortex is considered.

Summary and status of the Horizons ephemeris system

NASA Astrophysics Data System (ADS)

Giorgini, J.

2011-10-01

Since 1996, the Horizons system has provided searchable access to JPL ephemerides for all known solar system bodies, several dozen spacecraft, planetary system barycenters, and some libration points. Responding to 18 400 000 requests from 300 000 unique addresses, the system has recently averaged 420 000 ephemeris requests per month. Horizons is accessed and automated using three interfaces: interactive telnet, web-browser form, and e-mail command-file. Asteroid and comet ephemerides are numerically integrated from JPL's database of initial conditions. This small-body database is updated hourly by a separate process as new measurements and discoveries are reported by the Minor Planet Center and automatically incorporated into new JPL orbit solutions. Ephemerides for other objects are derived by interpolating previously developed solutions whose trajectories have been represented in a file. For asteroids and comets, such files may be dynamically created and transferred to users, effectively recording integrator output. These small-body SPK files may then be interpolated by user software to reproduce the trajectory without duplicating the numerically integrated n-body dynamical model or PPN equations of motion. Other Horizons output is numerical and in the form of plain-text observer, vector, osculating element, or close-approach tables, typically expected be read by other software as input. About one hundred quantities can be requested in various time-scales and coordinate systems. For JPL small-body solutions, this includes statistical uncertainties derived from measurement covariance and state transition matrices. With the exception of some natural satellites, Horizons is consistent with DE405/DE406, the IAU 1976 constants, ITRF93, and IAU2009 rotational models.

Shock formation in the dispersionless Kadomtsev-Petviashvili equation

NASA Astrophysics Data System (ADS)

Grava, T.; Klein, C.; Eggers, J.

2016-04-01

The dispersionless Kadomtsev-Petviashvili (dKP) equation {{≤ft({{u}t}+u{{u}x}\\right)}x}={{u}yy} is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation numerically we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation {{u}t}+u{{u}x}=0 . We show numerically that the solutions to the transformed equation stays regular for longer times than the solution of the dKP equation. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the (x, y) plane, where the solution of the dKP equation exists in a weak sense only, and a shock front develops. A local expansion reveals the universal scaling structure of the shock, which after a suitable change of coordinates corresponds to a generic cusp catastrophe. We provide a heuristic derivation of the shock front position near the critical point for the solution of the dKP equation, and study the solution of the dKP equation when a small amount of dissipation is added. Using multiple-scale analysis, we show that in the limit of small dissipation and near the critical point of the dKP solution, the solution of the dissipative dKP equation converges to a Pearcey integral. We test and illustrate our results by detailed comparisons with numerical simulations of both the regularized equation, the dKP equation, and the asymptotic description given in terms of the Pearcey integral.

Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems.

PubMed

Wang, Xiang-Sheng; He, Dongdong; Wylie, Jonathan J; Huang, Huaxiong

2014-02-01

We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention. Previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations. By adopting a methodology that preserves the symmetries of the PNP system, we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation. Due to the simple form of the transcendental equation, it can be solved numerically in a straightforward manner. Our methodology thus provides a standard procedure for solving the PNP system and we illustrate this by solving some practical examples. Despite the fact that for three ions, previous studies have indicated that multiple solutions may exist, we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the three-ion case.

Global collocation methods for approximation and the solution of partial differential equations

NASA Technical Reports Server (NTRS)

Solomonoff, A.; Turkel, E.

1986-01-01

Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.

Gravity Gradient Tensor of Arbitrary 3D Polyhedral Bodies with up to Third-Order Polynomial Horizontal and Vertical Mass Contrasts

NASA Astrophysics Data System (ADS)

Ren, Zhengyong; Zhong, Yiyuan; Chen, Chaojian; Tang, Jingtian; Kalscheuer, Thomas; Maurer, Hansruedi; Li, Yang

2018-03-01

During the last 20 years, geophysicists have developed great interest in using gravity gradient tensor signals to study bodies of anomalous density in the Earth. Deriving exact solutions of the gravity gradient tensor signals has become a dominating task in exploration geophysics or geodetic fields. In this study, we developed a compact and simple framework to derive exact solutions of gravity gradient tensor measurements for polyhedral bodies, in which the density contrast is represented by a general polynomial function. The polynomial mass contrast can continuously vary in both horizontal and vertical directions. In our framework, the original three-dimensional volume integral of gravity gradient tensor signals is transformed into a set of one-dimensional line integrals along edges of the polyhedral body by sequentially invoking the volume and surface gradient (divergence) theorems. In terms of an orthogonal local coordinate system defined on these edges, exact solutions are derived for these line integrals. We successfully derived a set of unified exact solutions of gravity gradient tensors for constant, linear, quadratic and cubic polynomial orders. The exact solutions for constant and linear cases cover all previously published vertex-type exact solutions of the gravity gradient tensor for a polygonal body, though the associated algorithms may differ in numerical stability. In addition, to our best knowledge, it is the first time that exact solutions of gravity gradient tensor signals are derived for a polyhedral body with a polynomial mass contrast of order higher than one (that is quadratic and cubic orders). Three synthetic models (a prismatic body with depth-dependent density contrasts, an irregular polyhedron with linear density contrast and a tetrahedral body with horizontally and vertically varying density contrasts) are used to verify the correctness and the efficiency of our newly developed closed-form solutions. Excellent agreements are obtained between our solutions and other published exact solutions. In addition, stability tests are performed to demonstrate that our exact solutions can safely be used to detect shallow subsurface targets.

Eigensensitivity analysis of rotating clamped uniform beams with the asymptotic numerical method

NASA Astrophysics Data System (ADS)

Bekhoucha, F.; Rechak, S.; Cadou, J. M.

2016-12-01

In this paper, free vibrations of a rotating clamped Euler-Bernoulli beams with uniform cross section are studied using continuation method, namely asymptotic numerical method. The governing equations of motion are derived using Lagrange's method. The kinetic and strain energy expression are derived from Rayleigh-Ritz method using a set of hybrid variables and based on a linear deflection assumption. The derived equations are transformed in two eigenvalue problems, where the first is a linear gyroscopic eigenvalue problem and presents the coupled lagging and stretch motions through gyroscopic terms. While the second is standard eigenvalue problem and corresponds to the flapping motion. Those two eigenvalue problems are transformed into two functionals treated by continuation method, the Asymptotic Numerical Method. New method proposed for the solution of the linear gyroscopic system based on an augmented system, which transforms the original problem to a standard form with real symmetric matrices. By using some techniques to resolve these singular problems by the continuation method, evolution curves of the natural frequencies against dimensionless angular velocity are determined. At high angular velocity, some singular points, due to the linear elastic assumption, are computed. Numerical tests of convergence are conducted and the obtained results are compared to the exact values. Results obtained by continuation are compared to those computed with discrete eigenvalue problem.

Solution of the lossy nonlinear Tricomi equation with application to sonic boom focusing

NASA Astrophysics Data System (ADS)

Salamone, Joseph A., III

Sonic boom focusing theory has been augmented with new terms that account for mean flow effects in the direction of propagation and also for atmospheric absorption/dispersion due to molecular relaxation due to oxygen and nitrogen. The newly derived model equation was numerically implemented using a computer code. The computer code was numerically validated using a spectral solution for nonlinear propagation of a sinusoid through a lossy homogeneous medium. An additional numerical check was performed to verify the linear diffraction component of the code calculations. The computer code was experimentally validated using measured sonic boom focusing data from the NASA sponsored Superboom Caustic and Analysis Measurement Program (SCAMP) flight test. The computer code was in good agreement with both the numerical and experimental validation. The newly developed code was applied to examine the focusing of a NASA low-boom demonstration vehicle concept. The resulting pressure field was calculated for several supersonic climb profiles. The shaping efforts designed into the signatures were still somewhat evident despite the effects of sonic boom focusing.

Analytic Formulation and Numerical Implementation of an Acoustic Pressure Gradient Prediction

NASA Technical Reports Server (NTRS)

Lee, Seongkyu; Brentner, Kenneth S.; Farassat, F.; Morris, Philip J.

2008-01-01

Two new analytical formulations of the acoustic pressure gradient have been developed and implemented in the PSU-WOPWOP rotor noise prediction code. The pressure gradient can be used to solve the boundary condition for scattering problems and it is a key aspect to solve acoustic scattering problems. The first formulation is derived from the gradient of the Ffowcs Williams-Hawkings (FW-H) equation. This formulation has a form involving the observer time differentiation outside the integrals. In the second formulation, the time differentiation is taken inside the integrals analytically. This formulation avoids the numerical time differentiation with respect to the observer time, which is computationally more efficient. The acoustic pressure gradient predicted by these new formulations is validated through comparison with available exact solutions for a stationary and moving monopole sources. The agreement between the predictions and exact solutions is excellent. The formulations are applied to the rotor noise problems for two model rotors. A purely numerical approach is compared with the analytical formulations. The agreement between the analytical formulations and the numerical method is excellent for both stationary and moving observer cases.

Numerical equilibrium analysis for structured consumer resource models.

PubMed

de Roos, A M; Diekmann, O; Getto, P; Kirkilionis, M A

2010-02-01

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for "Daphnia consuming algae" models in C-code. The results obtained by way of this implementation are shown in the form of graphs.

A mesh-free approach to acoustic scattering from multiple spheres nested inside a large sphere by using diagonal translation operators.

PubMed

Hesford, Andrew J; Astheimer, Jeffrey P; Greengard, Leslie F; Waag, Robert C

2010-02-01

A multiple-scattering approach is presented to compute the solution of the Helmholtz equation when a number of spherical scatterers are nested in the interior of an acoustically large enclosing sphere. The solution is represented in terms of partial-wave expansions, and a linear system of equations is derived to enforce continuity of pressure and normal particle velocity across all material interfaces. This approach yields high-order accuracy and avoids some of the difficulties encountered when using integral equations that apply to surfaces of arbitrary shape. Calculations are accelerated by using diagonal translation operators to compute the interactions between spheres when the operators are numerically stable. Numerical results are presented to demonstrate the accuracy and efficiency of the method.

A mesh-free approach to acoustic scattering from multiple spheres nested inside a large sphere by using diagonal translation operators

PubMed Central

Hesford, Andrew J.; Astheimer, Jeffrey P.; Greengard, Leslie F.; Waag, Robert C.

2010-01-01

A multiple-scattering approach is presented to compute the solution of the Helmholtz equation when a number of spherical scatterers are nested in the interior of an acoustically large enclosing sphere. The solution is represented in terms of partial-wave expansions, and a linear system of equations is derived to enforce continuity of pressure and normal particle velocity across all material interfaces. This approach yields high-order accuracy and avoids some of the difficulties encountered when using integral equations that apply to surfaces of arbitrary shape. Calculations are accelerated by using diagonal translation operators to compute the interactions between spheres when the operators are numerically stable. Numerical results are presented to demonstrate the accuracy and efficiency of the method. PMID:20136208

Nonlinear Waves in the Terrestrial Quasiparallel Foreshock.

PubMed

Hnat, B; Kolotkov, D Y; O'Connell, D; Nakariakov, V M; Rowlands, G

2016-12-02

We provide strongly conclusive evidence that the cubic nonlinearity plays an important part in the evolution of the large amplitude magnetic structures in the terrestrial foreshock. Large amplitude nonlinear wave trains at frequencies above the proton cyclotron frequency are identified after nonharmonic slow variations are filtered out by applying the empirical mode decomposition. Numerical solutions of the derivative nonlinear Schrödinger equation, predicted analytically by the use of a pseudopotential approach, are found to be consistent with the observed wave forms. The approximate phase speed of these nonlinear waves, indicated by the parameters of numerical solutions, is of the order of the local Alfvén speed. We suggest that the feedback of the large amplitude fluctuations on background plasma is reflected in the evolution of the pseudopotential.

Bessel function expansion to reduce the calculation time and memory usage for cylindrical computer-generated holograms.

PubMed

Sando, Yusuke; Barada, Daisuke; Jackin, Boaz Jessie; Yatagai, Toyohiko

2017-07-10

This study proposes a method to reduce the calculation time and memory usage required for calculating cylindrical computer-generated holograms. The wavefront on the cylindrical observation surface is represented as a convolution integral in the 3D Fourier domain. The Fourier transformation of the kernel function involving this convolution integral is analytically performed using a Bessel function expansion. The analytical solution can drastically reduce the calculation time and the memory usage without any cost, compared with the numerical method using fast Fourier transform to Fourier transform the kernel function. In this study, we present the analytical derivation, the efficient calculation of Bessel function series, and a numerical simulation. Furthermore, we demonstrate the effectiveness of the analytical solution through comparisons of calculation time and memory usage.

Homoclinic accretion solutions in the Schwarzschild-anti-de Sitter space-time

NASA Astrophysics Data System (ADS)

Mach, Patryk

2015-04-01

The aim of this paper is to clarify the distinction between homoclinic and standard (global) Bondi-type accretion solutions in the Schwarzschild-anti-de Sitter space-time. The homoclinic solutions have recently been discovered numerically for polytropic equations of state. Here I show that they exist also for certain isothermal (linear) equations of state, and an analytic solution of this type is obtained. It is argued that the existence of such solutions is generic, although for sufficiently relativistic matter models (photon gas, ultrahard equation of state) there exist global solutions that can be continued to infinity, similarly to standard Michel's solutions in the Schwarzschild space-time. In contrast to that global solutions should not exist for matter models with a nonvanishing rest-mass component, and this is demonstrated for polytropes. For homoclinic isothermal solutions I derive an upper bound on the mass of the black hole for which stationary transonic accretion is allowed.

On Chorin's Method for Stationary Solutions of the Oberbeck-Boussinesq Equation

NASA Astrophysics Data System (ADS)

Kagei, Yoshiyuki; Nishida, Takaaki

2017-06-01

Stability of stationary solutions of the Oberbeck-Boussinesq system (OB) and the corresponding artificial compressible system is considered. The latter system is obtained by adding the time derivative of the pressure with small parameter ɛ > 0 to the continuity equation of (OB), which was proposed by A. Chorin to find stationary solutions of (OB) numerically. Both systems have the same sets of stationary solutions and the system (OB) is obtained from the artificial compressible one as the limit ɛ \\to 0 which is a singular limit. It is proved that if a stationary solution of the artificial compressible system is stable for sufficiently small ɛ > 0, then it is also stable as a solution of (OB). The converse is proved provided that the velocity field of the stationary solution satisfies some smallness condition.

You Don't Need Richards'... A New General 1-D Vadose Zone Solution Method that is Reliable

NASA Astrophysics Data System (ADS)

Ogden, F. L.; Lai, W.; Zhu, J.; Steinke, R. C.; Talbot, C. A.

2015-12-01

Hydrologic modelers and mathematicians have strived to improve 1-D Richards' equation (RE) solution reliability for predicting vadose zone fluxes. Despite advances in computing power and the numerical solution of partial differential equations since Richards first published the RE in 1931, the solution remains unreliable. That is to say that there is no guarantee that for a particular set of soil constitutive relations, moisture profile conditions, or forcing input that a numerical RE solver will converge to an answer. This risk of non-convergence renders prohibitive the use of RE solvers in hydrological models that need perhaps millions of infiltration solutions. In lieu of using unreliable numerical RE solutions, researchers have developed a wide array of approximate solutions that more-or-less mimic the behavior of the RE, with some notable deficiencies such as parameter insensitivity or divergence over time. The improved Talbot-Ogden (T-O) finite water-content scheme was shown by Ogden et al. (2015) to be an extremely good approximation of the 1-D RE solution, with a difference in cumulative infiltration of only 0.2 percent over an 8 month simulation comparing the improved T-O scheme with a RE numerical solver. The reason is that the newly-derived fundamental flow equation that underpins the improved T-O method is equivalent to the RE minus a term that is equal to the diffusive flux divided by the slope of the wetting front. Because the diffusive flux has zero mean, this term is not important in calculating the mean flux. The wetting front slope is near infinite (sharp) in coarser soils that produce more significant hydrological interactions between surface and ground waters, which also makes this missing term 1) disappear in the limit, and, 2) create stability challenges for the numerical solution of RE. The improved T-O method is a replacement for the 1-D RE in soils that can be simulated as homogeneous layers, where the user is willing to neglect the effects of soil water diffusivity. This presentation emphasizes the transformative nature of the improved T-O finite water-content solution, and highlights the benefits of the methods' reliability in high-resolution large watershed simulations in the high performance computing environment, and discusses coupling of the soil matrix and non-Darcian macropores.

Numerical computation of gravitational field for general axisymmetric objects

NASA Astrophysics Data System (ADS)

Fukushima, Toshio

2016-10-01

We developed a numerical method to compute the gravitational field of a general axisymmetric object. The method (I) numerically evaluates a double integral of the ring potential by the split quadrature method using the double exponential rules, and (II) derives the acceleration vector by numerically differentiating the numerically integrated potential by Ridder's algorithm. Numerical comparison with the analytical solutions for a finite uniform spheroid and an infinitely extended object of the Miyamoto-Nagai density distribution confirmed the 13- and 11-digit accuracy of the potential and the acceleration vector computed by the method, respectively. By using the method, we present the gravitational potential contour map and/or the rotation curve of various axisymmetric objects: (I) finite uniform objects covering rhombic spindles and circular toroids, (II) infinitely extended spheroids including Sérsic and Navarro-Frenk-White spheroids, and (III) other axisymmetric objects such as an X/peanut-shaped object like NGC 128, a power-law disc with a central hole like the protoplanetary disc of TW Hya, and a tear-drop-shaped toroid like an axisymmetric equilibrium solution of plasma charge distribution in an International Thermonuclear Experimental Reactor-like tokamak. The method is directly applicable to the electrostatic field and will be easily extended for the magnetostatic field. The FORTRAN 90 programs of the new method and some test results are electronically available.

An adaptive finite element method for the inequality-constrained Reynolds equation

NASA Astrophysics Data System (ADS)

Gustafsson, Tom; Rajagopal, Kumbakonam R.; Stenberg, Rolf; Videman, Juha

2018-07-01

We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem and approximated by a residual-based stabilized method. Based on our recent results on the classical obstacle problem, we present optimal a priori estimates and derive novel a posteriori error estimators. The method is implemented as a Nitsche-type finite element technique and shown in numerical computations to be superior to the usually applied penalty methods.

Lookback Option Pricing with Fixed Proportional Transaction Costs under Fractional Brownian Motion.

PubMed

Sun, Jiao-Jiao; Zhou, Shengwu; Zhang, Yan; Han, Miao; Wang, Fei

2014-01-01

The pricing problem of lookback option with a fixed proportion of transaction costs is investigated when the underlying asset price follows a fractional Brownian motion process. Firstly, using Leland's hedging method a partial differential equation satisfied by the value of the lookback option is derived. Then we obtain its numerical solution by constructing a Crank-Nicolson format. Finally, the effectiveness of the proposed form is verified through a numerical example. Meanwhile, the impact of transaction cost rate and volatility on lookback option value is discussed.

Time-dependent difference theory for noise propagation in a two-dimensional duct. [of a turbofan engine

NASA Technical Reports Server (NTRS)

Baumeister, K. J.

1979-01-01

A time dependent numerical formulation was derived for sound propagation in a two dimensional straight soft-walled duct in the absence of mean flow. The time dependent governing acoustic-difference equations and boundary conditions were developed along with the maximum stable time increment. Example calculations were presented for sound attenuation in hard and soft wall ducts. The time dependent analysis were found to be superior to the conventional steady numerical analysis because of much shorter solution times and the elimination of matrix storage requirements.

Lookback Option Pricing with Fixed Proportional Transaction Costs under Fractional Brownian Motion

PubMed Central

Sun, Jiao-Jiao; Zhou, Shengwu; Zhang, Yan; Han, Miao; Wang, Fei

2014-01-01

The pricing problem of lookback option with a fixed proportion of transaction costs is investigated when the underlying asset price follows a fractional Brownian motion process. Firstly, using Leland's hedging method a partial differential equation satisfied by the value of the lookback option is derived. Then we obtain its numerical solution by constructing a Crank-Nicolson format. Finally, the effectiveness of the proposed form is verified through a numerical example. Meanwhile, the impact of transaction cost rate and volatility on lookback option value is discussed. PMID:27433525

Methodology of Numerical Optimization for Orbital Parameters of Binary Systems

NASA Astrophysics Data System (ADS)

Araya, I.; Curé, M.

2010-02-01

The use of a numerical method of maximization (or minimization) in optimization processes allows us to obtain a great amount of solutions. Therefore, we can find a global maximum or minimum of the problem, but this is only possible if we used a suitable methodology. To obtain the global optimum values, we use the genetic algorithm called PIKAIA (P. Charbonneau) and other four algorithms implemented in Mathematica. We demonstrate that derived orbital parameters of binary systems published in some papers, based on radial velocity measurements, are local minimum instead of global ones.

