A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation
NASA Astrophysics Data System (ADS)
Parand, Kourosh; Yousefi, Hossein; Delkhosh, Mehdi; Ghaderi, Amin
2016-07-01
In this paper, a new algorithm based on the fractional order of rational Euler functions (FRE) is introduced to study the Thomas-Fermi (TF) model which is a nonlinear singular ordinary differential equation on a semi-infinite interval. This problem, using the quasilinearization method (QLM), converts to the sequence of linear ordinary differential equations to obtain the solution. For the first time, the rational Euler (RE) and the FRE have been made based on Euler polynomials. In addition, the equation will be solved on a semi-infinite domain without truncating it to a finite domain by taking FRE as basic functions for the collocation method. This method reduces the solution of this problem to the solution of a system of algebraic equations. We demonstrated that the new proposed algorithm is efficient for obtaining the value of y'(0) , y(x) and y'(x) . Comparison with some numerical and analytical solutions shows that the present solution is highly accurate.
An accurate solution of elastodynamic problems by numerical local Green's functions
NASA Astrophysics Data System (ADS)
Loureiro, F. S.; Silva, J. E. A.; Mansur, W. J.
2015-09-01
Green's function based methodologies for elastodynamics in both time and frequency domains, which can be either numerical or analytical, appear in many branches of physics and engineering. Thus, the development of exact expressions for Green's functions is of great importance. Unfortunately, such expressions are known only for relatively few kinds of geometry, medium and boundary conditions. In this way, due to the difficulty in finding exact Green's functions, specially in the time domain, the present paper presents a solution of the transient elastodynamic equations by a time-stepping technique based on the Explicit Green's Approach method written in terms of the Green's and Step response functions, both being computed numerically by the finite element method. The major feature is the computation of these functions separately by the central difference time integration scheme and locally owing to the principle of causality. More precisely, Green's functions are computed only at t = Δt adopting two time substeps while Step response functions are computed directly without substeps. The proposed time-stepping method shows to be quite accurate with distinct numerical properties not presented in the standard central difference scheme as addressed in the numerical example.
Cobb, J.W.
1995-02-01
There is an increasing need for more accurate numerical methods for large-scale nonlinear magneto-fluid turbulence calculations. These methods should not only increase the current state of the art in terms of accuracy, but should also continue to optimize other desired properties such as simplicity, minimized computation, minimized memory requirements, and robust stability. This includes the ability to stably solve stiff problems with long time-steps. This work discusses a general methodology for deriving higher-order numerical methods. It also discusses how the selection of various choices can affect the desired properties. The explicit discussion focuses on third-order Runge-Kutta methods, including general solutions and five examples. The study investigates the linear numerical analysis of these methods, including their accuracy, general stability, and stiff stability. Additional appendices discuss linear multistep methods, discuss directions for further work, and exhibit numerical analysis results for some other commonly used lower-order methods.
NASA Astrophysics Data System (ADS)
Stecca, Guglielmo; Siviglia, Annunziato; Blom, Astrid
2016-07-01
We present an accurate numerical approximation to the Saint-Venant-Hirano model for mixed-sediment morphodynamics in one space dimension. Our solution procedure originates from the fully-unsteady matrix-vector formulation developed in [54]. The principal part of the problem is solved by an explicit Finite Volume upwind method of the path-conservative type, by which all the variables are updated simultaneously in a coupled fashion. The solution to the principal part is embedded into a splitting procedure for the treatment of frictional source terms. The numerical scheme is extended to second-order accuracy and includes a bookkeeping procedure for handling the evolution of size stratification in the substrate. We develop a concept of balancedness for the vertical mass flux between the substrate and active layer under bed degradation, which prevents the occurrence of non-physical oscillations in the grainsize distribution of the substrate. We suitably modify the numerical scheme to respect this principle. We finally verify the accuracy in our solution to the equations, and its ability to reproduce one-dimensional morphodynamics due to streamwise and vertical sorting, using three test cases. In detail, (i) we empirically assess the balancedness of vertical mass fluxes under degradation; (ii) we study the convergence to the analytical linearised solution for the propagation of infinitesimal-amplitude waves [54], which is here employed for the first time to assess a mixed-sediment model; (iii) we reproduce Ribberink's E8-E9 flume experiment [46].
On numerically accurate finite element
NASA Technical Reports Server (NTRS)
Nagtegaal, J. C.; Parks, D. M.; Rice, J. R.
1974-01-01
A general criterion for testing a mesh with topologically similar repeat units is given, and the analysis shows that only a few conventional element types and arrangements are, or can be made suitable for computations in the fully plastic range. Further, a new variational principle, which can easily and simply be incorporated into an existing finite element program, is presented. This allows accurate computations to be made even for element designs that would not normally be suitable. Numerical results are given for three plane strain problems, namely pure bending of a beam, a thick-walled tube under pressure, and a deep double edge cracked tensile specimen. The effects of various element designs and of the new variational procedure are illustrated. Elastic-plastic computation at finite strain are discussed.
NASA Technical Reports Server (NTRS)
Graves, R. A., Jr.
1975-01-01
The previously obtained second-order-accurate partial implicitization numerical technique used in the solution of fluid dynamic problems was modified with little complication to achieve fourth-order accuracy. The Von Neumann stability analysis demonstrated the unconditional linear stability of the technique. The order of the truncation error was deduced from the Taylor series expansions of the linearized difference equations and was verified by numerical solutions to Burger's equation. For comparison, results were also obtained for Burger's equation using a second-order-accurate partial-implicitization scheme, as well as the fourth-order scheme of Kreiss.
High order accurate solutions of viscous problems
NASA Technical Reports Server (NTRS)
Hayder, M. E.; Turkel, Eli
1993-01-01
We consider a fourth order extension to MacCormack's scheme. The original extension was fourth order only for the inviscid terms but was second order for the viscous terms. We show how to modify the viscous terms so that the scheme is uniformly fourth order in the spatial derivatives. Applications are given to some boundary layer flows. In addition, for applications to shear flows the effect of the outflow boundary conditions are very important. We compare the accuracy of several of these different boundary conditions for both boundary layer and shear flows. Stretching at the outflow usually increases the oscillations in the numerical solution but the addition of a filtered sponge layer (with or without stretching) reduces such oscillations. The oscillations are generated by insufficient resolution of the shear layer. When the shear layer is sufficiently resolved then oscillations are not generated and there is less of a need for a nonreflecting boundary condition.
Numerical solutions of nonlinear wave equations
Kouri, D.J.; Zhang, D.S.; Wei, G.W.; Konshak, T.; Hoffman, D.K.
1999-01-01
Accurate, stable numerical solutions of the (nonlinear) sine-Gordon equation are obtained with particular consideration of initial conditions that are exponentially close to the phase space homoclinic manifolds. Earlier local, grid-based numerical studies have encountered difficulties, including numerically induced chaos for such initial conditions. The present results are obtained using the recently reported distributed approximating functional method for calculating spatial derivatives to high accuracy and a simple, explicit method for the time evolution. The numerical solutions are chaos-free for the same conditions employed in previous work that encountered chaos. Moreover, stable results that are free of homoclinic-orbit crossing are obtained even when initial conditions are within 10{sup {minus}7} of the phase space separatrix value {pi}. It also is found that the present approach yields extremely accurate solutions for the Korteweg{endash}de Vries and nonlinear Schr{umlt o}dinger equations. Our results support Ablowitz and co-workers{close_quote} conjecture that ensuring high accuracy of spatial derivatives is more important than the use of symplectic time integration schemes for solving solitary wave equations. {copyright} {ital 1999} {ital The American Physical Society}
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…
Time-Accurate Numerical Simulations of Synthetic Jet Quiescent Air
NASA Technical Reports Server (NTRS)
Rupesh, K-A. B.; Ravi, B. R.; Mittal, R.; Raju, R.; Gallas, Q.; Cattafesta, L.
2007-01-01
The unsteady evolution of three-dimensional synthetic jet into quiescent air is studied by time-accurate numerical simulations using a second-order accurate mixed explicit-implicit fractional step scheme on Cartesian grids. Both two-dimensional and three-dimensional calculations of synthetic jet are carried out at a Reynolds number (based on average velocity during the discharge phase of the cycle V(sub j), and jet width d) of 750 and Stokes number of 17.02. The results obtained are assessed against PIV and hotwire measurements provided for the NASA LaRC workshop on CFD validation of synthetic jets.
Numerical solution methods for viscoelastic orthotropic materials
NASA Technical Reports Server (NTRS)
Gramoll, K. C.; Dillard, D. A.; Brinson, H. F.
1988-01-01
Numerical solution methods for viscoelastic orthotropic materials, specifically fiber reinforced composite materials, are examined. The methods include classical lamination theory using time increments, direction solution of the Volterra Integral, Zienkiewicz's linear Prony series method, and a new method called Nonlinear Differential Equation Method (NDEM) which uses a nonlinear Prony series. The criteria used for comparison of the various methods include the stability of the solution technique, time step size stability, computer solution time length, and computer memory storage. The Volterra Integral allowed the implementation of higher order solution techniques but had difficulties solving singular and weakly singular compliance function. The Zienkiewicz solution technique, which requires the viscoelastic response to be modeled by a Prony series, works well for linear viscoelastic isotropic materials and small time steps. The new method, NDEM, uses a modified Prony series which allows nonlinear stress effects to be included and can be used with orthotropic nonlinear viscoelastic materials. The NDEM technique is shown to be accurate and stable for both linear and nonlinear conditions with minimal computer time.
Can numerical simulations accurately predict hydrodynamic instabilities in liquid films?
NASA Astrophysics Data System (ADS)
Denner, Fabian; Charogiannis, Alexandros; Pradas, Marc; van Wachem, Berend G. M.; Markides, Christos N.; Kalliadasis, Serafim
2014-11-01
Understanding the dynamics of hydrodynamic instabilities in liquid film flows is an active field of research in fluid dynamics and non-linear science in general. Numerical simulations offer a powerful tool to study hydrodynamic instabilities in film flows and can provide deep insights into the underlying physical phenomena. However, the direct comparison of numerical results and experimental results is often hampered by several reasons. For instance, in numerical simulations the interface representation is problematic and the governing equations and boundary conditions may be oversimplified, whereas in experiments it is often difficult to extract accurate information on the fluid and its behavior, e.g. determine the fluid properties when the liquid contains particles for PIV measurements. In this contribution we present the latest results of our on-going, extensive study on hydrodynamic instabilities in liquid film flows, which includes direct numerical simulations, low-dimensional modelling as well as experiments. The major focus is on wave regimes, wave height and wave celerity as a function of Reynolds number and forcing frequency of a falling liquid film. Specific attention is paid to the differences in numerical and experimental results and the reasons for these differences. The authors are grateful to the EPSRC for their financial support (Grant EP/K008595/1).
Accurate spectral numerical schemes for kinetic equations with energy diffusion
NASA Astrophysics Data System (ADS)
Wilkening, Jon; Cerfon, Antoine J.; Landreman, Matt
2015-08-01
We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker-Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.
A time-accurate adaptive grid method and the numerical simulation of a shock-vortex interaction
NASA Technical Reports Server (NTRS)
Bockelie, Michael J.; Eiseman, Peter R.
1990-01-01
A time accurate, general purpose, adaptive grid method is developed that is suitable for multidimensional steady and unsteady numerical simulations. The grid point movement is performed in a manner that generates smooth grids which resolve the severe solution gradients and the sharp transitions in the solution gradients. The temporal coupling of the adaptive grid and the PDE solver is performed with a grid prediction correction method that is simple to implement and ensures the time accuracy of the grid. Time accurate solutions of the 2-D Euler equations for an unsteady shock vortex interaction demonstrate the ability of the adaptive method to accurately adapt the grid to multiple solution features.
ACCURATE CHEMICAL MASTER EQUATION SOLUTION USING MULTI-FINITE BUFFERS
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-01-01
The discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multi-scale nature of many networks where reaction rates have large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the Accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multi-finite buffers for reducing the state space by O(n!), exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes, and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be pre-computed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multi-scale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks. PMID:27761104
ACCURATE CHEMICAL MASTER EQUATION SOLUTION USING MULTI-FINITE BUFFERS.
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-01-01
The discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multi-scale nature of many networks where reaction rates have large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the Accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multi-finite buffers for reducing the state space by O(n!), exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes, and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be pre-computed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multi-scale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks.
NASA Astrophysics Data System (ADS)
Campforts, Benjamin; Schwanghart, Wolfgang; Govers, Gerard
2017-01-01
Landscape evolution models (LEMs) allow the study of earth surface responses to changing climatic and tectonic forcings. While much effort has been devoted to the development of LEMs that simulate a wide range of processes, the numerical accuracy of these models has received less attention. Most LEMs use first-order accurate numerical methods that suffer from substantial numerical diffusion. Numerical diffusion particularly affects the solution of the advection equation and thus the simulation of retreating landforms such as cliffs and river knickpoints. This has potential consequences for the integrated response of the simulated landscape. Here we test a higher-order flux-limiting finite volume method that is total variation diminishing (TVD-FVM) to solve the partial differential equations of river incision and tectonic displacement. We show that using the TVD-FVM to simulate river incision significantly influences the evolution of simulated landscapes and the spatial and temporal variability of catchment-wide erosion rates. Furthermore, a two-dimensional TVD-FVM accurately simulates the evolution of landscapes affected by lateral tectonic displacement, a process whose simulation was hitherto largely limited to LEMs with flexible spatial discretization. We implement the scheme in TTLEM (TopoToolbox Landscape Evolution Model), a spatially explicit, raster-based LEM for the study of fluvially eroding landscapes in TopoToolbox 2.
Fast and Accurate Learning When Making Discrete Numerical Estimates
Sanborn, Adam N.; Beierholm, Ulrik R.
2016-01-01
Many everyday estimation tasks have an inherently discrete nature, whether the task is counting objects (e.g., a number of paint buckets) or estimating discretized continuous variables (e.g., the number of paint buckets needed to paint a room). While Bayesian inference is often used for modeling estimates made along continuous scales, discrete numerical estimates have not received as much attention, despite their common everyday occurrence. Using two tasks, a numerosity task and an area estimation task, we invoke Bayesian decision theory to characterize how people learn discrete numerical distributions and make numerical estimates. Across three experiments with novel stimulus distributions we found that participants fell between two common decision functions for converting their uncertain representation into a response: drawing a sample from their posterior distribution and taking the maximum of their posterior distribution. While this was consistent with the decision function found in previous work using continuous estimation tasks, surprisingly the prior distributions learned by participants in our experiments were much more adaptive: When making continuous estimates, participants have required thousands of trials to learn bimodal priors, but in our tasks participants learned discrete bimodal and even discrete quadrimodal priors within a few hundred trials. This makes discrete numerical estimation tasks good testbeds for investigating how people learn and make estimates. PMID:27070155
NASA Technical Reports Server (NTRS)
Kiris, Cetin; Kwak, Dochan
2001-01-01
Two numerical procedures, one based on artificial compressibility method and the other pressure projection method, are outlined for obtaining time-accurate solutions of the incompressible Navier-Stokes equations. The performance of the two method are compared by obtaining unsteady solutions for the evolution of twin vortices behind a at plate. Calculated results are compared with experimental and other numerical results. For an un- steady ow which requires small physical time step, pressure projection method was found to be computationally efficient since it does not require any subiterations procedure. It was observed that the artificial compressibility method requires a fast convergence scheme at each physical time step in order to satisfy incompressibility condition. This was obtained by using a GMRES-ILU(0) solver in our computations. When a line-relaxation scheme was used, the time accuracy was degraded and time-accurate computations became very expensive.
PolyPole-1: An accurate numerical algorithm for intra-granular fission gas release
NASA Astrophysics Data System (ADS)
Pizzocri, D.; Rabiti, C.; Luzzi, L.; Barani, T.; Van Uffelen, P.; Pastore, G.
2016-09-01
The transport of fission gas from within the fuel grains to the grain boundaries (intra-granular fission gas release) is a fundamental controlling mechanism of fission gas release and gaseous swelling in nuclear fuel. Hence, accurate numerical solution of the corresponding mathematical problem needs to be included in fission gas behaviour models used in fuel performance codes. Under the assumption of equilibrium between trapping and resolution, the process can be described mathematically by a single diffusion equation for the gas atom concentration in a grain. In this paper, we propose a new numerical algorithm (PolyPole-1) to efficiently solve the fission gas diffusion equation in time-varying conditions. The PolyPole-1 algorithm is based on the analytic modal solution of the diffusion equation for constant conditions, combined with polynomial corrective terms that embody the information on the deviation from constant conditions. The new algorithm is verified by comparing the results to a finite difference solution over a large number of randomly generated operation histories. Furthermore, comparison to state-of-the-art algorithms used in fuel performance codes demonstrates that the accuracy of PolyPole-1 is superior to other algorithms, with similar computational effort. Finally, the concept of PolyPole-1 may be extended to the solution of the general problem of intra-granular fission gas diffusion during non-equilibrium trapping and resolution, which will be the subject of future work.
Accurate numerical simulation of short fiber optical parametric amplifiers.
Marhic, M E; Rieznik, A A; Kalogerakis, G; Braimiotis, C; Fragnito, H L; Kazovsky, L G
2008-03-17
We improve the accuracy of numerical simulations for short fiber optical parametric amplifiers (OPAs). Instead of using the usual coarse-step method, we adopt a model for birefringence and dispersion which uses fine-step variations of the parameters. We also improve the split-step Fourier method by exactly treating the nonlinear ellipse rotation terms. We find that results obtained this way for two-pump OPAs can be significantly different from those obtained by using the usual coarse-step fiber model, and/or neglecting ellipse rotation terms.
Numerical solution of the electron transport equation
NASA Astrophysics Data System (ADS)
Woods, Mark
The electron transport equation has been solved many times for a variety of reasons. The main difficulty in its numerical solution is that it is a very stiff boundary value problem. The most common numerical methods for solving boundary value problems are symmetric collocation methods and shooting methods. Both of these types of methods can only be applied to the electron transport equation if the boundary conditions are altered with unrealistic assumptions because they require too many points to be practical. Further, they result in oscillating and negative solutions, which are physically meaningless for the problem at hand. For these reasons, all numerical methods for this problem to date are a bit unusual because they were designed to try and avoid the problem of extreme stiffness. This dissertation shows that there is no need to introduce spurious boundary conditions or invent other numerical methods for the electron transport equation. Rather, there already exists methods for very stiff boundary value problems within the numerical analysis literature. We demonstrate one such method in which the fast and slow modes of the boundary value problem are essentially decoupled. This allows for an upwind finite difference method to be applied to each mode as is appropriate. This greatly reduces the number of points needed in the mesh, and we demonstrate how this eliminates the need to define new boundary conditions. This method is verified by showing that under certain restrictive assumptions, the electron transport equation has an exact solution that can be written as an integral. We show that the solution from the upwind method agrees with the quadrature evaluation of the exact solution. This serves to verify that the upwind method is properly solving the electron transport equation. Further, it is demonstrated that the output of the upwind method can be used to compute auroral light emissions.
Numerical solution of a tunneling equation
Wang, C.Y.; Carter, M.D.; Batchelor, D.B.; Jaeger, E.F.
1994-04-01
A numerical method is presented to solve mode conversion equations resulting from the use of radio frequency (rf) waves to heat plasmas. The solutions of the mode conversion equations contain exponentially growing modes, and ordinary numerical techniques give large errors. To avoid the unphysical growing modes, a set of boundary conditions are found, that eliminate the unphysical modes. The mode conversion equations are then solved with the boundary conditions as a standard two-point boundary value problem. A tunneling equation (one of the mode conversion equations without power absorption) is solved as a specific example of this numerical technique although the technique itself is very general and can be easily applied to solve any mode conversion equation. The results from the numerical calculation agree very well with those found from asymptotic analysis.
NASA Technical Reports Server (NTRS)
Bernstein, Ira B.; Brookshaw, Leigh; Fox, Peter A.
1992-01-01
The present numerical method for accurate and efficient solution of systems of linear equations proceeds by numerically developing a set of basis solutions characterized by slowly varying dependent variables. The solutions thus obtained are shown to have a computational overhead largely independent of the small size of the scale length which characterizes the solutions; in many cases, the technique obviates series solutions near singular points, and its known sources of error can be easily controlled without a substantial increase in computational time.
Accurate solutions for transonic viscous flow over finite wings
NASA Technical Reports Server (NTRS)
Vatsa, V. N.
1986-01-01
An explicit multistage Runge-Kutta type time-stepping scheme is used for solving the three-dimensional, compressible, thin-layer Navier-Stokes equations. A finite-volume formulation is employed to facilitate treatment of complex grid topologies encountered in three-dimensional calculations. Convergence to steady state is expedited through usage of acceleration techniques. Further numerical efficiency is achieved through vectorization of the computer code. The accuracy of the overall scheme is evaluated by comparing the computed solutions with the experimental data for a finite wing under different test conditions in the transonic regime. A grid refinement study ir conducted to estimate the grid requirements for adequate resolution of salient features of such flows.
Efficient and accurate numerical methods for the Klein-Gordon-Schroedinger equations
Bao, Weizhu . E-mail: bao@math.nus.edu.sg; Yang, Li . E-mail: yangli@nus.edu.sg
2007-08-10
In this paper, we present efficient, unconditionally stable and accurate numerical methods for approximations of the Klein-Gordon-Schroedinger (KGS) equations with/without damping terms. The key features of our methods are based on: (i) the application of a time-splitting spectral discretization for a Schroedinger-type equation in KGS (ii) the utilization of Fourier pseudospectral discretization for spatial derivatives in the Klein-Gordon equation in KGS (iii) the adoption of solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives. The numerical methods are either explicit or implicit but can be solved explicitly, unconditionally stable, and of spectral accuracy in space and second-order accuracy in time. Moreover, they are time reversible and time transverse invariant when there is no damping terms in KGS, conserve (or keep the same decay rate of) the wave energy as that in KGS without (or with a linear) damping term, keep the same dynamics of the mean value of the meson field, and give exact results for the plane-wave solution. Extensive numerical tests are presented to confirm the above properties of our numerical methods for KGS. Finally, the methods are applied to study solitary-wave collisions in one dimension (1D), as well as dynamics of a 2D problem in KGS.
Accurate numerical solutions for elastic-plastic models. [LMFBR
Schreyer, H. L.; Kulak, R. F.; Kramer, J. M.
1980-03-01
The accuracy of two integration algorithms is studied for the common engineering condition of a von Mises, isotropic hardening model under plane stress. Errors in stress predictions for given total strain increments are expressed with contour plots of two parameters: an angle in the pi plane and the difference between the exact and computed yield-surface radii. The two methods are the tangent-predictor/radial-return approach and the elastic-predictor/radial-corrector algorithm originally developed by Mendelson. The accuracy of a combined tangent-predictor/radial-corrector algorithm is also investigated.
NASA Astrophysics Data System (ADS)
Shishkin, G. I.; Shishkina, L. P.
2015-03-01
An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge ɛ-uniformly in the maximum norm (where ɛ is the perturbation parameter multiplying the highest order derivative, ɛ ∈ (0, 1]). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges ɛ-uniformly in the maximum norm at a rate of [InlineMediaObject not available: see fulltext.], where N + 1 and N 0 + 1 are the numbers of nodes in uniform meshes in x and t, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge ɛ-uniformly with an order close to the sixth in x and equal to the third in t.
High-order numerical solutions using cubic splines
NASA Technical Reports Server (NTRS)
Rubin, S. G.; Khosla, P. K.
1975-01-01
The cubic spline collocation procedure for the numerical solution of partial differential equations was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a nonuniform mesh and overall fourth-order accuracy for a uniform mesh. Application of the technique was made to the Burger's equation, to the flow around a linear corner, to the potential flow over a circular cylinder, and to boundary layer problems. The results confirmed the higher-order accuracy of the spline method and suggest that accurate solutions for more practical flow problems can be obtained with relatively coarse nonuniform meshes.
Comparison between analytical and numerical solution of mathematical drying model
NASA Astrophysics Data System (ADS)
Shahari, N.; Rasmani, K.; Jamil, N.
2016-02-01
Drying is often related to the food industry as a process of shifting heat and mass inside food, which helps in preserving food. Previous research using a mass transfer equation showed that the results were mostly concerned with the comparison between the simulation model and the experimental data. In this paper, the finite difference method was used to solve a mass equation during drying using different kinds of boundary condition, which are equilibrium and convective boundary conditions. The results of these two models provide a comparison between the analytical and the numerical solution. The result shows a close match between the two solution curves. It is concluded that the two proposed models produce an accurate solution to describe the moisture distribution content during the drying process. This analysis indicates that we have confidence in the behaviour of moisture in the numerical simulation. This result demonstrated that a combined analytical and numerical approach prove that the system is behaving physically. Based on this assumption, the model of mass transfer was extended to include the temperature transfer, and the result shows a similar trend to those presented in the simpler case.
A program for accurate solutions of two-electron atoms
NASA Astrophysics Data System (ADS)
Edvardsson, Sverker; Åberg, Daniel; Uddholm, Per
2005-02-01
We present a comprehensible computer program capable of treating non-relativistic ground and excited states for a two-electron atom having infinite nuclear mass. An iterative approach based on the implicitly restarted Arnoldi method (IRAM) is employed. The Hamiltonian matrix is never explicitly computed. Instead the action of the Hamiltonian operator on discrete pair functions is implemented. The finite difference method is applied and subsequent extrapolations gives the continuous grid result. The program is written in C and is highly optimized. All computations are made in double precision. Despite this relatively low degree of floating point precision (48 digits are not uncommon), the accuracy in the results can reach about 10 significant figures. Both serial and parallel versions are provided. The parallel program is particularly suitable for shared memory machines such as the Sun Starcat series. The serial version is simple to compile and should run on any platform. Program summaryTitle of program: corr2el Catalogue identifier: ADUX Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUX Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computer for which the program is designed and others on which it has been tested: Computers: Sun Fire 15K StarCat, Sun Ultra SPARC III, PC Operating systems or monitors under which the program has been tested: Sun Solaris 9, Linux Programming language used: ANSI C Memory required to execute with typical data: 3 Mwords or more No. bits in a word: 32 No. processors used: arbitrary Has the code been vectorized or parallelized: parallelized Number of lines in distributed program, including test data, etc.:5885 Number of bytes in distributed program, including test data, etc.: 26 199 Nature of physical problem: The Schrödinger equation for two-electron atoms is solved using finite differences. Method of solution: An iterative eigenvalue-solver that requires only
Trace hydrazines in aqueous solutions accurately determined by gas chromatography
NASA Technical Reports Server (NTRS)
Welz, E. A., Jr.
1967-01-01
Trace amounts of hydrazines in aqueous solutions can be determined by using polythyleneimine /PEI/ in conjunction with the gas chromatographic column. The PEI specifically retains water without altering the separability or elution order of the hydrazine and associated constituents.
Foundations for the numerical solution of the Euler equations
NASA Technical Reports Server (NTRS)
Salas, M. D.
1985-01-01
The Navier-Stokes equations represent an extremely good model of the physical phenomena encountered in most aeronautical problems. However, the computational resource needed to solve the Navier-Stokes equations are so large that even with today's supercomputers, it is necessary to make use of simpler models. A large number of external aerodynamic problems can be accurately described by a simpler model. This model consists of an outer inviscid flow plus a boundary-layer thickness correction for the vehicle shape. The outer inviscid model may be represented by the potential equation or by the Euler equation. The present paper provides the foundations for the numerical solution of the Euler equations. The governing equations are considered, taking into account conservation laws, the medium, the differential form of the conservation laws, generalized solutions, shock-fitting, and characteristics. Attention is also given to initial and boundary conditions, existence and uniqueness, and rotational phenomena.
Numerical solution of multiscale electromagnetic systems
NASA Astrophysics Data System (ADS)
Tobon Llano, Luis Eduardo
with different order of interpolation. In this case, the Riemman solver shows the best performance to treat interfaces between subdomains. A new spectral prismatic element, useful for modeling of layer structures, is also implemented for this approach. Furthermore, a new efficient and very accurate time integration method for sequential subdomains is implemented. The second approach for solving multidomain cases is based on E and B fields, which use curl- and divergence-conforming basis functions, respectively, with same order of interpolation. In this way, higher accuracy and lower memory consumption are obtained with respect to the first approach based on E and H fields. The centered flux is used to treat interfaces with non-conforming meshes, and both explicit Runge-Kutta method and implicit Crank-Nicholson method are implemented for time integration. Numerical examples and realistic cases are presented to verify that the proposed methods are non-spurious and efficient DGTD schemes.
Numerical integration of asymptotic solutions of ordinary differential equations
NASA Technical Reports Server (NTRS)
Thurston, Gaylen A.
1989-01-01
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
Numerical solution of a diffusion problem by exponentially fitted finite difference methods.
D'Ambrosio, Raffaele; Paternoster, Beatrice
2014-01-01
This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver.
Highly accurate boronimeter assay of concentrated boric acid solutions
Ball, R.M. )
1992-01-01
The Random-Walk Boronimeter has successfully been used as an on-line indicator of boric acid concentration in an operating commercial pressurized water reactor. The principle has been adapted for measurement of discrete samples to high accuracy and to concentrations up to 6000 ppm natural boron in light water. Boric acid concentration in an aqueous solution is a necessary measurement in many nuclear power plants, particularly those that use boric acid dissolved in the reactor coolant as a reactivity control system. Other nuclear plants use a high-concentration boric acid solution as a backup shutdown system. Such a shutdown system depends on rapid injection of the solution and frequent surveillance of the fluid to ensure the presence of the neutron absorber. The two methods typically used to measure boric acid are the chemical and the physical methods. The chemical method uses titration to determine the ionic concentration of the BO[sub 3] ions and infers the boron concentration. The physical method uses the attenuation of neutrons by the solution and infers the boron concentration from the neutron absorption properties. This paper describes the Random-Walk Boronimeter configured to measure discrete samples to high accuracy and high concentration.
Numerical Solutions for a Model of Tissue Invasion and Migration of Tumour Cells
Kolev, M.; Zubik-Kowal, B.
2011-01-01
The goal of this paper is to construct a new algorithm for the numerical simulations of the evolution of tumour invasion and metastasis. By means of mathematical model equations and their numerical solutions we investigate how cancer cells can produce and secrete matrix degradative enzymes, degrade extracellular matrix, and invade due to diffusion and haptotactic migration. For the numerical simulations of the interactions between the tumour cells and the surrounding tissue, we apply numerical approximations, which are spectrally accurate and based on small amounts of grid-points. Our numerical experiments illustrate the metastatic ability of tumour cells. PMID:21331265
A numerical dressing method for the nonlinear superposition of solutions of the KdV equation
NASA Astrophysics Data System (ADS)
Trogdon, Thomas; Deconinck, Bernard
2014-01-01
In this paper we present the unification of two existing numerical methods for the construction of solutions of the Korteweg-de Vries (KdV) equation. The first method is used to solve the Cauchy initial-value problem on the line for rapidly decaying initial data. The second method is used to compute finite-genus solutions of the KdV equation. The combination of these numerical methods allows for the computation of exact solutions that are asymptotically (quasi-)periodic finite-gap solutions and are a nonlinear superposition of dispersive, soliton and (quasi-)periodic solutions in the finite (x, t)-plane. Such solutions are referred to as superposition solutions. We compute these solutions accurately for all values of x and t.