Higher-derivative operators and effective field theory for general scalar-tensor theories

NASA Astrophysics Data System (ADS)

Solomon, Adam R.; Trodden, Mark

2018-02-01

We discuss the extent to which it is necessary to include higher-derivative operators in the effective field theory of general scalar-tensor theories. We explore the circumstances under which it is correct to restrict to second-order operators only, and demonstrate this using several different techniques, such as reduction of order and explicit field redefinitions. These methods are applied, in particular, to the much-studied Horndeski theories. The goal is to clarify the application of effective field theory techniques in the context of popular cosmological models, and to explicitly demonstrate how and when higher-derivative operators can be cast into lower-derivative forms suitable for numerical solution techniques.

First- and second-order sensitivity analysis of linear and nonlinear structures

NASA Technical Reports Server (NTRS)

Haftka, R. T.; Mroz, Z.

1986-01-01

This paper employs the principle of virtual work to derive sensitivity derivatives of structural response with respect to stiffness parameters using both direct and adjoint approaches. The computations required are based on additional load conditions characterized by imposed initial strains, body forces, or surface tractions. As such, they are equally applicable to numerical or analytical solution techniques. The relative efficiency of various approaches for calculating first and second derivatives is assessed. It is shown that for the evaluation of second derivatives the most efficient approach is one that makes use of both the first-order sensitivities and adjoint vectors. Two example problems are used for demonstrating the various approaches.

A new approximation for pore pressure accumulation in marine sediment due to water waves

NASA Astrophysics Data System (ADS)

Jeng, D.-S.; Seymour, B. R.; Li, J.

2007-01-01

The residual mechanism of wave-induced pore water pressure accumulation in marine sediments is re-examined. An analytical approximation is derived using a linear relation for pore pressure generation in cyclic loading, and mistakes in previous solutions (Int. J. Numer. Anal. Methods Geomech. 2001; 25:885-907; J. Offshore Mech. Arctic Eng. (ASME) 1989; 111(1):1-11) are corrected. A numerical scheme is then employed to solve the case with a non-linear relation for pore pressure generation. Both analytical and numerical solutions are verified with experimental data (Laboratory and field investigation of wave-sediment interaction. Joseph H. Defrees Hydraulics Laboratory, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY, 1983), and provide a better prediction of pore pressure accumulation than the previous solution (J. Offshore Mech. Arctic Eng. (ASME) 1989; 111(1):1-11). The parametric study concludes that the pore pressure accumulation and use of full non-linear relation of pore pressure become more important under the following conditions: (1) large wave amplitude, (2) longer wave period, (3) shallow water, (4) shallow soil and (5) softer soils with a low consolidation coefficient. Copyright

Dispersive shock waves in the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Demirci, Ali; Ma, Yi-Ping

2016-10-01

Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time (2 + 1) dimensions to finding DSW solutions of (1 + 1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the (2 + 1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1 + 1) dimensional equations.

Numerical solution of a logistic growth model for a population with Allee effect considering fuzzy initial values and fuzzy parameters

NASA Astrophysics Data System (ADS)

Amarti, Z.; Nurkholipah, N. S.; Anggriani, N.; Supriatna, A. K.

2018-03-01

Predicting the future of population number is among the important factors that affect the consideration in preparing a good management for the population. This has been done by various known method, one among them is by developing a mathematical model describing the growth of the population. The model usually takes form in a differential equation or a system of differential equations, depending on the complexity of the underlying properties of the population. The most widely used growth models currently are those having a sigmoid solution of time series, including the Verhulst logistic equation and the Gompertz equation. In this paper we consider the Allee effect of the Verhulst’s logistic population model. The Allee effect is a phenomenon in biology showing a high correlation between population size or density and the mean individual fitness of the population. The method used to derive the solution is the Runge-Kutta numerical scheme, since it is in general regarded as one among the good numerical scheme which is relatively easy to implement. Further exploration is done via the fuzzy theoretical approach to accommodate the impreciseness of the initial values and parameters in the model.

An interaction algorithm for prediction of mean and fluctuating velocities in two-dimensional aerodynamic wake flows

NASA Technical Reports Server (NTRS)

Baker, A. J.; Orzechowski, J. A.

1980-01-01

A theoretical analysis is presented yielding sets of partial differential equations for determination of turbulent aerodynamic flowfields in the vicinity of an airfoil trailing edge. A four phase interaction algorithm is derived to complete the analysis. Following input, the first computational phase is an elementary viscous corrected two dimensional potential flow solution yielding an estimate of the inviscid-flow induced pressure distribution. Phase C involves solution of the turbulent two dimensional boundary layer equations over the trailing edge, with transition to a two dimensional parabolic Navier-Stokes equation system describing the near-wake merging of the upper and lower surface boundary layers. An iteration provides refinement of the potential flow induced pressure coupling to the viscous flow solutions. The final phase is a complete two dimensional Navier-Stokes analysis of the wake flow in the vicinity of a blunt-bases airfoil. A finite element numerical algorithm is presented which is applicable to solution of all partial differential equation sets of inviscid-viscous aerodynamic interaction algorithm. Numerical results are discussed.

Strong shock implosion, approximate solution

NASA Astrophysics Data System (ADS)

Fujimoto, Y.; Mishkin, E. A.; Alejaldre, C.

1983-01-01

The self-similar, center-bound motion of a strong spherical, or cylindrical, shock wave moving through an ideal gas with a constant, γ= cp/ cv, is considered and a linearized, approximate solution is derived. An X, Y phase plane of the self-similar solution is defined and the representative curved of the system behind the shock front is replaced by a straight line connecting the mappings of the shock front with that of its tail. The reduced pressure P(ξ), density R(ξ) and velocity U1(ξ) are found in closed, quite accurate, form. Comparison with numerically obtained results, for γ= {5}/{3} and γ= {7}/{5}, is shown.

Cylindrical and spherical Akhmediev breather and freak waves in ultracold neutral plasmas

NASA Astrophysics Data System (ADS)

El-Tantawy, S. A.; El-Awady, E. I.

2018-01-01

The properties of cylindrical and spherical ion-acoustic breathers Akhmediev breather and freak waves in strongly coupled ultracold neutral plasmas (UNPs), whose constituents are inertial strongly coupled ions and weakly coupled Maxwellian electrons, are investigated numerically. Using the derivative expansion method, the basic set of fluid equations is reduced to a nonplanar (cylindrical and spherical)/modified nonlinear Schrödinger equation (mNLSE). The analytical solutions of the mNLSE were not possible until now, so their numerical solutions are obtained using the finite difference scheme with the help of the Dirichlet boundary conditions. Moreover, the criteria for the existence and propagation of breathers are discussed in detail. The geometrical effects due to the cylindrical and spherical geometries on the breather profile are studied numerically. It is found that the propagation of the ion-acoustic breathers in one-dimensional planar and nonplanar geometries is very different. Finally, our results may help to manipulate matter breathers experimentally in UNPs.

Nonlinear ion-acoustic cnoidal waves in a dense relativistic degenerate magnetoplasma.

PubMed

El-Shamy, E F

2015-03-01

The complex pattern and propagation characteristics of nonlinear periodic ion-acoustic waves, namely, ion-acoustic cnoidal waves, in a dense relativistic degenerate magnetoplasma consisting of relativistic degenerate electrons and nondegenerate cold ions are investigated. By means of the reductive perturbation method and appropriate boundary conditions for nonlinear periodic waves, a nonlinear modified Korteweg-de Vries (KdV) equation is derived and its cnoidal wave is analyzed. The various solutions of nonlinear ion-acoustic cnoidal and solitary waves are presented numerically with the Sagdeev potential approach. The analytical solution and numerical simulation of nonlinear ion-acoustic cnoidal waves of the nonlinear modified KdV equation are studied. Clearly, it is found that the features (amplitude and width) of nonlinear ion-acoustic cnoidal waves are proportional to plasma number density, ion cyclotron frequency, and direction cosines. The numerical results are applied to high density astrophysical situations, such as in superdense white dwarfs. This research will be helpful in understanding the properties of compact astrophysical objects containing cold ions with relativistic degenerate electrons.

Adiabatic pumping solutions in global AdS

NASA Astrophysics Data System (ADS)

Carracedo, Pablo; Mas, Javier; Musso, Daniele; Serantes, Alexandre

2017-05-01

We construct a family of very simple stationary solutions to gravity coupled to a massless scalar field in global AdS. They involve a constantly rising source for the scalar field at the boundary and thereby we name them pumping solutions. We construct them numerically in D = 4. They are regular and, generically, have negative mass. We perform a study of linear and nonlinear stability and find both stable and unstable branches. In the latter case, solutions belonging to different sub-branches can either decay to black holes or to limiting cycles. This observation motivates the search for non-stationary exactly timeperiodic solutions which we actually construct. We clarify the role of pumping solutions in the context of quasistatic adiabatic quenches. In D = 3 the pumping solutions can be related to other previously known solutions, like magnetic or translationally-breaking backgrounds. From this we derive an analytic expression.

The Harmonic Oscillator with a Gaussian Perturbation: Evaluation of the Integrals and Example Applications

ERIC Educational Resources Information Center

Earl, Boyd L.

2008-01-01

A general result for the integrals of the Gaussian function over the harmonic oscillator wavefunctions is derived using generating functions. Using this result, an example problem of a harmonic oscillator with various Gaussian perturbations is explored in order to compare the results of precise numerical solution, the variational method, and…

Application of Newtonian Physics to Predict the Speed of a Gravity Racer

ERIC Educational Resources Information Center

Driscoll, H. F.; Bullas, A. M.; King, C. E.; Senior, T.; Haake, S. J.; Hart, J.

2016-01-01

Gravity racing can be studied using numerical solutions to the equations of motion derived from Newton's second law. This allows students to explore the physics of gravity racing and to understand how design and course selection influences vehicle speed. Using Euler's method, we have developed a spreadsheet application that can be used to predict…

On the implementation of an accurate and efficient solver for convection-diffusion equations

NASA Astrophysics Data System (ADS)

Wu, Chin-Tien

In this dissertation, we examine several different aspects of computing the numerical solution of the convection-diffusion equation. The solution of this equation often exhibits sharp gradients due to Dirichlet outflow boundaries or discontinuities in boundary conditions. Because of the singular-perturbed nature of the equation, numerical solutions often have severe oscillations when grid sizes are not small enough to resolve sharp gradients. To overcome such difficulties, the streamline diffusion discretization method can be used to obtain an accurate approximate solution in regions where the solution is smooth. To increase accuracy of the solution in the regions containing layers, adaptive mesh refinement and mesh movement based on a posteriori error estimations can be employed. An error-adapted mesh refinement strategy based on a posteriori error estimations is also proposed to resolve layers. For solving the sparse linear systems that arise from discretization, goemetric multigrid (MG) and algebraic multigrid (AMG) are compared. In addition, both methods are also used as preconditioners for Krylov subspace methods. We derive some convergence results for MG with line Gauss-Seidel smoothers and bilinear interpolation. Finally, while considering adaptive mesh refinement as an integral part of the solution process, it is natural to set a stopping tolerance for the iterative linear solvers on each mesh stage so that the difference between the approximate solution obtained from iterative methods and the finite element solution is bounded by an a posteriori error bound. Here, we present two stopping criteria. The first is based on a residual-type a posteriori error estimator developed by Verfurth. The second is based on an a posteriori error estimator, using local solutions, developed by Kay and Silvester. Our numerical results show the refined mesh obtained from the iterative solution which satisfies the second criteria is similar to the refined mesh obtained from the finite element solution.

Code Verification of the HIGRAD Computational Fluid Dynamics Solver

DOE Office of Scientific and Technical Information (OSTI.GOV)

Van Buren, Kendra L.; Canfield, Jesse M.; Hemez, Francois M.

2012-05-04

The purpose of this report is to outline code and solution verification activities applied to HIGRAD, a Computational Fluid Dynamics (CFD) solver of the compressible Navier-Stokes equations developed at the Los Alamos National Laboratory, and used to simulate various phenomena such as the propagation of wildfires and atmospheric hydrodynamics. Code verification efforts, as described in this report, are an important first step to establish the credibility of numerical simulations. They provide evidence that the mathematical formulation is properly implemented without significant mistakes that would adversely impact the application of interest. Highly accurate analytical solutions are derived for four code verificationmore » test problems that exercise different aspects of the code. These test problems are referred to as: (i) the quiet start, (ii) the passive advection, (iii) the passive diffusion, and (iv) the piston-like problem. These problems are simulated using HIGRAD with different levels of mesh discretization and the numerical solutions are compared to their analytical counterparts. In addition, the rates of convergence are estimated to verify the numerical performance of the solver. The first three test problems produce numerical approximations as expected. The fourth test problem (piston-like) indicates the extent to which the code is able to simulate a 'mild' discontinuity, which is a condition that would typically be better handled by a Lagrangian formulation. The current investigation concludes that the numerical implementation of the solver performs as expected. The quality of solutions is sufficient to provide credible simulations of fluid flows around wind turbines. The main caveat associated to these findings is the low coverage provided by these four problems, and somewhat limited verification activities. A more comprehensive evaluation of HIGRAD may be beneficial for future studies.« less

The classical D-type expansion of spherical H II regions

NASA Astrophysics Data System (ADS)

Williams, Robin J. R.; Bibas, Thomas G.; Haworth, Thomas J.; Mackey, Jonathan

2018-06-01

Recent numerical and analytic work has highlighted some shortcomings in our understanding of the dynamics of H II region expansion, especially at late times, when the H II region approaches pressure equilibrium with the ambient medium. Here we reconsider the idealized case of a constant radiation source in a uniform and spherically symmetric ambient medium, with an isothermal equation of state. A thick-shell solution is developed which captures the stalling of the ionization front and the decay of the leading shock to a weak compression wave as it escapes to large radii. An acoustic approximation is introduced to capture the late-time damped oscillations of the H II region about the stagnation radius. Putting these together, a matched asymptotic equation is derived for the radius of the ionization front which accounts for both the inertia of the expanding shell and the finite temperature of the ambient medium. The solution to this equation is shown to agree very well with the numerical solution at all times, and is superior to all previously published solutions. The matched asymptotic solution can also accurately model the variation of H II region radius for a time-varying radiation source.

Numerical solutions of ideal quantum gas dynamical flows governed by semiclassical ellipsoidal-statistical distribution.

PubMed

Yang, Jaw-Yen; Yan, Chih-Yuan; Diaz, Manuel; Huang, Juan-Chen; Li, Zhihui; Zhang, Hanxin

2014-01-08

The ideal quantum gas dynamics as manifested by the semiclassical ellipsoidal-statistical (ES) equilibrium distribution derived in Wu et al. (Wu et al . 2012 Proc. R. Soc. A 468 , 1799-1823 (doi:10.1098/rspa.2011.0673)) is numerically studied for particles of three statistics. This anisotropic ES equilibrium distribution was derived using the maximum entropy principle and conserves the mass, momentum and energy, but differs from the standard Fermi-Dirac or Bose-Einstein distribution. The present numerical method combines the discrete velocity (or momentum) ordinate method in momentum space and the high-resolution shock-capturing method in physical space. A decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. Computations of two-dimensional Riemann problems are presented, and various contours of the quantities unique to this ES model are illustrated. The main flow features, such as shock waves, expansion waves and slip lines and their complex nonlinear interactions, are depicted and found to be consistent with existing calculations for a classical gas.

On similarity solutions of a boundary layer problem with an upstream moving wall

NASA Technical Reports Server (NTRS)

Hussaini, M. Y.; Lakin, W. D.; Nachman, A.

1986-01-01

The problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream is examined. It was previously shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. In particular, it was proved that a solution exists only if this parameter does not exceed a certain critical value, and numerical evidence was adduced to show that this solution is nonunique. Using Crocco formulation the present work proves this nonuniqueness. Also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter.

Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz-Musslimani equation with PT-symmetric potential.

PubMed

Yu, Fajun

2017-02-01

Starting from a discrete spectral problem, we derive a hierarchy of nonlinear discrete equations which include the Ablowitz-Ladik (AL) equation. We analytically study the discrete rogue-wave (DRW) solutions of AL equation with three free parameters. The trajectories of peaks and depressions of profiles for the first- and second-order DRWs are produced by means of analytical and numerical methods. In particular, we study the solutions with dispersion in parity-time ( PT) symmetric potential for Ablowitz-Musslimani equation. And we consider the non-autonomous DRW solutions, parameters controlling and their interactions with variable coefficients, and predict the long-living rogue wave solutions. Our results might provide useful information for potential applications of synthetic PT symmetric systems in nonlinear optics and condensed matter physics.

A numerical scheme to solve unstable boundary value problems

NASA Technical Reports Server (NTRS)

Kalnay-Rivas, E.

1977-01-01

The considered scheme makes it possible to determine an unstable steady state solution in cases in which, because of lack of symmetry, such a solution cannot be obtained analytically, and other time integration or relaxation schemes, because of instability, fail to converge. The iterative solution of a single complex equation is discussed and a nonlinear system of equations is considered. Described applications of the scheme are related to a steady state solution with shear instability, an unstable nonlinear Ekman boundary layer, and the steady state solution of a baroclinic atmosphere with asymmetric forcing. The scheme makes use of forward and backward time integrations of the original spatial differential operators and of an approximation of the adjoint operators. Only two computations of the time derivative per iteration are required.

Numerical methods for coupled fracture problems

NASA Astrophysics Data System (ADS)

Viesca, Robert C.; Garagash, Dmitry I.

2018-04-01

We consider numerical solutions in which the linear elastic response to an opening- or sliding-mode fracture couples with one or more processes. Classic examples of such problems include traction-free cracks leading to stress singularities or cracks with cohesive-zone strength requirements leading to non-singular stress distributions. These classical problems have characteristic square-root asymptotic behavior for stress, relative displacement, or their derivatives. Prior work has shown that such asymptotics lead to a natural quadrature of the singular integrals at roots of Chebyhsev polynomials of the first, second, third, or fourth kind. We show that such quadratures lead to convenient techniques for interpolation, differentiation, and integration, with the potential for spectral accuracy. We further show that these techniques, with slight amendment, may continue to be used for non-classical problems which lack the classical asymptotic behavior. We consider solutions to example problems of both the classical and non-classical variety (e.g., fluid-driven opening-mode fracture and fault shear rupture driven by thermal weakening), with comparisons to analytical solutions or asymptotes, where available.

Two-dimensional modulated ion-acoustic excitations in electronegative plasmas

NASA Astrophysics Data System (ADS)

Panguetna, Chérif S.; Tabi, Conrad B.; Kofané, Timoléon C.

2017-09-01

Two-dimensional modulated ion-acoustic waves are investigated in an electronegative plasma. Through the reductive perturbation expansion, the governing hydrodynamic equations are reduced to a Davey-Stewartson system with two-space variables. The latter is used to study the modulational instability of ion-acoustic waves along with the effect of plasma parameters, namely, the negative ion concentration ratio (α) and the electron-to-negative ion temperature ratio (σn). A parametric analysis of modulational instability is carried out, where regions of plasma parameters responsible for the emergence of modulated ion-acoustic waves are discussed, with emphasis on the behavior of the instability growth rate. Numerically, using perturbed plane waves as initial conditions, parameters from the instability regions give rise to series of dromion solitons under the activation of modulational instability. The sensitivity of the numerical solutions to plasma parameters is discussed. Some exact solutions in the form one- and two-dromion solutions are derived and their response to the effect of varying α and σn is discussed as well.

A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel

NASA Astrophysics Data System (ADS)

Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru

2018-02-01

The mathematical model of breaking of non-linear dispersive water waves with memory effect is very important in mathematical physics. In the present article, we examine a novel fractional extension of the non-linear Fornberg-Whitham equation occurring in wave breaking. We consider the most recent theory of differentiation involving the non-singular kernel based on the extended Mittag-Leffler-type function to modify the Fornberg-Whitham equation. We examine the existence of the solution of the non-linear Fornberg-Whitham equation of fractional order. Further, we show the uniqueness of the solution. We obtain the numerical solution of the new arbitrary order model of the non-linear Fornberg-Whitham equation with the aid of the Laplace decomposition technique. The numerical outcomes are displayed in the form of graphs and tables. The results indicate that the Laplace decomposition algorithm is a very user-friendly and reliable scheme for handling such type of non-linear problems of fractional order.

Generalized bipartite quantum state discrimination problems with sequential measurements

NASA Astrophysics Data System (ADS)

Nakahira, Kenji; Kato, Kentaro; Usuda, Tsuyoshi Sasaki

2018-02-01

We investigate an optimization problem of finding quantum sequential measurements, which forms a wide class of state discrimination problems with the restriction that only local operations and one-way classical communication are allowed. Sequential measurements from Alice to Bob on a bipartite system are considered. Using the fact that the optimization problem can be formulated as a problem with only Alice's measurement and is convex programming, we derive its dual problem and necessary and sufficient conditions for an optimal solution. Our results are applicable to various practical optimization criteria, including the Bayes criterion, the Neyman-Pearson criterion, and the minimax criterion. In the setting of the problem of finding an optimal global measurement, its dual problem and necessary and sufficient conditions for an optimal solution have been widely used to obtain analytical and numerical expressions for optimal solutions. Similarly, our results are useful to obtain analytical and numerical expressions for optimal sequential measurements. Examples in which our results can be used to obtain an analytical expression for an optimal sequential measurement are provided.