Numerical Solution of Optimal Control Problem under SPDE Constraints
2011-10-14
equations, Advances in Com- putational Mathematics. Vol. 33, 215–230, (2010). [3] Feng Bao, Yanzhao Cao and Weidong Zhao, Numerical solutions for forward...Visiting Scholar, Zhongshan University, China 4 Publications 1 Feng Bao, Yanzhao Cao and Weidong Zhao, Numerical solutions for forward backward doubly...Li Yin , Spectral method for nonlinear stochastic partial differ- ential equations of elliptic type, accepted by Numer. Math. Theo. Meth. App., Vol. 4
Multiresolution strategies for the numerical solution of optimal control problems
NASA Astrophysics Data System (ADS)
Jain, Sachin
There exist many numerical techniques for solving optimal control problems but less work has been done in the field of making these algorithms run faster and more robustly. The main motivation of this work is to solve optimal control problems accurately in a fast and efficient way. Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme. The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. The algorithm adapted dynamically to any existing or emerging irregularities in the solution by automatically allocating more grid points to the region where the solution exhibited sharp features and fewer points to the region where the solution was smooth. Thereby, the computational time and memory usage has been reduced significantly, while maintaining an accuracy equivalent to the one obtained using a fine uniform mesh. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a
A More Accurate Solution to the Elastic-Plastic Problem of Pressurized Thick-Walled Cylinders
1985-02-01
ACCURATE SOLUTION TO THE ELASTIC- PLASTIC PROBLEM OF PRESSURIZED THICK-WALLED CYLINDERS S. TYPE OF REPORT 4’ PERIOD COVERED Final 8. PERFORMING...o £ ) A MORE ACCURATE SOLUTION TO THE ELASTIC- PLASTIC PROBLEM OF PREr SURIZED THICK-WALLED CYLINDERS < • Peter C. T. Chen U.S. Army Armament...Watervllet, NY 12189 I iJSTRACT. A new method has been developed for solving the partially plastic problems of thlc’ -walled cylinders made of strain
NASA Technical Reports Server (NTRS)
VanZante, Dale E.; Strazisar, Anthony J.; Wood, Jerry R,; Hathaway, Michael D.; Okiishi, Theodore H.
2000-01-01
The tip clearance flows of transonic compressor rotors are important because they have a significant impact on rotor and stage performance. While numerical simulations of these flows are quite sophisticated. they are seldom verified through rigorous comparisons of numerical and measured data because these kinds of measurements are rare in the detail necessary to be useful in high-speed machines. In this paper we compare measured tip clearance flow details (e.g. trajectory and radial extent) with corresponding data obtained from a numerical simulation. Recommendations for achieving accurate numerical simulation of tip clearance flows are presented based on this comparison. Laser Doppler Velocimeter (LDV) measurements acquired in a transonic compressor rotor, NASA Rotor 35, are used. The tip clearance flow field of this transonic rotor was simulated using a Navier-Stokes turbomachinery solver that incorporates an advanced k-epsilon turbulence model derived for flows that are not in local equilibrium. Comparison between measured and simulated results indicates that simulation accuracy is primarily dependent upon the ability of the numerical code to resolve important details of a wall-bounded shear layer formed by the relative motion between the over-tip leakage flow and the shroud wall. A simple method is presented for determining the strength of this shear layer.
Seth A Veitzer
2008-10-21
Effects of stray electrons are a main factor limiting performance of many accelerators. Because heavy-ion fusion (HIF) accelerators will operate in regimes of higher current and with walls much closer to the beam than accelerators operating today, stray electrons might have a large, detrimental effect on the performance of an HIF accelerator. A primary source of stray electrons is electrons generated when halo ions strike the beam pipe walls. There is some research on these types of secondary electrons for the HIF community to draw upon, but this work is missing one crucial ingredient: the effect of grazing incidence. The overall goal of this project was to develop the numerical tools necessary to accurately model the effect of grazing incidence on the behavior of halo ions in a HIF accelerator, and further, to provide accurate models of heavy ion stopping powers with applications to ICF, WDM, and HEDP experiments.
Towards numerically accurate many-body perturbation theory: Short-range correlation effects
Gulans, Andris
2014-10-28
The example of the uniform electron gas is used for showing that the short-range electron correlation is difficult to handle numerically, while it noticeably contributes to the self-energy. Nonetheless, in condensed-matter applications studied with advanced methods, such as the GW and random-phase approximations, it is common to neglect contributions due to high-momentum (large q) transfers. Then, the short-range correlation is poorly described, which leads to inaccurate correlation energies and quasiparticle spectra. To circumvent this problem, an accurate extrapolation scheme is proposed. It is based on an analytical derivation for the uniform electron gas presented in this paper, and it provides an explanation why accurate GW quasiparticle spectra are easy to obtain for some compounds and very difficult for others.
The numerical solution of thermoporoelastoplasticity problems
NASA Astrophysics Data System (ADS)
Sivtsev, P. V.; Kolesov, A. E.; Sirditov, I. K.; Stepanov, S. P.
2016-10-01
Before constructing buildings in permafrost areas the careful study of stress-strain state of soils and building foundations must be performed in order to estimate their bearing capacity and stability to avoid issues with maintenance. To determine stress-strain state of frozen soils the numerical modeling of thermoporoelastoplasticity problems is used. The mathematical model of considered problems includes the elasto-plasticity equations and equations of heat and mass transfer with phase transition. The computational algorithm is based on the finite element approximation in space and the finite difference approximation in time. As the model problem we consider the deformation of soil under house weight and heating. Special attention is given to thawing of frozen soils, which can cause additional deformations and lead to loss of stability.
Numerical Solution of Ill Posed Problems in Partial Differential Equations
1988-06-30
periodic solutions of semilinear wave equations in exterior domains (breathers). Necessary and sufficient conditions for the existence of such solutions...numerically, that radial, global , positive solutions of the equation div grad u + uq u = 0 (X > 0, q > 1). ((1+1grad ul ) / exist for all X sufficiently... equation with a semilinear boundary condition , to appear in SIAM J. Math. Anal. 17] Levine, H.A. and Protter, M.H., The breakdown of solutions of
Evolution of midplate hotspot swells: Numerical solutions
NASA Technical Reports Server (NTRS)
Liu, Mian; Chase, Clement G.
1990-01-01
The evolution of midplate hotspot swells on an oceanic plate moving over a hot, upwelling mantle plume is numerically simulated. The plume supplies a Gaussian-shaped thermal perturbation and thermally-induced dynamic support. The lithosphere is treated as a thermal boundary layer with a strongly temperature-dependent viscosity. The two fundamental mechanisms of transferring heat, conduction and convection, during the interaction of the lithosphere with the mantle plume are considered. The transient heat transfer equations, with boundary conditions varying in both time and space, are solved in cylindrical coordinates using the finite difference ADI (alternating direction implicit) method on a 100 x 100 grid. The topography, geoid anomaly, and heat flow anomaly of the Hawaiian swell and the Bermuda rise are used to constrain the models. Results confirm the conclusion of previous works that the Hawaiian swell can not be explained by conductive heating alone, even if extremely high thermal perturbation is allowed. On the other hand, the model of convective thinning predicts successfully the topography, geoid anomaly, and the heat flow anomaly around the Hawaiian islands, as well as the changes in the topography and anomalous heat flow along the Hawaiian volcanic chain.
Higher-order numerical solutions using cubic splines
NASA Technical Reports Server (NTRS)
Rubin, S. G.; Khosla, P. K.
1976-01-01
A cubic spline collocation procedure was developed for the numerical solution of partial differential equations. This spline procedure is reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy of a nonuniform mesh. Solutions using both spline procedures, as well as three-point finite difference methods, are presented for several model problems.
NASA Astrophysics Data System (ADS)
El-Diasty, M.
2014-11-01
An accurate heading solution is required for many applications and it can be achieved by high grade (high cost) gyroscopes (gyros) which may not be suitable for such applications. Micro-Electro Mechanical Systems-based (MEMS) is an emerging technology, which has the potential of providing heading solution using a low cost MEMS-based gyro. However, MEMS-gyro-based heading solution drifts significantly over time. The heading solution can also be estimated using MEMS-based magnetometer by measuring the horizontal components of the Earth magnetic field. The MEMS-magnetometer-based heading solution does not drift over time, but are contaminated by high level of noise and may be disturbed by the presence of magnetic field sources such as metal objects. This paper proposed an accurate heading estimation procedure based on the integration of MEMS-based gyro and magnetometer measurements that correct gyro and magnetometer measurements where gyro angular rates of changes are estimated using magnetometer measurements and then integrated with the measured gyro angular rates of changes with a robust filter to estimate the heading. The proposed integration solution is implemented using two data sets; one was conducted in static mode without magnetic disturbances and the second was conducted in kinematic mode with magnetic disturbances. The results showed that the proposed integrated heading solution provides accurate, smoothed and undisturbed solution when compared with magnetometerbased and gyro-based heading solutions.
Numerical solution of Lane-Emden equation using neural network
NASA Astrophysics Data System (ADS)
Jalab, Hamid A.; Ibrahim, Rabha W.; Murad, Shayma A.; Melhum, Amera I.; Hadid, Samir B.
2012-09-01
This paper presents a numerical method based on neural network, for solving the Lane-Emden equations singular initial value problems. The numerical solution is given for integer case and non integer case. The non integer case is taken in the sense of Riemann-Liouville operators.
A numerical solution method for acoustic radiation from axisymmetric bodies
NASA Technical Reports Server (NTRS)
Caruthers, John E.; Raviprakash, G. K.
1995-01-01
A new and very efficient numerical method for solving equations of the Helmholtz type is specialized for problems having axisymmetric geometry. It is then demonstrated by application to the classical problem of acoustic radiation from a vibrating piston set in a stationary infinite plane. The method utilizes 'Green's Function Discretization', to obtain an accurate resolution of the waves using only 2-3 points per wave. Locally valid free space Green's functions, used in the discretization step, are obtained by quadrature. Results are computed for a range of grid spacing/piston radius ratios at a frequency parameter, omega R/c(sub 0), of 2 pi. In this case, the minimum required grid resolution appears to be fixed by the need to resolve a step boundary condition at the piston edge rather than by the length scale imposed by the wave length of the acoustic radiation. It is also demonstrated that a local near-field radiation boundary procedure allows the domain to be truncated very near the radiating source with little effect on the solution.
Robust numerical solution of the reservoir routing equation
NASA Astrophysics Data System (ADS)
Fiorentini, Marcello; Orlandini, Stefano
2013-09-01
The robustness of numerical methods for the solution of the reservoir routing equation is evaluated. The methods considered in this study are: (1) the Laurenson-Pilgrim method, (2) the fourth-order Runge-Kutta method, and (3) the fixed order Cash-Karp method. Method (1) is unable to handle nonmonotonic outflow rating curves. Method (2) is found to fail under critical conditions occurring, especially at the end of inflow recession limbs, when large time steps (greater than 12 min in this application) are used. Method (3) is computationally intensive and it does not solve the limitations of method (2). The limitations of method (2) can be efficiently overcome by reducing the time step in the critical phases of the simulation so as to ensure that water level remains inside the domains of the storage function and the outflow rating curve. The incorporation of a simple backstepping procedure implementing this control into the method (2) yields a robust and accurate reservoir routing method that can be safely used in distributed time-continuous catchment models.
NASA Technical Reports Server (NTRS)
Przekwas, A. J.; Athavale, M. M.; Hendricks, R. C.; Steinetz, B. M.
2006-01-01
Detailed information of the flow-fields in the secondary flowpaths and their interaction with the primary flows in gas turbine engines is necessary for successful designs with optimized secondary flow streams. Present work is focused on the development of a simulation methodology for coupled time-accurate solutions of the two flowpaths. The secondary flowstream is treated using SCISEAL, an unstructured adaptive Cartesian grid code developed for secondary flows and seals, while the mainpath flow is solved using TURBO, a density based code with capability of resolving rotor-stator interaction in multi-stage machines. An interface is being tested that links the two codes at the rim seal to allow data exchange between the two codes for parallel, coupled execution. A description of the coupling methodology and the current status of the interface development is presented. Representative steady-state solutions of the secondary flow in the UTRC HP Rig disc cavity are also presented.
Fullerton, G D; Keener, C R; Cameron, I L
1994-12-01
The authors describe empirical corrections to ideally dilute expressions for freezing point depression of aqueous solutions to arrive at new expressions accurate up to three molal concentration. The method assumes non-ideality is due primarily to solute/solvent interactions such that the correct free water mass Mwc is the mass of water in solution Mw minus I.M(s) where M(s) is the mass of solute and I an empirical solute/solvent interaction coefficient. The interaction coefficient is easily derived from the constant in the linear regression fit to the experimental plot of Mw/M(s) as a function of 1/delta T (inverse freezing point depression). The I-value, when substituted into the new thermodynamic expressions derived from the assumption of equivalent activity of water in solution and ice, provides accurate predictions of freezing point depression (+/- 0.05 degrees C) up to 2.5 molal concentration for all the test molecules evaluated; glucose, sucrose, glycerol and ethylene glycol. The concentration limit is the approximate monolayer water coverage limit for the solutes which suggests that direct solute/solute interactions are negligible below this limit. This is contrary to the view of many authors due to the common practice of including hydration forces (a soft potential added to the hard core atomic potential) in the interaction potential between solute particles. When this is recognized the two viewpoints are in fundamental agreement.
Numerical solution of an inverse conductive boundary value problem
NASA Astrophysics Data System (ADS)
Yaman, F.
2008-12-01
In this paper, we derive a numerical solution of an inverse obstacle scattering problem with conductive boundary condition. The aim of the direct problem is the computation of the scattered field for a given arbitrarily shaped cylinder with conductive boundary condition on its surface.The inverse problem considered here is the reconstruction of the conductivity function of the scatterer from meausurements of the far field. A potential approach is used to obtain boundary layer integral equations both for the solution of the direct and the inverse problem. The numerical solutions of the integral equations which contain logarithmically singular kernels are evaluated by a Nyström method and Tikhonov regularization is used to solve the first kind of integral equations occuring in the solution of the inverse problem. Finally, numerical simulations are carried out to test the applicability and the effectiveness of the method.
Takahashi, F; Endo, A
2007-01-01
A system utilising radiation transport codes has been developed to derive accurate dose distributions in a human body for radiological accidents. A suitable model is quite essential for a numerical analysis. Therefore, two tools were developed to setup a 'problem-dependent' input file, defining a radiation source and an exposed person to simulate the radiation transport in an accident with the Monte Carlo calculation codes-MCNP and MCNPX. Necessary resources are defined by a dialogue method with a generally used personal computer for both the tools. The tools prepare human body and source models described in the input file format of the employed Monte Carlo codes. The tools were validated for dose assessment in comparison with a past criticality accident and a hypothesized exposure.
Blackman, Jonathan; Field, Scott E; Galley, Chad R; Szilágyi, Béla; Scheel, Mark A; Tiglio, Manuel; Hemberger, Daniel A
2015-09-18
Simulating a binary black hole coalescence by solving Einstein's equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from nonspinning binary black hole coalescences with mass ratios in [1, 10] and durations corresponding to about 15 orbits before merger. We assess the model's uncertainty and show that our modeling strategy predicts NR waveforms not used for the surrogate's training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic _{-2}Y_{ℓm} waveform modes resolved by the NR code up to ℓ=8. We compare our surrogate model to effective one body waveforms from 50M_{⊙} to 300M_{⊙} for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases).
Keeping the edge: an accurate numerical method to solve the stream power law
NASA Astrophysics Data System (ADS)
Campforts, B.; Govers, G.
2015-12-01
Bedrock rivers set the base level of surrounding hill slopes and mediate the dynamic interplay between mountain building and denudation. The propensity of rivers to preserve pulses of increased tectonic uplift also allows to reconstruct long term uplift histories from longitudinal river profiles. An accurate reconstruction of river profile development at different timescales is therefore essential. Long term river development is typically modeled by means of the stream power law. Under specific conditions this equation can be solved analytically but numerical Finite Difference Methods (FDMs) are most frequently used. Nonetheless, FDMs suffer from numerical smearing, especially at knickpoint zones which are key to understand transient landscapes. Here, we solve the stream power law by means of a Finite Volume Method (FVM) which is Total Variation Diminishing (TVD). Total volume methods are designed to simulate sharp discontinuities making them very suitable to model river incision. In contrast to FDMs, the TVD_FVM is well capable of preserving knickpoints as illustrated for the fast propagating Niagara falls. Moreover, we show that the TVD_FVM performs much better when reconstructing uplift at timescales exceeding 100 Myr, using Eastern Australia as an example. Finally, uncertainty associated with parameter calibration is dramatically reduced when the TVD_FVM is applied. Therefore, the use of a TVD_FVM to understand long term landscape evolution is an important addition to the toolbox at the disposition of geomorphologists.
Quinci, Federico; Dressler, Matthew; Strickland, Anthony M; Limbert, Georges
2014-04-01
Considerable progress has been made in understanding implant wear and developing numerical models to predict wear for new orthopaedic devices. However any model of wear could be improved through a more accurate representation of the biomaterial mechanics, including time-varying dynamic and inelastic behaviour such as viscosity and plastic deformation. In particular, most computational models of wear of UHMWPE implement a time-invariant version of Archard's law that links the volume of worn material to the contact pressure between the metal implant and the polymeric tibial insert. During in-vivo conditions, however, the contact area is a time-varying quantity and is therefore dependent upon the dynamic deformation response of the material. From this observation one can conclude that creep deformations of UHMWPE may be very important to consider when conducting computational wear analyses, in stark contrast to what can be found in the literature. In this study, different numerical modelling techniques are compared with experimental creep testing on a unicondylar knee replacement system in a physiologically representative context. Linear elastic, plastic and time-varying visco-dynamic models are benchmarked using literature data to predict contact deformations, pressures and areas. The aim of this study is to elucidate the contributions of viscoelastic and plastic effects on these surface quantities. It is concluded that creep deformations have a significant effect on the contact pressure measured (experiment) and calculated (computational models) at the surface of the UHMWPE unicondylar insert. The use of a purely elastoplastic constitutive model for UHMWPE lead to compressive deformations of the insert which are much smaller than those predicted by a creep-capturing viscoelastic model (and those measured experimentally). This shows again the importance of including creep behaviour into a constitutive model in order to predict the right level of surface deformation
Numerical Solutions for Bayes Sequential Decision Approach to Bioequivalence Problem
1991-03-01
ADA3 707 2T1 -r Numerical Solutions for Bayes Sequential Decision Approach to Bioequivalence Problem Jing-Shiang Hwang Department of Statistics...Decision Approach to Bioequivalence Problem Jing-Shiang Hwang Department of Statistics Harvard University March 20, 1991 Abstract Bioequivalence is an...literatures. We address stop- ping rules for testing bioequivalence from a decision-theoretic point of view. The numerical techniques for Bayes
NASA Astrophysics Data System (ADS)
Ho, Kung-Chu; Su, Vin-Cent; Huang, Da-Yo; Lee, Ming-Lun; Chou, Nai-Kuan; Kuan, Chieh-Hsiung
2017-01-01
This paper reports the investigation of strong electrolytic solutions operated in low frequency regime through an accurate electrical impedance method realized with a specific microfluidic device and high resolution instruments. Experimental results show the better repeatability and accuracy of the proposed impedance method. Moreover, all electrolytic solutions appear the so-called relaxation frequency at each peak value of dielectric loss due to relaxing total polarization inside the device. The relaxation frequency of concentrated electrolytes becomes higher owing to the stronger total polarization behavior coming from the higher conductivity as well as the lower resistance in the electrolytic solutions.
Numerical Solution of a Nonlinear Integro-Differential Equation
NASA Astrophysics Data System (ADS)
Buša, Ján; Hnatič, Michal; Honkonen, Juha; Lučivjanský, Tomáš
2016-02-01
A discretization algorithm for the numerical solution of a nonlinear integrodifferential equation modeling the temporal variation of the mean number density a(t) in the single-species annihilation reaction A + A → 0 is discussed. The proposed solution for the two-dimensional case (where the integral entering the equation is divergent) uses regularization and then finite differences for the approximation of the differential operator together with a piecewise linear approximation of a(t) under the integral. The presented numerical results point to basic features of the behavior of the number density function a(t) and suggest further improvement of the proposed algorithm.
Numerical solution of inviscid and viscous flow around the profile
NASA Astrophysics Data System (ADS)
Slouka, Martin; Kozel, Karel; Prihoda, Jaromir
2015-05-01
This work deals with the 2D numerical solution of inviscid compressible flow and viscous compressible laminar and turbulent flow around the profile. In a case of turbulent flow algebraic Baldwin-Lomax model is used and compared with Wilcox's k-ω model. Calculations are done in GAMM channel computational domain with 10% DCA profile and in turbine cascade computational domain with 8% DCA profile. Numerical methods are based on a finite volume solution and compared with experimental measurements for 8% DCA profile.
Refined numerical solution of the transonic flow past a wedge
NASA Technical Reports Server (NTRS)
Liang, S.-M.; Fung, K.-Y.
1985-01-01
A numerical procedure combining the ideas of solving a modified difference equation and of adaptive mesh refinement is introduced. The numerical solution on a fixed grid is improved by using better approximations of the truncation error computed from local subdomain grid refinements. This technique is used to obtain refined solutions of steady, inviscid, transonic flow past a wedge. The effects of truncation error on the pressure distribution, wave drag, sonic line, and shock position are investigated. By comparing the pressure drag on the wedge and wave drag due to the shocks, a supersonic-to-supersonic shock originating from the wedge shoulder is confirmed.
Efficient numerical solution to vacuum decay with many fields
NASA Astrophysics Data System (ADS)
Masoumi, Ali; Olum, Ken D.; Shlaer, Benjamin
2017-01-01
Finding numerical solutions describing bubble nucleation is notoriously difficult in more than one field space dimension. Traditional shooting methods fail because of the extreme non-linearity of field evolution over a macroscopic distance as a function of initial conditions. Minimization methods tend to become either slow or imprecise for larger numbers of fields due to their dependence on the high dimensionality of discretized function spaces. We present a new method for finding solutions which is both very efficient and able to cope with the non-linearities. Our method directly integrates the equations of motion except at a small number of junction points, so we do not need to introduce a discrete domain for our functions. The method, based on multiple shooting, typically finds solutions involving three fields in around a minute, and can find solutions for eight fields in about an hour. We include a numerical package for Mathematica which implements the method described here.
Some studies of the numerical solution of ordinary differential equations
NASA Astrophysics Data System (ADS)
Mehdiyeva, G.; Ibrahimov, V.; Imanova, M.
2012-08-01
With the numerical solution of ordinary differential equations(ODE), scientists engaged in the Middle Ages, beginning with the work of Clairaut. The domain of the numerical methods involved in many famous mathematicians - Euler, Runge, Kutta, Adams, Laplace, and others. They have constructed methods with different properties. In this paper we consider the construction of numerical methods with high accuracy and to this end is proposed to use multi-step multi-derivative and hybrid methods. As well as specific methods are constructed with a certain accuracy.
Phoretic motion of soft vesicles and droplets: an XFEM/particle-based numerical solution
NASA Astrophysics Data System (ADS)
Shen, Tong; Vernerey, Franck
2017-03-01
When immersed in solution, surface-active particles interact with solute molecules and migrate along gradients of solute concentration. Depending on the conditions, this phenomenon could arise from either diffusiophoresis or the Marangoni effect, both of which involve strong interactions between the fluid and the particle surface. We introduce here a numerical approach that can accurately capture these interactions, and thus provide an efficient tool to understand and characterize the phoresis of soft particles. The model is based on a combination of the extended finite element—that enable the consideration of various discontinuities across the particle surface—and the particle-based moving interface method—that is used to measure and update the interface deformation in time. In addition to validating the approach with analytical solutions, the model is used to study the motion of deformable vesicles in solutions with spatial variations in both solute concentration and temperature.
Highly Accurate Beam Torsion Solutions Using the p-Version Finite Element Method
NASA Technical Reports Server (NTRS)
Smith, James P.
1996-01-01
A new treatment of the classical beam torsion boundary value problem is applied. Using the p-version finite element method with shape functions based on Legendre polynomials, torsion solutions for generic cross-sections comprised of isotropic materials are developed. Element shape functions for quadrilateral and triangular elements are discussed, and numerical examples are provided.
Numerical Solution for the Determination of Towboat Return Currents
1994-03-01
Numerical Solution of Potential Flow ...................... 7 Geometry and Grid Development ........................ 7 Boundaries...and size of the hull, and the channel geometry . h. Wake flow . The current produced as water fills in behind the stern to replace the water displaced...time-steps). Geometry and Grid Development Sensitivity tests were conducted using STREMR to determine the potential of modeling the flow field around
Error estimates of numerical solutions for a cyclic plasticity problem
NASA Astrophysics Data System (ADS)
Han, W.
A cyclic plasticity problem is numerically analyzed in [13], where a sub-optimal order error estimate is shown for a spatially discrete scheme. In this note, we prove an optimal order error estimate for the spatially discrete scheme under the same solution regularity condition. We also derive an error estimate for a fully discrete scheme for solving the plasticity problem.
Optimality conditions for the numerical solution of optimization problems with PDE constraints :
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.
NASA Technical Reports Server (NTRS)
Hagstrom, Thomas; Hariharan, S. I.; Maccamy, R. C.
1993-01-01
We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds.
Numerical solution-space analysis of satisfiability problems
NASA Astrophysics Data System (ADS)
Mann, Alexander; Hartmann, A. K.
2010-11-01
The solution-space structure of the three-satisfiability problem (3-SAT) is studied as a function of the control parameter α (ratio of the number of clauses to the number of variables) using numerical simulations. For this purpose one has to sample the solution space with uniform weight. It is shown here that standard stochastic local-search (SLS) algorithms like average satisfiability (ASAT) exhibit a sampling bias, as does “Metropolis-coupled Markov chain Monte Carlo” (MCMCMC) (also known as “parallel tempering”) when run for feasible times. Nevertheless, unbiased samples of solutions can be obtained using the “ballistic-networking approach,” which is introduced here. It is a generalization of “ballistic search” methods and yields also a cluster structure of the solution space. As application, solutions of 3-SAT instances are generated using ASAT plus ballistic networking. The numerical results are compatible with a previous analytical prediction of a simple solution-space structure for small values of α and a transition to a clustered phase at αc≈3.86 , where the solution space breaks up into several non-negligible clusters. Furthermore, in the thermodynamic limit there are, even for α=4.25 close to the SAT-UNSAT transition αs≈4.267 , always clusters without any frozen variables. This may explain why some SLS algorithms are able to solve very large 3-SAT instances close to the SAT-UNSAT transition.
Orbital Advection by Interpolation: A Fast and Accurate Numerical Scheme for Super-Fast MHD Flows
Johnson, B M; Guan, X; Gammie, F
2008-04-11
In numerical models of thin astrophysical disks that use an Eulerian scheme, gas orbits supersonically through a fixed grid. As a result the timestep is sharply limited by the Courant condition. Also, because the mean flow speed with respect to the grid varies with position, the truncation error varies systematically with position. For hydrodynamic (unmagnetized) disks an algorithm called FARGO has been developed that advects the gas along its mean orbit using a separate interpolation substep. This relaxes the constraint imposed by the Courant condition, which now depends only on the peculiar velocity of the gas, and results in a truncation error that is more nearly independent of position. This paper describes a FARGO-like algorithm suitable for evolving magnetized disks. Our method is second order accurate on a smooth flow and preserves {del} {center_dot} B = 0 to machine precision. The main restriction is that B must be discretized on a staggered mesh. We give a detailed description of an implementation of the code and demonstrate that it produces the expected results on linear and nonlinear problems. We also point out how the scheme might be generalized to make the integration of other supersonic/super-fast flows more efficient. Although our scheme reduces the variation of truncation error with position, it does not eliminate it. We show that the residual position dependence leads to characteristic radial variations in the density over long integrations.
Advanced numerical techniques for accurate unsteady simulations of a wingtip vortex
NASA Astrophysics Data System (ADS)
Ahmad, Shakeel
A numerical technique is developed to simulate the vortices associated with stationary and flapping wings. The Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations are used over an unstructured grid. The present work assesses the locations of the origins of vortex generation, models those locations and develops a systematic mesh refinement strategy to simulate vortices more accurately using the URANS model. The vortex center plays a key role in the analysis of the simulation data. A novel approach to locating a vortex center is also developed referred to as the Max-Max criterion. Experimental validation of the simulated vortex from a stationary NACA0012 wing is achieved. The tangential velocity along the core of the vortex falls within five percent of the experimental data in the case of the stationary NACA0012 simulation. The wing surface pressure coefficient also matches with the experimental data. The refinement techniques are then focused on unsteady simulations of pitching and dual-mode wing flapping. Tip vortex strength, location, and wing surface pressure are analyzed. Links to vortex behavior and wing motion are inferred. Key words: vortex, tangential velocity, Cp, vortical flow, unsteady vortices, URANS, Max-Max, Vortex center
NASA Astrophysics Data System (ADS)
Trogdon, Thomas; Deconinck, Bernard
2013-05-01
We derive a Riemann-Hilbert problem satisfied by the Baker-Akhiezer function for the finite-gap solutions of the Korteweg-de Vries (KdV) equation. As usual for Riemann-Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann-Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all periodic and quasi-periodic finite-genus solutions of the KdV equation.
Nanofluidic ionic diodes. Comparison of analytical and numerical solutions.
Vlassiouk, Ivan; Smirnov, Sergei; Siwy, Zuzanna
2008-08-01
Recently reported experimental and theoretical studies of nanofluidic nonlinear devices, such as bipolar and unipolar ionic diodes, have yet to answer the question about the possibility of their further miniaturization. In this Article, we theoretically investigate the effects of size reduction, applied bias, and solution ionic strength in such devices. We compare the numerical solutions of the Poisson, Nernst-Planck (PNP), and Navier-Stokes (NS) equations with their one-dimensional, analytical approximations. We demonstrate that the contribution of electroosmosis is insignificant and find analytical approximations to PNP for bipolar and unipolar diodes that are in good agreement with numerical 3D solutions. We identify the minimal dimensions for such diodes that demonstrate ion current rectification behavior and demonstrate the importance of the edge effect in very short diodes.
ASYMPTOTICALLY OPTIMAL HIGH-ORDER ACCURATE ALGORITHMS FOR THE SOLUTION OF CERTAIN ELLIPTIC PDEs
Leonid Kunyansky, PhD
2008-11-26
The main goal of the project, "Asymptotically Optimal, High-Order Accurate Algorithms for the Solution of Certain Elliptic PDE's" (DE-FG02-03ER25577) was to develop fast, high-order algorithms for the solution of scattering problems and spectral problems of photonic crystals theory. The results we obtained lie in three areas: (1) asymptotically fast, high-order algorithms for the solution of eigenvalue problems of photonics, (2) fast, high-order algorithms for the solution of acoustic and electromagnetic scattering problems in the inhomogeneous media, and (3) inversion formulas and fast algorithms for the inverse source problem for the acoustic wave equation, with applications to thermo- and opto- acoustic tomography.
Hassouna, M Sabry; Farag, A A
2007-09-01
A wide range of computer vision applications require an accurate solution of a particular Hamilton- Jacobi (HJ) equation, known as the Eikonal equation. In this paper, we propose an improved version of the fast marching method (FMM) that is highly accurate for both 2D and 3D Cartesian domains. The new method is called multi-stencils fast marching (MSFM), which computes the solution at each grid point by solving the Eikonal equation along several stencils and then picks the solution that satisfies the upwind condition. The stencils are centered at each grid point and cover its entire nearest neighbors. In 2D space, 2 stencils cover the 8-neighbors of the point, while in 3D space, 6 stencils cover its 26-neighbors. For those stencils that are not aligned with the natural coordinate system, the Eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes. The accuracy of the proposed method over the state-of-the-art FMM-based techniques has been demonstrated through comprehensive numerical experiments.
Stability of Inviscid Flow over Airfoils Admitting Multiple Numerical Solutions
NASA Astrophysics Data System (ADS)
Liu, Ya; Xiong, Juntao; Liu, Feng; Luo, Shijun
2012-11-01
Multiple numerical solutions at the same flight condition are found of inviscid transonic flow over certain airfoils (Jameson et al., AIAA 2011-3509) within some Mach number range. Both symmetric and asymmetric solutions exist for a symmetric airfoil at zero angle of attack. Global linear stability analysis of the multiple solutions is conducted. Linear perturbation equations of the Euler equations around a steady-state solution are formed and discretized numerically. An eigenvalue problem is then constructed using the modal analysis approach. Only a small portion of the eigen spectrum is needed and thus can be found efficiently by using Arnoldi's algorithm. The least stable or unstable mode corresponds to the eigenvalue with the largest real part. Analysis of the NACA 0012 airfoil indicates stability of symmetric solutions of the Euler equations at conditions where buffet is found from unsteady Navier-Stokes equations. Euler solutions of the same airfoil but modified to include the displacement thickness of the boundary layer computed from the Navier-Stokes equations, however, exhibit instability based on the present linear stability analysis. Graduate Student.
An accurate two-phase approximate solution to the acute viral infection model
Perelson, Alan S
2009-01-01
During an acute viral infection, virus levels rise, reach a peak and then decline. Data and numerical solutions suggest the growth and decay phases are linear on a log scale. While viral dynamic models are typically nonlinear with analytical solutions difficult to obtain, the exponential nature of the solutions suggests approximations can be found. We derive a two-phase approximate solution to the target cell limited influenza model and illustrate the accuracy using data and previously established parameter values of six patients infected with influenza A. For one patient, the subsequent fall in virus concentration was not consistent with our predictions during the decay phase and an alternate approximation is derived. We find expressions for the rate and length of initial viral growth in terms of the parameters, the extent each parameter is involved in viral peaks, and the single parameter responsible for virus decay. We discuss applications of this analysis in antiviral treatments and investigating host and virus heterogeneities.
Numerical solution of nonlinear Hammerstein fuzzy functional integral equations
NASA Astrophysics Data System (ADS)
Enkov, Svetoslav; Georgieva, Atanaska; Nikolla, Renato
2016-12-01
In this work we investigate nonlinear Hammerstein fuzzy functional integral equation. Our aim is to provide an efficient iterative method of successive approximations by optimal quadrature formula for classes of fuzzy number-valued functions of Lipschitz type to approximate the solution. We prove the convergence of the method by Banach's fixed point theorem and investigate the numerical stability of the presented method with respect to the choice of the first iteration. Finally, illustrative numerical experiment demonstrate the accuracy and the convergence of the proposed method.
Numerical time-dependent solutions of the Schrödinger equation with piecewise continuous potentials.
van Dijk, Wytse
2016-06-01
We consider accurate numerical solutions of the one-dimensional time-dependent Schrödinger equation when the potential is piecewise continuous. Spatial step sizes are defined for each of the regions between the discontinuities and a matching condition at the boundaries of the regions is employed. The Numerov method for spatial integration is particularly appropriate to this approach. By employing Padé approximants for the time-evolution operator, we obtain solutions with significantly improved precision without increased CPU time. This approach is also appropriate for adaptive changes in spatial step size even when there is no discontinuity of the potential.
Direct Coupling Method for Time-Accurate Solution of Incompressible Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Soh, Woo Y.
1992-01-01
A noniterative finite difference numerical method is presented for the solution of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The solution of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct.
Accurate numerical forward model for optimal retracking of SIRAL2 SAR echoes over open ocean
NASA Astrophysics Data System (ADS)
Phalippou, L.; Demeestere, F.
2011-12-01
The SAR mode of SIRAL-2 on board Cryosat-2 has been designed to measure primarily sea-ice and continental ice (Wingham et al. 2005). In 2005, K. Raney (KR, 2005) pointed out the improvements brought by SAR altimeter for open ocean. KR results were mostly based on 'rule of thumb' considerations on speckle noise reduction due to the higher PRF and to speckle decorrelation after SAR processing. In 2007, Phalippou and Enjolras (PE,2007) provided the theoretical background for optimal retracking of SAR echoes over ocean with a focus on the forward modelling of the power-waveforms. The accuracies of geophysical parameters (range, significant wave heights, and backscattering coefficient) retrieved from SAR altimeter data were derived accounting for SAR echo shape and speckle noise accurate modelling. The step forward to optimal retracking using numerical forward model (NFM) was also pointed out. NFM of the power waveform avoids analytical approximation, a warranty to minimise the geophysical dependent biases in the retrieval. NFM have been used for many years, in operational meteorology in particular, for retrieving temperature and humidity profiles from IR and microwave radiometers as the radiative transfer function is complex (Eyre, 1989). So far this technique was not used in the field of ocean conventional altimetry as analytical models (e.g. Brown's model for instance) were found to give sufficient accuracy. However, although NFM seems desirable even for conventional nadir altimetry, it becomes inevitable if one wish to process SAR altimeter data as the transfer function is too complex to be approximated by a simple analytical function. This was clearly demonstrated in PE 2007. The paper describes the background to SAR data retracking over open ocean. Since PE 2007 improvements have been brought to the forward model and it is shown that the altimeter on-ground and in flight characterisation (e.g antenna pattern range impulse response, azimuth impulse response
NASA Astrophysics Data System (ADS)
Wosnik, M.; Bachant, P.
2014-12-01
Cross-flow turbines, often referred to as vertical-axis turbines, show potential for success in marine hydrokinetic (MHK) and wind energy applications, ranging from small- to utility-scale installations in tidal/ocean currents and offshore wind. As turbine designs mature, the research focus is shifting from individual devices to the optimization of turbine arrays. It would be expensive and time-consuming to conduct physical model studies of large arrays at large model scales (to achieve sufficiently high Reynolds numbers), and hence numerical techniques are generally better suited to explore the array design parameter space. However, since the computing power available today is not sufficient to conduct simulations of the flow in and around large arrays of turbines with fully resolved turbine geometries (e.g., grid resolution into the viscous sublayer on turbine blades), the turbines' interaction with the energy resource (water current or wind) needs to be parameterized, or modeled. Models used today--a common model is the actuator disk concept--are not able to predict the unique wake structure generated by cross-flow turbines. This wake structure has been shown to create "constructive" interference in some cases, improving turbine performance in array configurations, in contrast with axial-flow, or horizontal axis devices. Towards a more accurate parameterization of cross-flow turbines, an extensive experimental study was carried out using a high-resolution turbine test bed with wake measurement capability in a large cross-section tow tank. The experimental results were then "interpolated" using high-fidelity Navier--Stokes simulations, to gain insight into the turbine's near-wake. The study was designed to achieve sufficiently high Reynolds numbers for the results to be Reynolds number independent with respect to turbine performance and wake statistics, such that they can be reliably extrapolated to full scale and used for model validation. The end product of
Numerical solution of quadratic matrix equations for free vibration analysis of structures
NASA Technical Reports Server (NTRS)
Gupta, K. K.
1975-01-01
This paper is concerned with the efficient and accurate solution of the eigenvalue problem represented by quadratic matrix equations. Such matrix forms are obtained in connection with the free vibration analysis of structures, discretized by finite 'dynamic' elements, resulting in frequency-dependent stiffness and inertia matrices. The paper presents a new numerical solution procedure of the quadratic matrix equations, based on a combined Sturm sequence and inverse iteration technique enabling economical and accurate determination of a few required eigenvalues and associated vectors. An alternative procedure based on a simultaneous iteration procedure is also described when only the first few modes are the usual requirement. The employment of finite dynamic elements in conjunction with the presently developed eigenvalue routines results in a most significant economy in the dynamic analysis of structures.
An efficient numerical technique for the solution of nonlinear singular boundary value problems
NASA Astrophysics Data System (ADS)
Singh, Randhir; Kumar, Jitendra
2014-04-01
In this work, a new technique based on Green's function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green's function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.
Numerical solution of flame sheet problems with and without multigrid methods
NASA Technical Reports Server (NTRS)
Douglas, Craig C.; Ern, Alexandre
1993-01-01
Flame sheet problems are on the natural route to the numerical solution of multidimensional flames, which, in turn, are important in many engineering applications. In order to model the structure of flames more accurately, we use the vorticity-velocity formulation of the fluid flow equations, as opposed to the streamfunction-vorticity approach. The numerical solution of the resulting nonlinear coupled elliptic partial differential equations involves a pseudo transient process and a steady state Newton iteration. Rather than working with dimensionless variables, we introduce scale factors that can yield significant savings in the execution time. In this context, we also investigate the applicability and performance of several multigrid methods, focusing on nonlinear damped Newton multigrid, using either one way or correction schemes.
Accurate integral equation theory for the central force model of liquid water and ionic solutions
NASA Astrophysics Data System (ADS)
Ichiye, Toshiko; Haymet, A. D. J.
1988-10-01
The atom-atom pair correlation functions and thermodynamics of the central force model of water, introduced by Lemberg, Stillinger, and Rahman, have been calculated accurately by an integral equation method which incorporates two new developments. First, a rapid new scheme has been used to solve the Ornstein-Zernike equation. This scheme combines the renormalization methods of Allnatt, and Rossky and Friedman with an extension of the trigonometric basis-set solution of Labik and co-workers. Second, by adding approximate ``bridge'' functions to the hypernetted-chain (HNC) integral equation, we have obtained predictions for liquid water in which the hydrogen bond length and number are in good agreement with ``exact'' computer simulations of the same model force laws. In addition, for dilute ionic solutions, the ion-oxygen and ion-hydrogen coordination numbers display both the physically correct stoichiometry and good agreement with earlier simulations. These results represent a measurable improvement over both a previous HNC solution of the central force model and the ex-RISM integral equation solutions for the TIPS and other rigid molecule models of water.
NASA Technical Reports Server (NTRS)
Weston, K. C.; Reynolds, A. C., Jr.; Alikhan, A.; Drago, D. W.
1974-01-01
Numerical solutions for radiative transport in a class of anisotropically scattering materials are presented. Conditions for convergence and divergence of the iterative method are given and supported by computed results. The relation of two flux theories to the equation of radiative transfer for isotropic scattering is discussed. The adequacy of the two flux approach for the reflectance, radiative flux and radiative flux divergence of highly scattering media is evaluated with respect to solutions of the radiative transfer equation.
Numerical Solution of Some Types of Fractional Optimal Control Problems
Sweilam, Nasser Hassan; Al-Ajami, Tamer Mostafa; Hoppe, Ronald H. W.
2013-01-01
We present two different approaches for the numerical solution of fractional optimal control problems (FOCPs) based on a spectral method using Chebyshev polynomials. The fractional derivative is described in the Caputo sense. The first approach follows the paradigm “optimize first, then discretize” and relies on the approximation of the necessary optimality conditions in terms of the associated Hamiltonian. In the second approach, the state equation is discretized first using the Clenshaw and Curtis scheme for the numerical integration of nonsingular functions followed by the Rayleigh-Ritz method to evaluate both the state and control variables. Two illustrative examples are included to demonstrate the validity and applicability of the suggested approaches. PMID:24385874
Numerical solution of plasma fluid equations using locally refined grids
Colella, P., LLNL
1997-01-26
This paper describes a numerical method for the solution of plasma fluid equations on block-structured, locally refined grids. The plasma under consideration is typical of those used for the processing of semiconductors. The governing equations consist of a drift-diffusion model of the electrons and an isothermal model of the ions coupled by Poisson's equation. A discretization of the equations is given for a uniform spatial grid, and a time-split integration scheme is developed. The algorithm is then extended to accommodate locally refined grids. This extension involves the advancement of the discrete system on a hierarchy of levels, each of which represents a degree of refinement, together with synchronization steps to ensure consistency across levels. A brief discussion of a software implementation is followed by a presentation of numerical results.
An algorithm for the numerical solution of linear differential games
Polovinkin, E S; Ivanov, G E; Balashov, M V; Konstantinov, R V; Khorev, A V
2001-10-31
A numerical algorithm for the construction of stable Krasovskii bridges, Pontryagin alternating sets, and also of piecewise program strategies solving two-person linear differential (pursuit or evasion) games on a fixed time interval is developed on the basis of a general theory. The aim of the first player (the pursuer) is to hit a prescribed target (terminal) set by the phase vector of the control system at the prescribed time. The aim of the second player (the evader) is the opposite. A description of numerical algorithms used in the solution of differential games of the type under consideration is presented and estimates of the errors resulting from the approximation of the game sets by polyhedra are presented.
Numerical solutions of thin-film equations for polymer flows.
Salez, Thomas; McGraw, Joshua D; Cormier, Sara L; Bäumchen, Oliver; Dalnoki-Veress, Kari; Raphaël, Elie
2012-11-01
We report on the numerical implementation of thin-film equations that describe the capillary-driven evolution of viscous films, in two-dimensional configurations. After recalling the general forms and features of these equations, we focus on two particular cases inspired by experiments: the leveling of a step at the free surface of a polymer film, and the leveling of a polymer droplet over an identical film. In each case, we first discuss the long-term self-similar regime reached by the numerical solution before comparing it to the experimental profile. The agreement between theory and experiment is excellent, thus providing a versatile probe for nanorheology of viscous liquids in thin-film geometries.
NUMERICAL SOLUTION FOR THE POTENTIAL AND DENSITY PROFILE OF A THERMAL EQUILIBRIUM SHEET BEAM
Lund, S M; Bazouin, G
2011-03-29
In a recent paper, S. M. Lund, A. Friedman, and G. Bazouin, Sheet beam model for intense space-charge: with application to Debye screening and the distribution of particle oscillation frequencies in a thermal equilibrium beam, in press, Phys. Rev. Special Topics - Accel. and Beams (2011), a 1D sheet beam model was extensively analyzed. In this complementary paper, we present details of a numerical procedure developed to construct the self-consistent electrostatic potential and density profile of a thermal equilibrium sheet beam distribution. This procedure effectively circumvents pathologies which can prevent use of standard numerical integration techniques when space-charge intensity is high. The procedure employs transformations and is straightforward to implement with standard numerical methods and produces accurate solutions which can be applied to thermal equilibria with arbitrarily strong space-charge intensity up to the applied focusing limit.
NUMERICAL SOLUTION FOR THE POTENTIAL AND DENSITY PROFILE OF A THERMAL EQUILIBRIUM SHEET BEAM
Bazouin, Steven M. Lund, Guillaume; Bazouin, Guillaume
2011-04-01
In a recent paper, S. M. Lund, A. Friedman, and G. Bazouin, Sheet beam model for intense space-charge: with application to Debye screening and the distribution of particle oscillation frequencies in a thermal equilibrium beam, in press, Phys. Rev. Special Topics - Accel. and Beams (2011), a 1D sheet beam model was extensively analyzed. In this complementary paper, we present details of a numerical procedure developed to construct the self-consistent electrostatic potential and density profile of a thermal equilibrium sheet beam distribution. This procedure effectively circumvents pathologies which can prevent use of standard numerical integration techniques when space-charge intensity is high. The procedure employs transformations and is straightforward to implement with standard numerical methods and produces accurate solutions which can be applied to thermal equilibria with arbitrarily strong space-charge intensity up to the applied focusing limit.
Bio-based lubricants for numerical solution of elastohydrodynamic lubrication
NASA Astrophysics Data System (ADS)
Cupu, Dedi Rosa Putra; Sheriff, Jamaluddin Md; Osman, Kahar
2012-06-01
This paper presents a programming code to provide numerical solution of elastohydrodynamic lubrication problem in line contacts which is modeled through an infinite cylinder on a plane to represent the application of roller bearing. In this simulation, vegetable oils will be used as bio-based lubricants. Temperature is assumed to be constant at 40°C. The results show that the EHL pressure for all vegetable oils was increasing from inlet flow until the center, then decrease a bit and rise to the peak pressure. The shapes of EHL film thickness for all tested vegetable oils are almost flat at contact region.
Sun, Hui; Zhou, Shenggao; Moore, David K; Cheng, Li-Tien; Li, Bo
2016-05-01
We design and implement numerical methods for the incompressible Stokes solvent flow and solute-solvent interface motion for nonpolar molecules in aqueous solvent. The balance of viscous force, surface tension, and van der Waals type dispersive force leads to a traction boundary condition on the solute-solvent interface. To allow the change of solute volume, we design special numerical boundary conditions on the boundary of a computational domain through a consistency condition. We use a finite difference ghost fluid scheme to discretize the Stokes equation with such boundary conditions. The method is tested to have a second-order accuracy. We combine this ghost fluid method with the level-set method to simulate the motion of the solute-solvent interface that is governed by the solvent fluid velocity. Numerical examples show that our method can predict accurately the blow up time for a test example of curvature flow and reproduce the polymodal (e.g., dry and wet) states of hydration of some simple model molecular systems.
Numerical solutions of the complete Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Hassan, H. A.
1988-01-01
The physical phenomena within supersonic flows that sustain chemical reactions are investigated. An earlier study to develop accurate physical models for supersonic reacting flowfields focused on 2-D laminar shear layers. The objective is to examine the mixing and subsequent combustion within turbulent reacting shear layers. To conduct this study, a computer program has been written to solve the axisymmetric Reynolds averaged Navier-Stokes equations. The numerical method uses a cell-centered finite volume approach and a Runge Kutta time stepping scheme. The Reynolds averaged equations are closed using the eddy viscosity concept. Several zero-equation models have been tested by making calculations for an H2-air nonreacting coaxial jet flow. Comparisons made with experimental data show that Cohen's eddy viscosity model provides best agreement. The finite rate chemistry model used in the study of 2-D laminar shear layers is incorporated into the computer program and data is compared from a recent experiment performed at NASA Langley.
Variable time-stepping in the pathwise numerical solution of the chemical Langevin equation.
Ilie, Silvana
2012-12-21
Stochastic modeling is essential for an accurate description of the biochemical network dynamics at the level of a single cell. Biochemically reacting systems often evolve on multiple time-scales, thus their stochastic mathematical models manifest stiffness. Stochastic models which, in addition, are stiff and computationally very challenging, therefore the need for developing effective and accurate numerical methods for approximating their solution. An important stochastic model of well-stirred biochemical systems is the chemical Langevin Equation. The chemical Langevin equation is a system of stochastic differential equation with multidimensional non-commutative noise. This model is valid in the regime of large molecular populations, far from the thermodynamic limit. In this paper, we propose a variable time-stepping strategy for the numerical solution of a general chemical Langevin equation, which applies for any level of randomness in the system. Our variable stepsize method allows arbitrary values of the time-step. Numerical results on several models arising in applications show significant improvement in accuracy and efficiency of the proposed adaptive scheme over the existing methods, the strategies based on halving/doubling of the stepsize and the fixed step-size ones.
A numerical solution for the diffusion equation in hydrogeologic systems
Ishii, A.L.; Healy, R.W.; Striegl, R.G.
1989-01-01
The documentation of a computer code for the numerical solution of the linear diffusion equation in one or two dimensions in Cartesian or cylindrical coordinates is presented. Applications of the program include molecular diffusion, heat conduction, and fluid flow in confined systems. The flow media may be anisotropic and heterogeneous. The model is formulated by replacing the continuous linear diffusion equation by discrete finite-difference approximations at each node in a block-centered grid. The resulting matrix equation is solved by the method of preconditioned conjugate gradients. The conjugate gradient method does not require the estimation of iteration parameters and is guaranteed convergent in the absence of rounding error. The matrixes are preconditioned to decrease the steps to convergence. The model allows the specification of any number of boundary conditions for any number of stress periods, and the output of a summary table for selected nodes showing flux and the concentration of the flux quantity for each time step. The model is written in a modular format for ease of modification. The model was verified by comparison of numerical and analytical solutions for cases of molecular diffusion, two-dimensional heat transfer, and axisymmetric radial saturated fluid flow. Application of the model to a hypothetical two-dimensional field situation of gas diffusion in the unsaturated zone is demonstrated. The input and output files are included as a check on program installation. The definition of variables, input requirements, flow chart, and program listing are included in the attachments. (USGS)
NASA Technical Reports Server (NTRS)
Wong, T. C.; Liu, C. H.; Geer, J.
1984-01-01
Approximate solutions for potential flow past an axisymmetric slender body and past a thin airfoil are calculated using a uniform perturbation method and then compared with either the exact analytical solution or the solution obtained using a purely numerical method. The perturbation method is based upon a representation of the disturbance flow as the superposition of singularities distributed entirely within the body, while the numerical (panel) method is based upon a distribution of singularities on the surface of the body. It is found that the perturbation method provides very good results for small values of the slenderness ratio and for small angles of attack. Moreover, for comparable accuracy, the perturbation method is simpler to implement, requires less computer memory, and generally uses less computation time than the panel method. In particular, the uniform perturbation method yields good resolution near the regions of the leading and trailing edges where other methods fail or require special attention.
Numerical solution of High-kappa model of superconductivity
Karamikhova, R.
1996-12-31
We present formulation and finite element approximations of High-kappa model of superconductivity which is valid in the high {kappa}, high magnetic field setting and accounts for applied magnetic field and current. Major part of this work deals with steady-state and dynamic computational experiments which illustrate our theoretical results numerically. In our experiments we use Galerkin discretization in space along with Backward-Euler and Crank-Nicolson schemes in time. We show that for moderate values of {kappa}, steady states of the model system, computed using the High-kappa model, are virtually identical with results computed using the full Ginzburg-Landau (G-L) equations. We illustrate numerically optimal rates of convergence in space and time for the L{sup 2} and H{sup 1} norms of the error in the High-kappa solution. Finally, our numerical approximations demonstrate some well-known experimentally observed properties of high-temperature superconductors, such as appearance of vortices, effects of increasing the applied magnetic field and the sample size, and the effect of applied constant current.
Accelerating numerical solution of stochastic differential equations with CUDA
NASA Astrophysics Data System (ADS)
Januszewski, M.; Kostur, M.
2010-01-01
hundreds of threads simultaneously makes it possible to speed up the computation by over two orders of magnitude, compared to a typical modern CPU. Solution method: The stochastic Runge-Kutta method of the second order is applied to integrate the equation of motion. Ensemble-averaged quantities of interest are obtained through averaging over multiple independent realizations of the system. Unusual features: The numerical solution of the stochastic differential equations in question is performed on a GPU using the CUDA environment. Running time: < 1 minute
NASA Astrophysics Data System (ADS)
Tang, Xiaojun
2016-04-01
The main purpose of this work is to provide multiple-interval integral Gegenbauer pseudospectral methods for solving optimal control problems. The latest developed single-interval integral Gauss/(flipped Radau) pseudospectral methods can be viewed as special cases of the proposed methods. We present an exact and efficient approach to compute the mesh pseudospectral integration matrices for the Gegenbauer-Gauss and flipped Gegenbauer-Gauss-Radau points. Numerical results on benchmark optimal control problems confirm the ability of the proposed methods to obtain highly accurate solutions.
AN ACCURATE AND EFFICIENT ALGORITHM FOR NUMERICAL SIMULATION OF CONDUCTION-TYPE PROBLEMS. (R824801)
A modification of the finite analytic numerical method for conduction-type (diffusion) problems is presented. The finite analytic discretization scheme is derived by means of the Fourier series expansion for the most general case of nonuniform grid and variabl...
Numerical solution of three-dimensional magnetic differential equations
Reiman, A.H.; Greenside, H.S.
1987-02-01
A computer code is described that solves differential equations of the form B . del f = h for a single-valued solution f, given a toroidal three-dimensional divergence-free field B and a single-valued function h. The code uses a new algorithm that Fourier decomposes a given function in a set of flux coordinates in which the field lines are straight. The algorithm automatically adjusts the required integration lengths to compensate for proximity to low order rational surfaces. Applying this algorithm to the Cartesian coordinates defines a transformation to magnetic coordinates, in which the magnetic differential equation can be accurately solved. Our method is illustrated by calculating the Pfirsch-Schlueter currents for a stellarator.
Determination of Solution Accuracy of Numerical Schemes as Part of Code and Calculation Verification
Blottner, F.G.; Lopez, A.R.
1998-10-01
This investigation is concerned with the accuracy of numerical schemes for solving partial differential equations used in science and engineering simulation codes. Richardson extrapolation methods for steady and unsteady problems with structured meshes are presented as part of the verification procedure to determine code and calculation accuracy. The local truncation error de- termination of a numerical difference scheme is shown to be a significant component of the veri- fication procedure as it determines the consistency of the numerical scheme, the order of the numerical scheme, and the restrictions on the mesh variation with a non-uniform mesh. Genera- tion of a series of co-located, refined meshes with the appropriate variation of mesh cell size is in- vestigated and is another important component of the verification procedure. The importance of mesh refinement studies is shown to be more significant than just a procedure to determine solu- tion accuracy. It is suggested that mesh refinement techniques can be developed to determine con- sistency of numerical schemes and to determine if governing equations are well posed. The present investigation provides further insight into the conditions and procedures required to effec- tively use Richardson extrapolation with mesh refinement studies to achieve confidence that sim- ulation codes are producing accurate numerical solutions.
Ohyanagi, Toshio; Sengoku, Yasuhito
2010-02-01
This article presents a new solution for measuring accurate reaction time (SMART) to visual stimuli. The SMART is a USB device realized with a Cypress Programmable System-on-Chip (PSoC) mixed-signal array programmable microcontroller. A brief overview of the hardware and firmware of the PSoC is provided, together with the results of three experiments. In Experiment 1, we investigated the timing accuracy of the SMART in measuring reaction time (RT) under different conditions of operating systems (OSs; Windows XP or Vista) and monitor displays (a CRT or an LCD). The results indicated that the timing error in measuring RT by the SMART was less than 2 msec, on average, under all combinations of OS and display and that the SMART was tolerant to jitter and noise. In Experiment 2, we tested the SMART with 8 participants. The results indicated that there was no significant difference among RTs obtained with the SMART under the different conditions of OS and display. In Experiment 3, we used Microsoft (MS) PowerPoint to present visual stimuli on the display. We found no significant difference in RTs obtained using MS DirectX technology versus using the PowerPoint file with the SMART. We are certain that the SMART is a simple and practical solution for measuring RTs accurately. Although there are some restrictions in using the SMART with RT paradigms, the SMART is capable of providing both researchers and health professionals working in clinical settings with new ways of using RT paradigms in their work.
Gridding guidelines for numerical solutions of simple flows
NASA Astrophysics Data System (ADS)
Vaughn, Milton Edward, Jr.
Numerical solutions are systematically computed for simple flat plate flows and compared with known analytical calculations and/or measurements to determine if quantitative relationships can be established between grid construction parameters and solution accuracy. Computational grids are created for plates at incompressible speeds without any pressure gradient for both laminar and turbulent types of boundary layers. Then the computed drag force coefficients, skin friction curves and velocity profiles are assessed against the known quantities. When examining turbulent flows, application is made of the Spalart-Allmaras, one-equation Reynolds-Averaged Navier-Stokes (RAMS) turbulence model; the Menter Shear Stress Transport, two-equation BANS turbulence model; and the Nichols-Nelson Hybrid (RANS/Large Eddy Simulation) turbulence technique. In addition, the structured, RANS Computational Fluid Dynamics (CFD) solver named Wind-US is used to perform the flowfield computations, while Gridgen(TM), interactive grid generation code, is used to create the computational grids and vary their parameters. It is found that quantitative relationships do in fact exist between grid parameters and solution accuracy. These relationships are formulated as rules of thumb that can be used to guide the generation of CFD grids. Although developed for simple flows, these guidelines will be helpful in creating grids for complex bodies with subregions characterized by simple flows.
NASA Technical Reports Server (NTRS)
Ellison, Donald; Conway, Bruce; Englander, Jacob
2015-01-01
A significant body of work exists showing that providing a nonlinear programming (NLP) solver with expressions for the problem constraint gradient substantially increases the speed of program execution and can also improve the robustness of convergence, especially for local optimizers. Calculation of these derivatives is often accomplished through the computation of spacecraft's state transition matrix (STM). If the two-body gravitational model is employed as is often done in the context of preliminary design, closed form expressions for these derivatives may be provided. If a high fidelity dynamics model, that might include perturbing forces such as the gravitational effect from multiple third bodies and solar radiation pressure is used then these STM's must be computed numerically. We present a method for the power hardward model and a full ephemeris model. An adaptive-step embedded eight order Dormand-Prince numerical integrator is discussed and a method for the computation of the time of flight derivatives in this framework is presented. The use of these numerically calculated derivatieves offer a substantial improvement over finite differencing in the context of a global optimizer. Specifically the inclusion of these STM's into the low thrust missiondesign tool chain in use at NASA Goddard Spaceflight Center allows for an increased preliminary mission design cadence.
NASA Astrophysics Data System (ADS)
Reis, C.; Clain, S.; Figueiredo, J.; Baptista, M. A.; Miranda, J. M. A.