Faraday waves in a Hele-Shaw cell

NASA Astrophysics Data System (ADS)

Li, Jing; Li, Xiaochen; Chen, Kaijie; Xie, Bin; Liao, Shijun

2018-04-01

We investigate Faraday waves in a Hele-Shaw cell via experimental, numerical, and theoretical studies. Inspired by the Kelvin-Helmholtz-Darcy theory, we develop the gap-averaged Navier-Stokes equations and end up with the stable standing waves with half frequency of the external forced vibration. To overcome the dependency of a numerical model on the experimental parameter of wave length, we take two-phase flow into consideration and a novel dispersion relation is derived. The numerical results compare well with our experimental data, which effectively validates our proposed mathematical model. Therefore, this model can produce robust solutions of Faraday wave patterns and resolve related physical phenomena, which demonstrates the practical importance of the present study.

For numerical differentiation, dimensionality can be a blessing!

NASA Astrophysics Data System (ADS)

Anderssen, Robert S.; Hegland, Markus

Finite difference methods, such as the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly amplified in the differentiation process, especially if small step-sizes are chosen and higher derivatives are required. A number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h. In this paper, it is initially shown how first (and higher) order single-variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one-dimensional data. The result is then extended to the multivariate differentiation of higher dimensional data. The nature of the trade-off between convergence and stability is explicitly characterized, and the complexity of various implementations is examined.

One-dimensional analytical solution for hydraulic head and numerical solution for solute transport through a horizontal fracture for submarine groundwater discharge.

PubMed

He, Cairong; Wang, Tongke; Zhao, Zhixue; Hao, Yonghong; Yeh, Tian-Chyi J; Zhan, Hongbin

2017-11-01

Submarine groundwater discharge (SGD) has been recognized as a major pathway of groundwater flow to coastal oceanic environments. It could affect water quality and marine ecosystems due to pollutants and trace elements transported through groundwater. Relations between different characteristics of aquifers and SGD have been investigated extensively before, but the role of fractures in SGD still remains unknown. In order to better understand the mechanism of groundwater flow and solute transport through fractures in SGD, one-dimensional analytical solutions of groundwater hydraulic head and velocity through a synthetic horizontal fracture with periodic boundary conditions were derived using a Laplace transform technique. Then, numerical solutions of solute transport associated with the given groundwater velocity were developed using a finite-difference method. The results indicated that SGD associated with groundwater flow and solute transport was mainly controlled by sea level periodic fluctuations, which altered the hydraulic head and the hydraulic head gradient in the fracture. As a result, the velocity of groundwater flow associated with SGD also fluctuated periodically. We found that the pollutant concentration associated with SGD oscillated around a constant value, and could not reach a steady state. This was particularly true at locations close to the seashore. This finding of the role of fracture in SGD will assist pollution remediation and marine conservation in coastal regions. Copyright © 2017 Elsevier B.V. All rights reserved.

One-dimensional analytical solution for hydraulic head and numerical solution for solute transport through a horizontal fracture for submarine groundwater discharge

NASA Astrophysics Data System (ADS)

He, Cairong; Wang, Tongke; Zhao, Zhixue; Hao, Yonghong; Yeh, Tian-Chyi J.; Zhan, Hongbin

2017-11-01

Submarine groundwater discharge (SGD) has been recognized as a major pathway of groundwater flow to coastal oceanic environments. It could affect water quality and marine ecosystems due to pollutants and trace elements transported through groundwater. Relations between different characteristics of aquifers and SGD have been investigated extensively before, but the role of fractures in SGD still remains unknown. In order to better understand the mechanism of groundwater flow and solute transport through fractures in SGD, one-dimensional analytical solutions of groundwater hydraulic head and velocity through a synthetic horizontal fracture with periodic boundary conditions were derived using a Laplace transform technique. Then, numerical solutions of solute transport associated with the given groundwater velocity were developed using a finite-difference method. The results indicated that SGD associated with groundwater flow and solute transport was mainly controlled by sea level periodic fluctuations, which altered the hydraulic head and the hydraulic head gradient in the fracture. As a result, the velocity of groundwater flow associated with SGD also fluctuated periodically. We found that the pollutant concentration associated with SGD oscillated around a constant value, and could not reach a steady state. This was particularly true at locations close to the seashore. This finding of the role of fracture in SGD will assist pollution remediation and marine conservation in coastal regions.

Three-dimensional semi-analytical solution to groundwater flow in confined and unconfined wedge-shaped aquifers

NASA Astrophysics Data System (ADS)

Sedghi, Mohammad Mahdi; Samani, Nozar; Sleep, Brent

2009-06-01

The Laplace domain solutions have been obtained for three-dimensional groundwater flow to a well in confined and unconfined wedge-shaped aquifers. The solutions take into account partial penetration effects, instantaneous drainage or delayed yield, vertical anisotropy and the water table boundary condition. As a basis, the Laplace domain solutions for drawdown created by a point source in uniform, anisotropic confined and unconfined wedge-shaped aquifers are first derived. Then, by the principle of superposition the point source solutions are extended to the cases of partially and fully penetrating wells. Unlike the previous solution for the confined aquifer that contains improper integrals arising from the Hankel transform [Yeh HD, Chang YC. New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions. Adv Water Resour 2006;26:471-80], numerical evaluation of our solution is relatively easy using well known numerical Laplace inversion methods. The effects of wedge angle, pumping well location and observation point location on drawdown and the effects of partial penetration, screen location and delay index on the wedge boundary hydraulic gradient in unconfined aquifers have also been investigated. The results are presented in the form of dimensionless drawdown-time and boundary gradient-time type curves. The curves are useful for parameter identification, calculation of stream depletion rates and the assessment of water budgets in river basins.

Stationary and moving solitons in spin-orbit-coupled spin-1 Bose-Einstein condensates

NASA Astrophysics Data System (ADS)

Li, Yu-E.; Xue, Ju-Kui

2018-04-01

We investigate the matter-wave solitons in a spin-orbit-coupled spin-1 Bose-Einstein condensate using a multiscale perturbation method. Beginning with the one-dimensional spin-orbit-coupled threecomponent Gross-Pitaevskii equations, we derive a single nonlinear Schrödinger equation, which allows determination of the analytical soliton solutions of the system. Stationary and moving solitons in the system are derived. In particular, a parameter space for different existing soliton types is provided. It is shown that there exist only dark or bright solitons when the spin-orbit coupling is weak, with the solitons depending on the atomic interactions. However, when the spin-orbit coupling is strong, both dark and bright solitons exist, being determined by the Raman coupling. Our analytical solutions are confirmed by direct numerical simulations.

Constraining some Horndeski gravity theories

NASA Astrophysics Data System (ADS)

Bhattacharya, Sourav; Chakraborty, Sumanta

2017-02-01

We discuss two spherically symmetric solutions admitted by the Horndeski (or scalar-tensor) theory in the context of Solar System and astrophysical scenarios. One of these solutions is derived for Einstein-Gauss-Bonnet gravity, while the other originates from the coupling of the Gauss-Bonnet invariant with a scalar field. Specifically, we discuss the perihelion precession and the bending angle of light for these two different spherically symmetric spacetimes derived in Maeda and Dadhich [Phys. Rev. D 75, 044007 (2007), 10.1103/PhysRevD.75.044007] and Sotiriou and Zhou [Phys. Rev. D 90, 124063 (2014), 10.1103/PhysRevD.90.124063], respectively. The latter, in particular, applies only to black-hole spacetimes. We further delineate on the numerical bounds of relevant parameters of these theories from such computations.

Damageable contact between an elastic body and a rigid foundation

NASA Astrophysics Data System (ADS)

Campo, M.; Fernández, J. R.; Silva, A.

2009-02-01

In this work, the contact problem between an elastic body and a rigid obstacle is studied, including the development of material damage which results from internal compression or tension. The variational problem is formulated as a first-kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field. The existence of a unique local weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, three two-dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme.

A derivation of the beam equation

NASA Astrophysics Data System (ADS)

Duque, Daniel

2016-01-01

The Euler-Bernoulli equation describing the deflection of a beam is a vital tool in structural and mechanical engineering. However, its derivation usually entails a number of intermediate steps that may confuse engineering or science students at the beginnig of their undergraduate studies. We explain how this equation may be deduced, beginning with an approximate expression for the energy, from which the forces and finally the equation itself may be obtained. The description is begun at the level of small ‘particles’, and the continuum level is taken later on. However, when a computational solution is sought, the description turns back to the discrete level again. We first consider the easier case of a string under tension, and then focus on the beam. Numerical solutions for several loads are obtained.

A Highly Accurate Technique for the Treatment of Flow Equations at the Polar Axis in Cylindrical Coordinates using Series Expansions. Appendix A

NASA Technical Reports Server (NTRS)

Constantinescu, George S.; Lele, S. K.

2001-01-01

Numerical methods for solving the flow equations in cylindrical or spherical coordinates should be able to capture the behavior of the exact solution near the regions where the particular form of the governing equations is singular. In this work we focus on the treatment of these numerical singularities for finite-differences methods by reinterpreting the regularity conditions developed in the context of pseudo-spectral methods. A generally applicable numerical method for treating the singularities present at the polar axis, when nonaxisymmetric flows are solved in cylindrical, coordinates using highly accurate finite differences schemes (e.g., Pade schemes) on non-staggered grids, is presented. Governing equations for the flow at the polar axis are derived using series expansions near r=0. The only information needed to calculate the coefficients in these equations are the values of the flow variables and their radial derivatives at the previous iteration (or time) level. These derivatives, which are multi-valued at the polar axis, are calculated without dropping the accuracy of the numerical method using a mapping of the flow domain from (0,R)*(0,2pi) to (-R,R)*(0,pi), where R is the radius of the computational domain. This allows the radial derivatives to be evaluated using high-order differencing schemes (e.g., compact schemes) at points located on the polar axis. The proposed technique is illustrated by results from simulations of laminar-forced jets and turbulent compressible jets using large eddy simulation (LES) methods. In term of the general robustness of the numerical method and smoothness of the solution close to the polar axis, the present results compare very favorably to similar calculations in which the equations are solved in Cartesian coordinates at the polar axis, or in which the singularity is removed by employing a staggered mesh in the radial direction without a mesh point at r=0, following the method proposed recently by Mohseni and Colonius (1). Extension of the method described here for incompressible flows or for any other set of equations that are solved on a non-staggered mesh in cylindrical or spherical coordinates with finite-differences schemes of various level of accuracy is immediate.

An analytic solution for numerical modeling validation in electromagnetics: the resistive sphere

NASA Astrophysics Data System (ADS)

Swidinsky, Andrei; Liu, Lifei

2017-11-01

We derive the electromagnetic response of a resistive sphere to an electric dipole source buried in a conductive whole space. The solution consists of an infinite series of spherical Bessel functions and associated Legendre polynomials, and follows the well-studied problem of a conductive sphere buried in a resistive whole space in the presence of a magnetic dipole. Our result is particularly useful for controlled-source electromagnetic problems using a grounded electric dipole transmitter and can be used to check numerical methods of calculating the response of resistive targets (such as finite difference, finite volume, finite element and integral equation). While we elect to focus on the resistive sphere in our examples, the expressions in this paper are completely general and allow for arbitrary source frequency, sphere radius, transmitter position, receiver position and sphere/host conductivity contrast so that conductive target responses can also be checked. Commonly used mesh validation techniques consist of comparisons against other numerical codes, but such solutions may not always be reliable or readily available. Alternatively, the response of simple 1-D models can be tested against well-known whole space, half-space and layered earth solutions, but such an approach is inadequate for validating models with curved surfaces. We demonstrate that our theoretical results can be used as a complementary validation tool by comparing analytic electric fields to those calculated through a finite-element analysis; the software implementation of this infinite series solution is made available for direct and immediate application.

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

NASA Astrophysics Data System (ADS)

Galucio, A. C.; Deü, J.-F.; Ohayon, R.

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler-Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.

Theoretical and experimental evidence of non-symmetric doubly localized rogue waves.

PubMed

He, Jingsong; Guo, Lijuan; Zhang, Yongshuai; Chabchoub, Amin

2014-11-08

We present determinant expressions for vector rogue wave (RW) solutions of the Manakov system, a two-component coupled nonlinear Schrödinger (NLS) equation. As a special case, we generate a family of exact and non-symmetric RW solutions of the NLS equation up to third order, localized in both space and time. The derived non-symmetric doubly localized second-order solution is generated experimentally in a water wave flume for deep-water conditions. Experimental results, confirming the characteristic non-symmetric pattern of the solution, are in very good agreement with theory as well as with numerical simulations, based on the modified NLS equation, known to model accurately the dynamics of weakly nonlinear wave packets in deep water.

Theoretical and experimental evidence of non-symmetric doubly localized rogue waves

PubMed Central

He, Jingsong; Guo, Lijuan; Zhang, Yongshuai; Chabchoub, Amin

2014-01-01

We present determinant expressions for vector rogue wave (RW) solutions of the Manakov system, a two-component coupled nonlinear Schrödinger (NLS) equation. As a special case, we generate a family of exact and non-symmetric RW solutions of the NLS equation up to third order, localized in both space and time. The derived non-symmetric doubly localized second-order solution is generated experimentally in a water wave flume for deep-water conditions. Experimental results, confirming the characteristic non-symmetric pattern of the solution, are in very good agreement with theory as well as with numerical simulations, based on the modified NLS equation, known to model accurately the dynamics of weakly nonlinear wave packets in deep water. PMID:25383023

Sensitivity of Simulated Warm Rain Formation to Collision and Coalescence Efficiencies, Breakup, and Turbulence: Comparison of Two Bin-Resolved Numerical Models

NASA Technical Reports Server (NTRS)

Fridlind, Ann; Seifert, Axel; Ackerman, Andrew; Jensen, Eric

2004-01-01

Numerical models that resolve cloud particles into discrete mass size distributions on an Eulerian grid provide a uniquely powerful means of studying the closely coupled interaction of aerosols, cloud microphysics, and transport that determine cloud properties and evolution. However, such models require many experimentally derived paramaterizations in order to properly represent the complex interactions of droplets within turbulent flow. Many of these parameterizations remain poorly quantified, and the numerical methods of solving the equations for temporal evolution of the mass size distribution can also vary considerably in terms of efficiency and accuracy. In this work, we compare results from two size-resolved microphysics models that employ various widely-used parameterizations and numerical solution methods for several aspects of stochastic collection.

A new method to simulate the effects of viscous fingering on miscible displacement processes in porous media

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vossoughi, S.; Green, D.W.; Smith, J.E.

This paper presents a new method to simulate the effects of viscous fingering on miscible displacement processes in porous media. The method is based on the numerical solution of a general form of the convection-dispersion equation. In this equation the convection term is represented by a fractional flow function. The fractional flow function is derived from Darcy's law using a concentration-dependent, average viscosity and relative flow area to each fluid at any point in the bed. The method was extended to the description of a polymer flood by including retention and inaccessible pore volume. A Langmuir-type model for polymer retentionmore » in the rock was used. The resulting convection-dispersion equation for displacement by polymer was then solved numerically by the use of a finite element method with linear basis functions and Crank-Nicholson derivative approximation. History matches were performed on four sets of laboratory data to verify the model. These were: an unfavorable viscosity ratio displacement, stable displacement of glycerol by polymer solution, unstable displacement of brine by a slug of polymer solution, and a favorable viscosity ratio displacement. In general, computed results from the model matched laboratory data closely. Good agreement of the model with experiments over a significant range of variables lends support to the analysis.« less

Exact solutions and conservation laws of the system of two-dimensional viscous Burgers equations

NASA Astrophysics Data System (ADS)

Abdulwahhab, Muhammad Alim

2016-10-01

Fluid turbulence is one of the phenomena that has been studied extensively for many decades. Due to its huge practical importance in fluid dynamics, various models have been developed to capture both the indispensable physical quality and the mathematical structure of turbulent fluid flow. Among the prominent equations used for gaining in-depth insight of fluid turbulence is the two-dimensional Burgers equations. Its solutions have been studied by researchers through various methods, most of which are numerical. Being a simplified form of the two-dimensional Navier-Stokes equations and its wide range of applicability in various fields of science and engineering, development of computationally efficient methods for the solution of the two-dimensional Burgers equations is still an active field of research. In this study, Lie symmetry method is used to perform detailed analysis on the system of two-dimensional Burgers equations. Optimal system of one-dimensional subalgebras up to conjugacy is derived and used to obtain distinct exact solutions. These solutions not only help in understanding the physical effects of the model problem but also, can serve as benchmarks for constructing algorithms and validation of numerical solutions of the system of Burgers equations under consideration at finite Reynolds numbers. Independent and nontrivial conserved vectors are also constructed.

DIFFUSED SOLUTE-SOLVENT INTERFACE WITH POISSON-BOLTZMANN ELECTROSTATICS: FREE-ENERGY VARIATION AND SHARP-INTERFACE LIMIT.

PubMed

Li, B O; Liu, Yuan

A phase-field free-energy functional for the solvation of charged molecules (e.g., proteins) in aqueous solvent (i.e., water or salted water) is constructed. The functional consists of the solute volumetric and solute-solvent interfacial energies, the solute-solvent van der Waals interaction energy, and the continuum electrostatic free energy described by the Poisson-Boltzmann theory. All these are expressed in terms of phase fields that, for low free-energy conformations, are close to one value in the solute phase and another in the solvent phase. A key property of the model is that the phase-field interpolation of dielectric coefficient has the vanishing derivative at both solute and solvent phases. The first variation of such an effective free-energy functional is derived. Matched asymptotic analysis is carried out for the resulting relaxation dynamics of the diffused solute-solvent interface. It is shown that the sharp-interface limit is exactly the variational implicit-solvent model that has successfully captured capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states of underlying biomolecular systems as found in experiment and molecular dynamics simulations. Our phase-field approach and analysis can be used to possibly couple the description of interfacial fluctuations for efficient numerical computations of biomolecular interactions.

A path integral approach to asset-liability management

NASA Astrophysics Data System (ADS)

Decamps, Marc; De Schepper, Ann; Goovaerts, Marc

2006-05-01

Functional integrals constitute a powerful tool in the investigation of financial models. In the recent econophysics literature, this technique was successfully used for the pricing of a number of derivative securities. In the present contribution, we introduce this approach to the field of asset-liability management. We work with a representation of cash flows by means of a two-dimensional delta-function perturbation, in the case of a Brownian model and a geometric Brownian model. We derive closed-form solutions for a finite horizon ALM policy. The results are numerically and graphically illustrated.

Exact solution of the hidden Markov processes.

PubMed

Saakian, David B

2017-11-01

We write a master equation for the distributions related to hidden Markov processes (HMPs) and solve it using a functional equation. Thus the solution of HMPs is mapped exactly to the solution of the functional equation. For a general case the latter can be solved only numerically. We derive an exact expression for the entropy of HMPs. Our expression for the entropy is an alternative to the ones given before by the solution of integral equations. The exact solution is possible because actually the model can be considered as a generalized random walk on a one-dimensional strip. While we give the solution for the two second-order matrices, our solution can be easily generalized for the L values of the Markov process and M values of observables: We should be able to solve a system of L functional equations in the space of dimension M-1.

Exact solution of the hidden Markov processes

NASA Astrophysics Data System (ADS)

Saakian, David B.

2017-11-01

We write a master equation for the distributions related to hidden Markov processes (HMPs) and solve it using a functional equation. Thus the solution of HMPs is mapped exactly to the solution of the functional equation. For a general case the latter can be solved only numerically. We derive an exact expression for the entropy of HMPs. Our expression for the entropy is an alternative to the ones given before by the solution of integral equations. The exact solution is possible because actually the model can be considered as a generalized random walk on a one-dimensional strip. While we give the solution for the two second-order matrices, our solution can be easily generalized for the L values of the Markov process and M values of observables: We should be able to solve a system of L functional equations in the space of dimension M -1 .