2015-12-01
Numerical tools turn to be very important for scenario evaluations of hazardous phenomena such as tsunami. Nevertheless, the predictions highly depends on the numerical tool quality and the design of efficient numerical schemes still receives important attention to provide robust and accurate solutions. In this study we propose a comparative study between the efficiency of two volume finite numerical codes with second-order discretization implemented with different method to solve the non-conservative shallow water equations, the MUSCL (Monotonic Upstream-Centered Scheme for Conservation Laws) and the MOOD methods (Multi-dimensional Optimal Order Detection) which optimize the accuracy of the approximation in function of the solution local smoothness. The MUSCL is based on a priori criteria where the limiting procedure is performed before updated the solution to the next time-step leading to non-necessary accuracy reduction. On the contrary, the new MOOD technique uses a posteriori detectors to prevent the solution from oscillating in the vicinity of the discontinuities. Indeed, a candidate solution is computed and corrections are performed only for the cells where non-physical oscillations are detected. Using a simple one-dimensional analytical benchmark, 'Single wave on a sloping beach', we show that the classical 1D shallow-water system can be accurately solved with the finite volume method equipped with the MOOD technique and provide better approximation with sharper shock and less numerical diffusion. For the code validation, we also use the Tohoku-Oki 2011 tsunami and reproduce two DART records, demonstrating that the quality of the solution may deeply interfere with the scenario one can assess. This work is funded by the Portugal-France research agreement, through the research project GEONUM FCT-ANR/MAT-NAN/0122/2012.Numerical tools turn to be very important for scenario evaluations of hazardous phenomena such as tsunami. Nevertheless, the predictions highly
NASA Astrophysics Data System (ADS)
Huerta, Eliu; Agarwal, Bhanu; Chua, Alvin; George, Daniel; Haas, Roland; Hinder, Ian; Kumar, Prayush; Moore, Christopher; Pfeiffer, Harald
2017-01-01
We recently constructed an inspiral-merger-ringdown (IMR) waveform model to describe the dynamical evolution of compact binaries on eccentric orbits, and used this model to constrain the eccentricity with which the gravitational wave transients currently detected by LIGO could be effectively recovered with banks of quasi-circular templates. We now present the second generation of this model, which is calibrated using a large catalog of eccentric numerical relativity simulations. We discuss the new features of this model, and show that its enhance accuracy makes it a powerful tool to detect eccentric signals with LIGO.
Willis, Catherine; Rubin, Jacob
1987-01-01
In this paper we consider examples of chemistry-affected transport processes in porous media. A moving boundary problem which arises during transport with precipitation-dissolution reactions is solved by three different numerical methods. Two of these methods (one explicit and one implicit) are based on an integral formulation of mass balance and lead to an approximation of a weak solution. These methods are compared to a front-tracking scheme. Although the two approaches are conceptually different, the numerical solutions showed good agreement. As the ratio of dispersion to convection decreases, the methods based on the integral formulation become computationally more efficient. Specific reactions were modeled to examine the dependence of the system on the physical and chemical parameters.
Numerical Solution of the k-Eigenvalue Problem
NASA Astrophysics Data System (ADS)
Hamilton, Steven Paul
2011-12-01
Obtaining solutions to the k-eigenvalue form of the radiation transport equation is an important topic in the design and analysis of nuclear reactors. Although this has been an area of active interest in the nuclear engineering community for several decades, to date no truly satisfactory solution strategies exist. In general, existing techniques are either slow to converge for difficult problems or suffer from stability and robustness issues that can cause solvers to diverge for some problems. This work provides a comparison between a variety of methods and introduces a new strategy based on the Davidson method that has been used in other fields for many years but never for this problem. The Davidson method offers an alternative to the nested iteration structure inherent to standard approaches and allows expensive linear solvers to be replaced by a potentially cheap preconditioner. To fill the role of this preconditioner, a strategy based on a multigrid treatment of the energy variable is developed. Numerical experiments using the 2-D NEWT transport package are presented, demonstrating the effectiveness of the proposed strategy.
On a numerical solution of the plastic buckling problem of structures
NASA Technical Reports Server (NTRS)
Gupta, K. K.
1978-01-01
An automated digital computer procedure is presented for the accurate and efficient solution of the plastic buckling problem of structures. This is achieved by a Sturm sequence method employing a bisection strategy, which eliminates the need for having to solve the buckling eigenvalue problem at each incremental (decremental) loading stage that is associated with the usual solution techniques. The plastic buckling mode shape is determined by a simple inverse iteration process, once the buckling load has been established. Numerical results are presented for plate problems with various edge conditions. The resulting computer program written in FORTRAN V for the JPL UNIVAC 1108 machine proves to be most economical in comparison with other existing methods of such analysis.
NASA Astrophysics Data System (ADS)
Zhu, Jun; Chen, Lijun; Ma, Lantao; Li, Dejian; Jiang, Wei; Pan, Lihong; Shen, Huiting; Jia, Hongmin; Hsiang, Chingyun; Cheng, Guojie; Ling, Li; Chen, Shijie; Wang, Jun; Liao, Wenkui; Zhang, Gary
2014-04-01
Defect review is a time consuming job. Human error makes result inconsistent. The defects located on don't care area would not hurt the yield and no need to review them such as defects on dark area. However, critical area defects can impact yield dramatically and need more attention to review them such as defects on clear area. With decrease in integrated circuit dimensions, mask defects are always thousands detected during inspection even more. Traditional manual or simple classification approaches are unable to meet efficient and accuracy requirement. This paper focuses on automatic defect management and classification solution using image output of Lasertec inspection equipment and Anchor pattern centric image process technology. The number of mask defect found during an inspection is always in the range of thousands or even more. This system can handle large number defects with quick and accurate defect classification result. Our experiment includes Die to Die and Single Die modes. The classification accuracy can reach 87.4% and 93.3%. No critical or printable defects are missing in our test cases. The missing classification defects are 0.25% and 0.24% in Die to Die mode and Single Die mode. This kind of missing rate is encouraging and acceptable to apply on production line. The result can be output and reloaded back to inspection machine to have further review. This step helps users to validate some unsure defects with clear and magnification images when captured images can't provide enough information to make judgment. This system effectively reduces expensive inline defect review time. As a fully inline automated defect management solution, the system could be compatible with current inspection approach and integrated with optical simulation even scoring function and guide wafer level defect inspection.
The use of experimental bending tests to more accurate numerical description of TBC damage process
NASA Astrophysics Data System (ADS)
Sadowski, T.; Golewski, P.
2016-04-01
Thermal barrier coatings (TBCs) have been extensively used in aircraft engines to protect critical engine parts such as blades and combustion chambers, which are exposed to high temperatures and corrosive environment. The blades of turbine engines are additionally exposed to high mechanical loads. These loads are created by the high rotational speed of the rotor (30 000 rot/min), causing the tensile and bending stresses. Therefore, experimental testing of coated samples is necessary in order to determine strength properties of TBCs. Beam samples with dimensions 50×10×2 mm were used in those studies. The TBC system consisted of 150 μm thick bond coat (NiCoCrAlY) and 300 μm thick top coat (YSZ) made by APS (air plasma spray) process. Samples were tested by three-point bending test with various loads. After bending tests, the samples were subjected to microscopic observation to determine the quantity of cracks and their depth. The above mentioned results were used to build numerical model and calibrate material data in Abaqus program. Brittle cracking damage model was applied for the TBC layer, which allows to remove elements after reaching criterion. Surface based cohesive behavior was used to model the delamination which may occur at the boundary between bond coat and top coat.
Fast and accurate numerical method for predicting gas chromatography retention time.
Claumann, Carlos Alberto; Wüst Zibetti, André; Bolzan, Ariovaldo; Machado, Ricardo A F; Pinto, Leonel Teixeira
2015-08-07
Predictive modeling for gas chromatography compound retention depends on the retention factor (ki) and on the flow of the mobile phase. Thus, different approaches for determining an analyte ki in column chromatography have been developed. The main one is based on the thermodynamic properties of the component and on the characteristics of the stationary phase. These models can be used to estimate the parameters and to optimize the programming of temperatures, in gas chromatography, for the separation of compounds. Different authors have proposed the use of numerical methods for solving these models, but these methods demand greater computational time. Hence, a new method for solving the predictive modeling of analyte retention time is presented. This algorithm is an alternative to traditional methods because it transforms its attainments into root determination problems within defined intervals. The proposed approach allows for tr calculation, with accuracy determined by the user of the methods, and significant reductions in computational time; it can also be used to evaluate the performance of other prediction methods.
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-04-01
truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks.
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-01-01
truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks. PMID:27105653
Advances in numerical solutions to integral equations in liquid state theory
NASA Astrophysics Data System (ADS)
Howard, Jesse J.
Solvent effects play a vital role in the accurate description of the free energy profile for solution phase chemical and structural processes. The inclusion of solvent effects in any meaningful theoretical model however, has proven to be a formidable task. Generally, methods involving Poisson-Boltzmann (PB) theory and molecular dynamic (MD) simulations are used, but they either fail to accurately describe the solvent effects or require an exhaustive computation effort to overcome sampling problems. An alternative to these methods are the integral equations (IEs) of liquid state theory which have become more widely applicable due to recent advancements in the theory of interaction site fluids and the numerical methods to solve the equations. In this work a new numerical method is developed based on a Newton-type scheme coupled with Picard/MDIIS routines. To extend the range of these numerical methods to large-scale data systems, the size of the Jacobian is reduced using basis functions, and the Newton steps are calculated using a GMRes solver. The method is then applied to calculate solutions to the 3D reference interaction site model (RISM) IEs of statistical mechanics, which are derived from first principles, for a solute model of a pair of parallel graphene plates at various separations in pure water. The 3D IEs are then extended to electrostatic models using an exact treatment of the long-range Coulomb interactions for negatively charged walls and DNA duplexes in aqueous electrolyte solutions to calculate the density profiles and solution thermodynamics. It is found that the 3D-IEs provide a qualitative description of the density distributions of the solvent species when compared to MD results, but at a much reduced computational effort in comparison to MD simulations. The thermodynamics of the solvated systems are also qualitatively reproduced by the IE results. The findings of this work show the IEs to be a valuable tool for the study and prediction of
Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation
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
NASA Astrophysics Data System (ADS)
Henniger, R.; Obrist, D.; Kleiser, L.
2010-05-01
The emergence of "petascale" supercomputers requires us to develop today's simulation codes for (incompressible) flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power. In that spirit, we present a massively parallel high-order Navier-Stokes solver for large incompressible flow problems in three dimensions. The governing equations are discretized with finite differences in space and a semi-implicit time integration scheme. This discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers. The iterative solver for the pressure uses a highly efficient commutation-based preconditioner which is robust with respect to grid stretching. The efficiency of the implementation is further enhanced by carefully setting the (adaptive) termination criteria for the different iterative solvers. The computational work is distributed to different processing units by a geometric data decomposition in all three dimensions. This decomposition scheme ensures a low communication overhead and excellent scaling capabilities. The discretization is thoroughly validated. First, we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow. Second, we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria. Third, Orr-Sommerfeld and Squire eigenmodes for plane Poiseuille flow are simulated and compared to analytical results. Fourth, the practical applicability of the implementation is tested for transitional and turbulent channel flow. The results are compared to solutions from a pseudospectral solver. Subsequently, the performance of the commutation-based preconditioner for the pressure iteration is demonstrated. Finally, the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to
NASA Technical Reports Server (NTRS)
Hu, Fang Q.
1994-01-01
It is known that the exact analytic solutions of wave scattering by a circular cylinder, when they exist, are not in a closed form but in infinite series which converges slowly for high frequency waves. In this paper, we present a fast number solution for the scattering problem in which the boundary integral equations, reformulated from the Helmholtz equation, are solved using a Fourier spectral method. It is shown that the special geometry considered here allows the implementation of the spectral method to be simple and very efficient. The present method differs from previous approaches in that the singularities of the integral kernels are removed and dealt with accurately. The proposed method preserves the spectral accuracy and is shown to have an exponential rate of convergence. Aspects of efficient implementation using FFT are discussed. Moreover, the boundary integral equations of combined single and double-layer representation are used in the present paper. This ensures the uniqueness of the numerical solution for the scattering problem at all frequencies. Although a strongly singular kernel is encountered for the Neumann boundary conditions, we show that the hypersingularity can be handled easily in the spectral method. Numerical examples that demonstrate the validity of the method are also presented.
Numerical solution of periodic vortical flows about a thin airfoil
NASA Technical Reports Server (NTRS)
Scott, James R.; Atassi, Hafiz M.
1989-01-01
A numerical method is developed for computing periodic, three-dimensional, vortical flows around isolated airfoils. The unsteady velocity is split into a vortical component which is a known function of the upstream flow conditions and the Lagrangian coordinates of the mean flow, and an irrotational field whose potential satisfies a nonconstant-coefficient, inhomogeneous, convective wave equation. Solutions for thin airfoils at zero degrees incidence to the mean flow are presented in this paper. Using an elliptic coordinate transformation, the computational domain is transformed into a rectangle. The Sommerfeld radiation condition is applied to the unsteady pressure on the grid line corresponding to the far field boundary. The results are compared with a Possio solver, and it is shown that for maximum accuracy the grid should depend on both the Mach number and reduced frequency. Finally, in order to assess the range of validity of the classical thin airfoil approximation, results for airfoils with zero thickness are compared with results for airfoils with small thickness.
Numerical solution of acoustic scattering by finite perforated elastic plates
NASA Astrophysics Data System (ADS)
Cavalieri, A. V. G.; Wolf, W. R.; Jaworski, J. W.
2016-04-01
We present a numerical method to compute the acoustic field scattered by finite perforated elastic plates. A boundary element method is developed to solve the Helmholtz equation subjected to boundary conditions related to the plate vibration. These boundary conditions are recast in terms of the vibration modes of the plate and its porosity, which enables a direct solution procedure. A parametric study is performed for a two-dimensional problem whereby a cantilevered perforated elastic plate scatters sound from a point quadrupole near the free edge. Both elasticity and porosity tend to diminish the scattered sound, in agreement with previous work considering semi-infinite plates. Finite elastic plates are shown to reduce acoustic scattering when excited at high Helmholtz numbers k0 based on the plate length. However, at low k0, finite elastic plates produce only modest reductions or, in cases related to structural resonance, an increase to the scattered sound level relative to the rigid case. Porosity, on the other hand, is shown to be more effective in reducing the radiated sound for low k0. The combined beneficial effects of elasticity and porosity are shown to be effective in reducing the scattered sound for a broader range of k0 for perforated elastic plates.
Numerical Solution of Viscoplastic Flow Problems by Augmented Lagrangians.
1984-05-01
Lagrange multipliers. RAIRO Anal. Numer. 8R2, p 129-151. CIARLET, P. G. (1978]: The finite element method for elliptic problems. Amsterdan, North...lineaires. RAIRO , serie rouge, Anal. Numer., 11, p 369-400. TANGUY, P. (19831: Numerical Simulation of a Pseudo 3-D Turbulent Flow in a iAplan Turbine
2007-12-06
problems studied in this project involve numerically solving partial differential equations with either discontinuous or rapidly changing solutions ...REPORT Algorithm Development and Application of High Order Numerical Methods for Shocked and Rapid Changing Solutions 14. ABSTRACT 16. SECURITY...discontinuous Galerkin finite element methods, for solving partial differential equations with discontinuous or rapidly changing solutions . Algorithm
NASA Astrophysics Data System (ADS)
Liu, Yuxiang; Barnett, Alex H.
2016-11-01
We present a high-order accurate boundary-based solver for three-dimensional (3D) frequency-domain scattering from a doubly-periodic grating of smooth axisymmetric sound-hard or transmission obstacles. We build the one-obstacle solution operator using separation into P azimuthal modes via the FFT, the method of fundamental solutions (with N proxy points lying on a curve), and dense direct least-squares solves; the effort is O (N3 P) with a small constant. Periodizing then combines fast multipole summation of nearest neighbors with an auxiliary global Helmholtz basis expansion to represent the distant contributions, and enforcing quasiperiodicity and radiation conditions on the unit cell walls. Eliminating the auxiliary coefficients, and preconditioning with the one-obstacle solution operator, leaves a well-conditioned square linear system that is solved iteratively. The solution time per incident wave is then O (NP) at fixed frequency. Our scheme avoids singular quadratures, periodic Green's functions, and lattice sums, and its convergence rate is unaffected by resonances within obstacles. We include numerical examples such as scattering from a grating of period 13 λ × 13 λ comprising highly-resonant sound-hard ;cups; each needing NP = 64800 surface unknowns, to 10-digit accuracy, in half an hour on a desktop.
Energy expenditure during level human walking: seeking a simple and accurate predictive solution.
Ludlow, Lindsay W; Weyand, Peter G
2016-03-01
Accurate prediction of the metabolic energy that walking requires can inform numerous health, bodily status, and fitness outcomes. We adopted a two-step approach to identifying a concise, generalized equation for predicting level human walking metabolism. Using literature-aggregated values we compared 1) the predictive accuracy of three literature equations: American College of Sports Medicine (ACSM), Pandolf et al., and Height-Weight-Speed (HWS); and 2) the goodness-of-fit possible from one- vs. two-component descriptions of walking metabolism. Literature metabolic rate values (n = 127; speed range = 0.4 to 1.9 m/s) were aggregated from 25 subject populations (n = 5-42) whose means spanned a 1.8-fold range of heights and a 4.2-fold range of weights. Population-specific resting metabolic rates (V̇o2 rest) were determined using standardized equations. Our first finding was that the ACSM and Pandolf et al. equations underpredicted nearly all 127 literature-aggregated values. Consequently, their standard errors of estimate (SEE) were nearly four times greater than those of the HWS equation (4.51 and 4.39 vs. 1.13 ml O2·kg(-1)·min(-1), respectively). For our second comparison, empirical best-fit relationships for walking metabolism were derived from the data set in one- and two-component forms for three V̇o2-speed model types: linear (∝V(1.0)), exponential (∝V(2.0)), and exponential/height (∝V(2.0)/Ht). We found that the proportion of variance (R(2)) accounted for, when averaged across the three model types, was substantially lower for one- vs. two-component versions (0.63 ± 0.1 vs. 0.90 ± 0.03) and the predictive errors were nearly twice as great (SEE = 2.22 vs. 1.21 ml O2·kg(-1)·min(-1)). Our final analysis identified the following concise, generalized equation for predicting level human walking metabolism: V̇o2 total = V̇o2 rest + 3.85 + 5.97·V(2)/Ht (where V is measured in m/s, Ht in meters, and V̇o2 in ml O2·kg(-1)·min(-1)).
Kawai, Soshi; Terashima, Hiroshi; Negishi, Hideyo
2015-11-01
This paper addresses issues in high-fidelity numerical simulations of transcritical turbulent flows at supercritical pressure. The proposed strategy builds on a tabulated look-up table method based on REFPROP database for an accurate estimation of non-linear behaviors of thermodynamic and fluid transport properties at the transcritical conditions. Based on the look-up table method we propose a numerical method that satisfies high-order spatial accuracy, spurious-oscillation-free property, and capability of capturing the abrupt variation in thermodynamic properties across the transcritical contact surface. The method introduces artificial mass diffusivity to the continuity and momentum equations in a physically-consistent manner in order to capture the steep transcritical thermodynamic variations robustly while maintaining spurious-oscillation-free property in the velocity field. The pressure evolution equation is derived from the full compressible Navier–Stokes equations and solved instead of solving the total energy equation to achieve the spurious pressure oscillation free property with an arbitrary equation of state including the present look-up table method. Flow problems with and without physical diffusion are employed for the numerical tests to validate the robustness, accuracy, and consistency of the proposed approach.
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.
Runkel, Robert L.; Chapra, Steven C.
1993-01-01
Several investigators have proposed solute transport models that incorporate the effects of transient storage. Transient storage occurs in small streams when portions of the transported solute become isolated in zones of water that are immobile relative to water in the main channel (e.g., pools, gravel beds). Transient storage is modeled by adding a storage term to the advection-dispersion equation describing conservation of mass for the main channel. In addition, a separate mass balance equation is written for the storage zone. Although numerous applications of the transient storage equations may be found in the literature, little attention has been paid to the numerical aspects of the approach. Of particular interest is the coupled nature of the equations describing mass conservation for the main channel and the storage zone. In the work described herein, an implicit finite difference technique is developed that allows for a decoupling of the governing differential equations. This decoupling method may be applied to other sets of coupled equations such as those describing sediment-water interactions for toxic contaminants. For the case at hand, decoupling leads to a 50% reduction in simulation run time. Computational costs may be further reduced through efficient application of the Thomas algorithm. These techniques may be easily incorporated into existing codes and new applications in which simulation run time is of concern.
A GENERAL MASS-CONSERVATIVE NUMERICAL SOLUTION FOR THE UNSATURATED FLOW EQUATION
Numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems. Numerical solution based on the standard h-based form of Richards equation generally yields poor results, ch...
NASA Astrophysics Data System (ADS)
Thompson, D. S.
1980-05-01
The full Navier-Stokes equations for incompressible turbulent flow must be solved to accurately represent all flow phenomena which occur in a high Reynolds number incompressible flow. A two layer algebraic eddy viscosity turbulence model is used to represent the Reynolds stress in the primitive variable formulation. The development of the boundary-fitted coordinate systems makes the numerical solution of these equations feasible for arbitrarily shaped bodies. The nondimensional time averaged Navier-Stokes equations, including the turbulence mode, are represented by finite difference approximations in the transformed plane. The resulting coupled system of nonlinear algebraic equations is solved using a point successive over relaxation iteration. The test case considered was a NACA 64A010 airfoil section at an angle of attack of two degrees and a Reynolds number of 2,000,000.
Numerical solution of potential flow about arbitrary 2-dimensional multiple bodies
NASA Technical Reports Server (NTRS)
Thompson, J. F.; Thames, F. C.
1982-01-01
A procedure for the finite-difference numerical solution of the lifting potential flow about any number of arbitrarily shaped bodies is given. The solution is based on a technique of automatic numerical generation of a curvilinear coordinate system having coordinate lines coincident with the contours of all bodies in the field, regardless of their shapes and number. The effects of all numerical parameters involved are analyzed and appropriate values are recommended. Comparisons with analytic solutions for single Karman-Trefftz airfoils and a circular cylinder pair show excellent agreement. The technique of application of the boundary-fitted coordinate systems to the numerical solution of partial differential equations is illustrated.
NASA Astrophysics Data System (ADS)
Shukla, K.; Wang, Y.; Jaiswal, P.
2014-12-01
In a porous medium the seismic energy not only propagates through matrix but also through pore-fluids. The differential movement between sediment grains of the matrix and interstitial fluid generates a diffusive wave which is commonly referred to as the slow P-wave. A combined system of equation which includes both elastic and diffusive phases is known as the poroelasticity. Analyzing seismic data through poroelastic modeling results in accurate interpretation of amplitude and separation of wave modes, leading to more accurate estimation of geomehanical properties of rocks. Despite its obvious multi-scale application, from sedimentary reservoir characterization to deep-earth fractured crust, poroelasticity remains under-developed primarily due to the complex nature of its constituent equations. We present a detail formulation of poroleastic wave equations for isotropic media by combining the Biot's and Newtonian mechanics. System of poroelastic wave equation constitutes for eight time dependent hyperbolic PDEs in 2D whereas in case of 3D number goes up to thirteen. Eigen decomposition of Jacobian of these systems confirms the presence of an additional slow-P wave phase with velocity lower than shear wave, posing stability issues on numerical scheme. To circumvent the issue, we derived a numerical scheme using nodal discontinuous Galerkin approach by adopting the triangular meshes in 2D which is extended to tetrahedral for 3D problems. In our nodal DG approach the basis function over a triangular element is interpolated using Legendre-Gauss-Lobatto (LGL) function leading to a more accurate local solutions than in the case of simple DG. We have tested the numerical scheme for poroelastic media in 1D and 2D case, and solution obtained for the systems offers high accuracy in results over other methods such as finite difference , finite volume and pseudo-spectral. The nodal nature of our approach makes it easy to convert the application into a multi-threaded algorithm
Numerical solution of the stochastic parabolic equation with the dependent operator coefficient
Ashyralyev, Allaberen; Okur, Ulker
2015-09-18
In the present paper, a single step implicit difference scheme for the numerical solution of the stochastic parabolic equation with the dependent operator coefficient is presented. Theorem on convergence estimates for the solution of this difference scheme is established. In applications, this abstract result permits us to obtain the convergence estimates for the solution of difference schemes for the numerical solution of initial boundary value problems for parabolic equations. The theoretical statements for the solution of this difference scheme are supported by the results of numerical experiments.
Analytical solutions of moisture flow equations and their numerical evaluation
Gibbs, A.G.
1981-04-01
The role of analytical solutions of idealized moisture flow problems is discussed. Some different formulations of the moisture flow problem are reviewed. A number of different analytical solutions are summarized, including the case of idealized coupled moisture and heat flow. The evaluation of special functions which commonly arise in analytical solutions is discussed, including some pitfalls in the evaluation of expressions involving combinations of special functions. Finally, perturbation theory methods are summarized which can be used to obtain good approximate analytical solutions to problems which are too complicated to solve exactly, but which are close to an analytically solvable problem.
NASA Astrophysics Data System (ADS)
Masson, Yder; Romanowicz, Barbara
2016-11-01
We derive a fast discrete solution to the scattering problem. This solution allows us to compute accurate synthetic seismograms or waveforms for arbitrary locations of sources and receivers within a medium containing localized perturbations. The key to efficiency is that wave propagation modeling does not need to be carried out in the entire volume that encompasses the sources and the receivers but only within the sub-volume containing the perturbations or scatterers. The proposed solution has important applications, for example, it permits the imaging of remote targets located in regions where no sources or receivers are present. Our solution relies on domain decomposition: within a small volume that contains the scatterers, wave propagation is modeled numerically, while in the surrounding volume, where the medium isn't perturbed, the response is obtained through wavefield extrapolation. The originality of this work is the derivation of discrete formulas for representation theorems and Kirchhoff-Helmholtz integrals that naturally adapt to the numerical scheme employed for modeling wave propagation. Our solution applies, for example, to finite difference methods or finite/spectral elements methods. The synthetic seismograms obtained with our solution can be considered "exact" as the total numerical error is comparable to that of the method employed for modeling wave propagation. We detail a basic implementation of our solution in the acoustic case using the finite difference method and present numerical examples that demonstrate the accuracy of the method. We show that ignoring some terms accounting for higher order scattering effects in our solution has a limited effect on the computed seismograms and significantly reduces the computational effort. Finally, we show that our solution can be used to compute localised sensitivity kernels and we discuss applications to target oriented imaging. Extension to the elastic case is straightforward and summarised in a
Numerical solution of the unsteady Navier-Stokes equation
NASA Technical Reports Server (NTRS)
Osher, Stanley J.; Engquist, Bjoern
1985-01-01
The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws are discussed. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper a uniformly second-order approximation is constructed, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.
Numerical solution of the Kolmogorov-Feller equation with singularities
NASA Astrophysics Data System (ADS)
Baranov, N. A.; Turchak, L. I.
2010-02-01
A method is proposed for solving the Kolmogorov-Feller integro-differential equation with kernels containing delta function singularities. The method is based on a decomposition of the solution into regular and singular parts.
Adaptive Grid Generation for Numerical Solution of Partial Differential Equations.
1983-12-01
RETURN 65 Bibliography 1. Thompson , J . F ., "A Survey of Grid Generation Tecniques in Computational Fluid Dynamics," AIAA Paper No. 83-0447, 1-36...edited by K. N. Ghia and U. Ghia. ASME FED, 5: 35-47 (1983). 3. Thompson , J . F ., Thames, F. C., and Mastin, C. W., "Automated Numerical Generation...Equations," Numerical Grid Generation, Edited by J. F. Thompson. New York: North Holland, 1982. 10. Thompson , J . F ., and Mastin, C. W., "Grid Generation
Numerical Solution of Boundary Layer MHD Flow with Viscous Dissipation
Mishra, S. R.; Jena, S.
2014-01-01
The present paper deals with a steady two-dimensional laminar flow of a viscous incompressible electrically conducting fluid over a shrinking sheet in the presence of uniform transverse magnetic field with viscous dissipation. Using suitable similarity transformations the governing partial differential equations are transformed into ordinary differential equations and then solved numerically by fourth-order Runge-Kutta method with shooting technique. Results for velocity and temperature profiles for different values of the governing parameters have been discussed in detail with graphical representation. The numerical evaluation of skin friction and Nusselt number are also given in this paper. PMID:24672367
Lewis, E.R.; Schwartz, S.
2010-03-15
Light scattering by aerosols plays an important role in Earth’s radiative balance, and quantification of this phenomenon is important in understanding and accounting for anthropogenic influences on Earth’s climate. Light scattering by an aerosol particle is determined by its radius and index of refraction, and for aerosol particles that are hygroscopic, both of these quantities vary with relative humidity RH. Here exact expressions are derived for the dependences of the radius ratio (relative to the volume-equivalent dry radius) and index of refraction on RH for aqueous solutions of single solutes. Both of these quantities depend on the apparent molal volume of the solute in solution and on the practical osmotic coefficient of the solution, which in turn depend on concentration and thus implicitly on RH. Simple but accurate approximations are also presented for the RH dependences of both radius ratio and index of refraction for several atmospherically important inorganic solutes over the entire range of RH values for which these substances can exist as solution drops. For all substances considered, the radius ratio is accurate to within a few percent, and the index of refraction to within ~0.02, over this range of RH. Such parameterizations will be useful in radiation transfer models and climate models.
NASA Technical Reports Server (NTRS)
Amitai, Dganit; Averbuch, Amir; Itzikowitz, Samuel; Turkel, Eli
1991-01-01
A major problem in achieving significant speed-up on parallel machines is the overhead involved with synchronizing the concurrent process. Removing the synchronization constraint has the potential of speeding up the computation. The authors present asynchronous (AS) and corrected-asynchronous (CA) finite difference schemes for the multi-dimensional heat equation. Although the discussion concentrates on the Euler scheme for the solution of the heat equation, it has the potential for being extended to other schemes and other parabolic partial differential equations (PDEs). These schemes are analyzed and implemented on the shared memory multi-user Sequent Balance machine. Numerical results for one and two dimensional problems are presented. It is shown experimentally that the synchronization penalty can be about 50 percent of run time: in most cases, the asynchronous scheme runs twice as fast as the parallel synchronous scheme. In general, the efficiency of the parallel schemes increases with processor load, with the time level, and with the problem dimension. The efficiency of the AS may reach 90 percent and over, but it provides accurate results only for steady-state values. The CA, on the other hand, is less efficient, but provides more accurate results for intermediate (non steady-state) values.
NASA Technical Reports Server (NTRS)
Harris, Julius E.; Iyer, Venkit; Radwan, Samir
1987-01-01
The application of stability theory in Laminar Flow Control (LFC) research requires that density and velocity profiles be specified throughout the viscous flow field of interest. These profile values must be as numerically accurate as possible and free of any numerically induced oscillations. Guidelines for the present research project are presented: develop an efficient and accurate procedure for solving the 3-D boundary layer equation for aerospace configurations; develop an interface program to couple selected 3-D inviscid programs that span the subsonic to hypersonic Mach number range; and document and release software to the LFC community. The interface program was found to be a dependable approach for developing a user friendly procedure for generating the boundary-layer grid and transforming an inviscid solution from a relatively coarse grid to a sufficiently fine boundary-layer grid. The boundary-layer program was shown to be fourth-order accurate in the direction normal to the wall boundary and second-order accurate in planes parallel to the boundary. The fourth-order accuracy allows accurate calculations with as few as one-fifth the number of grid points required for conventional second-order schemes.