New analytical solutions to the two-phase water faucet problem

DOE PAGES

Zou, Ling; Zhao, Haihua; Zhang, Hongbin

2016-06-17

Here, the one-dimensional water faucet problem is one of the classical benchmark problems originally proposed by Ransom to study the two-fluid two-phase flow model. With certain simplifications, such as massless gas phase and no wall and interfacial frictions, analytical solutions had been previously obtained for the transient liquid velocity and void fraction distribution. The water faucet problem and its analytical solutions have been widely used for the purposes of code assessment, benchmark and numerical verifications. In our previous study, the Ransom’s solutions were used for the mesh convergence study of a high-resolution spatial discretization scheme. It was found that, atmore » the steady state, an anticipated second-order spatial accuracy could not be achieved, when compared to the existing Ransom’s analytical solutions. A further investigation showed that the existing analytical solutions do not actually satisfy the commonly used two-fluid single-pressure two-phase flow equations. In this work, we present a new set of analytical solutions of the water faucet problem at the steady state, considering the gas phase density’s effect on pressure distribution. This new set of analytical solutions are used for mesh convergence studies, from which anticipated second-order of accuracy is achieved for the 2nd order spatial discretization scheme. In addition, extended Ransom’s transient solutions for the gas phase velocity and pressure are derived, with the assumption of decoupled liquid and gas pressures. Numerical verifications on the extended Ransom’s solutions are also presented.« less

Determination of the Fracture Parameters in a Stiffened Composite Panel

NASA Technical Reports Server (NTRS)

Lin, Chung-Yi

2000-01-01

A modified J-integral, namely the equivalent domain integral, is derived for a three-dimensional anisotropic cracked solid to evaluate the stress intensity factor along the crack front using the finite element method. Based on the equivalent domain integral method with auxiliary fields, an interaction integral is also derived to extract the second fracture parameter, the T-stress, from the finite element results. The auxiliary fields are the two-dimensional plane strain solutions of monoclinic materials with the plane of symmetry at x(sub 3) = 0 under point loads applied at the crack tip. These solutions are expressed in a compact form based on the Stroh formalism. Both integrals can be implemented into a single numerical procedure to determine the distributions of stress intensity factor and T-stress components, T11, T13, and thus T33, along a three-dimensional crack front. The effects of plate thickness and crack length on the variation of the stress intensity factor and T-stresses through the thickness are investigated in detail for through-thickness center-cracked plates (isotropic and orthotropic) and orthotropic stiffened panels under pure mode-I loading conditions. For all the cases studied, T11 remains negative. For plates with the same dimensions, a larger size of crack yields larger magnitude of the normalized stress intensity factor and normalized T-stresses. The results in orthotropic stiffened panels exhibit an opposite trend in general. As expected, for the thicker panels, the fracture parameters evaluated through the thickness, except the region near the free surfaces, approach two-dimensional plane strain solutions. In summary, the numerical methods presented in this research demonstrate their high computational effectiveness and good numerical accuracy in extracting these fracture parameters from the finite element results in three-dimensional cracked solids.

Quadrature errors in the partical derivatives required for the direct recovery of gravity anomalies from satellite observations

NASA Technical Reports Server (NTRS)

Hajela, D. P.

1972-01-01

The equations of motion of a geodetic satellite in the earth's gravitational field expressed by gravity anomalies require the evaluation, amongst others, of the partial derivatives of the disturbing force with respect to individual gravity anomalies. Data are discussed on how anomaly blocks should be subdivided so that the partial derivatives may be numerically evaluated for each subdivision, and then finally meaned to give the value representative of the whole blocks, with accuracies better than 2 to 3 percent for all blocks. The number of subdivisions is large for the blocks nearest to the satellite subpoint and decreases away from it. The actual values of this spherical distance and the actual subdivision of the mean gravity anomaly blocks was determined numerically for 184 15 deg x 15 deg equal area blocks. Satellite heights above the earth of 400 km, 800 km and 1600 km were considered. The computer times for the suggested scheme were compared with alternative solutions.

Scattering matrices of Lamb waves at irregular surface and void defects.

PubMed

Feng, Feilong; Shen, Jianzhong; Lin, Shuyu

2012-08-01

Time-harmonic solution of Lamb wave scattering in a plane-strain waveguide with irregular thickness is investigated based on stair-step discretization and stepwise mode matching. The transfer relations of the transmission matrices and reflection matrices are derived in both directions of the waveguide. With these, an explicit expression of the scattering matrix is derived. When the scattering region of an inner irregular defect is geometrically divided into several parts composed of sub-waveguides with variable thicknesses and void regions with vertical free edges corresponding to the plate surfaces, the scattering matrix of the whole region could then be derived by modal matching along the artificial boundaries, as explicit functions of all the scattering matrices of the sub-waveguides and reflection matrices of the free edges. The effectiveness of the formulation is examined by numerical examples; the calculated scattering coefficients are in good accordance with those obtained from numerical simulation models. Copyright © 2012 Elsevier B.V. All rights reserved.

Advancing MODFLOW Applying the Derived Vector Space Method

NASA Astrophysics Data System (ADS)

Herrera, G. S.; Herrera, I.; Lemus-García, M.; Hernandez-Garcia, G. D.

2015-12-01

The most effective domain decomposition methods (DDM) are non-overlapping DDMs. Recently a new approach, the DVS-framework, based on an innovative discretization method that uses a non-overlapping system of nodes (the derived-nodes), was introduced and developed by I. Herrera et al. [1, 2]. Using the DVS-approach a group of four algorithms, referred to as the 'DVS-algorithms', which fulfill the DDM-paradigm (i.e. the solution of global problems is obtained by resolution of local problems exclusively) has been derived. Such procedures are applicable to any boundary-value problem, or system of such equations, for which a standard discretization method is available and then software with a high degree of parallelization can be constructed. In a parallel talk, in this AGU Fall Meeting, Ismael Herrera will introduce the general DVS methodology. The application of the DVS-algorithms has been demonstrated in the solution of several boundary values problems of interest in Geophysics. Numerical examples for a single-equation, for the cases of symmetric, non-symmetric and indefinite problems were demonstrated before [1,2]. For these problems DVS-algorithms exhibited significantly improved numerical performance with respect to standard versions of DDM algorithms. In view of these results our research group is in the process of applying the DVS method to a widely used simulator for the first time, here we present the advances of the application of this method for the parallelization of MODFLOW. Efficiency results for a group of tests will be presented. References [1] I. Herrera, L.M. de la Cruz and A. Rosas-Medina. Non overlapping discretization methods for partial differential equations, Numer Meth Part D E, (2013). [2] Herrera, I., & Contreras Iván "An Innovative Tool for Effectively Applying Highly Parallelized Software To Problems of Elasticity". Geofísica Internacional, 2015 (In press)

Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature

NASA Astrophysics Data System (ADS)

Lotfy, K.; Sarkar, N.

2017-11-01

In this work, a novel generalized model of photothermal theory with two-temperature thermoelasticity theory based on memory-dependent derivative (MDD) theory is performed. A one-dimensional problem for an elastic semiconductor material with isotropic and homogeneous properties has been considered. The problem is solved with a new model (MDD) under the influence of a mechanical force with a photothermal excitation. The Laplace transform technique is used to remove the time-dependent terms in the governing equations. Moreover, the general solutions of some physical fields are obtained. The surface taken into consideration is free of traction and subjected to a time-dependent thermal shock. The numerical Laplace inversion is used to obtain the numerical results of the physical quantities of the problem. Finally, the obtained results are presented and discussed graphically.

Properties of bright solitons in averaged and unaveraged models for SDG fibres

NASA Astrophysics Data System (ADS)

Kumar, Ajit; Kumar, Atul

1996-04-01

Using the slowly varying envelope approximation and averaging over the fibre cross-section the evolution equation for optical pulses in semiconductor-doped glass (SDG) fibres is derived from the nonlinear wave equation. Bright soliton solutions of this equation are obtained numerically and their properties are studied and compared with those of the bright solitons in the unaveraged model.

Free energy change of off-eutectic binary alloys on solidification

NASA Technical Reports Server (NTRS)

Ohsaka, K.; Trinh, E. H.; Lin, J.-C.; Perepezko, J. H.

1991-01-01

A formula for the free energy difference between the undercooled liquid phase and the stable solid phase is derived for off-eutectic binary alloys in which the equilibrium solid/liquid transition takes place over a certain temperature range. The free energy change is then evaluated numerically for a Bi-25 at. pct Cd alloy modeled as a sub-subregular solution.

Surface phenomena and the evolution of radiating fluid spheres in general relativity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Herrera, L.; Jimenez, J.; Esculpi, M.

1989-10-01

A method used to study the evolution of radiating spheres (Herrera, Jimenez, and Ruggeri) is extended to the case in which surface phenomena are taken into account. The equations have been integrated numerically for a model derived from the Schwarzschild interior solution, bringing out the effects of surface tension on the evolution of the spheres. 17 refs.

Optimal tactics for close support operations. III - Degraded intelligence and communications

NASA Astrophysics Data System (ADS)

Hess, J.; Kalaba, R.; Kagiwada, H.; Spingarn, K.; Tsokos, C.

1980-04-01

A new generation of C3 (command, control, and communication) models for military cybernetics is developed. Recursive equations for the solution of the C3 problem are derived for an amphibious campaign with linear time-varying dynamics. Air and ground commanders are assumed to have no intelligence and no communications. Numerical results are given for the optimal decision rules.

Chandrasekhar equations and computational algorithms for distributed parameter systems

NASA Technical Reports Server (NTRS)

Burns, J. A.; Ito, K.; Powers, R. K.

1984-01-01

The Chandrasekhar equations arising in optimal control problems for linear distributed parameter systems are considered. The equations are derived via approximation theory. This approach is used to obtain existence, uniqueness, and strong differentiability of the solutions and provides the basis for a convergent computation scheme for approximating feedback gain operators. A numerical example is presented to illustrate these ideas.

Mimetic compact stars

NASA Astrophysics Data System (ADS)

Momeni, D.; Moraes, P. H. R. S.; Gholizade, H.; Myrzakulov, R.

Modified gravity models have been constantly proposed with the purpose of evading some standard gravity shortcomings. Recently proposed by Chamseddine and Mukhanov, the Mimetic Gravity arises as an optimistic alternative. Our purpose in this work is to derive Tolman-Oppenheimer-Volkoff equations and solutions for such a gravity theory. We solve them numerically for quark star and neutron star cases. The results are carefully discussed.

Generalized analytical solutions to multispecies transport equations with scale-dependent dispersion coefficients subject to time-dependent boundary conditions

NASA Astrophysics Data System (ADS)

Chen, J. S.; Chiang, S. Y.; Liang, C. P.

2017-12-01

It is essential to develop multispecies transport analytical models based on a set of advection-dispersion equations (ADEs) coupled with sequential first-order decay reactions for the synchronous prediction of plume migrations of both parent and its daughter species of decaying contaminants such as radionuclides, dissolved chlorinated organic compounds, pesticides and nitrogen. Although several analytical models for multispecies transport have already been reported, those currently available in the literature have primarily been derived based on ADEs with constant dispersion coefficients. However, there have been a number of studies demonstrating that the dispersion coefficients increase with the solute travel distance as a consequence of variation in the hydraulic properties of the porous media. This study presents novel analytical models for multispecies transport with distance-dependent dispersion coefficients. The correctness of the derived analytical models is confirmed by comparing them against the numerical models. Results show perfect agreement between the analytical and numerical models. Comparison of our new analytical model for multispecies transport with scale-dependent dispersion to an analytical model with constant dispersion is made to illustrate the effects of the dispersion coefficients on the multispecies transport of decaying contaminants.

Tidal, Residual, Intertidal Mudflat (TRIM) Model and its Applications to San Francisco Bay, California

USGS Publications Warehouse

Cheng, R.T.; Casulli, V.; Gartner, J.W.

1993-01-01

A numerical model using a semi-implicit finite-difference method for solving the two-dimensional shallow-water equations is presented. The gradient of the water surface elevation in the momentum equations and the velocity divergence in the continuity equation are finite-differenced implicitly, the remaining terms are finite-differenced explicitly. The convective terms are treated using an Eulerian-Lagrangian method. The combination of the semi-implicit finite-difference solution for the gravity wave propagation, and the Eulerian-Lagrangian treatment of the convective terms renders the numerical model unconditionally stable. When the baroclinic forcing is included, a salt transport equation is coupled to the momentum equations, and the numerical method is subject to a weak stability condition. The method of solution and the properties of the numerical model are given. This numerical model is particularly suitable for applications to coastal plain estuaries and tidal embayments in which tidal currents are dominant, and tidally generated residual currents are important. The model is applied to San Francisco Bay, California where extensive historical tides and current-meter data are available. The model calibration is considered by comparing time-series of the field data and of the model results. Alternatively, and perhaps more meaningfully, the model is calibrated by comparing the harmonic constants of tides and tidal currents derived from field data with those derived from the model. The model is further verified by comparing the model results with an independent data set representing the wet season. The strengths and the weaknesses of the model are assessed based on the results of model calibration and verification. Using the model results, the properties of tides and tidal currents in San Francisco Bay are characterized and discussed. Furthermore, using the numerical model, estimates of San Francisco Bay's volume, surface area, mean water depth, tidal prisms, and tidal excursions at spring and neap tides are computed. Additional applications of the model reveal, qualitatively the spatial distribution of residual variables. ?? 1993 Academic Press. All rights reserved.

Fluid-structure interaction with pipe-wall viscoelasticity during water hammer

NASA Astrophysics Data System (ADS)

Keramat, A.; Tijsseling, A. S.; Hou, Q.; Ahmadi, A.

2012-01-01

Fluid-structure interaction (FSI) due to water hammer in a pipeline which has viscoelastic wall behaviour is studied. Appropriate governing equations are derived and numerically solved. In the numerical implementation of the hydraulic and structural equations, viscoelasticity is incorporated using the Kelvin-Voigt mechanical model. The equations are solved by two different approaches, namely the Method of Characteristics-Finite Element Method (MOC-FEM) and full MOC. In both approaches two important effects of FSI in fluid-filled pipes, namely Poisson and junction coupling, are taken into account. The study proposes a more comprehensive model for studying fluid transients in pipelines as compared to previous works, which take into account either FSI or viscoelasticity. To verify the proposed mathematical model and its numerical solutions, the following problems are investigated: axial vibration of a viscoelastic bar subjected to a step uniaxial loading, FSI in an elastic pipe, and hydraulic transients in a pressurised polyethylene pipe without FSI. The results of each case are checked with available exact and experimental results. Then, to study the simultaneous effects of FSI and viscoelasticity, which is the new element of the present research, one problem is solved by the two different numerical approaches. Both numerical methods give the same results, thus confirming the correctness of the solutions.

Algorithms for Computing the Magnetic Field, Vector Potential, and Field Derivatives for a Thin Solenoid with Uniform Current Density

DOE Office of Scientific and Technical Information (OSTI.GOV)

Walstrom, Peter Lowell

A numerical algorithm for computing the field components B r and B z and their r and z derivatives with open boundaries in cylindrical coordinates for radially thin solenoids with uniform current density is described in this note. An algorithm for computing the vector potential A θ is also described. For the convenience of the reader, derivations of the final expressions from their defining integrals are given in detail, since their derivations are not all easily found in textbooks. Numerical calculations are based on evaluation of complete elliptic integrals using the Bulirsch algorithm cel. The (apparently) new feature of themore » algorithms described in this note applies to cases where the field point is outside of the bore of the solenoid and the field-point radius approaches the solenoid radius. Since the elliptic integrals of the third kind normally used in computing B z and A θ become infinite in this region of parameter space, fields for points with the axial coordinate z outside of the ends of the solenoid and near the solenoid radius are treated by use of elliptic integrals of the third kind of modified argument, derived by use of an addition theorem. Also, the algorithms also avoid the numerical difficulties the textbook solutions have for points near the axis arising from explicit factors of 1/r or 1/r 2 in the some of the expressions.« less

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations That Arise from Nonlinear Two-Point Boundary Value Problems

NASA Technical Reports Server (NTRS)

Sidi, Avram; Pennline, James A.

1999-01-01

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + definite integral of g(x, t)F(t,y(t))dt with limits between 0 and 1,0 less than or equal to x les than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integral equations arise, e.g., when one applied Green's function techniques to nonlinear two-point boundary value problems of the form y "(x) =f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and y(l) = y(sub l), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trepezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal rule, thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations that Arise from Nonlinear Two-Point Boundary Value Problems

NASA Technical Reports Server (NTRS)

Sidi, Avram; Pennline, James A.

1999-01-01

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + integral(0 to 1) g(x,t) F(t, y(t)) dt, 0 less than or equal to x less than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integrals equations arise, e.g., when one applies Green's function techniques to nonlinear two-point boundary value problems of the form U''(x) = f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and g(l) = y(sub 1), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trapezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zou, Ling; Zhao, Haihua; Zhang, Hongbin

Here, the one-dimensional water faucet problem is one of the classical benchmark problems originally proposed by Ransom to study the two-fluid two-phase flow model. With certain simplifications, such as massless gas phase and no wall and interfacial frictions, analytical solutions had been previously obtained for the transient liquid velocity and void fraction distribution. The water faucet problem and its analytical solutions have been widely used for the purposes of code assessment, benchmark and numerical verifications. In our previous study, the Ransom’s solutions were used for the mesh convergence study of a high-resolution spatial discretization scheme. It was found that, atmore » the steady state, an anticipated second-order spatial accuracy could not be achieved, when compared to the existing Ransom’s analytical solutions. A further investigation showed that the existing analytical solutions do not actually satisfy the commonly used two-fluid single-pressure two-phase flow equations. In this work, we present a new set of analytical solutions of the water faucet problem at the steady state, considering the gas phase density’s effect on pressure distribution. This new set of analytical solutions are used for mesh convergence studies, from which anticipated second-order of accuracy is achieved for the 2nd order spatial discretization scheme. In addition, extended Ransom’s transient solutions for the gas phase velocity and pressure are derived, with the assumption of decoupled liquid and gas pressures. Numerical verifications on the extended Ransom’s solutions are also presented.« less

Evaluation of Proteus as a Tool for the Rapid Development of Models of Hydrologic Systems

NASA Astrophysics Data System (ADS)

Weigand, T. M.; Farthing, M. W.; Kees, C. E.; Miller, C. T.

2013-12-01

Models of modern hydrologic systems can be complex and involve a variety of operators with varying character. The goal is to implement approximations of such models that are both efficient for the developer and computationally efficient, which is a set of naturally competing objectives. Proteus is a Python-based toolbox that supports prototyping of model formulations as well as a wide variety of modern numerical methods and parallel computing. We used Proteus to develop numerical approximations for three models: Richards' equation, a brine flow model derived using the Thermodynamically Constrained Averaging Theory (TCAT), and a multiphase TCAT-based tumor growth model. For Richards' equation, we investigated discontinuous Galerkin solutions with higher order time integration based on the backward difference formulas. The TCAT brine flow model was implemented using Proteus and a variety of numerical methods were compared to hand coded solutions. Finally, an existing tumor growth model was implemented in Proteus to introduce more advanced numerics and allow the code to be run in parallel. From these three example models, Proteus was found to be an attractive open-source option for rapidly developing high quality code for solving existing and evolving computational science models.

Couple of the Variational Iteration Method and Fractional-Order Legendre Functions Method for Fractional Differential Equations

PubMed Central

Song, Junqiang; Leng, Hongze; Lu, Fengshun

2014-01-01

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique. PMID:24511303

Strongly localized dark modes in binary discrete media with cubic-quintic nonlinearity within the anti-continuum limit

NASA Astrophysics Data System (ADS)

Taib, L. Abdul; Hadi, M. S. Abdul; Umarov, B. A.

2017-12-01

The existence of dark strongly localized modes of binary discrete media with cubic-quintic nonlinearity is numerically demonstrated by solving the relevant discrete nonlinear Schrödinger equations. In the model, the coupling coefficients between adjacent sites are set to be relatively small representing the anti-continuum limit. In addition, approximated analytical solutions for vectorial solitons with various topologies are derived. Stability analysis of the localized states was performed using the standard linearized eigenfrequency problem. The prediction from the stability analysis are furthermore verified by direct numerical integrations.

Numerical computation of diffusion on a surface.

PubMed

Schwartz, Peter; Adalsteinsson, David; Colella, Phillip; Arkin, Adam Paul; Onsum, Matthew

2005-08-09

We present a numerical method for computing diffusive transport on a surface derived from image data. Our underlying discretization method uses a Cartesian grid embedded boundary method for computing the volume transport in a region consisting of all points a small distance from the surface. We obtain a representation of this region from image data by using a front propagation computation based on level set methods for solving the Hamilton-Jacobi and eikonal equations. We demonstrate that the method is second-order accurate in space and time and is capable of computing solutions on complex surface geometries obtained from image data of cells.

Effective equations and the inverse cascade theory for Kolmogorov flows

NASA Technical Reports Server (NTRS)

Weinan, E.; Shu, Chi-Wang

1992-01-01

We study the two dimensional Kolmogorov flows in the limit as the forcing frequency goes to infinity. Direct numerical simulation indicates that the low frequency energy spectrum evolves to a universal kappa (exp -4) decay law. We derive effective equations governing the behavior of the large scale flow quantities. We then present numerical evidence that with smooth initial data, the solution to the effective equation develops a kappa (exp -4) type singularity at a finite time. This gives a convenient explanation for the kappa (exp -4) decay law exhibited by the original Kolmogorov flows.