NASA Astrophysics Data System (ADS)
Soltani, Peyman; Darudi, Ahmad; Moradi, Ali Reza; Amiri, Javad; Nehmetallah, Georges
2016-05-01
In this paper, the Transport of Intensity Equation (TIE) for testing of an aspheric surface is verified experimentally. Using simulation, a proper defocus distance Δ𝑧 that leads to an accurate solution of TIE is estimated whenever the conic constant and configuration of the experiment are known. To verify this procedure a non-nulled experiment for testing an aspheric is used. For verification of the solution, the results are compared with the Shack-Hartmann sensor. The theoretical method and experimental results are compared to validate the results.
Numerical solution of control problems governed by nonlinear differential equations
Heinkenschloss, M.
1994-12-31
In this presentation the author investigates an iterative method for the solution of optimal control problems. These problems are formulated as constrained optimization problems with constraints arising from the state equation and in the form of bound constraints on the control. The method for the solution of these problems uses the special structure of the problem arising from the bound constraint and the state equation. It is derived from SQP methods and projected Newton methods and combines the advantages of both methods. The bound constraint is satisfied by all iterates using a projection, the nonlinear state equation is satisfied in the limit. Only a linearized state equation has to be solved in every iteration. The solution of the linearized problems are done using multilevel methods and GMRES.
Numerical Solution of Incompressible Navier-Stokes Equations Using a Fractional-Step Approach
NASA Technical Reports Server (NTRS)
Kiris, Cetin; Kwak, Dochan
1999-01-01
A fractional step method for the solution of steady and unsteady incompressible Navier-Stokes equations is outlined. The method is based on a finite volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (3rd and 5th) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds Numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when 5th-order upwind differencing and a modified production term in the Baldwin-Barth one-equation turbulence model are used with adequate grid resolution.
Mathematics: Numerical Solution of Inverse Problems in Acoustics
1992-04-30
Camerino Dipartimento di Matematica e Fisica 62032 Camerino (MC) Italy Index I Introduction pag. 3 2 Statement of the work accomplished pag. 4 3...Scientific Iles-areh under contract so AFOSR . 200.0228 with the Universit& di Carnerino tDipartimento di Matematica e Fisica UnIversitk dl Camerio- .2032...for time harmonic acoustic waves: a numerical method* Luciano Misici Dipartimento di Matematica e Fisica Universita di Camerino 62032 Camerino (MC
NASA Astrophysics Data System (ADS)
Nassar, Mohamed K.; Ginn, Timothy R.
2014-08-01
We investigate the effect of computational error on the inversion of a density-dependent flow and transport model, using SEAWAT and UCODE-2005 in an inverse identification of hydraulic conductivity and dispersivity using head and concentration data from a 2-D laboratory experiment. We investigated inversions using three different solution schemes including variation of number of particles and time step length, in terms of the three aspects: the shape and smoothness of the objective function surface, the consequent impacts to the optimization, and the resulting Pareto analyses. This study demonstrates that the inversion is very sensitive to the choice of the forward model solution scheme. In particular, standard finite difference methods provide the smoothest objective function surface; however, this is obtained at the cost of numerical artifacts that can lead to erroneous warping of the objective function surface. Total variation diminishing (TVD) schemes limit these impacts at the cost of more computation time, while the hybrid method of characteristics (HMOC) approach with increased particle numbers and/or reduced time step gives both smoothed and accurate objective function surface. Use of the most accurate methods (TVD and HMOC) did lead to successful inversion of the two parameters; however, with distinct results for Pareto analyses. These results illuminate the sensitivity of the inversion to a number of aspects of the forward solution of the density-driven flow problem and reveal that parameter values may result that are erroneous but that counteract numerical errors in the solution.
An Integrated Numerical Hydrodynamic Shallow Flow-Solute Transport Model for Urban Area
NASA Astrophysics Data System (ADS)
Alias, N. A.; Mohd Sidek, L.
2016-03-01
The rapidly changing on land profiles in the some urban areas in Malaysia led to the increasing of flood risk. Extensive developments on densely populated area and urbanization worsen the flood scenario. An early warning system is really important and the popular method is by numerically simulating the river and flood flows. There are lots of two-dimensional (2D) flood model predicting the flood level but in some circumstances, still it is difficult to resolve the river reach in a 2D manner. A systematic early warning system requires a precisely prediction of flow depth. Hence a reliable one-dimensional (1D) model that provides accurate description of the flow is essential. Research also aims to resolve some of raised issues such as the fate of pollutant in river reach by developing the integrated hydrodynamic shallow flow-solute transport model. Presented in this paper are results on flow prediction for Sungai Penchala and the convection-diffusion of solute transports simulated by the developed model.
Numerical Solution of Hamilton-Jacobi Equations in High Dimension
2012-11-23
high dimension FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA-Universita di Roma P. Aldo Moro, 2 00185 ROMA AH930...Università di Roma P. Aldo Moro, 2 - 00185 ROMA email: falcone@mat.uniroma1.it November 23, 2012 Abstract The solution of nonlinear optimal control problems
On the numerical study of the rational solutions of the nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Islas, A.; Schober, C. M.
2016-10-01
The stability of the rational solutions of the nonlinear Schrödinger (NLS) equation has only recently began to be addressed. In this paper we develop a Chebyshev pseudo-spectral method for the NLS equation to study the stability of the rational solutions. Using the map x = cot θ and the Fast Fourier Transform (FFT) to approximate uxx, the Chebyshev scheme effectively handles the infinite line boundary conditions. An extensive numerical study, involving large ensembles of perturbed initial data for the Peregrine solution (the lowest order rational solution), indicates it is linearly unstable. Working with unstable solutions is numerically challenging. In the current literature, numerical experiments related to the Peregrine solution frequently employ standard Fourier methods without a discussion of the related numerical issues. We examine the performance of a Fourier pseudo-spectral method (FPS4) using Peregrine initial data. Applying FPS4, tiny Gibbs oscillations occur in the first few steps of the numerical solution. These oscillations grow to O(1), providing further evidence of the instability of the Peregrine solution. We modify the FPS4 method using a spectral-splitting technique which resolves the Gibbs oscillations and significantly improves the numerical solution.
Improved numerical solutions for chaotic-cancer-model
NASA Astrophysics Data System (ADS)
Yasir, Muhammad; Ahmad, Salman; Ahmed, Faizan; Aqeel, Muhammad; Akbar, Muhammad Zubair
2017-01-01
In biological sciences, dynamical system of cancer model is well known due to its sensitivity and chaoticity. Present work provides detailed computational study of cancer model by counterbalancing its sensitive dependency on initial conditions and parameter values. Cancer chaotic model is discretized into a system of nonlinear equations that are solved using the well-known Successive-Over-Relaxation (SOR) method with a proven convergence. This technique enables to solve large systems and provides more accurate approximation which is illustrated through tables, time history maps and phase portraits with detailed analysis.
Isomorphism and solid solution as shown by an accurate high-resolution diffraction experiment.
Poulain, Agnieszka; Kubicki, Maciej; Lecomte, Claude
2014-12-01
High-resolution crystal structure determination and spherical and multipolar refinement enabled an organic solid solution of 1-(4'-chlorophenyl)-2-methyl-4-nitro-1H-imidazole-5-carbonitrile and 5-bromo-1-(4'-chlorophenyl)-2-methyl-4-nitro-1H-imidazole to be found, which would not normally be revealed using only standard resolution data (ca 0.8 Å), as the disordered part is only visible at high resolution. Therefore, this new structure would have been reported as just another polymorphic form, even more reasonably as isostructural with other derivatives. To the best of our knowledge this is the first example of organic solid solution modelled via charge density Hansen-Coppens formalism and analysed by means of quantum theory of atoms in molecules (QTAIM) theory.
High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities
2015-03-31
restoring the design accuracy of the scheme in the presence of singularities at the boundary. While this method is well studied for low order methods...boundary. While this method is well studied for low order methods and for problems in which singularities arise from the geometry (e.g., corners), we adapt...Solution of multiple problems at low cost . . . . . . . . . . . . . . . . . . 56 3.3.2 Parameters of the computational setting
The accurate solution of Poisson's equation by expansion in Chebyshev polynomials
NASA Technical Reports Server (NTRS)
Haidvogel, D. B.; Zang, T.
1979-01-01
A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. Solutions are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.
Numerical solutions of the complete Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Hassan, H. A.
1993-01-01
The objective of this study is to compare the use of assumed pdf (probability density function) approaches for modeling supersonic turbulent reacting flowfields with the more elaborate approach where the pdf evolution equation is solved. Assumed pdf approaches for averaging the chemical source terms require modest increases in CPU time typically of the order of 20 percent above treating the source terms as 'laminar.' However, it is difficult to assume a form for these pdf's a priori that correctly mimics the behavior of the actual pdf governing the flow. Solving the evolution equation for the pdf is a theoretically sound approach, but because of the large dimensionality of this function, its solution requires a Monte Carlo method which is computationally expensive and slow to coverage. Preliminary results show both pdf approaches to yield similar solutions for the mean flow variables.
Numerical Solution of Ill Posed Problems in Partial Differential Equations.
1987-09-01
periodic solutions of semilinear wave equations in exterior domains (breathers). Necessary and sufficient conditions for the existence of such...Crandall, M.G., and Sacks, P.E., Some L1 existence and depandence results for semilinear elliptic equations under nonlinear boundary conditions , to...the former case, a convective diffusion equation with a semilinear source in the boundary condition was analyzed. A fairly complete picture of the
Numerical solutions of the three-dimensional magnetohydrodynamic alpha model.
Mininni, Pablo D; Montgomery, David C; Pouquet, Annick
2005-04-01
We present direct numerical simulations and alpha -model simulations of four familiar three-dimensional magnetohydrodynamic (MHD) turbulence effects: selective decay, dynamic alignment, inverse cascade of magnetic helicity, and the helical dynamo effect. The MHD alpha model is shown to capture the long-wavelength spectra in all these problems, allowing for a significant reduction of computer time and memory at the same kinetic and magnetic Reynolds numbers. In the helical dynamo, not only does the alpha model correctly reproduce the growth rate of magnetic energy during the kinematic regime, it also captures the nonlinear saturation level and the late generation of a large scale magnetic field by the helical turbulence.
Numerical solutions for unsteady subsonic vortical flows around loaded cascades
NASA Technical Reports Server (NTRS)
Fang, J.; Atassi, H. M.
1992-01-01
A frequency domain linearized unsteady aerodynamic analysis is presented for three-dimensional unsteady vortical flows around a cascade of loaded airfoils. The analysis fully accounts for the distortion of the impinging vortical disturbances by the mean flow. The entire unsteady flow field is calculated in response to upstream three-dimensional harmonic disturbances. Numerical results are presented for two standard cascade configurations representing turbine and compressor bladings for a reduced frequency range from 0.1 to 5. Results show that the upstream gust conditions and blade sweep strongly affect the unsteady blade response.
Numerical solutions for patterns statistics on Markov chains.
Nuel, Gregory
2006-01-01
We propose here a review of the methods available to compute pattern statistics on text generated by a Markov source. Theoretical, but also numerical aspects are detailed for a wide range of techniques (exact, Gaussian, large deviations, binomial and compound Poisson). The SPatt package (Statistics for Pattern, free software available at http://stat.genopole.cnrs.fr/spatt) implementing all these methods is then used to compare all these approaches in terms of computational time and reliability in the most complete pattern statistics benchmark available at the present time.
NASA Astrophysics Data System (ADS)
Sarojkumar, K.; Krishna, S.
2016-08-01
Online dynamic security assessment (DSA) is a computationally intensive task. In order to reduce the amount of computation, screening of contingencies is performed. Screening involves analyzing the contingencies with the system described by a simpler model so that computation requirement is reduced. Screening identifies those contingencies which are sure to not cause instability and hence can be eliminated from further scrutiny. The numerical method and the step size used for screening should be chosen with a compromise between speed and accuracy. This paper proposes use of energy function as a measure of error in the numerical solution used for screening contingencies. The proposed measure of error can be used to determine the most accurate numerical method satisfying the time constraint of online DSA. Case studies on 17 generator system are reported.
NASA Astrophysics Data System (ADS)
Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Van Gorder, Robert A.
2014-03-01
A Jacobi-Gauss-Lobatto collocation (J-GL-C) method, used in combination with the implicit Runge-Kutta method of fourth order, is proposed as a numerical algorithm for the approximation of solutions to nonlinear Schrödinger equations (NLSE) with initial-boundary data in 1+1 dimensions. Our procedure is implemented in two successive steps. In the first one, the J-GL-C is employed for approximating the functional dependence on the spatial variable, using (N-1) nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon two general Jacobi parameters. The resulting equations together with the two-point boundary conditions induce a system of 2(N-1) first-order ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve this temporal system. The proposed J-GL-C method, used in combination with the implicit Runge-Kutta method of fourth order, is employed to obtain highly accurate numerical approximations to four types of NLSE, including the attractive and repulsive NLSE and a Gross-Pitaevskii equation with space-periodic potential. The numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively few nodes used, the absolute error in our numerical solutions is sufficiently small.
NASA Astrophysics Data System (ADS)
Ray, Sudipta; Saha, Sandeep
2016-11-01
Numerical solution of engineering problems with interfacial discontinuities requires exact implementation of the jump conditions else the accuracy deteriorates significantly; particularly, achieving spectral accuracy has been limited due to complex interface geometry and Gibbs phenomenon. We adopt a novel implementation of the immersed-interface method that satisfies the jump conditions at the interfaces exactly, in conjunction with the Chebyshev-collocation method. We consider solutions to linear second order ordinary and partial differential equations having a discontinuity in their zeroth and first derivatives across an interface traced by a complex curve. The solutions obtained demonstrate the ability of the proposed method to achieve spectral accuracy for discontinuous solutions across tortuous interfaces. The solution methodology is illustrated using two model problems: (i) an ordinary differential equation with jump conditions forced by an infinitely differentiable function, (ii) Poisson's equation having a discontinuous solution across interfaces that are ellipses of varying aspect ratio. The use of more polynomials in the direction of the major axis than the minor axis of the ellipse increases the convergence rate of the solution.
NASA Astrophysics Data System (ADS)
Murashige, Sunao
This paper considers numerical methods for stability analyses of periodic solutions of ordinary differential equations. Stability of a periodic solution can be determined by the corresponding monodromy matrix and its eigenvalues. Some commonly used numerical methods can produce inaccurate results of them in some cases, for example, near bifurcation points or when one of the eigenvalues is very large or very small. This work proposes a numerical method using a periodic boundary condition for vector fields, which preserves a critical property of the monodromy matrix. Numerical examples demonstrate effectiveness and a drawback of this method.
MHD micropumping of power-law fluids: A numerical solution
NASA Astrophysics Data System (ADS)
Moghaddam, Saied
2013-02-01
The performance of MHD micropumps is studied numerically assuming that the viscosity of the fluid is shear-dependent. Using power-law model to represent the fluid of interest, the effect of power-law exponent, N, is investigated on the volumetric flow rate in a rectangular channel. Assuming that the flow is laminar, incompressible, two-dimensional, but (approximately) unidirectional, finite difference method (FDM) is used to solve the governing equations. It is found that shear-thinning fluids provide a larger flow rate as compared to Newtonian fluids provided that the Hartmann number is above a critical value. There exists also an optimum Hartmann number (which is larger than the critical Hartmann number) at which the flow rate is maximum. The power-law exponent, N, strongly affects the optimum geometry depending on the Hartmann number being smaller or larger than the critical Hartmann number.
Numerical solutions of atmospheric flow over semielliptical simulated hills
NASA Technical Reports Server (NTRS)
Shieh, C. F.; Frost, W.
1981-01-01
Atmospheric motion over obstacles on plane surfaces to compute simulated wind fields over terrain features was studied. Semielliptical, two dimensional geometry and numerical simulation of flow over rectangular geometries is also discussed. The partial differential equations for the vorticity, stream function, turbulence kinetic energy, and turbulence length scale were solved by a finite difference technique. The mechanism of flow separation induced by a semiellipse is the same as flow over a gradually sloping surface for which the flow separation is caused by the interaction between the viscous force, the pressure force, and the turbulence level. For flow over bluff bodies, a downstream recirculation bubble is created which increases the aspect ratio and/or the turbulence level results in flow reattachment close behind the obstacle.
Mutelet, Fabrice; Jaubert, Jean-Noël
2006-01-13
Activity coefficients at infinite dilution of 29 organic compounds in two room temperature ionic liquids were determined using inverse gas chromatography. The measurements were carried out at different temperatures between 323.15 and 343.15K. To establish the influence of concurrent retention mechanisms on the accuracy of activity coefficients at infinite dilution for 1-butyl-3-methylimidazolium octyl sulfate and 1-ethyl-3-methylimidazolium tosylate, phase loading studies of the net retention volume per gram of packing as a function of the percent phase loading were used. It is shown that most of the solutes are retained largely by partition with a small contribution from adsorption on 1-butyl-3-methylimidazolium octyl sulfate and that the n-alkanes are retained predominantly by interfacial adsorption on 1-ethyl-3-methylimidazolium tosylate.
Design and Construction Solutions in the Accurate Realization of NCSX Magnetic Fields
Heitzenroeder, P.; Dudek, Lawrence E.; Brooks, Arthur W.; Viola, Michael E.; Brown, Thomas; Neilson, George H.; Zarnstorff, Michael C.; Rej, Donald; Cole,Michael J.; Freudenberg, Kevin D.; Harris J. H.; McGinnis, Gary
2008-09-29
The National Compact Stellarator Experiment, NCSX, is being constructed at the Princeton Plasma Physics Laboratory (PPPL) in partnership with the Oak Ridge national Laboratory. The goal of NCSX is to provide the understanding necessary to develop an attractive, disruption free, steady state compact stellaratorbased reactor design. This paper describes the recently revised designs of the critical interfaces between the modular coils, the construction solutions developed to meet assembly tolerances, and the recently revised trim coil system that provides the required compensation to correct for the “as built” conditions and to allow flexibility in the disposition of as-built conditions. In May, 2008, the sponsor decided to terminate the NCSX project due to growth in the project’s cost and schedule estimates. However significant technical challenges in design and construction were overcome, greatly reducing the risk in the remaining work to complete the project.
Zaidel, Jacob
2013-01-01
Known analytical solutions of groundwater flow equations are routinely used for verification of computer codes. However, these analytical solutions (e.g., the Dupuit solution for the steady-state unconfined unidirectional flow in a uniform aquifer with a flat bottom) represent smooth and continuous water table configurations, simulating which does not pose any significant problems for the numerical groundwater flow models, like MODFLOW. One of the most challenging numerical cases for MODFLOW arises from drying-rewetting problems often associated with abrupt changes in the elevations of impervious base of a thin unconfined aquifer. Numerical solutions of groundwater flow equations cannot be rigorously verified for such cases due to the lack of corresponding exact analytical solutions. Analytical solutions of the steady-state Boussinesq equation, associated with the discontinuous water table configurations over a stairway impervious base, are presented in this article. Conditions resulting in such configurations are analyzed and discussed. These solutions appear to be well suited for testing and verification of computer codes. Numerical solutions, obtained by the latest versions of MODFLOW (MODFLOW-2005 and MODFLOW-NWT), are compared with the presented discontinuous analytical solutions. It is shown that standard MODFLOW-2005 code (as well as MODFLOW-2000 and older versions) has significant convergence problems simulating such cases. The problems manifest themselves either in a total convergence failure or erroneous results. Alternatively, MODFLOW-NWT, providing a good match to the presented discontinuous analytical solutions, appears to be a more reliable and appropriate code for simulating abrupt changes in water table elevations.
Hong Xinguo; Hao Quan
2009-01-15
In this paper, we report a method of precise in situ x-ray scattering measurements on protein solutions using small stationary sample cells. Although reduction in the radiation damage induced by intense synchrotron radiation sources is indispensable for the correct interpretation of scattering data, there is still a lack of effective methods to overcome radiation-induced aggregation and extract scattering profiles free from chemical or structural damage. It is found that radiation-induced aggregation mainly begins on the surface of the sample cell and grows along the beam path; the diameter of the damaged region is comparable to the x-ray beam size. Radiation-induced aggregation can be effectively avoided by using a two-dimensional scan (2D mode), with an interval as small as 1.5 times the beam size, at low temperature (e.g., 4 deg. C). A radiation sensitive protein, bovine hemoglobin, was used to test the method. A standard deviation of less than 5% in the small angle region was observed from a series of nine spectra recorded in 2D mode, in contrast to the intensity variation seen using the conventional stationary technique, which can exceed 100%. Wide-angle x-ray scattering data were collected at a standard macromolecular diffraction station using the same data collection protocol and showed a good signal/noise ratio (better than the reported data on the same protein using a flow cell). The results indicate that this method is an effective approach for obtaining precise measurements of protein solution scattering.
NASA Astrophysics Data System (ADS)
Hong, Xinguo; Hao, Quan
2009-01-01
In this paper, we report a method of precise in situ x-ray scattering measurements on protein solutions using small stationary sample cells. Although reduction in the radiation damage induced by intense synchrotron radiation sources is indispensable for the correct interpretation of scattering data, there is still a lack of effective methods to overcome radiation-induced aggregation and extract scattering profiles free from chemical or structural damage. It is found that radiation-induced aggregation mainly begins on the surface of the sample cell and grows along the beam path; the diameter of the damaged region is comparable to the x-ray beam size. Radiation-induced aggregation can be effectively avoided by using a two-dimensional scan (2D mode), with an interval as small as 1.5 times the beam size, at low temperature (e.g., 4 °C). A radiation sensitive protein, bovine hemoglobin, was used to test the method. A standard deviation of less than 5% in the small angle region was observed from a series of nine spectra recorded in 2D mode, in contrast to the intensity variation seen using the conventional stationary technique, which can exceed 100%. Wide-angle x-ray scattering data were collected at a standard macromolecular diffraction station using the same data collection protocol and showed a good signal/noise ratio (better than the reported data on the same protein using a flow cell). The results indicate that this method is an effective approach for obtaining precise measurements of protein solution scattering.
Numerical solutions of the complete Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Hassan, H. A.
1986-01-01
Using ideas from the kinetic theory, the Navier-Stokes equations are modified in such a way that they can be cast as a set of first order hyperbolic equations. This is achieved by incorporating time dependent terms into the definition of the stress tensor and the heat flux vectors. The boundary conditions are then determined from the theory of characteristics. Because the resulting equations reduce to the traditional Navier-Stokes equations when the steady state is reached, the present approach provides a straightforward scheme for the determination of inflow and outflow boundary conditions. The method is validated by comparing its predictions with known exact solutions of the steady Navier-Stokes equations.
Numerical diagnostics of solution blowup in differential equations
NASA Astrophysics Data System (ADS)
Belov, A. A.
2017-01-01
New simple and robust methods have been proposed for detecting poles, logarithmic poles, and mixed-type singularities in systems of ordinary differential equations. The methods produce characteristics of these singularities with a posteriori asymptotically precise error estimates. This approach is applicable to an arbitrary parametrization of integral curves, including the arc length parametrization, which is optimal for stiff and ill-conditioned problems. The method can be used to detect solution blowup for a broad class of important nonlinear partial differential equations, since they can be reduced to huge-order systems of ordinary differential equations by applying the method of lines. The method is superior in robustness and simplicity to previously known methods.
Numerical solution of differential equations by artificial neural networks
NASA Technical Reports Server (NTRS)
Meade, Andrew J., Jr.
1995-01-01
Conventionally programmed digital computers can process numbers with great speed and precision, but do not easily recognize patterns or imprecise or contradictory data. Instead of being programmed in the conventional sense, artificial neural networks (ANN's) are capable of self-learning through exposure to repeated examples. However, the training of an ANN can be a time consuming and unpredictable process. A general method is being developed by the author to mate the adaptability of the ANN with the speed and precision of the digital computer. This method has been successful in building feedforward networks that can approximate functions and their partial derivatives from examples in a single iteration. The general method also allows the formation of feedforward networks that can approximate the solution to nonlinear ordinary and partial differential equations to desired accuracy without the need of examples. It is believed that continued research will produce artificial neural networks that can be used with confidence in practical scientific computing and engineering applications.
Numerical solution of an edge flame boundary value problem
NASA Astrophysics Data System (ADS)
Shields, Benjamin; Freund, Jonathan; Pantano, Carlos
2016-11-01
We study edge flames for modeling extinction, reignition, and flame lifting in turbulent non-premixed combustion. An adaptive resolution finite element method is developed for solving a strained laminar edge flame in the intrinsic moving frame of reference of a spatially evolving shear layer. The variable-density zero Mach Navier-Stokes equations are used to solve for both advancing and retreating edge flames. The eigenvalues of the system are determined simultaneously (implicitly) with the scalar fields using a Schur complement strategy. A homotopy transformation over density is used to transition from constant- to variable-density, and pseudo arc-length continuation is used for parametric tracing of solutions. Full details of the edge flames as a function of strain and Lewis numbers will be discussed. This material is based upon work supported [in part] by the Department of Energy, National Nuclear Security Administration, under Award Number DE-NA0002374.
NASA Astrophysics Data System (ADS)
Tocci, Michael D.; Kelley, C. T.; Miller, Cass T.
The pressure-head form of Richards' equation (RE) is difficult to solve accurately using standard time integration methods. For example, mass balance errors grow as the integration progresses unless very small time steps are taken. Further, RE may be solved for many problems more economically and robustly with variable-size time steps rather than with a constant time-step size, but variable step-size methods applied to date have relied upon empirical approaches to control step size, which do not explicitly control temporal truncation error of the solution. We show how a differential algebrain equation implementation of the method of lines can give solutions to RE that are accurate, have good mass balance properties, explicitly control temporal truncation error, and are more economical than standard approaches for a wide range of solution accuracy. We detail changes to a standard integrator, DASPK, that improves efficiency for the test problems considered, and we advocate the use of this approach for both RE and other problems involving subsurface flow and transport phenomena.
Finite-difference scheme for the numerical solution of the Schroedinger equation
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.; Ramadhani, Issa
1992-01-01
A finite-difference scheme for numerical integration of the Schroedinger equation is constructed. Asymptotically (r goes to infinity), the method gives the exact solution correct to terms of order r exp -2.
Numerical solution of the Navier-Stokes equations for arbitrary 2-dimensional multi-element airfoils
NASA Technical Reports Server (NTRS)
Thompson, J. F.
1983-01-01
Numerical solutions of the Navier-Stokes equations, with an algebraic turbulence model, for time-dependent two dimensional flow about multi-element airfoils were developed. Fundamental to these solutions was the use of numerically-generated boundary-conforming curvilinear coordinate systems to allow bodies of arbitrary shape to be treated. A general two dimensional grid generation code for multiple-body configuration was written as a part of this project and made available through the COSMIC code library.
Numerical Solution of Inviscid Compressible Steady Flows around the RAE 2822 Airfoil
NASA Astrophysics Data System (ADS)
Kryštůfek, P.; Kozel, K.
2015-05-01
The article presents results of a numerical solution of subsonic, transonic and supersonic flows described by the system of Euler equations in 2D compressible flows around the RAE 2822 airfoil. Authors used FVM multistage Runge-Kutta method to numerically solve the flows around the RAE 2822 airfoil. The results are compared with the solution using the software Ansys Fluent 15.0.7.
Numerical solution of multiparameter spectral problems by high order finite different schemes
NASA Astrophysics Data System (ADS)
Amodio, Pierluigi; Settanni, Giuseppina
2016-10-01
We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spectral problems for ordinary differential equations. We describe how to obtain a discrete problem by means of High Order Finite Difference Schemes and discuss its numerical solution. Based on this approach, we also define a recursive algorithm to compute approximations of the parameters by means of the solution of a set of problems converging to the original one.
A numerical inversion of a the Laplace transform solution to radial dispersion in a porous medium.
Moench, A.F.; Ogata, A.
1981-01-01
A special form of the numerical inversion of the Laplace transform described by Stehfest (1970) is applied to the transformed solution of dispersion in a radial flow system in a porous medium. The inversion is extremely simple to use because the weighting coefficients depend only on the number of terms used in the computation and not upon the transform solution as required by most numerical inversion techniques.-from Authors
The Numerical Solution of Boundary Value Problems on ’Long’ Intervals.
1981-04-01
paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. These problems have the...following form. We look for the solution of a system of ordinary differential equations which is defined on the interval [I,-] which fulfills a...J. Numer. Anal. 10, 637-669. 4. F. R. de Hoog and R. Weiss (1980a). On the Boundary Value Problem for Systems of Ordinary Differential equations With
NASA Technical Reports Server (NTRS)
Gossard, Myron L
1952-01-01
An iterative transformation procedure suggested by H. Wielandt for numerical solution of flutter and similar characteristic-value problems is presented. Application of this procedure to ordinary natural-vibration problems and to flutter problems is shown by numerical examples. Comparisons of computed results with experimental values and with results obtained by other methods of analysis are made.
Numerical Solution of Compressible Steady Flows around the RAE 2822 Airfoil
NASA Astrophysics Data System (ADS)
Kryštůfek, P.; Kozel, K.
2014-03-01
The article presents results of a numerical solution of subsonic, transonic and supersonic flows described by the system of Navier-Stokes equations in 2D laminar compressible flows around the RAE 2822 airfoil. Authors used FVM multistage Runge-Kutta method to numerically solve the flows around the RAE 2822 airfoil.
NASA Astrophysics Data System (ADS)
Karafyllis, Iasson; Grüne, Lars
2009-09-01
In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations, by means of tools from nonlinear control theory. Lyapunov-based stabilization methods are exploited.
Numerical solution of 2D-vector tomography problem using the method of approximate inverse
NASA Astrophysics Data System (ADS)
Svetov, Ivan; Maltseva, Svetlana; Polyakova, Anna
2016-08-01
We propose a numerical solution of reconstruction problem of a two-dimensional vector field in a unit disk from the known values of the longitudinal and transverse ray transforms. The algorithm is based on the method of approximate inverse. Numerical simulations confirm that the proposed method yields good results of reconstruction of vector fields.
A numerical solution to the cattaneo-mindlin problem for viscoelastic materials
NASA Astrophysics Data System (ADS)
Spinu, S.; Cerlinca, D.
2016-08-01
The problem of the frictional mechanical contact with slip and stick, also referred to as the Cattaneo-Mindlin problem, is an important topic in engineering, with applications in the modeling of particle-flow simulations or in the study of contact between rough surfaces. In the frame of Linear Theory of Elasticity, accurate description of the slip-stick contact can only be achieved numerically, due to mutual interaction between normal and shear contact tractions. Additional difficulties arise when considering a viscoelastic constitutive law, as the mechanical response of the contacting materials depends explicitly on time. To overcome this obstacle, an existing algorithm for the purely elastic slip-stick contact is coupled with a semi-analytical method for viscoelastic displacement computation. The main advantage of this approach is that the contact model can be divided in subunits having the same structure as that of the purely elastic frictionless contact model, for which a well-established solution is readily available. In each time step, the contact solver assesses the contact area, the pressure distribution, the stick area and the shear tractions that satisfy the contact compatibility conditions and the static force equilibrium in both normal and tangential directions. A temporal discretization of the simulation windows assures that the memory effect, specific to both viscoelasticity and friction as a path-dependent processes, is properly replicated.