Theory and Circuit Model for Lossy Coaxial Transmission Line

DOE Office of Scientific and Technical Information (OSTI.GOV)

Genoni, T. C.; Anderson, C. N.; Clark, R. E.

2017-04-01

The theory of signal propagation in lossy coaxial transmission lines is revisited and new approximate analytic formulas for the line impedance and attenuation are derived. The accuracy of these formulas from DC to 100 GHz is demonstrated by comparison to numerical solutions of the exact field equations. Based on this analysis, a new circuit model is described which accurately reproduces the line response over the entire frequency range. Circuit model calculations are in excellent agreement with the numerical and analytic results, and with finite-difference-time-domain simulations which resolve the skindepths of the conducting walls.

Derivation of Formulations 1 and 1A of Farassat

NASA Technical Reports Server (NTRS)

Farassat, F.

2007-01-01

Formulations 1 and 1A are the solutions of the Ffowcs Williams-Hawkings (FW-H) equation with surface sources only when the surface moves at subsonic speed. Both formulations have been successfully used for helicopter rotor and propeller noise prediction for many years although we now recommend using Formulation 1A for this purpose. Formulation 1 has an observer time derivative that is taken numerically, and thus, increasing execution time on a computer and reducing the accuracy of the results. After some discussion of the Green's function of the wave equation, we derive Formulation 1 which is the basis of deriving Formulation 1A. We will then show how to take this observer time derivative analytically to get Formulation 1A. We give here the most detailed derivation of these formulations. Once you see the whole derivation, you will ask yourself why you did not do it yourself!

A remark on fractional differential equation involving I-function

NASA Astrophysics Data System (ADS)

Mishra, Jyoti

2018-02-01

The present paper deals with the solution of the fractional differential equation using the Laplace transform operator and its corresponding properties in the fractional calculus; we derive an exact solution of a complex fractional differential equation involving a special function known as I-function. The analysis of the some fractional integral with two parameters is presented using the suggested Theorem 1. In addition, some very useful corollaries are established and their proofs presented in detail. Some obtained exact solutions are depicted to see the effect of each fractional order. Owing to the wider applicability of the I-function, we can conclude that, the obtained results in our work generalize numerous well-known results obtained by specializing the parameters.

Diagrammatic theory of transition of pendulum like systems. [orbit-orbit and spin-orbit gravitational resonance interactions

NASA Technical Reports Server (NTRS)

Yoder, C. F.

1979-01-01

Orbit-orbit and spin-orbit gravitational resonances are analyzed using the model of a rigid pendulum subject to both a time-dependent periodic torque and a constant applied torque. First, a descriptive model of passage through resonance is developed from an examination of the polynomial equation that determines the extremes of the momentum variable. From this study, a probability estimate for capture into libration is derived. Second, a lowest order solution is constructed and compared with the solution obtained from numerical integration. The steps necessary to systematically improve this solution are also discussed. Finally, the effect of a dissipative term in the pendulum equation is analyzed.

Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-01-01

The propagation of three-dimensional nonlinear irrotational flow of an inviscid and incompressible fluid of the long waves in dispersive shallow-water approximation is analyzed. The problem formulation of the long waves in dispersive shallow-water approximation lead to fifth-order Kadomtsev-Petviashvili (KP) dynamical equation by applying the reductive perturbation theory. By using an extended auxiliary equation method, the solitary travelling-wave solutions of the two-dimensional nonlinear fifth-order KP dynamical equation are derived. An analytical as well as a numerical solution of the two-dimensional nonlinear KP equation are obtained and analyzed with the effects of external pressure flow.

Water movement through plant roots - exact solutions of the water flow equation in roots with linear or exponential piecewise hydraulic properties

NASA Astrophysics Data System (ADS)

Meunier, Félicien; Couvreur, Valentin; Draye, Xavier; Zarebanadkouki, Mohsen; Vanderborght, Jan; Javaux, Mathieu

2017-12-01

In 1978, Landsberg and Fowkes presented a solution of the water flow equation inside a root with uniform hydraulic properties. These properties are root radial conductivity and axial conductance, which control, respectively, the radial water flow between the root surface and xylem and the axial flow within the xylem. From the solution for the xylem water potential, functions that describe the radial and axial flow along the root axis were derived. These solutions can also be used to derive root macroscopic parameters that are potential input parameters of hydrological and crop models. In this paper, novel analytical solutions of the water flow equation are developed for roots whose hydraulic properties vary along their axis, which is the case for most plants. We derived solutions for single roots with linear or exponential variations of hydraulic properties with distance to root tip. These solutions were subsequently combined to construct single roots with complex hydraulic property profiles. The analytical solutions allow one to verify numerical solutions and to get a generalization of the hydric behaviour with the main influencing parameters of the solutions. The resulting flow distributions in heterogeneous roots differed from those in uniform roots and simulations led to more regular, less abrupt variations of xylem suction or radial flux along root axes. The model could successfully be applied to maize effective root conductance measurements to derive radial and axial hydraulic properties. We also show that very contrasted root water uptake patterns arise when using either uniform or heterogeneous root hydraulic properties in a soil-root model. The optimal root radius that maximizes water uptake under a carbon cost constraint was also studied. The optimal radius was shown to be highly dependent on the root hydraulic properties and close to observed properties in maize roots. We finally used the obtained functions for evaluating the impact of root maturation versus root growth on water uptake. Very diverse uptake strategies arise from the analysis. These solutions open new avenues to investigate for optimal genotype-environment-management interactions by optimization, for example, of plant-scale macroscopic hydraulic parameters used in ecohydrogolocial models.

Analytical solution for the transient wave propagation of a buried cylindrical P-wave line source in a semi-infinite elastic medium with a fluid surface layer

NASA Astrophysics Data System (ADS)

Shan, Zhendong; Ling, Daosheng

2018-02-01

This article develops an analytical solution for the transient wave propagation of a cylindrical P-wave line source in a semi-infinite elastic solid with a fluid layer. The analytical solution is presented in a simple closed form in which each term represents a transient physical wave. The Scholte equation is derived, through which the Scholte wave velocity can be determined. The Scholte wave is the wave that propagates along the interface between the fluid and solid. To develop the analytical solution, the wave fields in the fluid and solid are defined, their analytical solutions in the Laplace domain are derived using the boundary and interface conditions, and the solutions are then decomposed into series form according to the power series expansion method. Each item of the series solution has a clear physical meaning and represents a transient wave path. Finally, by applying Cagniard's method and the convolution theorem, the analytical solutions are transformed into the time domain. Numerical examples are provided to illustrate some interesting features in the fluid layer, the interface and the semi-infinite solid. When the P-wave velocity in the fluid is higher than that in the solid, two head waves in the solid, one head wave in the fluid and a Scholte wave at the interface are observed for the cylindrical P-wave line source.

Electroneutral models for dynamic Poisson-Nernst-Planck systems

NASA Astrophysics Data System (ADS)

Song, Zilong; Cao, Xiulei; Huang, Huaxiong

2018-01-01

The Poisson-Nernst-Planck (PNP) system is a standard model for describing ion transport. In many applications, e.g., ions in biological tissues, the presence of thin boundary layers poses both modeling and computational challenges. In this paper, we derive simplified electroneutral (EN) models where the thin boundary layers are replaced by effective boundary conditions. There are two major advantages of EN models. First, it is much cheaper to solve them numerically. Second, EN models are easier to deal with compared to the original PNP system; therefore, it would also be easier to derive macroscopic models for cellular structures using EN models. Even though the approach used here is applicable to higher-dimensional cases, this paper mainly focuses on the one-dimensional system, including the general multi-ion case. Using systematic asymptotic analysis, we derive a variety of effective boundary conditions directly applicable to the EN system for the bulk region. This EN system can be solved directly and efficiently without computing the solution in the boundary layer. The derivation is based on matched asymptotics, and the key idea is to bring back higher-order contributions into the effective boundary conditions. For Dirichlet boundary conditions, the higher-order terms can be neglected and the classical results (continuity of electrochemical potential) are recovered. For flux boundary conditions, higher-order terms account for the accumulation of ions in boundary layer and neglecting them leads to physically incorrect solutions. To validate the EN model, numerical computations are carried out for several examples. Our results show that solving the EN model is much more efficient than the original PNP system. Implemented with the Hodgkin-Huxley model, the computational time for solving the EN model is significantly reduced without sacrificing the accuracy of the solution due to the fact that it allows for relatively large mesh and time-step sizes.

Compressed sensing with gradient total variation for low-dose CBCT reconstruction

NASA Astrophysics Data System (ADS)

Seo, Chang-Woo; Cha, Bo Kyung; Jeon, Seongchae; Huh, Young; Park, Justin C.; Lee, Byeonghun; Baek, Junghee; Kim, Eunyoung

2015-06-01

This paper describes the improvement of convergence speed with gradient total variation (GTV) in compressed sensing (CS) for low-dose cone-beam computed tomography (CBCT) reconstruction. We derive a fast algorithm for the constrained total variation (TV)-based a minimum number of noisy projections. To achieve this task we combine the GTV with a TV-norm regularization term to promote an accelerated sparsity in the X-ray attenuation characteristics of the human body. The GTV is derived from a TV and enforces more efficient computationally and faster in convergence until a desired solution is achieved. The numerical algorithm is simple and derives relatively fast convergence. We apply a gradient projection algorithm that seeks a solution iteratively in the direction of the projected gradient while enforcing a non-negatively of the found solution. In comparison with the Feldkamp, Davis, and Kress (FDK) and conventional TV algorithms, the proposed GTV algorithm showed convergence in ≤18 iterations, whereas the original TV algorithm needs at least 34 iterations in reducing 50% of the projections compared with the FDK algorithm in order to reconstruct the chest phantom images. Future investigation includes improving imaging quality, particularly regarding X-ray cone-beam scatter, and motion artifacts of CBCT reconstruction.

Numerical solution of modified differential equations based on symmetry preservation.

PubMed

Ozbenli, Ersin; Vedula, Prakash

2017-12-01

In this paper, we propose a method to construct invariant finite-difference schemes for solution of partial differential equations (PDEs) via consideration of modified forms of the underlying PDEs. The invariant schemes, which preserve Lie symmetries, are obtained based on the method of equivariant moving frames. While it is often difficult to construct invariant numerical schemes for PDEs due to complicated symmetry groups associated with cumbersome discrete variable transformations, we note that symmetries associated with more convenient transformations can often be obtained by appropriately modifying the original PDEs. In some cases, modifications to the original PDEs are also found to be useful in order to avoid trivial solutions that might arise from particular selections of moving frames. In our proposed method, modified forms of PDEs can be obtained either by addition of perturbation terms to the original PDEs or through defect correction procedures. These additional terms, whose primary purpose is to enable symmetries with more convenient transformations, are then removed from the system by considering moving frames for which these specific terms go to zero. Further, we explore selection of appropriate moving frames that result in improvement in accuracy of invariant numerical schemes based on modified PDEs. The proposed method is tested using the linear advection equation (in one- and two-dimensions) and the inviscid Burgers' equation. Results obtained for these tests cases indicate that numerical schemes derived from the proposed method perform significantly better than existing schemes not only by virtue of improvement in numerical accuracy but also due to preservation of qualitative properties or symmetries of the underlying differential equations.

Spherical Pendulum Small Oscillations for Slewing Crane Motion

PubMed Central

Perig, Alexander V.; Stadnik, Alexander N.; Deriglazov, Alexander I.

2014-01-01

The present paper focuses on the Lagrange mechanics-based description of small oscillations of a spherical pendulum with a uniformly rotating suspension center. The analytical solution of the natural frequencies' problem has been derived for the case of uniform rotation of a crane boom. The payload paths have been found in the inertial reference frame fixed on earth and in the noninertial reference frame, which is connected with the rotating crane boom. The numerical amplitude-frequency characteristics of the relative payload motion have been found. The mechanical interpretation of the terms in Lagrange equations has been outlined. The analytical expression and numerical estimation for cable tension force have been proposed. The numerical computational results, which correlate very accurately with the experimental observations, have been shown. PMID:24526891

Parsec-Scale Obscuring Accretion Disk with Large-Scale Magnetic Field in AGNs

NASA Technical Reports Server (NTRS)

Dorodnitsyn, A.; Kallman, T.

2017-01-01

A magnetic field dragged from the galactic disk, along with inflowing gas, can provide vertical support to the geometrically and optically thick pc (parsec) -scale torus in AGNs (Active Galactic Nuclei). Using the Soloviev solution initially developed for Tokamaks, we derive an analytical model for a rotating torus that is supported and confined by a magnetic field. We further perform three-dimensional magneto-hydrodynamic simulations of X-ray irradiated, pc-scale, magnetized tori. We follow the time evolution and compare models that adopt initial conditions derived from our analytic model with simulations in which the initial magnetic flux is entirely contained within the gas torus. Numerical simulations demonstrate that the initial conditions based on the analytic solution produce a longer-lived torus that produces obscuration that is generally consistent with observed constraints.

A weak Hamiltonian finite element method for optimal guidance of an advanced launch vehicle

NASA Technical Reports Server (NTRS)

Hodges, Dewey H.; Calise, Anthony J.; Bless, Robert R.; Leung, Martin

1989-01-01

A temporal finite-element method based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables, which are expanded in terms of nodal values and simple shape functions. Time derivatives of the states and costates do not appear in the governing variational equation; the only quantities whose time derivatives appear therein are virtual states and virtual costates. Numerical results are presented for an elementary trajectory optimization problem; they show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The feasibility of this approach for real-time guidance applications is evaluated. A simplified model for an advanced launch vehicle application that is suitable for finite-element solution is presented.

Analytic model for the long-term evolution of circular Earth satellite orbits including lunar node regression

NASA Astrophysics Data System (ADS)

Zhu, Ting-Lei; Zhao, Chang-Yin; Zhang, Ming-Jiang

2017-04-01

This paper aims to obtain an analytic approximation to the evolution of circular orbits governed by the Earth's J2 and the luni-solar gravitational perturbations. Assuming that the lunar orbital plane coincides with the ecliptic plane, Allan and Cook (Proc. R. Soc. A, Math. Phys. Eng. Sci. 280(1380):97, 1964) derived an analytic solution to the orbital plane evolution of circular orbits. Using their result as an intermediate solution, we establish an approximate analytic model with lunar orbital inclination and its node regression be taken into account. Finally, an approximate analytic expression is derived, which is accurate compared to the numerical results except for the resonant cases when the period of the reference orbit approximately equals the integer multiples (especially 1 or 2 times) of lunar node regression period.

A general radiation model for sound fields and nearfield acoustical holography in wedge propagation spaces.

PubMed

Hoffmann, Falk-Martin; Fazi, Filippo Maria; Williams, Earl G; Fontana, Simone

2017-09-01

In this work an expression for the solution of the Helmholtz equation for wedge spaces is derived. Such propagation spaces represent scenarios for many acoustical problems where a free field assumption is not eligible. The proposed sound field model is derived from the general solution of the wave equation in cylindrical coordinates, using sets of orthonormal basis functions. The latter are modified to satisfy several boundary conditions representing the reflective behaviour of wedge-shaped propagation spaces. This formulation is then used in the context of nearfield acoustical holography (NAH) and to obtain the expression of the Neumann Green function. The model and its suitability for NAH is demonstrated through both numerical simulations and measured data, where the latter was acquired for the specific case of a loudspeaker on a hemi-cylindrical rigid baffle.

Parsec-scale Obscuring Accretion Disk with Large-scale Magnetic Field in AGNs

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dorodnitsyn, A.; Kallman, T.

A magnetic field dragged from the galactic disk, along with inflowing gas, can provide vertical support to the geometrically and optically thick pc-scale torus in AGNs. Using the Soloviev solution initially developed for Tokamaks, we derive an analytical model for a rotating torus that is supported and confined by a magnetic field. We further perform three-dimensional magneto-hydrodynamic simulations of X-ray irradiated, pc-scale, magnetized tori. We follow the time evolution and compare models that adopt initial conditions derived from our analytic model with simulations in which the initial magnetic flux is entirely contained within the gas torus. Numerical simulations demonstrate thatmore » the initial conditions based on the analytic solution produce a longer-lived torus that produces obscuration that is generally consistent with observed constraints.« less

Age-of-Air, Tape Recorder, and Vertical Transport Schemes

NASA Technical Reports Server (NTRS)

Lin, S.-J.; Einaudi, Franco (Technical Monitor)

2000-01-01

A numerical-analytic investigation of the impacts of vertical transport schemes on the model simulated age-of-air and the so-called 'tape recorder' will be presented using an idealized 1-D column transport model as well as a more realistic 3-D dynamical model. By comparing to the 'exact' solutions of 'age-of-air' and the 'tape recorder' obtainable in the 1-D setting, useful insight is gained on the impacts of numerical diffusion and dispersion of numerical schemes used in global models. Advantages and disadvantages of Eulerian, semi-Lagrangian, and Lagrangian transport schemes will be discussed. Vertical resolution requirement for numerical schemes as well as observing systems for capturing the fine details of the 'tape recorder' or any upward propagating wave-like structures can potentially be derived from the 1-D analytic model.

New algorithms to compute the nearness symmetric solution of the matrix equation.

PubMed

Peng, Zhen-Yun; Fang, Yang-Zhi; Xiao, Xian-Wei; Du, Dan-Dan

2016-01-01

In this paper we consider the nearness symmetric solution of the matrix equation AXB = C to a given matrix [Formula: see text] in the sense of the Frobenius norm. By discussing equivalent form of the considered problem, we derive some necessary and sufficient conditions for the matrix [Formula: see text] is a solution of the considered problem. Based on the idea of the alternating variable minimization with multiplier method, we propose two iterative methods to compute the solution of the considered problem, and analyze the global convergence results of the proposed algorithms. Numerical results illustrate the proposed methods are more effective than the existing two methods proposed in Peng et al. (Appl Math Comput 160:763-777, 2005) and Peng (Int J Comput Math 87: 1820-1830, 2010).

A variational data assimilation system for the range dependent acoustic model using the representer method: Theoretical derivations.

PubMed

Ngodock, Hans; Carrier, Matthew; Fabre, Josette; Zingarelli, Robert; Souopgui, Innocent

2017-07-01

This study presents the theoretical framework for variational data assimilation of acoustic pressure observations into an acoustic propagation model, namely, the range dependent acoustic model (RAM). RAM uses the split-step Padé algorithm to solve the parabolic equation. The assimilation consists of minimizing a weighted least squares cost function that includes discrepancies between the model solution and the observations. The minimization process, which uses the principle of variations, requires the derivation of the tangent linear and adjoint models of the RAM. The mathematical derivations are presented here, and, for the sake of brevity, a companion study presents the numerical implementation and results from the assimilation simulated acoustic pressure observations.

Analysis of Discontinuities in a Rectangular Waveguide Using Dyadic Green's Function Approach in Conjunction with Method of Moments

NASA Technical Reports Server (NTRS)

Deshpande, M. D.

1997-01-01

The dyadic Green's function for an electric current source placed in a rectangular waveguide is derived using a magnetic vector potential approach. A complete solution for the electric and magnetic fields including the source location is obtained by simple differentiation of the vector potential around the source location. The simple differentiation approach which gives electric and magnetic fields identical to an earlier derivation is overlooked by the earlier workers in the derivation of the dyadic Green's function particularly around the source location. Numerical results obtained using the Green's function approach are compared with the results obtained using the Finite Element Method (FEM).

Canonical Nonlinear Viscous Core Solution in pipe and elliptical geometry

NASA Astrophysics Data System (ADS)

Ozcakir, Ozge

2016-11-01

In an earlier paper (Ozcakir et al. (2016)), two new nonlinear traveling wave solutions were found with collapsing structure towards the center of the pipe as Reynolds number R -> ∞ , which were called Nonlinear Viscous Core (NVC) states. Asymptotic scaling arguments suggested that the NVC state collapse rate scales as R - 1 / 4 where axial, radial and azimuthal velocity perturbations from Hagen-Poiseuille flow scale as R - 1 / 2, R - 3 / 4 and R - 3 / 4 respectively, while (1 - c) = O (R - 1 / 2) where c is the traveling wave speed. The theoretical scaling results were roughly consistent with full Navier-Stokes numerical computations in the range 105 < R <106 . In the present paper, through numerical solutions, we show that the scaled parameter free canonical differential equations derived in Ozcakir et al. (2016) indeed has solution that satisfies requisite far-field conditions. We also show that these are in good agreement with full Navier-Stokes calculations in a larger R range than previously calculated (R upto 106). Further, we extend our study to NVC states for pipes with elliptical cross-section and identify similar canonical structure in these cases. National Science Foundation NSF-DMS-1515755, EPSRC Grant EP/1037948/1.

Finite element modeling of borehole heat exchanger systems. Part 1. Fundamentals

NASA Astrophysics Data System (ADS)

Diersch, H.-J. G.; Bauer, D.; Heidemann, W.; Rühaak, W.; Schätzl, P.

2011-08-01

Single borehole heat exchanger (BHE) and arrays of BHE are modeled by using the finite element method. The first part of the paper derives the fundamental equations for BHE systems and their finite element representations, where the thermal exchange between the borehole components is modeled via thermal transfer relations. For this purpose improved relationships for thermal resistances and capacities of BHE are introduced. Pipe-to-grout thermal transfer possesses multiple grout points for double U-shape and single U-shape BHE to attain a more accurate modeling. The numerical solution of the final 3D problems is performed via a widely non-sequential (essentially non-iterative) coupling strategy for the BHE and porous medium discretization. Four types of vertical BHE are supported: double U-shape (2U) pipe, single U-shape (1U) pipe, coaxial pipe with annular (CXA) and centred (CXC) inlet. Two computational strategies are used: (1) The analytical BHE method based on Eskilson and Claesson's (1988) solution, (2) numerical BHE method based on Al-Khoury et al.'s (2005) solution. The second part of the paper focusses on BHE meshing aspects, the validation of BHE solutions and practical applications for borehole thermal energy store systems.

Heat Transfer to Surfaces of Finite Catalytic Activity in Frozen Dissociated Hypersonic Flow

NASA Technical Reports Server (NTRS)

Chung, Paul M.; Anderson, Aemer D.

1961-01-01

The heat transfer due to catalytic recombination of a partially dissociated diatomic gas along the surfaces of two-dimensional and axisymmetric bodies with finite catalytic efficiencies is studied analytically. An integral method is employed resulting in simple yet relatively complete solutions for the particular configurations considered. A closed form solution is derived which enables one to calculate atom mass-fraction distribution, therefore catalytic heat transfer distribution, along the surface of a flat plate in frozen compressible flow with and without transpiration. Numerical calculations are made to determine the atom mass-fraction distribution along an axisymmetric conical body with spherical nose in frozen hypersonic compressible flow. A simple solution based on a local similarity concept is found to be in good agreement with these numerical calculations. The conditions are given for which the local similarity solution is expected to be satisfactory. The limitations on the practical application of the analysis to the flight of the blunt bodies in the atmosphere are discussed. The use of boundary-layer theory and the assumption of frozen flow restrict application of the analysis to altitudes between about 150,000 and 250,000 feet.

Exact Solutions of Linear Reaction-Diffusion Processes on a Uniformly Growing Domain: Criteria for Successful Colonization

PubMed Central

Simpson, Matthew J

2015-01-01

Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction-diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction-diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction—diffusion process on 0

Exact solutions of linear reaction-diffusion processes on a uniformly growing domain: criteria for successful colonization.

PubMed

Simpson, Matthew J

2015-01-01

Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction-diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction-diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction-diffusion process on 0

An Extended Trajectory Mechanics Approach for Calculating the Path of a Pressure Transient: Derivation and Illustration

NASA Astrophysics Data System (ADS)

Vasco, D. W.

2018-04-01

Following an approach used in quantum dynamics, an exponential representation of the hydraulic head transforms the diffusion equation governing pressure propagation into an equivalent set of ordinary differential equations. Using a reservoir simulator to determine one set of dependent variables leaves a reduced set of equations for the path of a pressure transient. Unlike the current approach for computing the path of a transient, based on a high-frequency asymptotic solution, the trajectories resulting from this new formulation are valid for arbitrary spatial variations in aquifer properties. For a medium containing interfaces and layers with sharp boundaries, the trajectory mechanics approach produces paths that are compatible with travel time fields produced by a numerical simulator, while the asymptotic solution produces paths that bend too strongly into high permeability regions. The breakdown of the conventional asymptotic solution, due to the presence of sharp boundaries, has implications for model parameter sensitivity calculations and the solution of the inverse problem. For example, near an abrupt boundary, trajectories based on the asymptotic approach deviate significantly from regions of high sensitivity observed in numerical computations. In contrast, paths based on the new trajectory mechanics approach coincide with regions of maximum sensitivity to permeability changes.

Review of Thawing Time Prediction Models Depending
on Process Conditions and Product Characteristics

PubMed Central

Kluza, Franciszek; Spiess, Walter E. L.; Kozłowicz, Katarzyna

2016-01-01

Summary Determining thawing times of frozen foods is a challenging problem as the thermophysical properties of the product change during thawing. A number of calculation models and solutions have been developed. The proposed solutions range from relatively simple analytical equations based on a number of assumptions to a group of empirical approaches that sometimes require complex calculations. In this paper analytical, empirical and graphical models are presented and critically reviewed. The conditions of solution, limitations and possible applications of the models are discussed. The graphical and semi--graphical models are derived from numerical methods. Using the numerical methods is not always possible as running calculations takes time, whereas the specialized software and equipment are not always cheap. For these reasons, the application of analytical-empirical models is more useful for engineering. It is demonstrated that there is no simple, accurate and feasible analytical method for thawing time prediction. Consequently, simplified methods are needed for thawing time estimation of agricultural and food products. The review reveals the need for further improvement of the existing solutions or development of new ones that will enable accurate determination of thawing time within a wide range of practical conditions of heat transfer during processing. PMID:27904387

Rapid analytical assessment of the mechanical perturbations induced by non-isothermal injection into a subsurface formation.

NASA Astrophysics Data System (ADS)

De Simone, Silvia; Carrera, Jesús; María Gómez Castro, Berta

2016-04-01

Fluid injection into geological formations is required for several engineering operations, e.g. geothermal energy production, hydrocarbon production and storage, CO2 storage, wastewater disposal, etc. Non-isothermal fluid injection causes alterations of the pressure and temperature fields, which affect the mechanical stability of the reservoir. This coupled thermo-hydro-mechanical behavior has become a matter of special interest because of public concern about induced seismicity. The response is complex and its evaluation often requires numerical modeling. Nevertheless, analytical solutions are useful in improving our understanding of interactions, identifying the controlling parameters, testing codes and in providing a rapid assessment of the system response to an alteration. We present an easy-to-use solution to the transient advection-conduction heat transfer problem for parallel and radial flow. The solution is then applied to derive analytical expressions for hydraulic and thermal driven displacements and stresses. The validity is verified by comparison with numerical simulations and yields fairly accurate results. The solution is then used to illustrate some features of the poroelastic and thermoelastic response and, in particular, the sensitivity to the external mechanical constraints and to the reservoir dimension.

Stochastic study of solute transport in a nonstationary medium.

PubMed

Hu, Bill X

2006-01-01

A Lagrangian stochastic approach is applied to develop a method of moment for solute transport in a physically and chemically nonstationary medium. Stochastic governing equations for mean solute flux and solute covariance are analytically obtained in the first-order accuracy of log conductivity and/or chemical sorption variances and solved numerically using the finite-difference method. The developed method, the numerical method of moments (NMM), is used to predict radionuclide solute transport processes in the saturated zone below the Yucca Mountain project area. The mean, variance, and upper bound of the radionuclide mass flux through a control plane 5 km downstream of the footprint of the repository are calculated. According to their chemical sorption capacities, the various radionuclear chemicals are grouped as nonreactive, weakly sorbing, and strongly sorbing chemicals. The NMM method is used to study their transport processes and influence factors. To verify the method of moments, a Monte Carlo simulation is conducted for nonreactive chemical transport. Results indicate the results from the two methods are consistent, but the NMM method is computationally more efficient than the Monte Carlo method. This study adds to the ongoing debate in the literature on the effect of heterogeneity on solute transport prediction, especially on prediction uncertainty, by showing that the standard derivation of solute flux is larger than the mean solute flux even when the hydraulic conductivity within each geological layer is mild. This study provides a method that may become an efficient calculation tool for many environmental projects.

A Riemann solver for RANS

NASA Astrophysics Data System (ADS)

Chuvakhov, P. V.

2014-01-01

An exact expression for a system of both eigenvalues and right/left eigenvectors of a Jacobian matrix for a convective two-equation differential closure RANS operator split along a curvilinear coordinate is derived. It is shown by examples of numerical modeling of supersonic flows over a flat plate and a compression corner with separation that application of the exact system of eigenvalues and eigenvectors to the Roe approach for approximate solution of the Riemann problem gives rise to an increase in the convergence rate, better stability and higher accuracy of a steady-state solution in comparison with those in the case of an approximate system.

Analytical solutions for the motion of a charged particle in electric and magnetic fields via non-singular fractional derivatives

NASA Astrophysics Data System (ADS)

Morales-Delgado, V. F.; Gómez-Aguilar, J. F.; Taneco-Hernandez, M. A.

2017-12-01

In this work we propose fractional differential equations for the motion of a charged particle in electric, magnetic and electromagnetic fields. Exact solutions are obtained for the fractional differential equations by employing the Laplace transform method. The temporal fractional differential equations are considered in the Caputo-Fabrizio-Caputo and Atangana-Baleanu-Caputo sense. Application examples consider constant, ramp and harmonic fields. In addition, we present numerical results for different values of the fractional order. In all cases, when α = 1, we recover the standard electrodynamics.

Laplace-transform-based method to calculate back-reflected radiance from an isotropically scattering half-space

NASA Astrophysics Data System (ADS)

Rinzema, K.; Hoenders, B. J.; Ferwerda, H. A.

1997-07-01

We present a method to determine the back-reflected radiance from an isotropically scattering half-space with matched boundary. This method has the advantage that it leads very quickly to the relevant equations, the numerical solution of which is also quite easy. Essentially, the method is derived from a mathematical criterion that effectively forbids the existence of solutions to the transport equation which grow exponentially as one moves away from the surface and deeper into the medium. Preliminary calculations for infinitely wide beams yield results which agree very well with what is found in the literature.

A reaction-diffusion malaria model with seasonality and incubation period.

PubMed

Bai, Zhenguo; Peng, Rui; Zhao, Xiao-Qiang

2018-07-01

In this paper, we propose a time-periodic reaction-diffusion model which incorporates seasonality, spatial heterogeneity and the extrinsic incubation period (EIP) of the parasite. The basic reproduction number [Formula: see text] is derived, and it is shown that the disease-free periodic solution is globally attractive if [Formula: see text], while there is an endemic periodic solution and the disease is uniformly persistent if [Formula: see text]. Numerical simulations indicate that prolonging the EIP may be helpful in the disease control, while spatial heterogeneity of the disease transmission coefficient may increase the disease burden.

Non-oscillatory and non-diffusive solution of convection problems by the iteratively reweighted least-squares finite element method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan

1993-01-01

A comparative description is presented for the least-squares FEM (LSFEM) for 2D steady-state pure convection problems. In addition to exhibiting better control of the streamline derivative than the streamline upwinding Petrov-Galerkin method, numerical convergence rates are obtained which show the LSFEM to be virtually optimal. The LSFEM is used as a framework for an iteratively reweighted LSFEM yielding nonoscillatory and nondiffusive solutions for problems with contact discontinuities; this method is shown to convect contact discontinuities without error when using triangular and bilinear elements.

Reduced and simplified chemical kinetics for air dissociation using Computational Singular Perturbation

NASA Technical Reports Server (NTRS)

Goussis, D. A.; Lam, S. H.; Gnoffo, P. A.

1990-01-01

The Computational Singular Perturbation CSP methods is employed (1) in the modeling of a homogeneous isothermal reacting system and (2) in the numerical simulation of the chemical reactions in a hypersonic flowfield. Reduced and simplified mechanisms are constructed. The solutions obtained on the basis of these approximate mechanisms are shown to be in very good agreement with the exact solution based on the full mechanism. Physically meaningful approximations are derived. It is demonstrated that the deduction of these approximations from CSP is independent of the complexity of the problem and requires no intuition or experience in chemical kinetics.

A study of various methods for calculating locations of lightning events

NASA Technical Reports Server (NTRS)

Cannon, John R.

1995-01-01

This article reports on the results of numerical experiments on finding the location of lightning events using different numerical methods. The methods include linear least squares, nonlinear least squares, statistical estimations, cluster analysis and angular filters and combinations of such techniques. The experiments involved investigations of methods for excluding fake solutions which are solutions that appear to be reasonable but are in fact several kilometers distant from the actual location. Some of the conclusions derived from the study are that bad data produces fakes, that no fool-proof method of excluding fakes was found, that a short base-line interferometer under development at Kennedy Space Center to measure the direction cosines of an event shows promise as a filter for excluding fakes. The experiments generated a number of open questions, some of which are discussed at the end of the report.

Turbulence and deterministic chaos. [computational fluid dynamics

NASA Technical Reports Server (NTRS)

Deissler, Robert G.

1992-01-01

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low Reynolds number fully developed turbulence are compared. Several flows are noted: fully chaotic, complex periodic, weakly chaotic, simple periodic, and fixed-point. Of these, only fully chaotic is classified as turbulent. Besides the sustained flows, a flow which decays as it becomes turbulent is examined. For the finest grid, 128(exp 3) points, the spatial resolution appears to be quite good. As a final note, the variation of the velocity derivatives skewness of a Navier-Stokes flow as the Reynolds number goes to zero is calculated numerically. The value of the skewness is shown to become small at low Reynolds numbers, in agreement with intuitive arguments that nonlinear terms should be negligible.

Finite element solution of optimal control problems with inequality constraints

NASA Technical Reports Server (NTRS)

Bless, Robert R.; Hodges, Dewey H.

1990-01-01

A finite-element method based on a weak Hamiltonian form of the necessary conditions is summarized for optimal control problems. Very crude shape functions (so simple that element numerical quadrature is not necessary) can be used to develop an efficient procedure for obtaining candidate solutions (i.e., those which satisfy all the necessary conditions) even for highly nonlinear problems. An extension of the formulation allowing for discontinuities in the states and derivatives of the states is given. A theory that includes control inequality constraints is fully developed. An advanced launch vehicle (ALV) model is presented. The model involves staging and control constraints, thus demonstrating the full power of the weak formulation to date. Numerical results are presented along with total elapsed computer time required to obtain the results. The speed and accuracy in obtaining the results make this method a strong candidate for a real-time guidance algorithm.

Temporal and spatial foliations of spacetimes.

NASA Astrophysics Data System (ADS)

Herold, H.

For the solution of initial-value problems in numerical relativity usually the (3+1) splitting of Einstein's equations is employed. An important part of this splitting is the choice of the temporal gauge condition. In order to estimate the quality of time-evolution schemes, different time slicings of given well-known spherically symmetric spacetimes have been studied. Besides the maximal slicing condition the harmonic slicing prescription has been used to calculate temporal foliations of the Schwarzschild and the Oppenheimer-Snyder spacetime. Additionally, the author has studied a recently proposed, geometrically motivated spatial gauge condition, which is defined by considering the foliations of the three-dimensional space-like hypersurfaces by 2-surfaces of constant mean extrinsic curvature. Apart from the equations for the shift vector, which can be derived for this gauge condition, he has investigated such spatial foliations for well-known stationary axially symmetric spacetimes, namely for the Kerr metric and for numerically determined solutions for rapidly rotating neutron stars.

Dynamics and elastic interactions of the discrete multi-dark soliton solutions for the Kaup-Newell lattice equation

NASA Astrophysics Data System (ADS)

Liu, Nan; Wen, Xiao-Yong

2018-03-01

Under consideration in this paper is the Kaup-Newell (KN) lattice equation which is an integrable discretization of the KN equation. Infinitely, many conservation laws and discrete N-fold Darboux transformation (DT) for this system are constructed and established based on its Lax representation. Via the resulting N-fold DT, the discrete multi-dark soliton solutions in terms of determinants are derived from non-vanishing background. Propagation and elastic interaction structures of such solitons are shown graphically. Overtaking interaction phenomena between/among the two, three and four solitons are discussed. Numerical simulations are used to explore their dynamical behaviors of such multi-dark solitons. Numerical results show that their evolutions are stable against a small noise. Results in this paper might be helpful for understanding the propagation of nonlinear Alfvén waves in plasmas.

FEAST fundamental framework for electronic structure calculations: Reformulation and solution of the muffin-tin problem

NASA Astrophysics Data System (ADS)

Levin, Alan R.; Zhang, Deyin; Polizzi, Eric

2012-11-01

In a recent article Polizzi (2009) [15], the FEAST algorithm has been presented as a general purpose eigenvalue solver which is ideally suited for addressing the numerical challenges in electronic structure calculations. Here, FEAST is presented beyond the “black-box” solver as a fundamental modeling framework which can naturally address the original numerical complexity of the electronic structure problem as formulated by Slater in 1937 [3]. The non-linear eigenvalue problem arising from the muffin-tin decomposition of the real-space domain is first derived and then reformulated to be solved exactly within the FEAST framework. This new framework is presented as a fundamental and practical solution for performing both accurate and scalable electronic structure calculations, bypassing the various issues of using traditional approaches such as linearization and pseudopotential techniques. A finite element implementation of this FEAST framework along with simulation results for various molecular systems is also presented and discussed.

Efficient field-theoretic simulation of polymer solutions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Villet, Michael C.; Fredrickson, Glenn H., E-mail: ghf@mrl.ucsb.edu; Department of Materials, University of California, Santa Barbara, California 93106

2014-12-14

We present several developments that facilitate the efficient field-theoretic simulation of polymers by complex Langevin sampling. A regularization scheme using finite Gaussian excluded volume interactions is used to derive a polymer solution model that appears free of ultraviolet divergences and hence is well-suited for lattice-discretized field theoretic simulation. We show that such models can exhibit ultraviolet sensitivity, a numerical pathology that dramatically increases sampling error in the continuum lattice limit, and further show that this pathology can be eliminated by appropriate model reformulation by variable transformation. We present an exponential time differencing algorithm for integrating complex Langevin equations for fieldmore » theoretic simulation, and show that the algorithm exhibits excellent accuracy and stability properties for our regularized polymer model. These developments collectively enable substantially more efficient field-theoretic simulation of polymers, and illustrate the importance of simultaneously addressing analytical and numerical pathologies when implementing such computations.« less

Water and chloride transport in a fine-textured soil in a feedlot pen.

PubMed

Veizaga, E A; Rodríguez, L; Ocampo, C J

2015-11-01

Cattle feeding in feedlot pens produces large amounts of manure and animal urine. Manure solutions resulting from surface runoff are composed of numerous chemical constituents whose leaching causes salinization of the soil profile. There is a relatively large number of studies on preferential flow characterization and modeling in clayed soils. However, research on water flow and solute transport derived from cattle feeding operations in fine-textured soils under naturally occurring precipitation events is less frequent. A field monitoring and modeling investigation was conducted at two plots on a fine-textured soil near a feedlot pen in Argentina to assess the potential of solute leaching into the soil profile. Soil pressure head and chloride concentration of the soil solution were used in combination with HYDRUS-1D numerical model to simulate water flow and chloride transport resorting to the concept of mobile/immobile-MIM water for solute transport. Pressure head sensors located at different depths registered a rapid response to precipitation suggesting the occurrence of preferential flow-paths for infiltrating water. Cracks and small fissures were documented at the field site where the % silt and % clay combined is around 94%. Chloride content increased with depth for various soil pressure head conditions, although a dilution process was observed as precipitation increased. The MIM approach improved numerical results at one of the tested sites where the development of cracks and macropores is likely, obtaining a more dynamic response in comparison with the advection-dispersion equation. Copyright © 2015 Elsevier B.V. All rights reserved.

Geopotential coefficient determination and the gravimetric boundary value problem: A new approach

NASA Technical Reports Server (NTRS)

Sjoeberg, Lars E.

1989-01-01

New integral formulas to determine geopotential coefficients from terrestrial gravity and satellite altimetry data are given. The formulas are based on the integration of data over the non-spherical surface of the Earth. The effect of the topography to low degrees and orders of coefficients is estimated numerically. Formulas for the solution of the gravimetric boundary value problem are derived.

Updates to Simulation of a Single-Element Lean-Direct Injection Combustor Using a Polyhedral Mesh Derived From Hanging-Node Elements

NASA Technical Reports Server (NTRS)

Wey, Changju Thomas; Liu, Nan-Suey

2014-01-01

This paper summarizes the procedures of inserting a thin-layer mesh to existing inviscid polyhedral mesh either with or without hanging-node elements as well as presents sample results from its applications to the numerical solution of a single-element LDI combustor using a releasable edition of the National Combustion Code (NCC).

Updates to Simulation of a Single-Element Lean-Direct Injection Combustor Using a Polyhedral Mesh Derived from Hanging-Node Elements

NASA Technical Reports Server (NTRS)

Wey, Thomas; Liu, Nan-Suey

2014-01-01

This paper summarizes the procedures of inserting a thin-layer mesh to existing inviscid polyhedral mesh either with or without hanging-node elements as well as presents sample results from its applications to the numerical solution of a single-element LDI combustor using a releasable edition of the National Combustion Code (NCC).

Flux Jacobian Matrices For Equilibrium Real Gases

NASA Technical Reports Server (NTRS)

Vinokur, Marcel

1990-01-01

Improved formulation includes generalized Roe average and extension to three dimensions. Flux Jacobian matrices derived for use in numerical solutions of conservation-law differential equations of inviscid flows of ideal gases extended to real gases. Real-gas formulation of these matrices retains simplifying assumptions of thermodynamic and chemical equilibrium, but adds effects of vibrational excitation, dissociation, and ionization of gas molecules via general equation of state.