Code and Solution Verification of 3D Numerical Modeling of Flow in the Gust Erosion Chamber
NASA Astrophysics Data System (ADS)
Yuen, A.; Bombardelli, F. A.
2014-12-01
Erosion microcosms are devices commonly used to investigate the erosion and transport characteristics of sediments at the bed of rivers, lakes, or estuaries. In order to understand the results these devices provide, the bed shear stress and flow field need to be accurately described. In this research, the UMCES Gust Erosion Microcosm System (U-GEMS) is numerically modeled using Finite Volume Method. The primary aims are to simulate the bed shear stress distribution at the surface of the sediment core/bottom of the microcosm, and to validate the U-GEMS produces uniform bed shear stress at the bottom of the microcosm. The mathematical model equations are solved by on a Cartesian non-uniform grid. Multiple numerical runs were developed with different input conditions and configurations. Prior to developing the U-GEMS model, the General Moving Objects (GMO) model and different momentum algorithms in the code were verified. Code verification of these solvers was done via simulating the flow inside the top wall driven square cavity on different mesh sizes to obtain order of convergence. The GMO model was used to simulate the top wall in the top wall driven square cavity as well as the rotating disk in the U-GEMS. Components simulated with the GMO model were rigid bodies that could have any type of motion. In addition cross-verification was conducted as results were compared with numerical results by Ghia et al. (1982), and good agreement was found. Next, CFD results were validated by simulating the flow within the conventional microcosm system without suction and injection. Good agreement was found when the experimental results by Khalili et al. (2008) were compared. After the ability of the CFD solver was proved through the above code verification steps. The model was utilized to simulate the U-GEMS. The solution was verified via classic mesh convergence study on four consecutive mesh sizes, in addition to that Grid Convergence Index (GCI) was calculated and based on
On the numerical solution of the dynamically loaded hydrodynamic lubrication of the point contact
NASA Technical Reports Server (NTRS)
Lim, Sang G.; Prahl, Joseph M.; Brewe, David E.
1991-01-01
A numerical transient solution of the hydrodynamically lubricated point contact problem is obtained using the ball-on-plane model. Results, which include the variation of the minimum film thickness and phase-lag with time as functions of excitation frequency, are compared with the analytic solution of the transient step bearing problem with the same dynamic loading function.
New numerical methods for open-loop and feedback solutions to dynamic optimization problems
NASA Astrophysics Data System (ADS)
Ghosh, Pradipto
The topic of the first part of this research is trajectory optimization of dynamical systems via computational swarm intelligence. Particle swarm optimization is a nature-inspired heuristic search method that relies on a group of potential solutions to explore the fitness landscape. Conceptually, each particle in the swarm uses its own memory as well as the knowledge accumulated by the entire swarm to iteratively converge on an optimal or near-optimal solution. It is relatively straightforward to implement and unlike gradient-based solvers, does not require an initial guess or continuity in the problem definition. Although particle swarm optimization has been successfully employed in solving static optimization problems, its application in dynamic optimization, as posed in optimal control theory, is still relatively new. In the first half of this thesis particle swarm optimization is used to generate near-optimal solutions to several nontrivial trajectory optimization problems including thrust programming for minimum fuel, multi-burn spacecraft orbit transfer, and computing minimum-time rest-to-rest trajectories for a robotic manipulator. A distinct feature of the particle swarm optimization implementation in this work is the runtime selection of the optimal solution structure. Optimal trajectories are generated by solving instances of constrained nonlinear mixed-integer programming problems with the swarming technique. For each solved optimal programming problem, the particle swarm optimization result is compared with a nearly exact solution found via a direct method using nonlinear programming. Numerical experiments indicate that swarm search can locate solutions to very great accuracy. The second half of this research develops a new extremal-field approach for synthesizing nearly optimal feedback controllers for optimal control and two-player pursuit-evasion games described by general nonlinear differential equations. A notable revelation from this development
Application of symbolic/numeric matrix solution techniques to the NASTRAN program
NASA Technical Reports Server (NTRS)
Buturla, E. M.; Burroughs, S. H.
1977-01-01
The matrix solving algorithm of any finite element algorithm is extremely important since solution of the matrix equations requires a large amount of elapse time due to null calculations and excessive input/output operations. An alternate method of solving the matrix equations is presented. A symbolic processing step followed by numeric solution yields the solution very rapidly and is especially useful for nonlinear problems.
Computational experiment on the numerical solution of some inverse problems of mathematical physics
NASA Astrophysics Data System (ADS)
Vasil'ev, V. I.; Kardashevsky, A. M.; Sivtsev, PV
2016-11-01
In this article the computational experiment on the numerical solution of the most popular linear inverse problems for equations of mathematical physics are presented. The discretization of retrospective inverse problem for parabolic equation is performed using difference scheme with non-positive weight multiplier. Similar difference scheme is also used for the numerical solution of Cauchy problem for two-dimensional Laplace equation. The results of computational experiment, performed on model problems with exact solution, including ones with randomly perturbed input data are presented and discussed.
NASA Technical Reports Server (NTRS)
Daso, E. O.
1986-01-01
An implicit approximate factorization algorithm is employed to quantify the parametric effects of Courant number and artificial smoothing on numerical solutions of the unsteady 3-D Euler equations for a windmilling propeller (low speed) flow field. The results show that propeller global or performance chracteristics vary strongly with Courant number and artificial dissipation parameters, though the variation is such less severe at high Courant numbers. Candidate sets of Courant number and dissipation parameters could result in parameter-dependent solutions. Parameter-independent numerical solutions can be obtained if low values of the dissipation parameter-time step ratio are used in the computations. Furthermore, it is realized that too much artificial damping can degrade numerical stability. Finally, it is demonstrated that highly resolved meshes may, in some cases, delay convergence, thereby suggesting some optimum cell size for a given flow solution. It is suspected that improper boundary treatment may account for the cell size constraint.
Numerical solution of DGLAP equations using Laguerre polynomials expansion and Monte Carlo method.
Ghasempour Nesheli, A; Mirjalili, A; Yazdanpanah, M M
2016-01-01
We investigate the numerical solutions of the DGLAP evolution equations at the LO and NLO approximations, using the Laguerre polynomials expansion. The theoretical framework is based on Furmanski et al.'s articles. What makes the content of this paper different from other works, is that all calculations in the whole stages to extract the evolved parton distributions, are done numerically. The employed techniques to do the numerical solutions, based on Monte Carlo method, has this feature that all the results are obtained in a proper wall clock time by computer. The algorithms are implemented in FORTRAN and the employed coding ideas can be used in other numerical computations as well. Our results for the evolved parton densities are in good agreement with some phenomenological models. They also indicate better behavior with respect to the results of similar numerical calculations.
NASA Astrophysics Data System (ADS)
Kong, Dali; Zhang, Keke; Schubert, Gerald
2017-02-01
It is expected that the Juno spacecraft will provide an accurate spectrum of the Jovian zonal gravitational coefficients that would be affected by both the deep zonal flow, if it exists, and the basic rotational distortion. We derive the first analytical solution, under the spheroidal-shape approximation, for the density anomaly induced by an internal zonal flow in rapidly rotating Jupiter-like planets. We compare the density anomaly of the analytical solution to that obtained from a fully numerical solution based on a three-dimensional finite element method; the two show excellent agreement. We apply the analytical solution to a rapidly rotating Jupiter-like planet and show that there exists a close relationship between the spatial structure of the zonal flow and the spectrum of zonal gravitational coefficients. We check the accuracy of the spheroidal-shape approximation by computing both the spheroidal and non-spheroidal solutions with exactly the same physical parameters. We also discuss implications of the new analytical solution for interpreting the future high-precision gravitational measurements of the Juno spacecraft.
Symmetry-plane model of 3D Euler flows: Mapping to regular systems and numerical solutions of blowup
NASA Astrophysics Data System (ADS)
Mulungye, Rachel M.; Lucas, Dan; Bustamante, Miguel D.
2014-11-01
We introduce a family of 2D models describing the dynamics on the so-called symmetry plane of the full 3D Euler fluid equations. These models depend on a free real parameter and can be solved analytically. For selected representative values of the free parameter, we apply the method introduced in [M.D. Bustamante, Physica D: Nonlinear Phenom. 240, 1092 (2011)] to map the fluid equations bijectively to globally regular systems. By comparing the analytical solutions with the results of numerical simulations, we establish that the numerical simulations of the mapped regular systems are far more accurate than the numerical simulations of the original systems, at the same spatial resolution and CPU time. In particular, the numerical integrations of the mapped regular systems produce robust estimates for the growth exponent and singularity time of the main blowup quantity (vorticity stretching rate), converging well to the analytically-predicted values even beyond the time at which the flow becomes under-resolved (i.e. the reliability time). In contrast, direct numerical integrations of the original systems develop unstable oscillations near the reliability time. We discuss the reasons for this improvement in accuracy, and explain how to extend the analysis to the full 3D case. Supported under the programme for Research in Third Level Institutions (PRTLI) Cycle 5 and co-funded by the European Regional Development Fund.
Numerical Investigations of Vadose Zone Transport of Saturated Sodium Thiosulfate Solutions
NASA Astrophysics Data System (ADS)
White, M. D.; Ward, A. L.
2001-12-01
Compared with water, hypersaline liquid wastes ([NaNO3] > 10 N) from the reduction-oxidation (REDOX) process at the Hanford site have elevated viscosity (μ > 1.2 cP), density (ρ > 1.4 gm/cm3), and surface tension (σ > 100 dyn/cm). Such liquids have infiltrated into the vadose zone at Hanford from leaking underground storage tanks. The migration behavior of saturated or hypersaline salt solutions through unsaturated soils is largely unknown. Laboratory tests with tank-waste simulants suggest that the elevated density, viscosity, and surface tension properties of these liquids can influence the wetting front behavior, altering its shape and migration rate. Conditions under which these mechanisms are active in the field and the extent to which they contribute to transport through the vadose zone are largely unknown, making it impossible to accurately predict the post-leak distribution of these fluids in the field. To investigate the effects of fluid properties on subsurface migration of hypersaline saline solutions, numerical simulations were conducted of a field-scale, tank-leak experiment. The field experiments consisted of five 4000-L injections, at a depth of 5 m, of saturated sodium thiosulfate brine (used as a surrogate for REDOX type wastes) over a 5-week period, followed by three 4000-L injections of Columbia River water. Pre-test modeling of river water injections at this Hanford field site predicted significant lateral spreading of the moisture plume and were confirmed by geophysical logging. A series of three-dimensional, multifluid (i.e., aqueous and gas phases) numerical simulations were conducted that systematically considered the effects of elevated density, viscosity, and surface tension, and reduced vapor pressure on vadose-zone transport. Hydrologic properties were determined from cores collected at the field site and calibrated using river-water injection experiments. Isothermal conditions were assumed for the simulations, however, the effects of
Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift
Zhao, Lei; Yue, Xingye; Waxman, David
2013-01-01
A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size. PMID:23749318
Liu, Li; Lai, Choi-Hong; Zhou, Shao-Dan; Xie, Fen; Rui, Lu
2011-04-01
In order to predict variations of drug concentration during a given period of time, numerical solutions of pharmacokinetic models need to be obtained efficiently. Analytical solutions of linear pharmacokinetic models are usually obtained using the Laplace transform and inverse Laplace tables. The derivations of solutions to complex nonlinear models are tedious, and such solution process may be difficult to implement as a robust software code. For nonlinear models, the fourth-order Runge-Kutta (RK4) is the most classical numerical method in obtaining approximate numerical solutions, which is impossible to be implemented in distributed computing environments without much modification. The reason is that numerical solutions obtained by using RK4 can only be computed in sequential time steps. In this paper, time-domain decomposition methods are adapted for nonlinear pharmacokinetic models. The numerical Inverse Laplace method for PharmacoKinetic models (ILPK) is implemented to solve pharmacokinetic models with iterative inverse Laplace transform in each time interval. The distributed ILPK algorithm, which is based on a two-level time-domain decomposition concept, is proposed to improve its efficiency. Solutions on the coarser temporal mesh at the top level are obtained one by one, and then those on the finer temporal mesh at the bottom level are calculated concurrently by using those initial solutions that have been obtained at the top level decomposition. Accuracy and efficiency of the proposed algorithm and its distributed equivalent are investigated by using several test models. Results indicate that the ILPK algorithm and its distributed equivalent are good candidates for both linear and nonlinear pharmacokinetic models.
Linearized numerical solution method for rotating coaxial disk flows at moderate Reynolds numbers
NASA Astrophysics Data System (ADS)
Wu, J.; Delgado, A.; Rath, H. J.
A linearized solution method for rotating coaxial disk flows at moderate Reynolds numbers is discussed below. The analytical or numerical linearized similarity solutions agree with the nonlinear ones for infinite disk flows of the Stewartson-type as well as of the Batchelor-type with a small difference between angular velocities of both the disks. Over the inner portion of shrouded flows the computed results of the linearized partial differential equations have, overall, a good agreement with the solutions of the nonlinear von Karman similarity one and also with the complete Navier-Stokes solution.
Numerical solution of a class of integral equations arising in two-dimensional aerodynamics
NASA Technical Reports Server (NTRS)
Fromme, J.; Golberg, M. A.
1978-01-01
We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This allows the problem to be studied in terms of an equivalent integral equation of the second kind. A convergent numerical algorithm for its solution is derived by using Galerkin's method. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as the method of computation. Extensive numerical calculations are presented establishing the validity of the theory.
Numerical solution of random singular integral equation appearing in crack problems
NASA Technical Reports Server (NTRS)
Sambandham, M.; Srivatsan, T. S.; Bharucha-Reid, A. T.
1986-01-01
The solution of several elasticity problems, and particularly crack problems, can be reduced to the solution of one-dimensional singular integral equations with a Cauchy-type kernel or to a system of uncoupled singular integral equations. Here a method for the numerical solution of random singular integral equations of Cauchy type is presented. The solution technique involves a Chebyshev series approximation, the coefficients of which are the solutions of a system of random linear equations. This method is applied to the problem of periodic array of straight cracks inside an infinite isotropic elastic medium and subjected to a nonuniform pressure distribution along the crack edges. The statistical properties of the random solution are evaluated numerically, and the random solution is used to determine the values of the stress-intensity factors at the crack tips. The error, expressed as the difference between the mean of the random solution and the deterministic solution, is established. Values of stress-intensity factors at the crack tip for different random input functions are presented.
NASA Technical Reports Server (NTRS)
Rogers, Stuart E.
1990-01-01
The current work is initiated in an effort to obtain an efficient, accurate, and robust algorithm for the numerical solution of the incompressible Navier-Stokes equations in two- and three-dimensional generalized curvilinear coordinates for both steady-state and time-dependent flow problems. This is accomplished with the use of the method of artificial compressibility and a high-order flux-difference splitting technique for the differencing of the convective terms. Time accuracy is obtained in the numerical solutions by subiterating the equations in psuedo-time for each physical time step. The system of equations is solved with a line-relaxation scheme which allows the use of very large pseudo-time steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. Numerous laminar test flow problems are computed and presented with a comparison against analytically known solutions or experimental results. These include the flow in a driven cavity, the flow over a backward-facing step, the steady and unsteady flow over a circular cylinder, flow over an oscillating plate, flow through a one-dimensional inviscid channel with oscillating back pressure, the steady-state flow through a square duct with a 90 degree bend, and the flow through an artificial heart configuration with moving boundaries. An adequate comparison with the analytical or experimental results is obtained in all cases. Numerical comparisons of the upwind differencing with central differencing plus artificial dissipation indicates that the upwind differencing provides a much more robust algorithm, which requires significantly less computing time. The time-dependent problems require on the order of 10 to 20 subiterations, indicating that the elliptical nature of the problem does require a substantial amount of computing effort.
Grid cell distortion and MODFLOW's integrated finite-difference numerical solution.
Romero, Dave M; Silver, Steven E
2006-01-01
The ground water flow model MODFLOW inherently implements a nongeneralized integrated finite-difference (IFD) numerical scheme. The IFD numerical scheme allows for construction of finite-difference model grids with curvilinear (piecewise linear) rows. The resulting grid comprises model cells in the shape of trapezoids and is distorted in comparison to a traditional MODFLOW finite-difference grid. A version of MODFLOW-88 (herein referred to as MODFLOW IFD) with the code adapted to make the one-dimensional DELR and DELC arrays two dimensional, so that equivalent conductance between distorted grid cells can be calculated, is described. MODFLOW IFD is used to inspect the sensitivity of the numerical head and velocity solutions to the level of distortion in trapezoidal grid cells within a converging radial flow domain. A test problem designed for the analysis implements a grid oriented such that flow is parallel to columns with converging widths. The sensitivity analysis demonstrates MODFLOW IFD's capacity to numerically derive a head solution and resulting intercell volumetric flow when the internal calculation of equivalent conductance accounts for the distortion of the grid cells. The sensitivity of the velocity solution to grid cell distortion indicates criteria for distorted grid design. In the radial flow test problem described, the numerical head solution is not sensitive to grid cell distortion. The accuracy of the velocity solution is sensitive to cell distortion with error <1% if the angle between the nonparallel sides of trapezoidal cells is <12.5 degrees. The error of the velocity solution is related to the degree to which the spatial discretization of a curve is approximated with piecewise linear segments. Curvilinear finite-difference grid construction adds versatility to spatial discretization of the flow domain. MODFLOW-88's inherent IFD numerical scheme and the test problem results imply that more recent versions of MODFLOW 2000, with minor
Numerical solution of inviscid and viscous laminar and turbulent flow around the airfoil
NASA Astrophysics Data System (ADS)
Slouka, Martin; Kozel, Karel
2016-03-01
This work deals with the 2D numerical solution of inviscid compressible flow and viscous compressible laminar and turbulent flow around the profile. In a case of turbulent flow algebraic Baldwin-Lomax model is used and compared with Wilcox k-omega model. Calculations are done for NACA 0012 and RAE 2822 airfoil profile for the different angles of upstream flow. Numerical results are compared and discussed with experimental data.
Numerical solution of fluid-structure interaction represented by human vocal folds in airflow
NASA Astrophysics Data System (ADS)
Valášek, J.; Sváček, P.; Horáček, J.
2016-03-01
The paper deals with the human vocal folds vibration excited by the fluid flow. The vocal fold is modelled as an elastic body assuming small displacements and therefore linear elasticity theory is used. The viscous incompressible fluid flow is considered. For purpose of numerical solution the arbitrary Lagrangian-Euler method (ALE) is used. The whole problem is solved by the finite element method (FEM) based solver. Results of numerical experiments with different boundary conditions are presented.
Applying integrals of motion to the numerical solution of differential equations
NASA Technical Reports Server (NTRS)
Vezewski, D. J.
1980-01-01
A method is developed for using the integrals of systems of nonlinear, ordinary, differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scalar or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.
Applying integrals of motion to the numerical solution of differential equations
NASA Technical Reports Server (NTRS)
Jezewski, D. J.
1979-01-01
A method is developed for using the integrals of systems of nonlinear, ordinary differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scaler or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.
Higher-order numerical solutions using cubic splines. [for partial differential equations
NASA Technical Reports Server (NTRS)
Rubin, S. G.; Khosla, P. K.
1975-01-01
A cubic spline collocation procedure has recently been developed for the numerical solution of partial differential equations. In the present paper, this spline procedure is reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a non-uniform mesh and overall fourth-order accuracy for a uniform mesh. Solutions using both spline procedures, as well as three-point finite difference methods, will be presented for several model problems.-
Solution of prey-predator problem by numeric-analytic technique
NASA Astrophysics Data System (ADS)
Chowdhury, M. S. H.; Hashim, I.; Mawa, S.
2009-04-01
In this paper, an analytical expression for the solution of the prey-predator problem by an adaptation of the classical Adomian decomposition method (ADM). The ADM is treated as an algorithm for approximating the solution of the problem in a sequence of time intervals, i.e. the classical ADM is converted into a hybrid numeric-analytic method called the multistage ADM (MADM). Numerical comparisons with the classical ADM, and the classical fourth-order Rungge-Kutta (RK4) methods are presented.
Method for Numerical Solution of the Stationary Schrödinger Equation
NASA Astrophysics Data System (ADS)
Knyazev, S. Yu.; Shcherbakova, E. E.
2017-02-01
The aim of this work is to describe a method of numerical solution of the stationary Schrödinger equation based on the integral equation that is identical to the Schrödinger equation. The method considered here allows one to find the eigenvalues and eigensolutions for quantum-mechanical problems of different dimensionality. The method is tested by solving problems for one-dimensional and two-dimensional quantum oscillators, and results of these tests are presented. Satisfactory agreement of the results obtained using this numerical method with well-known analytical solutions is demonstrated.
Xie, Jiaquan; Huang, Qingxue; Yang, Xia
2016-01-01
In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent.
NASA Astrophysics Data System (ADS)
Chauhan, D. S.; Agrawal, R.
2011-05-01
A viscous incompressible electrically conducting fluid flow through a porous medium over a stretching sheet is considered in the presence of a magnetic field. Such flow problems have relevance in the process of a polymer sheet extrusion from a dye, and the numerical and approximate solutions of these problems are of great interest as these solutions serve practical purposes. By using the technique of stretching variables of the flow concern in porous medium and minimizing the residual of the resulting governing differential equations by the least squares method, we obtained an approximate solution for this problem of flow through porous medium near a stretching sheet. The results are also compared to a numerical solution determined by using the shooting method along with the Runge-Kutta method. The effects of various pertinent parameters on the velocity distribution and the residual function are investigated. The results are depicted graphically and discussed.
NASA Technical Reports Server (NTRS)
Wang, Gang
2003-01-01
A multi grid solution procedure for the numerical simulation of turbulent flows in complex geometries has been developed. A Full Multigrid-Full Approximation Scheme (FMG-FAS) is incorporated into the continuity and momentum equations, while the scalars are decoupled from the multi grid V-cycle. A standard kappa-Epsilon turbulence model with wall functions has been used to close the governing equations. The numerical solution is accomplished by solving for the Cartesian velocity components either with a traditional grid staggering arrangement or with a multiple velocity grid staggering arrangement. The two solution methodologies are evaluated for relative computational efficiency. The solution procedure with traditional staggering arrangement is subsequently applied to calculate the flow and temperature fields around a model Short Take-off and Vertical Landing (STOVL) aircraft hovering in ground proximity.
NASA Astrophysics Data System (ADS)
Varado, N.; Braud, I.; Ross, P. J.
2003-04-01
A new numerical method for solving the 1D Richard's equation has been proposed by P. Ross (Agronomy J., 2003, in press). The Kirchhoff transform or degree of saturation is used instead of the classical matrix potential. The solution can be used both for saturated or non saturated soils. Hydraulic properties are described using the Brooks and Corey model. The soil is discretized into layers. Their thickness can be larger than in classical matrix potential methods, due to the use of a time and space varying weighing procedure for the calculation of fluxes between layers. This allows the use of a non iterative procedure, ensuring a very fast numerical solution. Extensive tests showed that the new method was very accurate for bare soils. The next step was the addition of a root extraction module in order to account for plant transpiration. Two root water uptake modules with compensation mechanisms in case of water stress were chosen from the literature. They express the transpiration source term in the Richards equation as a linear function of a potential transpiration and take into account water stress and its effects on plant transpiration. These modules were proposed first by Lai and Katul (Adv. Water Resour., 2000) and Li et al. (J. Hydrol., 2001). The new version of the model has been tested in a systematic way with several soils characteristics, climate forcings, and evapotranspiration calculation. Like the tests without vegetation, the SiSPAT (Simple Soil Plant Atmosphere Transfer) model was considered as a reference after implementation of the same roots modules. The numerical solution was also tested using a soybean data set. The variations and the cumulative values like drainage, water content, real transpiration and real evapotranspiration were in a good agreement with the SiSPAT modelling, with a relative error of less than 3%. The error on soil evaporation remained important (about 20%) on low cumulative values (less than 20mm), i.e. when LAI was close to
NASA Astrophysics Data System (ADS)
Moore, Christopher; Hopkins, Matthew; Moore, Stan; Boerner, Jeremiah; Cartwright, Keith
2015-09-01
Simulation of breakdown is important for understanding and designing a variety of applications such as mitigating undesirable discharge events. Such simulations need to be accurate through early time arc initiation to late time stable arc behavior. Here we examine constraints on the timestep and mesh size required for arc simulations using the particle-in-cell (PIC) method with direct simulation Monte Carlo (DMSC) collisions. Accurate simulation of electron avalanche across a fixed voltage drop and constant neutral density (reduced field of 1000 Td) was found to require a timestep ~ 1/100 of the mean time between collisions and a mesh size ~ 1/25 the mean free path. These constraints are much smaller than the typical PIC-DSMC requirements for timestep and mesh size. Both constraints are related to the fact that charged particles are accelerated by the external field. Thus gradients in the electron energy distribution function can exist at scales smaller than the mean free path and these must be resolved by the mesh size for accurate collision rates. Additionally, the timestep must be small enough that the particle energy change due to the fields be small in order to capture gradients in the cross sections versus energy. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. DOE's National Nuclear Security Administration under Contract DE-AC04-94AL85000.
On the numerical solution of two-dimensional, laminar compressible flows with imbedded shock waves.
NASA Technical Reports Server (NTRS)
Goodrich, W. D.; Lamb, J. P.; Bertin, J. J.
1972-01-01
The complete, time-dependent Navier-Stokes equations are expressed in conservation form and solved by employing an explicit finite difference numerical technique which incorporates artificial viscosity terms of the form first suggested by Rusanov for numerical stability in the vicinity of shock waves. Surface boundary conditions are developed in a consistent and unique manner through the use of a physically oriented extrapolation procedure. From numerical experimentation an extended range for the explicit stability parameter is established. Also employed is an additional convergence parameter which relates incremental spatial steps. Convergence of the transient solution to a steady state flow was obtained after 400 to 500 time steps.
Error propagation in the numerical solutions of the differential equations of orbital mechanics
NASA Technical Reports Server (NTRS)
Bond, V. R.
1982-01-01
The relationship between the eigenvalues of the linearized differential equations of orbital mechanics and the stability characteristics of numerical methods is presented. It is shown that the Cowell, Encke, and Encke formulation with an independent variable related to the eccentric anomaly all have a real positive eigenvalue when linearized about the initial conditions. The real positive eigenvalue causes an amplification of the error of the solution when used in conjunction with a numerical integration method. In contrast an element formulation has zero eigenvalues and is numerically stable.
Two Different Methods for Numerical Solution of the Modified Burgers' Equation
Karakoç, Seydi Battal Gazi; Başhan, Ali; Geyikli, Turabi
2014-01-01
A numerical solution of the modified Burgers' equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing L 2 and L ∞ error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM. PMID:25162064
NASA Technical Reports Server (NTRS)
Reese, O. W.
1972-01-01
The numerical calculation is described of the steady-state flow of electrons in an axisymmetric, spherical, electrostatic collector for a range of boundary conditions. The trajectory equations of motion are solved alternately with Poisson's equation for the potential field until convergence is achieved. A direct (noniterative) numerical technique is used to obtain the solution to Poisson's equation. Space charge effects are included for initial current densities as large as 100 A/sq cm. Ways of dealing successfully with the difficulties associated with these high densities are discussed. A description of the mathematical model, a discussion of numerical techniques, results from two typical runs, and the FORTRAN computer program are included.
Two different methods for numerical solution of the modified Burgers' equation.
Karakoç, Seydi Battal Gazi; Başhan, Ali; Geyikli, Turabi
2014-01-01
A numerical solution of the modified Burgers' equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing L 2 and L ∞ error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM.
Use of Green's functions in the numerical solution of two-point boundary value problems
NASA Technical Reports Server (NTRS)
Gallaher, L. J.; Perlin, I. E.
1974-01-01
This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.
NASA Astrophysics Data System (ADS)
Stukel, Michael R.; Landry, Michael R.; Ohman, Mark D.; Goericke, Ralf; Samo, Ty; Benitez-Nelson, Claudia R.
2012-03-01
Despite the increasing use of linear inverse modeling techniques to elucidate fluxes in undersampled marine ecosystems, the accuracy with which they estimate food web flows has not been resolved. New Markov Chain Monte Carlo (MCMC) solution methods have also called into question the biases of the commonly used L2 minimum norm (L 2MN) solution technique. Here, we test the abilities of MCMC and L 2MN methods to recover field-measured ecosystem rates that are sequentially excluded from the model input. For data, we use experimental measurements from process cruises of the California Current Ecosystem (CCE-LTER) Program that include rate estimates of phytoplankton and bacterial production, micro- and mesozooplankton grazing, and carbon export from eight study sites varying from rich coastal upwelling to offshore oligotrophic conditions. Both the MCMC and L 2MN methods predicted well-constrained rates of protozoan and mesozooplankton grazing with reasonable accuracy, but the MCMC method overestimated primary production. The MCMC method more accurately predicted the poorly constrained rate of vertical carbon export than the L 2MN method, which consistently overestimated export. Results involving DOC and bacterial production were equivocal. Overall, when primary production is provided as model input, the MCMC method gives a robust depiction of ecosystem processes. Uncertainty in inverse ecosystem models is large and arises primarily from solution under-determinacy. We thus suggest that experimental programs focusing on food web fluxes expand the range of experimental measurements to include the nature and fate of detrital pools, which play large roles in the model.
NASA Astrophysics Data System (ADS)
Ahmed, E.; El-Sayed, A. M. A.; El-Saka, H. A. A.
2007-01-01
In this paper we are concerned with the fractional-order predator-prey model and the fractional-order rabies model. Existence and uniqueness of solutions are proved. The stability of equilibrium points are studied. Numerical solutions of these models are given. An example is given where the equilibrium point is a centre for the integer order system but locally asymptotically stable for its fractional-order counterpart.
TLC scheme for numerical solution of the transport equation on equilateral triangular meshes
Walters, W.F.
1983-01-01
A new triangular linear characteristic TLC scheme for numerically solving the transport equation on equilateral triangular meshes has been developed. This scheme uses the analytic solution of the transport equation in the triangle as its basis. The data on edges of the triangle are assumed linear as is the source representation. A characteristic approach or nodal approach is used to obtain the analytic solution. Test problems indicate that the new TLC is superior to the widely used DITRI scheme for accuracy.
Analytical solutions and numerical modeling for a dam-break problem in inclined channels
NASA Astrophysics Data System (ADS)
Pelinovsky, Efim; Didenkulova, Ira; Didenkulov, Oleg; Rodin, Artem
2016-04-01
Here we obtain different analytical solutions of the shallow-water equations for inviscid nonlinear waves in inclined channels. (i) The first solution describes Riemann wave moving up or down alone the channel slope. It requires the initial fluid flow, which often accompanies waves generated by landslides. This solution is valid for a finite time before the wave breaks. (ii) The second solution generalizes the classical dam-break problem for the case of a dam located in the inclined channel. In this case the cross-section of the channel influences the speed of wave propagation inside the channel, and therefore changes wave dynamics inside the channel compare to the plane beach. (iii) The third solution describes the intermediate stage of the wave front dynamics for a dam of a large height. This solution is derived with the use of generalized Carrier-Greenspan approach developed early by Didenkulova & Pelinovsky (2011) and Rybkin et al (2014). Some of the analytical solutions are tested with the means of numerical modeling. The numerical modeling is carried out using the CLAWPACK software based on nonlinear shallow water equations. Application of the described solutions to possible laboratory experiments is discussed.
Deterministic numerical solutions of the Boltzmann equation using the fast spectral method
NASA Astrophysics Data System (ADS)
Wu, Lei; White, Craig; Scanlon, Thomas J.; Reese, Jason M.; Zhang, Yonghao
2013-10-01
The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard-Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev-Krook-Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss-Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard-Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases
Anderson, O.A.
2007-01-31
The well-known Kapchinskij-Vladimirskij (KV) equations are difficult to solve in general, but the problem is simplified for the matched-beam case with sufficient symmetry. They show that the interdependence of the two KV equations is eliminated, so that only one needs to be solved--a great simplification. They present an iterative method of solution which can potentially yield any desired level of accuracy. The lowest level, the well-known smooth approximation, yields simple, explicit results with good accuracy for weak or moderate focusing fields. The next level improves the accuracy for high fields; they previously showed [Part. Accel. 52, 133 (1996)] how to maintain a simple explicit format for the results. That paper used expansion in a small parameter to obtain results of second-level accuracy. The present paper, using straightforward iteration, obtains equations of first, second, and third levels of accuracy. For a periodic lattice with beam matched to lattice, they use the lattice and beam parameters as input and solve for phase advances and envelope functions. They find excellent agreement with numerical solutions over a wide range of beam emittances and intensities.
Anderson, Oscar A.
2006-08-06
The well-known Kapchinskij-Vladimirskij (KV) equations are difficult to solve in general, but the problem is simplified for the matched-beam case with sufficient symmetry. They show that the interdependence of the two KV equations is eliminated, so that only one needs to be solved--a great simplification. They present an iterative method of solution which can potentially yield any desired level of accuracy. The lowest level, the well-known smooth approximation, yields simple, explicit results with good accuracy for weak or moderate focusing fields. The next level improves the accuracy for high fields; they previously showed how to maintain a simple explicit format for the results. That paper used expansion in a small parameter to obtain the second level. The present paper, using straightforward iteration, obtains equations of first, second, and third levels of accuracy. For a periodic lattice with beam matched to lattice, they use the lattice and beam parameters as input and solve for phase advances and envelope waveforms. They find excellent agreement with numerical solutions over a wide range of beam emittances and intensities.
NASA Astrophysics Data System (ADS)
Mulungye, Rachel M.; Lucas, Dan; Bustamante, Miguel D.
2016-02-01
We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point-type introduced by Gibbon et al. (1999). By employing the method of mapping to regular systems, presented in Bustamante (2011) and extended to the symmetry-plane case by Mulungye et al. (2015), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^2.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t < T^* - 10^{-9000}$, where $T^*$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating point double precision ($\\approx 10^{-16}$) creates a `machine-epsilon' barrier. Due to this limitation on the \\emph{original} independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^*- t \\approx 10^{-140}.$
A numerical method for finding sign-changing solutions of superlinear Dirichlet problems
Neuberger, J.M.
1996-12-31
In a recent result it was shown via a variational argument that a class of superlinear elliptic boundary value problems has at least three nontrivial solutions, a pair of one sign and one which sign changes exactly once. These three and all other nontrivial solutions are saddle points of an action functional, and are characterized as local minima of that functional restricted to a codimension one submanifold of the Hilbert space H-0-1-2, or an appropriate higher codimension subset of that manifold. In this paper, we present a numerical Sobolev steepest descent algorithm for finding these three solutions.
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
NASA Technical Reports Server (NTRS)
Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)
2002-01-01
We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.
Several numerical and analytical solutions of the radiative transfer equation (RTE) for plane albedo were compared for solar light reflection by sea water. The study incorporated the simplest case, that being a semi-infinite one-dimensional plane-parallel absorbing and scattering...
Three numerical algorithms were compared to provide a solution of a radiative transfer equation (RTE) for plane albedo (hemispherical reflectance) in semi-infinite one-dimensional plane-parallel layer. Algorithms were based on the invariant imbedding method and two different var...
Numerical solutions of nonlinear STIFF initial value problems by perturbed functional iterations
NASA Technical Reports Server (NTRS)
Dey, S. K.
1982-01-01
Numerical solution of nonlinear stiff initial value problems by a perturbed functional iterative scheme is discussed. The algorithm does not fully linearize the system and requires only the diagonal terms of the Jacobian. Some examples related to chemical kinetics are presented.
Malik, Suheel Abdullah; Qureshi, Ijaz Mansoor; Amir, Muhammad; Malik, Aqdas Naveed; Haq, Ihsanul
2015-01-01
In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems. PMID:25811858
Malik, Suheel Abdullah; Qureshi, Ijaz Mansoor; Amir, Muhammad; Malik, Aqdas Naveed; Haq, Ihsanul
2015-01-01
In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems.
Ida, Masato; Taniguchi, Nobuyuki
2003-09-01
This paper introduces a candidate for the origin of the numerical instabilities in large eddy simulation repeatedly observed in academic and practical industrial flow computations. Without resorting to any subgrid-scale modeling, but based on a simple assumption regarding the streamwise component of flow velocity, it is shown theoretically that in a channel-flow computation, the application of the Gaussian filtering to the incompressible Navier-Stokes equations yields a numerically unstable term, a cross-derivative term, which is similar to one appearing in the Gaussian filtered Vlasov equation derived by Klimas [J. Comput. Phys. 68, 202 (1987)] and also to one derived recently by Kobayashi and Shimomura [Phys. Fluids 15, L29 (2003)] from the tensor-diffusivity subgrid-scale term in a dynamic mixed model. The present result predicts that not only the numerical methods and the subgrid-scale models employed but also only the applied filtering process can be a seed of this numerical instability. An investigation concerning the relationship between the turbulent energy scattering and the unstable term shows that the instability of the term does not necessarily represent the backscatter of kinetic energy which has been considered a possible origin of numerical instabilities in large eddy simulation. The present findings raise the question whether a numerically stable subgrid-scale model can be ideally accurate.
Novokhatski, A.; /SLAC
2011-08-17
We present and discuss the properties of the coherent electromagnetic fields of a very short, ultrarelativistic bunch in a rectangular vacuum chamber inside a bending magnet. The analysis is based on the results of a direct numerical solution of Maxwell's equations together with Newton's equations. We use a new dispersion-free time-domain algorithm which employs a more efficient use of finite element mesh techniques and, hence, produces self-consistent and stable solutions for very short bunches. We investigate the fine structure of the coherent synchrotron radiation fields. We also discuss coherent edge radiation. We present a clear picture of the field using the electric field lines constructed from the numerical solutions. This method should be useful in the study of existing and future concepts of particle accelerators and ultrafast coherent light sources, where high peak currents and very short bunches are envisioned.
Numerical solution of the full potential equation using a chimera grid approach
NASA Technical Reports Server (NTRS)
Holst, Terry L.
1995-01-01
A numerical scheme utilizing a chimera zonal grid approach for solving the full potential equation in two spatial dimensions is described. Within each grid zone a fully-implicit approximate factorization scheme is used to advance the solution one interaction. This is followed by the explicit advance of all common zonal grid boundaries using a bilinear interpolation of the velocity potential. The presentation is highlighted with numerical results simulating the flow about a two-dimensional, nonlifting, circular cylinder. For this problem, the flow domain is divided into two parts: an inner portion covered by a polar grid and an outer portion covered by a Cartesian grid. Both incompressible and compressible (transonic) flow solutions are included. Comparisons made with an analytic solution as well as single grid results indicate that the chimera zonal grid approach is a viable technique for solving the full potential equation.
Numerical Solution of Time-Dependent Gravitational Schr"odinger Equation
NASA Astrophysics Data System (ADS)
Christianto, Vic; Rapoport, Diego L.; Smarandache, Florentin
2009-11-01
In recent years, there are attempts to describe quantization of planetary distance based on time-independent gravitational Schr"odinger equation, including Rubcic & Rubcic's method and also Nottale's Scale Relativity method. Nonetheless, there is no solution yet for time-dependent gravitational Schr"odinger equation (TDGSE). In the present paper, a numerical solution of time-dependent gravitational Schr"odinger equation is presented, apparently for the first time. This numerical solution leads to gravitational Bohr-radius, as expected. In the subsequent section, we also discuss plausible extension of this gravitational Schr"odinger equation to include the effect of phion condensate via Gross-Pitaevskii equation, as described recently by Moffat. Alternatively one can consider this condensate from the viewpoint of BogoliubovdeGennes theory, which can be approximated with coupled time-independent gravitational Schr"odinger equation. Further observation is of course recommended in order to refute or verify this proposition.
NASA Astrophysics Data System (ADS)
Gaudreault, Stéphane; Pudykiewicz, Janusz A.
2016-10-01
The exponential propagation methods were applied in the past for accurate integration of the shallow water equations on the sphere. Despite obvious advantages related to the exact solution of the linear part of the system, their use for the solution of practical problems in geophysics has been limited because efficiency of the traditional algorithm for evaluating the exponential of Jacobian matrix is inadequate. In order to circumvent this limitation, we modify the existing scheme by using the Incomplete Orthogonalization Method instead of the Arnoldi iteration. We also propose a simple strategy to determine the initial size of the Krylov space using information from previous time instants. This strategy is ideally suited for the integration of fluid equations where the structure of the system Jacobian does not change rapidly between the subsequent time steps. A series of standard numerical tests performed with the shallow water model on a geodesic icosahedral grid shows that the new scheme achieves efficiency comparable to the semi-implicit methods. This fact, combined with the accuracy and the mass conservation of the exponential propagation scheme, makes the presented method a good candidate for solving many practical problems, including numerical weather prediction.
NASA Technical Reports Server (NTRS)
Thompson, D. S.
1980-01-01
The full Navier-Stokes equations for incompressible turbulent flow must be solved to accurately represent all flow phenomena which occur in a high Reynolds number incompressible flow. A two layer algebraic eddy viscosity turbulence model is used to represent the Reynolds stress in the primitive variable formulation. The development of the boundary-fitted coordinate systems makes the numerical solution of these equations feasible for arbitrarily shaped bodies. The nondimensional time averaged Navier-Stokes equations, including the turbulence mode, are represented by finite difference approximations in the transformed plane. The resulting coupled system of nonlinear algebraic equations is solved using a point successive over relaxation iteration. The test case considered was a NACA 64A010 airfoil section at an angle of attack of two degrees and a Reynolds number of 2,000,000.
The space-time solution element method: A new numerical approach for the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Scott, James R.; Chang, Sin-Chung
1995-01-01
This paper is one of a series of papers describing the development of a new numerical method for the Navier-Stokes equations. Unlike conventional numerical methods, the current method concentrates on the discrete simulation of both the integral and differential forms of the Navier-Stokes equations. Conservation of mass, momentum, and energy in space-time is explicitly provided for through a rigorous enforcement of both the integral and differential forms of the governing conservation laws. Using local polynomial expansions to represent the discrete primitive variables on each cell, fluxes at cell interfaces are evaluated and balanced using exact functional expressions. No interpolation or flux limiters are required. Because of the generality of the current method, it applies equally to the steady and unsteady Navier-Stokes equations. In this paper, we generalize and extend the authors' 2-D, steady state implicit scheme. A general closure methodology is presented so that all terms up through a given order in the local expansions may be retained. The scheme is also extended to nonorthogonal Cartesian grids. Numerous flow fields are computed and results are compared with known solutions. The high accuracy of the scheme is demonstrated through its ability to accurately resolve developing boundary layers on coarse grids. Finally, we discuss applications of the current method to the unsteady Navier-Stokes equations.
Comparison of input parameters regarding rock mass in analytical solution and numerical modelling
NASA Astrophysics Data System (ADS)
Yasitli, N. E.
2016-12-01
Characteristics of stress redistribution around a tunnel excavated in rock are of prime importance for an efficient tunnelling operation and maintaining stability. As it is a well known fact that rock mass properties are the most important factors affecting stability together with in-situ stress field and tunnel geometry. Induced stresses and resultant deformation around a tunnel can be approximated by means of analytical solutions and application of numerical modelling. However, success of these methods depends on assumptions and input parameters which must be representative for the rock mass. However, mechanical properties of intact rock can be found by laboratory testing. The aim of this paper is to demonstrate the importance of proper representation of rock mass properties as input data for analytical solution and numerical modelling. For this purpose, intact rock data were converted into rock mass data by using the Hoek-Brown failure criterion and empirical relations. Stress-deformation analyses together with yield zone thickness determination have been carried out by using analytical solutions and numerical analyses by using FLAC3D programme. Analyses results have indicated that incomplete and incorrect design causes stability and economic problems in the tunnel. For this reason during the tunnel design analytical data and rock mass data should be used together. In addition, this study was carried out to prove theoretically that numerical modelling results should be applied to the tunnel design for the stability and for the economy of the support.
NASA Astrophysics Data System (ADS)
Alfonso, Lester; Zamora, Jose; Cruz, Pedro
2015-04-01
The stochastic approach to coagulation considers the coalescence process going in a system of a finite number of particles enclosed in a finite volume. Within this approach, the full description of the system can be obtained from the solution of the multivariate master equation, which models the evolution of the probability distribution of the state vector for the number of particles of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain type of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels and initial conditions is introduced. The performance of the method was checked by comparing the numerically calculated particle mass spectrum with analytical solutions obtained for the constant and sum kernels, with an excellent correspondence between the analytical and numerical solutions. In order to increase the speedup of the algorithm, software parallelization techniques with OpenMP standard were used, along with an implementation in order to take advantage of new accelerator technologies. Simulations results show an important speedup of the parallelized algorithms. This study was funded by a grant from Consejo Nacional de Ciencia y Tecnologia de Mexico SEP-CONACYT CB-131879. The authors also thanks LUFAC® Computacion SA de CV for CPU time and all the support provided.
A numerical solution for thermoacoustic convection of fluids in low gravity
NASA Technical Reports Server (NTRS)
Spradley, L. W.; Bourgeois, S. V., Jr.; Fan, C.; Grodzka, P. G.
1973-01-01
A finite difference numerical technique for solving the differential equations which describe thermal convection of compressible fluids in low gravity are reported. Results of one-dimensional calculations are presented, and comparisons are made to previous solutions. The primary result presented is a one-dimensional radial model of the Apollo 14 heat flow and convection demonstration flight experiment. The numerical calculations show that thermally induced convective motion in a confined fluid can have significant effects on heat transfer in a low gravity environment.
Numerical solutions for steady thermal convection from a concentrated source in a porous medium
Hickox, C.E.; Watts, H.A.
1980-06-01
Solutions for the steady, axisymmetric velocity and temperature fields associated with a point source of thermal energy in a fluid-saturated porous medium are obtained numerically through use of similarity transformations. The two cases considered are those of a point source located on the lower, insulated boundary of a semi-infinite region and a point source embedded in an infinite region. Numerical results are presented from which complete descriptions of the velocity and temperature fields can be constructed for Rayleigh numbers ranging from 10/sup -3/ to 10/sup 2/.
A fifth order implicit method for the numerical solution of differential-algebraic equations
NASA Astrophysics Data System (ADS)
Skvortsov, L. M.
2015-06-01
An implicit two-step Runge-Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.
NASA Astrophysics Data System (ADS)
Martin, I.; Tirado, F.; Vazquez, L.
We present a process to achieve the solution of the two dimensional nonlinear Schrödinger equation using a multigrid technique on a distributed memory machine. Some features about the multigrid technique as its good convergence and parallel properties are explained in this paper. This makes multigrid method the optimal one to solve the systems of equations arising at each time step from an implicit numerical scheme. We give some experimental results about the parallel numerical simulation of this equation on a message passing parallel machine.
Numerical Modeling Tools for the Prediction of Solution Migration Applicable to Mining Site
Martell, M.; Vaughn, P.
1999-01-06
Mining has always had an important influence on cultures and traditions of communities around the globe and throughout history. Today, because mining legislation places heavy emphasis on environmental protection, there is great interest in having a comprehensive understanding of ancient mining and mining sites. Multi-disciplinary approaches (i.e., Pb isotopes as tracers) are being used to explore the distribution of metals in natural environments. Another successful approach is to model solution migration numerically. A proven method to simulate solution migration in natural rock salt has been applied to project through time for 10,000 years the system performance and solution concentrations surrounding a proposed nuclear waste repository. This capability is readily adaptable to simulate solution migration around mining.
NASA Technical Reports Server (NTRS)
Rosenfeld, Moshe; Israeli, Moshe; Wolfshtein, Micha
1987-01-01
A marching iterative method for the solution of the three dimensional, incompressibhle, steady and parabolized Navier-Stokes equations is described. The equations are written in primitive variables and discretized in general axisymmetric orthogonal coordinate systems. The coupled set of finite-difference equations are solved without any splitting or factorization errors. Moreover, the continuity equation and the two crossflow momentum equations are exactly satisfied at every step of the iterative process. The solution scheme is equivalent to the solution of one Poisson equation by the Successive Plane Over Relaxation method and has good convergence properties. Other existing solution methods resemble a Jacobi-type iterative scheme and therefore are less efficient. Numerical experiments include the laminar, incompressible flow over prolate spheroids at incidence.
Impact of 3D root uptake on solute transport: a numerical study
NASA Astrophysics Data System (ADS)
Schröder, N.; Javaux, M.; Vanderborght, J.; Steffen, B.; Vereecken, H.
2011-12-01
Plant transpiration is an important component of the hydrological cycle. Through root water uptake, plants do not only affect the 3D soil water flow velocity distribution, but also solute movement in soil. This numerical study aims at investigating how solute fate is impacted by root uptake using the 3D biophysical model R-SWMS (Javaux et al., 2008). This model solves the Richards equation in 3D in the soil and the flow equation within the plant root xylem vessels. Furthermore, for solute transport simulations, the 3D particle tracker PARTRACE (Bechtold et al., 2011) was used. . We generated 3D virtual steady-state breakthrough curves (BTC) experiments in soils with transpiring plants. The averaged BTCs were then fitted with a 1D numerical flow model under steady-state conditions to obtain apparent CDE parameters. Two types of root architecture, a fibrous and a taprooted structure, were compared in virtual 3D experiments. The solute uptake type or the transpiration rate were also modified and we analyzed how these parameters affected apparent disperisivity and velocity profiles. Our simulation results show, that both, apparent velocity and dispersivity length are affected by water and solute root uptake. In addition, under high exclusion processes (slight or no active uptake), solute accumulates around roots and generates a long tailing to the breakthrough curves, which cannot be reproduced by 1D models that simulate root water uptake with solute exclusion. This observation may have an important impact on how to model pollutant mass transfer to groundwater at larger scales. Javaux, M., T. Schröder, J. Vanderborght, and H. Vereecken. 2008. Use of a three-dimensional detailed modeling approach for predicting root water uptake. Vadose Zone J. 7:1079-1088.doi: 10.2136/vzj2007.0115. Bechtold, M., S. Haber-Pohlmeier, J. Vanderborght, A. Pohlmeier, P.A. Ferre, and H. Vereecken. 2011. Near-surface solute redistribution during evaporation. Submitted to Geophys. Res. Lett
NASA Astrophysics Data System (ADS)
Elman, Howard C.; Forstall, Virginia
2017-04-01
Reduced-order modeling is an efficient approach for solving parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem at significantly lower cost, producing an accurate estimate of the solution. For nonlinear problems, however, standard methods do not achieve the desired cost savings. Empirical interpolation methods represent a modification of this methodology used for cases of nonlinear operators or nonaffine parameter dependence. These methods identify points in the discretization necessary for representing the nonlinear component of the reduced model accurately, and they incur online computational costs that are independent of the spatial dimension $N$. We will show that empirical interpolation methods can be used to significantly reduce the costs of solving parameterized versions of the Navier-Stokes equations, and that iterative solution methods can be used in place of direct methods to further reduce the costs of solving the algebraic systems arising from reduced-order models.
Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation.
Sánchez-Rey, Bernardo; Johansson, Magnus
2005-03-01
In this paper we study numerically existence and stability of exact dark waves on the (nonintegrable) discrete nonlinear Schrödinger equation for a finite one-dimensional lattice. These are solutions that bifurcate from stationary dark modes with constant background intensity and zero intensity at a site, and whose initial state translates exactly one site each period of the internal oscillations. We show that exact dark waves are characterized by an oscillatory background whose wavelength is closely related with the velocity. Faster dark waves require smaller wavelengths. For slow enough velocity dark waves are linearly stable, but when trying to continue numerically a solution towards higher velocities bifurcations appear, due to rearrangements in the oscillatory tail in order to make possible a decreasing of the wavelength. However, in principle, one might control the stability of an exact dark wave adjusting a phase factor which plays the role of a discreteness parameter. In addition, we also study the regimes of existence and stability for stationary discrete gray modes, which are exact solutions with phase-twisted constant-amplitude background and nonzero minimum intensity. Also such solutions develop envelope oscillations on top of the homogeneous background when continued into moving phase-twisted solutions.
Numerical Solutions for Supersonic Flow of an Ideal Gas Around Blunt Two-Dimensional Bodies
NASA Technical Reports Server (NTRS)
Fuller, Franklyn B.
1961-01-01
The method described is an inverse one; the shock shape is chosen and the solution proceeds downstream to a body. Bodies blunter than circular cylinders are readily accessible, and any adiabatic index can be chosen. The lower limit to the free-stream Mach number available in any case is determined by the extent of the subsonic field, which in turn depends upon the body shape. Some discussion of the stability of the numerical processes is given. A set of solutions for flows about circular cylinders at several Mach numbers and several values of the adiabatic index is included.
The gravitational field equations of a twisted skyrmion string: numerical solution
NASA Astrophysics Data System (ADS)
Hadi1, Miftachul; Anderson, Malcolm; Husein, Andri
2017-01-01
We study nonlinear sigma model, especially Skyrme model with twist: twisted Skyrmion string where twist term, mkz, is indicated in vortex solution. To add gravity, we replace gμv in Lagrangian system with a space-time metric tensor, gμv which in view of the time-independence and cylindrical symmetry of the assumed vortex solution is taken to be a function of r alone. We use ode45 for numerical calculation, i.e. a tool box in Matlab to solve coupled Einstein field equations which have ordinary differential equations (ODE) form.
Numerical solution of shock and ramp compression for general material properties
Swift, D C
2009-01-28
A general formulation was developed to represent material models for applications in dynamic loading. Numerical methods were devised to calculate response to shock and ramp compression, and ramp decompression, generalizing previous solutions for scalar equations of state. The numerical methods were found to be flexible and robust, and matched analytic results to a high accuracy. The basic ramp and shock solution methods were coupled to solve for composite deformation paths, such as shock-induced impacts, and shock interactions with a planar interface between different materials. These calculations capture much of the physics of typical material dynamics experiments, without requiring spatially-resolving simulations. Example calculations were made of loading histories in metals, illustrating the effects of plastic work on the temperatures induced in quasi-isentropic and shock-release experiments, and the effect of a phase transition.
Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials
Khan, Rahmat Ali; Tajadodi, Haleh; Johnston, Sarah Jane
2014-01-01
In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques. PMID:25485293
A note on a corrector formula for the numerical solution of ordinary differential equations
NASA Technical Reports Server (NTRS)
Chien, Y.-C.; Agrawal, K. M.
1979-01-01
A new corrector formula for predictor-corrector methods for numerical solutions of ordinary differential equations is presented. Two considerations for choosing corrector formulas are given: (1) the coefficient in the error term and (2) its stability properties. The graph of the roots of an equation plotted against its stability region, of different values, is presented along with the tables that correspond to various corrector equations, including Hamming's and Milne and Reynolds'.
Far-Field Boundary Conditions in Numerical Solutions of the Navier-Stokes Equations.
2014-09-26
nonlinear system of mixed parabolic- hyperbolic type in two space dimensions and time, with four independent variables must be solved in an exterior...conditions. * III THE NAVIER-STOKES EQUATIONS AND CHARACTERISTIC VARIABLES : We now begin our discussion of the equations of gas dynamics. We will neglect...8217 Far-Field Boundary Conditions in Numerical Solutions of the Navier-Stokes Equations L°O * (. P.J. McKenna LA. DTIC * E.LECTE Final Report AFOSR Grant
NASA Astrophysics Data System (ADS)
Nguyen, T. T.; Laurent, F.; Fox, R. O.; Massot, M.
2016-11-01
The accurate description and robust simulation, at relatively low cost, of global quantities (e.g. number density or volume fraction) as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, two types of methods are investigated for solving the population balance equation with aggregation, continuous particle size change (growth and size reduction), and nucleation: the extended quadrature method of moments (EQMOM) based on the work of Yuan et al. [52] and a hybrid method (TSM) between the sectional and moment methods, considering two moments per section based on the work of Laurent et al. [30]. For both methods, the closure employs a continuous reconstruction of the number density function of the particles from its moments, thus allowing evaluation of all the unclosed terms in the moment equations, including the negative flux due to the disappearance of particles. Here, new robust and efficient algorithms are developed for this reconstruction step and two kinds of reconstruction are tested for each method. Moreover, robust and accurate numerical methods are developed, ensuring the realizability of the moments. The robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebraic constraints thanks to a tailored overall strategy. EQMOM and TSM are compared to a sectional method for various simple but relevant test cases, showing their ability to describe accurately the fine-particle population with a much lower number of variables. These results demonstrate the efficiency of the modeling and numerical choices, and their potential for the simulation of real-world applications.
Numerical solution of the nonlinear Schrödinger equation using smoothed-particle hydrodynamics.
Mocz, Philip; Succi, Sauro
2015-05-01
We formulate a smoothed-particle hydrodynamics numerical method, traditionally used for the Euler equations for fluid dynamics in the context of astrophysical simulations, to solve the nonlinear Schrödinger equation in the Madelung formulation. The probability density of the wave function is discretized into moving particles, whose properties are smoothed by a kernel function. The traditional fluid pressure is replaced by a quantum pressure tensor, for which a robust discretization is found. We demonstrate our numerical method on a variety of numerical test problems involving the simple harmonic oscillator, soliton-soliton collision, Bose-Einstein condensates, collapsing singularities, and dark matter halos governed by the Gross-Pitaevskii-Poisson equation. Our method is conservative, applicable to unbounded domains, and is automatically adaptive in its resolution, making it well suited to study problems with collapsing solutions.
Evaluation of a transfinite element numerical solution method for nonlinear heat transfer problems
NASA Technical Reports Server (NTRS)
Cerro, J. A.; Scotti, S. J.
1991-01-01
Laplace transform techniques have been widely used to solve linear, transient field problems. A transform-based algorithm enables calculation of the response at selected times of interest without the need for stepping in time as required by conventional time integration schemes. The elimination of time stepping can substantially reduce computer time when transform techniques are implemented in a numerical finite element program. The coupling of transform techniques with spatial discretization techniques such as the finite element method has resulted in what are known as transfinite element methods. Recently attempts have been made to extend the transfinite element method to solve nonlinear, transient field problems. This paper examines the theoretical basis and numerical implementation of one such algorithm, applied to nonlinear heat transfer problems. The problem is linearized and solved by requiring a numerical iteration at selected times of interest. While shown to be acceptable for weakly nonlinear problems, this algorithm is ineffective as a general nonlinear solution method.
Evaluation of a transfinite element numerical solution method for nonlinear heat transfer problems
NASA Astrophysics Data System (ADS)
Cerro, J. A.; Scotti, S. J.
1991-07-01
Laplace transform techniques have been widely used to solve linear, transient field problems. A transform-based algorithm enables calculation of the response at selected times of interest without the need for stepping in time as required by conventional time integration schemes. The elimination of time stepping can substantially reduce computer time when transform techniques are implemented in a numerical finite element program. The coupling of transform techniques with spatial discretization techniques such as the finite element method has resulted in what are known as transfinite element methods. Recently attempts have been made to extend the transfinite element method to solve nonlinear, transient field problems. This paper examines the theoretical basis and numerical implementation of one such algorithm, applied to nonlinear heat transfer problems. The problem is linearized and solved by requiring a numerical iteration at selected times of interest. While shown to be acceptable for weakly nonlinear problems, this algorithm is ineffective as a general nonlinear solution method.
Numerical solution of the nonlinear Schrödinger equation using smoothed-particle hydrodynamics
NASA Astrophysics Data System (ADS)
Mocz, Philip; Succi, Sauro
2015-05-01
We formulate a smoothed-particle hydrodynamics numerical method, traditionally used for the Euler equations for fluid dynamics in the context of astrophysical simulations, to solve the nonlinear Schrödinger equation in the Madelung formulation. The probability density of the wave function is discretized into moving particles, whose properties are smoothed by a kernel function. The traditional fluid pressure is replaced by a quantum pressure tensor, for which a robust discretization is found. We demonstrate our numerical method on a variety of numerical test problems involving the simple harmonic oscillator, soliton-soliton collision, Bose-Einstein condensates, collapsing singularities, and dark matter halos governed by the Gross-Pitaevskii-Poisson equation. Our method is conservative, applicable to unbounded domains, and is automatically adaptive in its resolution, making it well suited to study problems with collapsing solutions.
The numerical solution of the boundary inverse problem for a parabolic equation
NASA Astrophysics Data System (ADS)
Vasil'ev, V. V.; Vasilyeva, M. V.; Kardashevsky, A. M.
2016-10-01
Boundary inverse problems occupy an important place among the inverse problems of mathematical physics. They are connected with the problems of diagnosis, when additional measurements on one of the borders or inside the computational domain are necessary to restore the boundary regime in the other border, inaccessible to direct measurements. The boundary inverse problems belong to a class of conditionally correct problems, and therefore, their numerical solution requires the development of special computational algorithms. The paper deals with the solution of the boundary inverse problem for one-dimensional second-order parabolic equations, consisting in the restoration of boundary regime according to measurements inside the computational domain. For the numerical solution of the inverse problem it is proposed to use an analogue of a computational algorithm, proposed and developed to meet the challenges of identification of the right side of the parabolic equations in the works P.N.Vabishchevich and his students based on a special decomposition of solving the problem at each temporal layer. We present and discuss the results of a computational experiment conducted on model problems with quasi-solutions, including with random errors in the input data.
NASA Astrophysics Data System (ADS)
González, A. Ojeda; Domingues, M. O.; Mendes, O.; Kaibara, M. K.; Prestes, A.
2015-10-01
The Grad-Shafranov equation is a Poisson's equation, i.e., a partial differential equation of elliptic type. The problem is depending on the initial condition and can be treated as a Cauchy problem. Although it is ill-posed or ill-conditioned, it can be integrated numerically. In the integration of the GS equation, singularities with large values of the potential arise after a certain number of integration steps away from the original data line, and a filter should be used. The Grad-Shafranov reconstruction (GSR) technique was developed from 1996 to 2000 for recovering two-dimensional structures in the magnetopause in an ideal MHD formulation. Other works have used the GSR techniques to study magnetic flux ropes in the solar wind and in the magnetotail from a single spacecraft dataset; posteriorly, it was extended to treat measurements from multiple satellites. From Vlasov equation, it is possible to arrive at the GS-equation in function of the normalized vector potential. A general solution is obtained using complex variable theory. A specific solution was chosen as benchmark case to solve numerically the GS equation. We propose some changes in the resolution scheme of the GS equation to improve the solution. The result of each method is compared with the solution proposed by Hau and Sonnerup (J. Geophys. Res. 104(A4), 6899-6917 (1999)). The main improvement found in the GS resolution was the need to filter B x values at each y value.