Effects of radial envelope modulations on the collisionless trapped-electron mode in tokamak plasmas

NASA Astrophysics Data System (ADS)

Chen, Hao-Tian; Chen, Liu

2018-05-01

Adopting the ballooning-mode representation and including the effects of radial envelope modulations, we have derived the corresponding linear eigenmode equation for the collisionless trapped-electron mode in tokamak plasmas. Numerical solutions of the eigenmode equation indicate that finite radial envelope modulations can affect the linear stability properties both quantitatively and qualitatively via the significant modifications in the corresponding eigenmode structures.

Analytic Approximations to the Free Boundary and Multi-dimensional Problems in Financial Derivatives Pricing

NASA Astrophysics Data System (ADS)

Lau, Chun Sing

This thesis studies two types of problems in financial derivatives pricing. The first type is the free boundary problem, which can be formulated as a partial differential equation (PDE) subject to a set of free boundary condition. Although the functional form of the free boundary condition is given explicitly, the location of the free boundary is unknown and can only be determined implicitly by imposing continuity conditions on the solution. Two specific problems are studied in details, namely the valuation of fixed-rate mortgages and CEV American options. The second type is the multi-dimensional problem, which involves multiple correlated stochastic variables and their governing PDE. One typical problem we focus on is the valuation of basket-spread options, whose underlying asset prices are driven by correlated geometric Brownian motions (GBMs). Analytic approximate solutions are derived for each of these three problems. For each of the two free boundary problems, we propose a parametric moving boundary to approximate the unknown free boundary, so that the original problem transforms into a moving boundary problem which can be solved analytically. The governing parameter of the moving boundary is determined by imposing the first derivative continuity condition on the solution. The analytic form of the solution allows the price and the hedging parameters to be computed very efficiently. When compared against the benchmark finite-difference method, the computational time is significantly reduced without compromising the accuracy. The multi-stage scheme further allows the approximate results to systematically converge to the benchmark results as one recasts the moving boundary into a piecewise smooth continuous function. For the multi-dimensional problem, we generalize the Kirk (1995) approximate two-asset spread option formula to the case of multi-asset basket-spread option. Since the final formula is in closed form, all the hedging parameters can also be derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark prices obtained by numerical integration or Monte Carlo simulation. By exploiting an explicit relationship between the option price and the underlying probability distribution, we further derive an approximate distribution function for the general basket-spread variable. It can be used to approximate the transition probability distribution of any linear combination of correlated GBMs. Finally, an implicit perturbation is applied to reduce the pricing errors by factors of up to 100. When compared against the existing methods, the basket-spread option formula coupled with the implicit perturbation turns out to be one of the most robust and accurate approximation methods.

Optimal implicit 2-D finite differences to model wave propagation in poroelastic media

NASA Astrophysics Data System (ADS)

Itzá, Reymundo; Iturrarán-Viveros, Ursula; Parra, Jorge O.

2016-08-01

Numerical modeling of seismic waves in heterogeneous porous reservoir rocks is an important tool for the interpretation of seismic surveys in reservoir engineering. We apply globally optimal implicit staggered-grid finite differences (FD) to model 2-D wave propagation in heterogeneous poroelastic media at a low-frequency range (<10 kHz). We validate the numerical solution by comparing it to an analytical-transient solution obtaining clear seismic wavefields including fast P and slow P and S waves (for a porous media saturated with fluid). The numerical dispersion and stability conditions are derived using von Neumann analysis, showing that over a wide range of porous materials the Courant condition governs the stability and this optimal implicit scheme improves the stability of explicit schemes. High-order explicit FD can be replaced by some lower order optimal implicit FD so computational cost will not be as expensive while maintaining the accuracy. Here, we compute weights for the optimal implicit FD scheme to attain an accuracy of γ = 10-8. The implicit spatial differentiation involves solving tridiagonal linear systems of equations through Thomas' algorithm.

Numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains.

PubMed

Li, Hongwei; Guo, Yue

2017-12-01

The numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains is considered by applying the artificial boundary method in this paper. In order to design the local absorbing boundary conditions for the coupled nonlinear Schrödinger equations, we generalize the unified approach previously proposed [J. Zhang et al., Phys. Rev. E 78, 026709 (2008)PLEEE81539-375510.1103/PhysRevE.78.026709]. Based on the methodology underlying the unified approach, the original problem is split into two parts, linear and nonlinear terms, and we then achieve a one-way operator to approximate the linear term to make the wave out-going, and finally we combine the one-way operator with the nonlinear term to derive the local absorbing boundary conditions. Then we reduce the original problem into an initial boundary value problem on the bounded domain, which can be solved by the finite difference method. The stability of the reduced problem is also analyzed by introducing some auxiliary variables. Ample numerical examples are presented to verify the accuracy and effectiveness of our proposed method.

Numerical solution of a conspicuous consumption model with constant control delay☆

PubMed Central

Huschto, Tony; Feichtinger, Gustav; Hartl, Richard F.; Kort, Peter M.; Sager, Sebastian; Seidl, Andrea

2011-01-01

We derive optimal pricing strategies for conspicuous consumption products in periods of recession. To that end, we formulate and investigate a two-stage economic optimal control problem that takes uncertainty of the recession period length and delay effects of the pricing strategy into account. This non-standard optimal control problem is difficult to solve analytically, and solutions depend on the variable model parameters. Therefore, we use a numerical result-driven approach. We propose a structure-exploiting direct method for optimal control to solve this challenging optimization problem. In particular, we discretize the uncertainties in the model formulation by using scenario trees and target the control delays by introduction of slack control functions. Numerical results illustrate the validity of our approach and show the impact of uncertainties and delay effects on optimal economic strategies. During the recession, delayed optimal prices are higher than the non-delayed ones. In the normal economic period, however, this effect is reversed and optimal prices with a delayed impact are smaller compared to the non-delayed case. PMID:22267871

A New Formulation of Time Domain Boundary Integral Equation for Acoustic Wave Scattering in the Presence of a Uniform Mean Flow

NASA Technical Reports Server (NTRS)

Hu, Fang; Pizzo, Michelle E.; Nark, Douglas M.

2017-01-01

It has been well-known that under the assumption of a constant uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation, in both the time domain and the frequency domain. Compared with solving partial differential equations, numerical methods based on the boundary integral equation have the advantage of a reduced spatial dimension and, hence, requiring only a surface mesh. However, the constant uniform mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the uniform mean flow. In this paper, we argue that the proper boundary condition for the acoustic wave should not have its normal velocity be zero everywhere on the solid surfaces, as has been applied in the literature. A careful study of the acoustic energy conservation equation is presented that shows such a boundary condition in fact leads to erroneous source or sink points on solid surfaces not aligned with the mean flow. A new solid wall boundary condition is proposed that conserves the acoustic energy and a new time domain boundary integral equation is derived. In addition to conserving the acoustic energy, another significant advantage of the new equation is that it is considerably simpler than previous formulations. In particular, tangential derivatives of the solution on the solid surfaces are no longer needed in the new formulation, which greatly simplifies numerical implementation. Furthermore, stabilization of the new integral equation by Burton-Miller type reformulation is presented. The stability of the new formulation is studied theoretically as well as numerically by an eigenvalue analysis. Numerical solutions are also presented that demonstrate the stability of the new formulation.

Modeling the controllable pH-responsive swelling and pore size of networked alginate based biomaterials.

PubMed

Chan, Ariel W; Neufeld, Ronald J

2009-10-01

Semisynthetic network alginate polymer (SNAP), synthesized by acetalization of linear alginate with di-aldehyde, is a pH-responsive tetrafunctionally linked 3D gel network, and has potential application in oral delivery of protein therapeutics and active biologicals, and as tissue bioscaffold for regenerative medicine. A constitutive polyelectrolyte gel model based on non-Gaussian polymer elasticity, Flory-Huggins liquid lattice theory, and non-ideal Donnan membrane equilibria was derived, to describe SNAP gel swelling in dilute and ionic solutions containing uni-univalent, uni-bivalent, bi-univalent or bi-bi-valent electrolyte solutions. Flory-Huggins interaction parameters as a function of ionic strength and characteristic ratio of alginates of various molecular weights were determined experimentally to numerically predict SNAP hydrogel swelling. SNAP hydrogel swells pronouncedly to 1000 times in dilute solution, compared to its compact polymer volume, while behaving as a neutral polymer with limited swelling in high ionic strength or low pH solutions. The derived model accurately describes the pH-responsive swelling of SNAP hydrogel in acid and alkaline solutions of wide range of ionic strength. The pore sizes of the synthesized SNAP hydrogels of various crosslink densities were estimated from the derived model to be in the range of 30-450 nm which were comparable to that measured by thermoporometry, and diffusion of bovine serum albumin. The derived equilibrium swelling model can characterize hydrogel structure such as molecular weight between crosslinks and crosslinking density, or can be used as predictive model for swelling, pore size and mechanical properties if gel structural information is known, and can potentially be applied to other point-link network polyelectrolytes such as hyaluronic acid gel.

Nonlocal electrical diffusion equation

NASA Astrophysics Data System (ADS)

Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.

2016-07-01

In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0<β≤1 and for the time domain is 0<γ≤2. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.

Differential geometry based solvation model I: Eulerian formulation

NASA Astrophysics Data System (ADS)

Chen, Zhan; Baker, Nathan A.; Wei, G. W.

2010-11-01

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the solvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By optimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second-order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature.

Differential geometry based solvation model I: Eulerian formulation

PubMed Central

Chen, Zhan; Baker, Nathan A.; Wei, G. W.

2010-01-01

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature. PMID:20938489

ACCURATE SOLUTION AND GRADIENT COMPUTATION FOR ELLIPTIC INTERFACE PROBLEMS WITH VARIABLE COEFFICIENTS

PubMed Central

LI, ZHILIN; JI, HAIFENG; CHEN, XIAOHONG

2016-01-01

A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is not only to get a second order accurate solution but also a second order accurate gradient from each side of the interface. The key of the new method is to introduce the jump in the normal derivative of the solution as an augmented variable and re-write the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivatives jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points near or on the interface so that the resulting coefficient matrix is an M-matrix. A multi-grid solver is used to solve the linear system of equations and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface has also been proved in this paper. Numerical examples for general elliptic interface problems have confirmed the theoretical analysis and efficiency of the new method. PMID:28983130

Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions

NASA Astrophysics Data System (ADS)

Barré, J.; Carrillo, J. A.; Degond, P.; Peurichard, D.; Zatorska, E.

2018-02-01

We provide a numerical study of the macroscopic model of Barré et al. (Multiscale Model Simul, 2017, to appear) derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodeling process is very fast, the macroscopic model takes the form of a single aggregation-diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the one-dimensional case, we show numerically that the stationary solutions of the microscopic model undergo the same phase transitions and bifurcation types as the macroscopic model. In the two-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement with the predicted theoretical values. This study provides a partial validation of the formal derivation of the macroscopic model from a microscopic formulation and shows that the former is a consistent approximation of an underlying particle dynamics, making it a powerful tool for the modeling of dynamical networks at a large scale.

Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator

NASA Astrophysics Data System (ADS)

Owolabi, Kolade M.

2018-03-01

In this work, we are concerned with the solution of non-integer space-fractional reaction-diffusion equations with the Riemann-Liouville space-fractional derivative in high dimensions. We approximate the Riemann-Liouville derivative with the Fourier transform method and advance the resulting system in time with any time-stepping solver. In the numerical experiments, we expect the travelling wave to arise from the given initial condition on the computational domain (-∞, ∞), which we terminate in the numerical experiments with a large but truncated value of L. It is necessary to choose L large enough to allow the waves to have enough space to distribute. Experimental results in high dimensions on the space-fractional reaction-diffusion models with applications to biological models (Fisher and Allen-Cahn equations) are considered. Simulation results reveal that fractional reaction-diffusion equations can give rise to a range of physical phenomena when compared to non-integer-order cases. As a result, most meaningful and practical situations are found to be modelled with the concept of fractional calculus.

Finding the numerical compensation in multiple criteria decision-making problems under fuzzy environment

NASA Astrophysics Data System (ADS)

Gupta, Mahima; Mohanty, B. K.

2017-04-01

In this paper, we have developed a methodology to derive the level of compensation numerically in multiple criteria decision-making (MCDM) problems under fuzzy environment. The degree of compensation is dependent on the tranquility and anxiety level experienced by the decision-maker while taking the decision. Higher tranquility leads to the higher realisation of the compensation whereas the increased level of anxiety reduces the amount of compensation in the decision process. This work determines the level of tranquility (or anxiety) using the concept of fuzzy sets and its various level sets. The concepts of indexing of fuzzy numbers, the risk barriers and the tranquility level of the decision-maker are used to derive his/her risk prone or risk averse attitude of decision-maker in each criterion. The aggregation of the risk levels in each criterion gives us the amount of compensation in the entire MCDM problem. Inclusion of the compensation leads us to model the MCDM problem as binary integer programming problem (BIP). The solution to BIP gives us the compensatory decision to MCDM. The proposed methodology is illustrated through a numerical example.

Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions.

PubMed

Barré, J; Carrillo, J A; Degond, P; Peurichard, D; Zatorska, E

2018-01-01

We provide a numerical study of the macroscopic model of Barré et al. (Multiscale Model Simul, 2017, to appear) derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodeling process is very fast, the macroscopic model takes the form of a single aggregation-diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the one-dimensional case, we show numerically that the stationary solutions of the microscopic model undergo the same phase transitions and bifurcation types as the macroscopic model. In the two-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement with the predicted theoretical values. This study provides a partial validation of the formal derivation of the macroscopic model from a microscopic formulation and shows that the former is a consistent approximation of an underlying particle dynamics, making it a powerful tool for the modeling of dynamical networks at a large scale.

Numerical solutions of ideal quantum gas dynamical flows governed by semiclassical ellipsoidal-statistical distribution

PubMed Central

Yang, Jaw-Yen; Yan, Chih-Yuan; Diaz, Manuel; Huang, Juan-Chen; Li, Zhihui; Zhang, Hanxin

2014-01-01

The ideal quantum gas dynamics as manifested by the semiclassical ellipsoidal-statistical (ES) equilibrium distribution derived in Wu et al. (Wu et al. 2012 Proc. R. Soc. A 468, 1799–1823 (doi:10.1098/rspa.2011.0673)) is numerically studied for particles of three statistics. This anisotropic ES equilibrium distribution was derived using the maximum entropy principle and conserves the mass, momentum and energy, but differs from the standard Fermi–Dirac or Bose–Einstein distribution. The present numerical method combines the discrete velocity (or momentum) ordinate method in momentum space and the high-resolution shock-capturing method in physical space. A decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. Computations of two-dimensional Riemann problems are presented, and various contours of the quantities unique to this ES model are illustrated. The main flow features, such as shock waves, expansion waves and slip lines and their complex nonlinear interactions, are depicted and found to be consistent with existing calculations for a classical gas. PMID:24399919

Numerical Simulation of Subsonic and Transonic Propeller Flow. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Snyder, Aaron

1988-01-01

The numerical simulation of 3-D transonic flow about a system of propeller blades is investigated. In particular, it is shown that the use of helical coordinates significantly simplifies the form of the governing equation when the propeller system is assumed to be surrounded by an irrotational flow field of an inviscid fluid. The unsteady small disturbance equation, valid for lightly loaded blades and expressed in helical coordinates, is derived from the general blade-fixed potential equation, given for an arbitrary coordinate system. The use of a coordinate system which inherently adapts to the mean flow results in a disturbance equation requiring relatively few terms to accurately model the physics of the flow. Furthermore, the helical coordinate system presented here is novel in that it is periodic in the circumferential direction while, simultaneously, maintaining orthogonal properties at the mean blade locations. The periodic characteristic allows a complete cascade of blades to be treated, and the orthogonality property affords straightforward treatment of blade boundary conditions. An ADI numerical scheme is used to compute the solution of the steady flow as an asymptotic limit of an unsteady flow. As an example of the method, solutions are presented for subsonic and transonic flow about a 5 percent thick bicircular arc blade of an 8-bladed cascade. Both high and low advance ratio cases are computed and include a lifting as well as nonlifting cases. The nonlifting solutions obtained are compared to solutions from a Euler code.

Revealing Numerical Solutions of a Differential Equation

ERIC Educational Resources Information Center

Glaister, P.

2006-01-01

In this article, the author considers a student exercise that involves determining the exact and numerical solutions of a particular differential equation. He shows how a typical student solution is at variance with a numerical solution, suggesting that the numerical solution is incorrect. However, further investigation shows that this numerical…

Shallow-water sloshing in a moving vessel with variable cross-section and wetting-drying using an extension of George's well-balanced finite volume solver

NASA Astrophysics Data System (ADS)

Alemi Ardakani, Hamid; Bridges, Thomas J.; Turner, Matthew R.

2016-06-01

A class of augmented approximate Riemann solvers due to George (2008) [12] is extended to solve the shallow-water equations in a moving vessel with variable bottom topography and variable cross-section with wetting and drying. A class of Roe-type upwind solvers for the system of balance laws is derived which respects the steady-state solutions. The numerical solutions of the new adapted augmented f-wave solvers are validated against the Roe-type solvers. The theory is extended to solve the shallow-water flows in moving vessels with arbitrary cross-section with influx-efflux boundary conditions motivated by the shallow-water sloshing in the ocean wave energy converter (WEC) proposed by Offshore Wave Energy Ltd. (OWEL) [1]. A fractional step approach is used to handle the time-dependent forcing functions. The numerical solutions are compared to an extended new Roe-type solver for the system of balance laws with a time-dependent source function. The shallow-water sloshing finite volume solver can be coupled to a Runge-Kutta integrator for the vessel motion.

Nonequilibrium electrophoresis of an ion-selective microgranule for weak and moderate external electric fields

NASA Astrophysics Data System (ADS)

Frants, E. A.; Ganchenko, G. S.; Shelistov, V. S.; Amiroudine, S.; Demekhin, E. A.

2018-02-01

Electrokinetics and the movement of charge-selective micro-granules in an electrolyte solution under the influence of an external electric field are investigated theoretically. Straightforward perturbation analysis is applied to a thin electric double layer and a weak external field, while a numerical solution is used for moderate electric fields. The asymptotic solution enables the determination of the salt concentration, electric charge distribution, and electro-osmotic velocity fields. It may also be used to obtain a simple analytical formula for the electrophoretic velocity in the case of quasi-equilibrium electrophoresis (electrophoresis of the first kind). This formula differs from the famous Helmholtz-Smoluchowski relation, which applies to dielectric microparticles, but not to ion-selective granules. Numerical calculations are used to validate the derived formula for weak external electric fields, but for moderate fields, nonlinear effects lead to a significant increase in electrophoretic mobility and to a transition from quasi-equilibrium electrophoresis of the first kind to nonequilibrium electrophoresis of the second kind. Theoretical results are successfully compared with experimental data.

Existence and global exponential stability of periodic solution of memristor-based BAM neural networks with time-varying delays.

PubMed

Li, Hongfei; Jiang, Haijun; Hu, Cheng

2016-03-01

In this paper, we investigate a class of memristor-based BAM neural networks with time-varying delays. Under the framework of Filippov solutions, boundedness and ultimate boundedness of solutions of memristor-based BAM neural networks are guaranteed by Chain rule and inequalities technique. Moreover, a new method involving Yoshizawa-like theorem is favorably employed to acquire the existence of periodic solution. By applying the theory of set-valued maps and functional differential inclusions, an available Lyapunov functional and some new testable algebraic criteria are derived for ensuring the uniqueness and global exponential stability of periodic solution of memristor-based BAM neural networks. The obtained results expand and complement some previous work on memristor-based BAM neural networks. Finally, a numerical example is provided to show the applicability and effectiveness of our theoretical results. Copyright © 2015 Elsevier Ltd. All rights reserved.

An hp-adaptivity and error estimation for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Bey, Kim S.

1995-01-01

This paper presents an hp-adaptive discontinuous Galerkin method for linear hyperbolic conservation laws. A priori and a posteriori error estimates are derived in mesh-dependent norms which reflect the dependence of the approximate solution on the element size (h) and the degree (p) of the local polynomial approximation. The a posteriori error estimate, based on the element residual method, provides bounds on the actual global error in the approximate solution. The adaptive strategy is designed to deliver an approximate solution with the specified level of error in three steps. The a posteriori estimate is used to assess the accuracy of a given approximate solution and the a priori estimate is used to predict the mesh refinements and polynomial enrichment needed to deliver the desired solution. Numerical examples demonstrate the reliability of the a posteriori error estimates and the effectiveness of the hp-adaptive strategy.

A General Solution for Groundwater Flow in Estuarine Leaky Aquifer System with Considering Aquifer Anisotropy

NASA Astrophysics Data System (ADS)

Chen, Po-Chia; Chuang, Mo-Hsiung; Tan, Yih-Chi

2014-05-01

In recent years the urban and industrial developments near the coastal area are rapid and therefore the associated population grows dramatically. More and more water demand for human activities, agriculture irrigation, and aquaculture relies on heavy pumping in coastal area. The decline of groundwater table may result in the problems of seawater intrusion and/or land subsidence. Since the 1950s, numerous studies focused on the effect of tidal fluctuation on the groundwater flow in the coastal area. Many studies concentrated on the developments of one-dimensional (1D) and two-dimensional (2D) analytical solutions describing the tide-induced head fluctuations. For example, Jacob (1950) derived an analytical solution of 1D groundwater flow in a confined aquifer with a boundary condition subject to sinusoidal oscillation. Jiao and Tang (1999) derived a 1D analytical solution of a leaky confined aquifer by considered a constant groundwater head in the overlying unconfined aquifer. Jeng et al. (2002) studied the tidal propagation in a coupled unconfined and confined costal aquifer system. Sun (1997) presented a 2D solution for groundwater response to tidal loading in an estuary. Tang and Jiao (2001) derived a 2D analytical solution in a leaky confined aquifer system near open tidal water. This study aims at developing a general analytical solution describing the head fluctuations in a 2D estuarine aquifer system consisted of an unconfined aquifer, a confined aquifer, and an aquitard between them. Both the confined and unconfined aquifers are considered to be anisotropic. The predicted head fluctuations from this solution will compare with the simulation results from the MODFLOW program. In addition, the solutions mentioned above will be shown to be special cases of the present solution. Some hypothetical cases regarding the head fluctuation in costal aquifers will be made to investigate the dynamic effects of water table fluctuation, hydrogeological conditions, and characteristics of soil on the groundwater level fluctuations in the 2D estuarine leaky aquifer system.

Numerical solution of system of boundary value problems using B-spline with free parameter

NASA Astrophysics Data System (ADS)

Gupta, Yogesh

2017-01-01

This paper deals with method of B-spline solution for a system of boundary value problems. The differential equations are useful in various fields of science and engineering. Some interesting real life problems involve more than one unknown function. These result in system of simultaneous differential equations. Such systems have been applied to many problems in mathematics, physics, engineering etc. In present paper, B-spline and B-spline with free parameter methods for the solution of a linear system of second-order boundary value problems are presented. The methods utilize the values of cubic B-spline and its derivatives at nodal points together with the equations of the given system and boundary conditions, ensuing into the linear matrix equation.

Displacement potential solution of a guided deep beam of composite materials under symmetric three-point bending

NASA Astrophysics Data System (ADS)

Rahman, M. Muzibur; Ahmad, S. Reaz

2017-12-01

An analytical investigation of elastic fields for a guided deep beam of orthotropic composite material having three point symmetric bending is carried out using displacement potential boundary modeling approach. Here, the formulation is developed as a single function of space variables defined in terms of displacement components, which has to satisfy the mixed type of boundary conditions. The relevant displacement and stress components are derived into infinite series using Fourier integral along with suitable polynomials coincided with boundary conditions. The results are presented mainly in the form of graphs and verified with finite element solutions using ANSYS. This study shows that the analytical and numerical solutions are in good agreement and thus enhances reliability of the displacement potential approach.

Homoclinic snaking in the discrete Swift-Hohenberg equation

NASA Astrophysics Data System (ADS)

Kusdiantara, R.; Susanto, H.

2017-12-01

We consider the discrete Swift-Hohenberg equation with cubic and quintic nonlinearity, obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences. We investigate the discretization effect on the bifurcation behavior, where we identify three regions of the coupling parameter, i.e., strong, weak, and intermediate coupling. Within the regions, the discrete Swift-Hohenberg equation behaves either similarly or differently from the continuum limit. In the intermediate coupling region, multiple Maxwell points can occur for the periodic solutions and may cause irregular snaking and isolas. Numerical continuation is used to obtain and analyze localized and periodic solutions for each case. Theoretical analysis for the snaking and stability of the corresponding solutions is provided in the weak coupling region.

On computing the geoelastic response to a disk load

NASA Astrophysics Data System (ADS)

Bevis, M.; Melini, D.; Spada, G.

2016-06-01

We review the theory of the Earth's elastic and gravitational response to a surface disk load. The solutions for displacement of the surface and the geoid are developed using expansions of Legendre polynomials, their derivatives and the load Love numbers. We provide a MATLAB function called diskload that computes the solutions for both uncompensated and compensated disk loads. In order to numerically implement the Legendre expansions, it is necessary to choose a harmonic degree, nmax, at which to truncate the series used to construct the solutions. We present a rule of thumb (ROT) for choosing an appropriate value of nmax, describe the consequences of truncating the expansions prematurely and provide a means to judiciously violate the ROT when that becomes a practical necessity.

Stress waves in transversely isotropic media: The homogeneous problem

NASA Technical Reports Server (NTRS)

Marques, E. R. C.; Williams, J. H., Jr.

1986-01-01

The homogeneous problem of stress wave propagation in unbounded transversely isotropic media is analyzed. By adopting plane wave solutions, the conditions for the existence of the solution are established in terms of phase velocities and directions of particle displacements. Dispersion relations and group velocities are derived from the phase velocity expressions. The deviation angles (e.g., angles between the normals to the adopted plane waves and the actual directions of their propagation) are numerically determined for a specific fiber-glass epoxy composite. A graphical method is introduced for the construction of the wave surfaces using magnitudes of phase velocities and deviation angles. The results for the case of isotropic media are shown to be contained in the solutions for the transversely isotropic media.

A high order accurate finite element algorithm for high Reynolds number flow prediction

NASA Technical Reports Server (NTRS)

Baker, A. J.

1978-01-01

A Galerkin-weighted residuals formulation is employed to establish an implicit finite element solution algorithm for generally nonlinear initial-boundary value problems. Solution accuracy, and convergence rate with discretization refinement, are quantized in several error norms, by a systematic study of numerical solutions to several nonlinear parabolic and a hyperbolic partial differential equation characteristic of the equations governing fluid flows. Solutions are generated using selective linear, quadratic and cubic basis functions. Richardson extrapolation is employed to generate a higher-order accurate solution to facilitate isolation of truncation error in all norms. Extension of the mathematical theory underlying accuracy and convergence concepts for linear elliptic equations is predicted for equations characteristic of laminar and turbulent fluid flows at nonmodest Reynolds number. The nondiagonal initial-value matrix structure introduced by the finite element theory is determined intrinsic to improved solution accuracy and convergence. A factored Jacobian iteration algorithm is derived and evaluated to yield a consequential reduction in both computer storage and execution CPU requirements while retaining solution accuracy.

Groundwater flow to a horizontal or slanted well in an unconfined aquifer

NASA Astrophysics Data System (ADS)

Zhan, Hongbin; Zlotnik, Vitaly A.

2002-07-01

New semianalytical solutions for evaluation of the drawdown near horizontal and slanted wells with finite length screens in unconfined aquifers are presented. These fully three-dimensional solutions consider instantaneous drainage or delayed yield and aquifer anisotropy. As a basis, solution for the drawdown created by a point source in a uniform anisotropic unconfined aquifer is derived in Laplace domain. Using superposition, the point source solution is extended to the cases of the horizontal and slanted wells. The previous solutions for vertical wells can be described as a special case of the new solutions. Numerical Laplace inversion allows effective evaluation of the drawdown in real time. Examples illustrate the effects of well geometry and the aquifer parameters on drawdown. Results can be used to generate type curves from observations in piezometers and partially or fully penetrating observation wells. The proposed solutions and software are useful for the parameter identification, design of remediation systems, drainage, and mine dewatering.

An unconditionally stable method for numerically solving solar sail spacecraft equations of motion

NASA Astrophysics Data System (ADS)

Karwas, Alex

Solar sails use the endless supply of the Sun's radiation to propel spacecraft through space. The sails use the momentum transfer from the impinging solar radiation to provide thrust to the spacecraft while expending zero fuel. Recently, the first solar sail spacecraft, or sailcraft, named IKAROS completed a successful mission to Venus and proved the concept of solar sail propulsion. Sailcraft experimental data is difficult to gather due to the large expenses of space travel, therefore, a reliable and accurate computational method is needed to make the process more efficient. Presented in this document is a new approach to simulating solar sail spacecraft trajectories. The new method provides unconditionally stable numerical solutions for trajectory propagation and includes an improved physical description over other methods. The unconditional stability of the new method means that a unique numerical solution is always determined. The improved physical description of the trajectory provides a numerical solution and time derivatives that are continuous throughout the entire trajectory. The error of the continuous numerical solution is also known for the entire trajectory. Optimal control for maximizing thrust is also provided within the framework of the new method. Verification of the new approach is presented through a mathematical description and through numerical simulations. The mathematical description provides details of the sailcraft equations of motion, the numerical method used to solve the equations, and the formulation for implementing the equations of motion into the numerical solver. Previous work in the field is summarized to show that the new approach can act as a replacement to previous trajectory propagation methods. A code was developed to perform the simulations and it is also described in this document. Results of the simulations are compared to the flight data from the IKAROS mission. Comparison of the two sets of data show that the new approach is capable of accurately simulating sailcraft motion. Sailcraft and spacecraft simulations are compared to flight data and to other numerical solution techniques. The new formulation shows an increase in accuracy over a widely used trajectory propagation technique. Simulations for two-dimensional, three-dimensional, and variable attitude trajectories are presented to show the multiple capabilities of the new technique. An element of optimal control is also part of the new technique. An additional equation is added to the sailcraft equations of motion that maximizes thrust in a specific direction. A technical description and results of an example optimization problem are presented. The spacecraft attitude dynamics equations take the simulation a step further by providing control torques using the angular rate and acceleration outputs of the numerical formulation.

Iterative methods for mixed finite element equations

NASA Technical Reports Server (NTRS)

Nakazawa, S.; Nagtegaal, J. C.; Zienkiewicz, O. C.

1985-01-01

Iterative strategies for the solution of indefinite system of equations arising from the mixed finite element method are investigated in this paper with application to linear and nonlinear problems in solid and structural mechanics. The augmented Hu-Washizu form is derived, which is then utilized to construct a family of iterative algorithms using the displacement method as the preconditioner. Two types of iterative algorithms are implemented. Those are: constant metric iterations which does not involve the update of preconditioner; variable metric iterations, in which the inverse of the preconditioning matrix is updated. A series of numerical experiments is conducted to evaluate the numerical performance with application to linear and nonlinear model problems.

Discrete-time model reduction in limited frequency ranges

NASA Technical Reports Server (NTRS)

Horta, Lucas G.; Juang, Jer-Nan; Longman, Richard W.

1991-01-01

A mathematical formulation for model reduction of discrete time systems such that the reduced order model represents the system in a particular frequency range is discussed. The algorithm transforms the full order system into balanced coordinates using frequency weighted discrete controllability and observability grammians. In this form a criterion is derived to guide truncation of states based on their contribution to the frequency range of interest. Minimization of the criterion is accomplished without need for numerical optimization. Balancing requires the computation of discrete frequency weighted grammians. Close form solutions for the computation of frequency weighted grammians are developed. Numerical examples are discussed to demonstrate the algorithm.

Propagation of singularities for linearised hybrid data impedance tomography

NASA Astrophysics Data System (ADS)

Bal, Guillaume; Hoffmann, Kristoffer; Knudsen, Kim

2018-02-01

For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit propagating singularities under certain non-elliptic conditions, and the associated directions of propagation are precisely identified relative to the directions in which ellipticity is lost. The same result is found in the setting for the corresponding normal formulation of the scalar pseudo-differential equations. A numerical reconstruction procedure based of the least squares finite element method is derived, and a series of numerical experiments visualise exactly how the loss of ellipticity manifests itself as propagating singularities.

Quantum weak turbulence with applications to semiconductor lasers

NASA Astrophysics Data System (ADS)

Lvov, Yuri Victorovich

Based on a model Hamiltonian appropriate for the description of fermionic systems such as semiconductor lasers, we describe a natural asymptotic closure of the BBGKY hierarchy in complete analogy with that derived for classical weak turbulence. The main features of the interaction Hamiltonian are the inclusion of full Fermi statistics containing Pauli blocking and a simple, phenomenological, uniformly weak two particle interaction potential equivalent to the static screening approximation. The resulting asymytotic closure and quantum kinetic Boltzmann equation are derived in a self consistent manner without resorting to a priori statistical hypotheses or cumulant discard assumptions. We find a new class of solutions to the quantum kinetic equation which are analogous to the Kolmogorov spectra of hydrodynamics and classical weak turbulence. They involve finite fluxes of particles and energy across momentum space and are particularly relevant for describing the behavior of systems containing sources and sinks. We explore these solutions by using differential approximation to collision integral. We make a prima facie case that these finite flux solutions can be important in the context of semiconductor lasers. We show that semiconductor laser output efficiency can be improved by exciting these finite flux solutions. Numerical simulations of the semiconductor Maxwell Bloch equations support the claim.

From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Angstmann, C.N.; Donnelly, I.C.; Henry, B.I., E-mail: B.Henry@unsw.edu.au

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also showmore » that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.« less

A time reversal algorithm in acoustic media with Dirac measure approximations

NASA Astrophysics Data System (ADS)

Bretin, Élie; Lucas, Carine; Privat, Yannick

2018-04-01

This article is devoted to the study of a photoacoustic tomography model, where one is led to consider the solution of the acoustic wave equation with a source term writing as a separated variables function in time and space, whose temporal component is in some sense close to the derivative of the Dirac distribution at t  =  0. This models a continuous wave laser illumination performed during a short interval of time. We introduce an algorithm for reconstructing the space component of the source term from the measure of the solution recorded by sensors during a time T all along the boundary of a connected bounded domain. It is based at the same time on the introduction of an auxiliary equivalent Cauchy problem allowing to derive explicit reconstruction formula and then to use of a deconvolution procedure. Numerical simulations illustrate our approach. Finally, this algorithm is also extended to elasticity wave systems.

Influence of non-integer-order derivatives on unsteady unidirectional motions of an Oldroyd-B fluid with generalized boundary conditions

NASA Astrophysics Data System (ADS)

Zafar, A. A.; Riaz, M. B.; Shah, N. A.; Imran, M. A.

2018-03-01

The objective of this article is to study some unsteady Couette flows of an Oldroyd-B fluid with non-integer derivatives. The fluid fills an annular region of two infinite co-axial circular cylinders. Flows are due to the motion of the outer cylinder, that rotates about its axis with an arbitrary time-dependent velocity while the inner cylinder is held fixed. Closed form solutions of dimensionless velocity field and tangential tension are obtained by means of the finite Hankel transform and the theory of Laplace transform for fractional calculus. Several results in the literature including the rotational flows through an infinite cylinder can be obtained as limiting cases of our general solutions. Finally, the control of the fractional framework on the dynamics of fluid is analyzed by numerical simulations and graphical illustrations.

Finite difference methods for transient signal propagation in stratified dispersive media

NASA Technical Reports Server (NTRS)

Lam, D. H.

1975-01-01

Explicit difference equations are presented for the solution of a signal of arbitrary waveform propagating in an ohmic dielectric, a cold plasma, a Debye model dielectric, and a Lorentz model dielectric. These difference equations are derived from the governing time-dependent integro-differential equations for the electric fields by a finite difference method. A special difference equation is derived for the grid point at the boundary of two different media. Employing this difference equation, transient signal propagation in an inhomogeneous media can be solved provided that the medium is approximated in a step-wise fashion. The solutions are generated simply by marching on in time. It is concluded that while the classical transform methods will remain useful in certain cases, with the development of the finite difference methods described, an extensive class of problems of transient signal propagating in stratified dispersive media can be effectively solved by numerical methods.

Continual approach at T=0 in the mean field theory of incommensurate magnetic states in the frustrated Heisenberg ferromagnet with an easy axis anisotropy

NASA Astrophysics Data System (ADS)

Martynov, S. N.; Tugarinov, V. I.; Martynov, A. S.

2017-10-01

The algorithm of approximate solution was developed for the differential equation describing the anharmonical change of the spin orientation angle in the model of ferromagnet with the exchange competition between nearest and next nearest magnetic neighbors and the easy axis exchange anisotropy. The equation was obtained from the collinearity constraint on the discrete lattice. In the low anharmonicity approximation the equation is resulted to an autonomous form and is integrated in quadratures. The obvious dependence of the angle velocity and second derivative of angle from angle and initial condition was derived by expanding the first integral of the equation in the Taylor series in vicinity of initial condition. The ground state of the soliton solutions was calculated by a numerical minimization of the energy integral. The evaluation of the used approximation was made for a triple point of the phase diagram.

H2, fixed architecture, control design for large scale systems. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Mercadal, Mathieu

1990-01-01

The H2, fixed architecture, control problem is a classic linear quadratic Gaussian (LQG) problem whose solution is constrained to be a linear time invariant compensator with a decentralized processing structure. The compensator can be made of p independent subcontrollers, each of which has a fixed order and connects selected sensors to selected actuators. The H2, fixed architecture, control problem allows the design of simplified feedback systems needed to control large scale systems. Its solution becomes more complicated, however, as more constraints are introduced. This work derives the necessary conditions for optimality for the problem and studies their properties. It is found that the filter and control problems couple when the architecture constraints are introduced, and that the different subcontrollers must be coordinated in order to achieve global system performance. The problem requires the simultaneous solution of highly coupled matrix equations. The use of homotopy is investigated as a numerical tool, and its convergence properties studied. It is found that the general constrained problem may have multiple stabilizing solutions, and that these solutions may be local minima or saddle points for the quadratic cost. The nature of the solution is not invariant when the parameters of the system are changed. Bifurcations occur, and a solution may continuously transform into a nonstabilizing compensator. Using a modified homotopy procedure, fixed architecture compensators are derived for models of large flexible structures to help understand the properties of the constrained solutions and compare them to the corresponding unconstrained ones.

Exact finite element method analysis of viscoelastic tapered structures to transient loads

NASA Technical Reports Server (NTRS)

Spyrakos, Constantine Chris

1987-01-01

A general method is presented for determining the dynamic torsional/axial response of linear structures composed of either tapered bars or shafts to transient excitations. The method consists of formulating and solving the dynamic problem in the Laplace transform domain by the finite element method and obtaining the response by a numerical inversion of the transformed solution. The derivation of the torsional and axial stiffness matrices is based on the exact solution of the transformed governing equation of motion, and it consequently leads to the exact solution of the problem. The solution permits treatment of the most practical cases of linear tapered bars and shafts, and employs modeling of structures with only one element per member which reduces the number of degrees of freedom involved. The effects of external viscous or internal viscoelastic damping are also taken into account.

Finite element procedures for coupled linear analysis of heat transfer, fluid and solid mechanics

NASA Technical Reports Server (NTRS)

Sutjahjo, Edhi; Chamis, Christos C.

1993-01-01

Coupled finite element formulations for fluid mechanics, heat transfer, and solid mechanics are derived from the conservation laws for energy, mass, and momentum. To model the physics of interactions among the participating disciplines, the linearized equations are coupled by combining domain and boundary coupling procedures. Iterative numerical solution strategy is presented to solve the equations, with the partitioning of temporal discretization implemented.

Exact finite elements for conduction and convection

NASA Technical Reports Server (NTRS)

Thornton, E. A.; Dechaumphai, P.; Tamma, K. K.

1981-01-01

An approach for developing exact one dimensional conduction-convection finite elements is presented. Exact interpolation functions are derived based on solutions to the governing differential equations by employing a nodeless parameter. Exact interpolation functions are presented for combined heat transfer in several solids of different shapes, and for combined heat transfer in a flow passage. Numerical results demonstrate that exact one dimensional elements offer advantages over elements based on approximate interpolation functions.

Viscous Rayleigh-Taylor instability in spherical geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mikaelian, Karnig O.

We consider viscous fluids in spherical geometry, a lighter fluid supporting a heavier one. Chandrasekhar [Q. J. Mech. Appl. Math. 8, 1 (1955)] analyzed this unstable configuration providing the equations needed to find, numerically, the exact growth rates for the ensuing Rayleigh-Taylor instability. He also derived an analytic but approximate solution. We point out a weakness in his approximate dispersion relation (DR) and offer one that is to some extent improved.

Viscous Rayleigh-Taylor instability in spherical geometry

DOE PAGES

Mikaelian, Karnig O.

2016-02-08

We consider viscous fluids in spherical geometry, a lighter fluid supporting a heavier one. Chandrasekhar [Q. J. Mech. Appl. Math. 8, 1 (1955)] analyzed this unstable configuration providing the equations needed to find, numerically, the exact growth rates for the ensuing Rayleigh-Taylor instability. He also derived an analytic but approximate solution. We point out a weakness in his approximate dispersion relation (DR) and offer one that is to some extent improved.