NASA Technical Reports Server (NTRS)
Neilson, D. G.; Incropera, F. D.; Bennon, W. D.
1990-01-01
A computational study of solidification of a binary Na2CO3 solution in a horizontal cylindrical annulus is performed using a continuum formulation with a control-volume based, finite-difference scheme. The initial conditions were selected to facilitate the study of counter thermal and solutal convection, accompanied by extensive mushy region growth. Numerical results are compared with experimental data with mixed success. Qualitative agreement is obtained for the overall solidification process and associated physical phenomena. However, the plume thickness calculated for the solutally-driven convective upflow is substantially smaller than the observed value. Evolution of double-diffusive layers is predicted, but over a time scale much smaller than that observed experimentally. Good agreement is obtained between predicted and measured results for solid growth, but the mushy region thickness is significantly overpredicted.
NASA Astrophysics Data System (ADS)
minatti, L.
2013-12-01
A finite volume model solving the shallow water equations coupled with the sediments continuity equation in composite channels with irregular geometry is presented. The model is essentially 1D but can handle composite cross-sections in which bedload transport is considered to occur inside the main channel only. This assumption is coherent with the observed behavior of rivers on short time scales where main channel areas exhibit more relevant morphological variations than overbanks. Furthermore, such a model allows a more precise prediction of thalweg elevation and cross section shape variations than fully 1D models where bedload transport is considered to occur uniformly over the entire cross section. The coupling of the equations describing water and sediments dynamics results in a hyperbolic non-conservative system that cannot be solved numerically with the use of a conservative scheme. Therefore, a path-conservative scheme, based on the approach proposed by Pares and Castro (2004) has been devised in order to account for the coupling with the sediments continuity equation and for the concurrent presence of bottom elevation and breadth variations of the cross section. In order to correctly compute numerical fluxes related to bedload transport in main channel areas, a special treatment of the equations is employed in the model. The resulting scheme is well balanced and fully coupled and can accurately model abrupt time variations of flow and bedload transport conditions in wide rivers, characterized by the presence of overbank areas that are less active than the main channel. The accuracy of the model has been first tested in fixed bed conditions by solving problems with a known analytical solution: in these tests the model proved to be able to handle shocks and supercritical flow conditions properly(see Fig. 01). A practical application of the model to the Ombrone river, southern Tuscany (Italy) is shown. The river has shown relevant morphological changes during
1989-12-01
Numerical Solution ....... ....................... 113 E.1 Gauss- Jordan .......................................... 113 E.2 L-U Decomposition...2.4 Numerical Solution of the Boltzmann Equation Four numerical were used to solve equation (39). These were: : Gauss- Jordan , L-U Decom- position...both the Gauss- Jordan and L-U decomposition methods. 2.5 Transport Coefficiente In order to provide the required input to the laser design program CO2OSC
ADI FD schemes for the numerical solution of the three-dimensional Heston-Cox-Ingersoll-Ross PDE
NASA Astrophysics Data System (ADS)
Haentjens, Tinne
2012-09-01
This paper deals with the numerical solution of the time-dependent, three-dimensional Heston-Cox-Ingersoll- Ross PDE, with all correlations nonzero, for the fair pricing of European call options. We apply a finite difference dis-cretization on non-uniform spatial grids and then numerically solve the semi-discrete system in time by using an Alternating Direction Implicit scheme. We show that this leads to a highly efficient and stable numerical solution method.
NASA Astrophysics Data System (ADS)
Eliason, Donald E.; Bourgeois, Alfred J.
2001-11-01
An analytical solution is presented for the case of a stratified, tidally forced lagoon. This solution, especially its energy and mass balances, is useful for the validation of numerical shallow water models under stratified, tidally forced conditions. The utility of the analytical solution for validation is demonstrated for a simple finite difference numerical model. The energy, mass, and their balances from the numerical model are shown to converge to the analytical solution with a power law dependence as spatial and temporal resolution are increased. The power law convergence of the numerical results to the analytical solution validates both the finite difference scheme and the open boundary conditions used for the tidal forcing. The open boundary conditions work well because they are consistent with the characteristics of the analytical solution.
Analytical and Numerical Solutions of Generalized Fokker-Planck Equations - Final Report
Prinja, Anil K.
2000-12-31
The overall goal of this project was to develop advanced theoretical and numerical techniques to quantitatively describe the spreading of a collimated beam of charged particles in space, in angle, and in energy, as a result of small deflection, small energy transfer Coulomb collisions with the target nuclei and electrons. Such beams arise in several applications of great interest in nuclear engineering, and include electron and ion radiotherapy, ion beam modification of materials, accelerator transmutation of waste, and accelerator production of tritium, to name some important candidates. These applications present unique and difficult modeling challenges, but from the outset are amenable to the language of ''transport theory'', which is very familiar to nuclear engineers and considerably less-so to physicists and material scientists. Thus, our approach has been to adopt a fundamental description based on transport equations, but the forward peakedness associated with charged particle interactions precludes a direct application of solution methods developed for neutral particle transport. Unique problem formulations and solution techniques are necessary to describe the transport and interaction of charged particles. In particular, we have developed the Generalized Fokker-Planck (GFP) approach to describe the angular and radial spreading of a collimated beam and a renormalized transport model to describe the energy-loss straggling of an initially monoenergetic distribution. Both analytic and numerical solutions have been investigated and in particular novel finite element numerical methods have been developed. In the first phase of the project, asymptotic methods were used to develop closed form solutions to the GFP equation for different orders of expansion, and was described in a previous progress report. In this final report we present a detailed description of (i) a novel energy straggling model based on a Fokker-Planck approximation but which is adapted for a
Numerical Solution of Multi-Dimensional Hyperbolic Conservation Laws on Unstructured Meshes
NASA Technical Reports Server (NTRS)
Barth, Timothy J.; Kwak, Dochan (Technical Monitor)
1995-01-01
The lecture material will discuss the application of one-dimensional approximate Riemann solutions and high order accurate data reconstruction as building blocks for solving multi-dimensional hyperbolic equations. This building block procedure is well-documented in the nationally available literature. The relevant stability and convergence theory using positive operator analysis will also be presented. All participants in the minisymposium will be asked to solve one or more generic test problems so that a critical comparison of accuracy can be made among differing approaches.
Solution of stochastic media transport problems using a numerical quadrature-based method
Pautz, S. D.; Franke, B. C.; Prinja, A. K.; Olson, A. J.
2013-07-01
We present a new conceptual framework for analyzing transport problems in random media. We decompose such problems into stratified subproblems according to the number of material pseudo-interfaces within realizations. For a given subproblem we assign pseudo-interface locations in each realization according to product quadrature rules, which allows us to deterministically generate a fixed number of realizations. Quadrature integration of the solutions of these realizations thus approximately solves each subproblem; the weighted superposition of solutions of the subproblems approximately solves the general stochastic media transport problem. We revisit some benchmark problems to determine the accuracy and efficiency of this approach in comparison to randomly generated realizations. We find that this method is very accurate and fast when the number of pseudo-interfaces in a problem is generally low, but that these advantages quickly degrade as the number of pseudo-interfaces increases. (authors)
Numerical solution of the Penna model of biological aging with age-modified mutation rate
NASA Astrophysics Data System (ADS)
Magdoń-Maksymowicz, M. S.; Maksymowicz, A. Z.
2009-06-01
In this paper we present results of numerical calculation of the Penna bit-string model of biological aging, modified for the case of a -dependent mutation rate m(a) , where a is the parent’s age. The mutation rate m(a) is the probability per bit of an extra bad mutation introduced in offspring inherited genome. We assume that m(a) increases with age a . As compared with the reference case of the standard Penna model based on a constant mutation rate m , the dynamics of the population growth shows distinct changes in age distribution of the population. Here we concentrate on mortality q(a) , a fraction of items eliminated from the population when we go from age (a) to (a+1) in simulated transition from time (t) to next time (t+1) . The experimentally observed q(a) dependence essentially follows the Gompertz exponential law for a above the minimum reproduction age. Deviation from the Gompertz law is however observed for the very old items, close to the maximal age. This effect may also result from an increase in mutation rate m with age a discussed in this paper. The numerical calculations are based on analytical solution of the Penna model, presented in a series of papers by Coe [J. B. Coe, Y. Mao, and M. E. Cates, Phys. Rev. Lett. 89, 288103 (2002)]. Results of the numerical calculations are supported by the data obtained from computer simulation based on the solution by Coe
Numerical implementation of the integral-transform solution to Lamb's point-load problem
NASA Astrophysics Data System (ADS)
Georgiadis, H. G.; Vamvatsikos, D.; Vardoulakis, I.
The present work describes a procedure for the numerical evaluation of the classical integral-transform solution of the transient elastodynamic point-load (axisymmetric) Lamb's problem. This solution involves integrals of rapidly oscillatory functions over semi-infinite intervals and inversion of one-sided (time) Laplace transforms. These features introduce difficulties for a numerical treatment and constitute a challenging problem in trying to obtain results for quantities (e.g. displacements) in the interior of the half-space. To deal with the oscillatory integrands, which in addition may take very large values (pseudo-pole behavior) at certain points, we follow the concept of Longman's method but using as accelerator in the summation procedure a modified Epsilon algorithm instead of the standard Euler's transformation. Also, an adaptive procedure using the Gauss 32-point rule is introduced to integrate in the vicinity of the pseudo-pole. The numerical Laplace-transform inversion is based on the robust Fourier-series technique of Dubner/Abate-Crump-Durbin. Extensive results are given for sub-surface displacements, whereas the limit-case results for the surface displacements compare very favorably with previous exact results.
Numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity
NASA Astrophysics Data System (ADS)
Korepanov, V. V.; Matveenko, V. P.; Fedorov, A. Yu.; Shardakov, I. N.
2013-07-01
An algorithm for the numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity is considered. The algorithm is based on separation of a power-law dependence from the finite-element solution in a neighborhood of singular points in the domain under study, where singular solutions are possible. The obtained power-law dependencies allow one to conclude whether the stresses have singularities and what the character of these singularities is. The algorithm was tested for problems of classical elasticity by comparing the stress singularity exponents obtained by the proposed method and from known analytic solutions. Problems with various cases of singular points, namely, body surface points at which either the smoothness of the surface is violated, or the type of boundary conditions is changed, or distinct materials are in contact, are considered as applications. The stress singularity exponents obtained by using the models of classical and asymmetric elasticity are compared. It is shown that, in the case of cracks, the stress singularity exponents are the same for the elasticity models under study, but for other cases of singular points, the stress singularity exponents obtained on the basis of asymmetric elasticity have insignificant quantitative distinctions from the solutions of the classical elasticity.
Numerical solutions of turbulent models for flow over a flat plate with angle of attack
Truncellito, N.T.; Yeh, H.; Lior, N.
1985-03-01
Numerical solutions of the two-dimensional boundary layer equations were developed as applied to flow over a flat plate at various angles of attack. Three methods of approach were examined. An integral solution was constructed for laminar and turbulent flow, as well as finite difference solutions for zeroth- and first-order turbulence models. The models also account for buoyancy effects. A three part mixing length model was employed in the zeroth-order model, and an additional turbulent kinetic energy equation was utilized for the first-order model. The computational method utilized Patankar-Spalding coordinates and differs from other methods in that no matching procedure is required for the inner and outer flow regions. The Falkner-Skan velocity profile is applied as an edge boundary condition while variable wall temperature conditions can be imposed. The effects of freestream velocity and angle of attack on skin friction and heat transfer were established, and the velocity and temperature fields were determined. Results of the zeroth-order solution are in excellent agreement with the Colburn equation and several other data sources. These solutions provide correlations in terms of Nusselt number and skin friction coefficient versus local Reynolds number which can be used for estimating heat transfer and wind loadings on a flat plate. Results generated are especially useful in predicting the performance of solar system designs.
NASA Astrophysics Data System (ADS)
Zabihi, F.; Saffarian, M.
2016-07-01
The aim of this article is to obtain the numerical solution of the two-dimensional KdV-Burgers equation. We construct the solution by using a different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. We use a predictor-corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.
Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks
NASA Astrophysics Data System (ADS)
Mai-Duy, Nam; Tran-Cong, Thanh
2001-09-01
A numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a universal approximator based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of finite element-type discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re = 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of , Re = 400 for the driven cavity flow with a density of and Rayleigh number Ra = 1 000 000 for the natural convection flow with a density of . Copyright
NASA Technical Reports Server (NTRS)
Chang, Sin-Chung; To, Wai-Ming
1992-01-01
A new numerical method for solving conservation laws is being developed. It differs substantially from the well established methods, i.e., finite difference, finite volume, finite element, and spectral methods, in both concept and methodology. It is much simpler than a typical high resolution method. No flux limiter or any technique related to characteristics is involved. No artificial viscosity or smoothing is introduced, and no moving mesh is used. Yet this method is capable of generating highly accurate shock tube solutions. The slight numerical overshoot and/or oscillations generated can be removed if a simple averaging formula initially used is replaced by a weighted formula. This modification has little effect on other parts of the solution. Because of its simplicity, generalization of this new method for multi-dimensional problems is straightforward.
NASA Technical Reports Server (NTRS)
Przekwas, A. J.; Yang, H. Q.
1989-01-01
The capability of accurate nonlinear flow analysis of resonance systems is essential in many problems, including combustion instability. Classical numerical schemes are either too diffusive or too dispersive especially for transient problems. In the last few years, significant progress has been made in the numerical methods for flows with shocks. The objective was to assess advanced shock capturing schemes on transient flows. Several numerical schemes were tested including TVD, MUSCL, ENO, FCT, and Riemann Solver Godunov type schemes. A systematic assessment was performed on scalar transport, Burgers' and gas dynamic problems. Several shock capturing schemes are compared on fast transient resonant pipe flow problems. A system of 1-D nonlinear hyperbolic gas dynamics equations is solved to predict propagation of finite amplitude waves, the wave steepening, formation, propagation, and reflection of shocks for several hundred wave cycles. It is shown that high accuracy schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time.
Physiology driven adaptivity for the numerical solution of the bidomain equations.
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.
Numerical solution of flow problems using body-fitted coordinate systems
NASA Technical Reports Server (NTRS)
Thompson, J. F.
1980-01-01
The paper deals with numerically generated boundary-fitted coordinate systems. This procedure eliminates the shape of the boundaries as a complicating factor and allows the flow about arbitrary boundaries to be treated essentially as easily as that about simple boundaries. The technique of boundary-fitted coordinate systems is based on a method of automatic numerical generation of a general curvilinear coordinate system having a coordinate line coincident with each boundary of a general multiconnected region involving any number of arbitrarily shaped boundaries. Once the curvilinear coordinate system is generated, any partial differential system of interest may be solved on the coordinate system by transforming the equations and solving the resulting system in finite-difference approximation on the rectangular transformed plane. Attention is given to the types of boundary-fitted coordinate systems, coordinate system control, operation of the coordinate codes, solution of partial differential equations, application to free-surface flow, and other applications of interest.
NASA Technical Reports Server (NTRS)
Winterstein, R.; Hafez, M.
1993-01-01
A finite volume method is used to calculate compressible inviscid flows over blunt bodies using, in general, unstructured grids. Artificial viscosity forms are derived based on a simplified least squares procedure. The extra second order terms are consistent with the governing equations, hence a systematic treatment of the numerical boundary conditions can be easily implemented. A special treatment of blunt bodies may be required. The discrete equations are linearized and the resulting system is solved by a relaxation method. Preliminary results indicate that the effect of the numerical dissipation is minimal. For subsonic flows over smooth bodies, the solution is practically vorticity-free and the total pressure loss is of the same order as the truncation error. Finally, some extensions of the present method are briefly discussed.
Numerical solution of the Rosenau-KdV-RLW equation by using RBFs collocation method
NASA Astrophysics Data System (ADS)
Korkmaz, Bahar; Dereli, Yilmaz
2016-04-01
In this study, a meshfree method based on the collocation with radial basis functions (RBFs) is proposed to solve numerically an initial-boundary value problem of Rosenau-KdV-regularized long-wave (RLW) equation. Numerical values of invariants of the motion are computed to examine the fundamental conservative properties of the equation. Computational experiments for the simulation of solitary waves examine the accuracy of the scheme in terms of error norms L2 and L∞. Linear stability analysis is investigated to determine whether the present method is stable or unstable. The scheme gives unconditionally stable, and second-order convergent. The obtained results are compared with analytical solution and some other earlier works in the literature. The presented results indicate the accuracy and efficiency of the method.
Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines
NASA Astrophysics Data System (ADS)
Bermúdez, Alfredo; López, Xián; Vázquez-Cendón, M. Elena
2016-10-01
A finite volume scheme for the numerical solution of a mathematical model for non-isothermal non-adiabatic compressible flow of a real gas in a pipeline is introduced. In order to make an upwind discretization of the flux, the Q-scheme of van Leer is used. Unlike standard Euler equations, the model takes into account wall friction, variable height and heat transfer between the pipe and the environment. Since all these terms are sources, in order to get a well-balanced scheme they are discretized by making a similar upwinding to the one in the flux term. The performance of the overall method has been shown for some usual numerical tests. The final goal, which is beyond the scope of this paper, is to consider a network including several pipelines connected at junctions, as those employed for natural gas transport.
CSR Fields: Direct Numerical Solution of the Maxwell___s Equation
Novokhatski, A.; /SLAC
2011-06-22
We discuss the properties of the coherent electromagnetic fields of a very short, ultra-relativistic bunch in a rectangular vacuum chamber inside a bending magnet. The analysis is based on the results of a direct numerical solution of Maxwell's equations together with Newton's equations. We use a new dispersion-free time-domain algorithm which employs a more efficient use of finite element mesh techniques and hence produces self-consistent and stable solutions for very short bunches. We investigate the fine structure of the CSR fields including coherent edge radiation. This approach should be useful in the study of existing and future concepts of particle accelerators and ultrafast coherent light sources. The coherent synchrotron radiation (CSR) fields have a strong action on the beam dynamics of very short bunches, which are moving in the bends of all kinds of magnetic elements. They are responsible for additional energy loss and energy spread; micro bunching and beam emittance growth. These fields may bound the efficiency of damping rings, electron-positron colliders and ultrafast coherent light sources, where high peak currents and very short bunches are envisioned. This is relevant to most high-brightness beam applications. On the other hand these fields together with transition radiation fields can be used for beam diagnostics or even as a powerful resource of THz radiation. A history of the study of CSR and a good collection of references can be found in [1]. Electromagnetic theory suggests several methods on how to calculate CSR fields. The most popular method is to use Lienard-Wiechert potentials. Other approach is to solve numerically the approximate equations, which are a Schrodinger type equation. These numerical methods are described in [2]. We suggest that a direct solution of Maxwell's equations together with Newton's equations can describe the detailed structure of the CSR fields [3].
NASA Astrophysics Data System (ADS)
Watanabe, N.; Sun, Y.; Taron, J.; Shao, H.; Kolditz, O.
2013-12-01
Modeling fracture permeability evolution is of great interest in various geotechnical applications including underground waste repositories, carbon capture and storage, and engineered geothermal systems where fractures dominate transport behaviors. In this study, a numerical model is presented to simulate fracture permeability evolution due to reactive transport and pressure solution processes in single fractures. The model was developed within the international benchmarking project for radioactive waste disposals, DECOVALEX 2015 (Task C1). The model combines bulk behavior in pore spaces with intergranular process at asperity contacts. Hydraulic flow and reactive transport including mineral dissolution and precipitation in fracture pore space are simulated using the Galerkin finite element method. A pressure solution model developed by Taron and Elsworth (2010 JGR) is applied to simulating stress-enhanced dissolution, solute exchange with pore space, and volume removal at grain contacts. Fracture aperture and contact area ratio are updated as a result of the pore-space reaction and intergranular dissolution. In order to increase robustness and time step size, relevant processes are monolithically coupled with the simulations. The model is implemented in a scientific open-source project OpenGeoSys (www.opengeosys.org) for numerical simulation of thermo-hydro-mechanical/chemical processes in porous and fractured media. Numerical results are compared to previous experiment performed by Yasuhara et al. (2006) on flow through fractures in the Arkansas novaculite sample. The novaculite is approximated as pure quartz aggregates. Only with fitted quartz dissolution rate constants and solubility is the current model capable of reproducing observed hydraulic aperture reduction and aqueous silicate concentrations. Future work will examine reaction parameters and further validate the model against experimental results.
A non-grey analytical model for irradiated atmospheres. II. Analytical vs. numerical solutions
NASA Astrophysics Data System (ADS)
Parmentier, Vivien; Guillot, Tristan; Fortney, Jonathan J.; Marley, Mark S.
2015-02-01
Context. The recent discovery and characterization of the diversity of the atmospheres of exoplanets and brown dwarfs calls for the development of fast and accurate analytical models. Aims: We wish to assess the goodness of the different approximations used to solve the radiative transfer problem in irradiated atmospheres analytically, and we aim to provide a useful tool for a fast computation of analytical temperature profiles that remains correct over a wide range of atmospheric characteristics. Methods: We quantify the accuracy of the analytical solution derived in paper I for an irradiated, non-grey atmosphere by comparing it to a state-of-the-art radiative transfer model. Then, using a grid of numerical models, we calibrate the different coefficients of our analytical model for irradiated solar-composition atmospheres of giant exoplanets and brown dwarfs. Results: We show that the so-called Eddington approximation used to solve the angular dependency of the radiation field leads to relative errors of up to ~5% on the temperature profile. For grey or semi-grey atmospheres (i.e., when the visible and thermal opacities, respectively, can be considered independent of wavelength), we show that the presence of a convective zone has a limited effect on the radiative atmosphere above it and leads to modifications of the radiative temperature profile of approximately ~2%. However, for realistic non-grey planetary atmospheres, the presence of a convective zone that extends to optical depths smaller than unity can lead to changes in the radiative temperature profile on the order of 20% or more. When the convective zone is located at deeper levels (such as for strongly irradiated hot Jupiters), its effect on the radiative atmosphere is again on the same order (~2%) as in the semi-grey case. We show that the temperature inversion induced by a strong absorber in the optical, such as TiO or VO is mainly due to non-grey thermal effects reducing the ability of the upper
Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem
NASA Technical Reports Server (NTRS)
Ceccobello, C.; Farinelli, R.; Titarchuk, L.
2014-01-01
We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the
On numerical solution of multipoint NBVP for hyperbolic-parabolic equations with Neumann condition
NASA Astrophysics Data System (ADS)
Ashyralyev, Allaberen; Ozdemir, Yildirim
2012-08-01
A numerical method is proposed for solving multi-dimensional hyperbolic-parabolic differential equations with the nonlocal boundary condition in t and Neumann condition in space variables. The first and second orders of accuracy difference schemes are presented. The stability estimates for the solution and its first and second orders difference derivatives are established. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of a one-dimensional hyperbolic-parabolic differential equations with variable in x coefficients.
Numerical solutions of the time-dependent Schrödinger equation in two dimensions
NASA Astrophysics Data System (ADS)
van Dijk, Wytse; Vanderwoerd, Trevor; Prins, Sjirk-Jan
2017-02-01
The generalized Crank-Nicolson method is employed to obtain numerical solutions of the two-dimensional time-dependent Schrödinger equation. An adapted alternating-direction implicit method is used, along with a high-order finite-difference scheme in space. Extra care has to be taken for the needed precision of the time development. The method permits a systematic study of the accuracy and efficiency in terms of powers of the spatial and temporal step sizes. To illustrate its utility the method is applied to several two-dimensional systems.
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.
NASA Technical Reports Server (NTRS)
Nixon, D.
1978-01-01
The linear transonic perturbation integral equation previously derived for nonlifting airfoils is formulated for lifting cases. In order to treat shock wave motions, a strained coordinate system is used in which the shock location is invariant. The tangency boundary conditions are either formulated using the thin airfoil approximation or by using the analytic continuation concept. A direct numerical solution to this equation is derived in contrast to the iterative scheme initially used, and results of both lifting and nonlifting examples indicate that the method is satisfactory.
Numerical solutions of the time-dependent Schrödinger equation in two dimensions.
van Dijk, Wytse; Vanderwoerd, Trevor; Prins, Sjirk-Jan
2017-02-01
The generalized Crank-Nicolson method is employed to obtain numerical solutions of the two-dimensional time-dependent Schrödinger equation. An adapted alternating-direction implicit method is used, along with a high-order finite-difference scheme in space. Extra care has to be taken for the needed precision of the time development. The method permits a systematic study of the accuracy and efficiency in terms of powers of the spatial and temporal step sizes. To illustrate its utility the method is applied to several two-dimensional systems.
Numerical Solution of Hydrodynamics Lubrications with Non-Newtonian Fluid Flow
NASA Astrophysics Data System (ADS)
Osman, Kahar; Sheriff, Jamaluddin Md; Bahak, Mohd. Zubil; Bahari, Adli; Asral
2010-06-01
This paper focuses on solution of numerical model for fluid film lubrication problem related to hydrodynamics with non-Newtonian fluid. A programming code is developed to investigate the effect of bearing design parameter such as pressure. A physical problem is modeled by a contact point of sphere on a disc with certain assumption. A finite difference method with staggered grid is used to improve the accuracy. The results show that the fluid characteristics as defined by power law fluid have led to a difference in the fluid pressure profile. Therefore a lubricant with special viscosity can reduced the pressure near the contact area of bearing.
NASA Technical Reports Server (NTRS)
Jameson, A.
1976-01-01
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
WATSFAR: numerical simulation of soil WATer and Solute fluxes using a FAst and Robust method
NASA Astrophysics Data System (ADS)
Crevoisier, David; Voltz, Marc
2013-04-01
To simulate the evolution of hydro- and agro-systems, numerous spatialised models are based on a multi-local approach and improvement of simulation accuracy by data-assimilation techniques are now used in many application field. The latest acquisition techniques provide a large amount of experimental data, which increase the efficiency of parameters estimation and inverse modelling approaches. In turn simulations are often run on large temporal and spatial domains which requires a large number of model runs. Eventually, despite the regular increase in computing capacities, the development of fast and robust methods describing the evolution of saturated-unsaturated soil water and solute fluxes is still a challenge. Ross (2003, Agron J; 95:1352-1361) proposed a method, solving 1D Richards' and convection-diffusion equation, that fulfil these characteristics. The method is based on a non iterative approach which reduces the numerical divergence risks and allows the use of coarser spatial and temporal discretisations, while assuring a satisfying accuracy of the results. Crevoisier et al. (2009, Adv Wat Res; 32:936-947) proposed some technical improvements and validated this method on a wider range of agro- pedo- climatic situations. In this poster, we present the simulation code WATSFAR which generalises the Ross method to other mathematical representations of soil water retention curve (i.e. standard and modified van Genuchten model) and includes a dual permeability context (preferential fluxes) for both water and solute transfers. The situations tested are those known to be the less favourable when using standard numerical methods: fine textured and extremely dry soils, intense rainfall and solute fluxes, soils near saturation, ... The results of WATSFAR have been compared with the standard finite element model Hydrus. The analysis of these comparisons highlights two main advantages for WATSFAR, i) robustness: even on fine textured soil or high water and solute
NASA Technical Reports Server (NTRS)
Rosenbaum, J. S.
1971-01-01
Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.
An Implicit Semianalytic Numerical Method for the Solution of Nonequilibrium Chemistry Problems.
1974-07-01
NONEQUILIBRIUM CHEMISTRY PROBLEMS By R. A. Graves, Jr., P. A. Gnoffo and R. E. Boughner _A- O a TIC AW• d Sex p . , I llELECTE 94-15969 JTu1974Y1r7 This Informal...Report Dow An Implicit Semianalytical Numerical Method for the July 1974 Solution of Nonequilibrium Chemistry Problems 6. P ,-oSrming OfpiZatlOOCode...Absract Most nonequilibrium chemistry problems, and many other physical phenomena, are modeled by systems of first order nonlinear ordinary or partial
A high-resolution numerical technique for inviscid gas-dynamic problems with weak solutions
NASA Technical Reports Server (NTRS)
Yee, H. C.; Warming, R. F.; Harten, A.
1982-01-01
The shock resolution of Harten's (1982) second-order explicit method for one-dimensional hyperbolic conservation laws is investigated for a two-dimensional gas-dynamic problem. The possible extension to a high resolution implicit method for both one- and two-dimensional problems is also investigated. Applications of Harten's method to the quasi-one-dimensional nozzle problem with two nozzle shapes (divergent and convergent-divergent) and the two-dimensional shock-reflection problem resulted in high shock resolution steady-state numerical solutions.
Numerical solution of the Navier-Stokes equations for arbitrary blunt bodies in supersonic flows
NASA Technical Reports Server (NTRS)
Warsi, Z. U. A.; Devarayalu, K.; Thompson, J. F.
1978-01-01
A time-dependent, two-dimensional Navier-Stokes code employing the body-fitted coordinate technique has been developed for supersonic flows past blunt bodies of arbitrary shape. The computer program is based on the finite-difference approximation of the compressible Navier-Stokes equations transformed to nonorthogonal curvilinear coordinates with the contravariant components of the velocity vector as dependent variables. The bow shock ahead of the body is obtained as part of the solution, by 'shock capturing'. Numerical solutions of the complete equations are presented in detail for free-stream Mach number 4.6, Reynolds number 10,000, and an isothermal wall temperature of 556 K for a circular cylinder with the free-stream outer boundaries forming a hyperbola in the front and a circular arc in the back.
A Numerical Algorithm for the Solution of a Phase-Field Model of Polycrystalline Materials
Dorr, M R; Fattebert, J; Wickett, M E; Belak, J F; Turchi, P A
2008-12-04
We describe an algorithm for the numerical solution of a phase-field model (PFM) of microstructure evolution in polycrystalline materials. The PFM system of equations includes a local order parameter, a quaternion representation of local orientation and a species composition parameter. The algorithm is based on the implicit integration of a semidiscretization of the PFM system using a backward difference formula (BDF) temporal discretization combined with a Newton-Krylov algorithm to solve the nonlinear system at each time step. The BDF algorithm is combined with a coordinate projection method to maintain quaternion unit length, which is related to an important solution invariant. A key element of the Newton-Krylov algorithm is the selection of a preconditioner to accelerate the convergence of the Generalized Minimum Residual algorithm used to solve the Jacobian linear system in each Newton step. Results are presented for the application of the algorithm to 2D and 3D examples.
NASA Astrophysics Data System (ADS)
Dominguez, D. R. C.; Maravall, M.; Turiel, A.; Ciria, J. C.; Parga, N.
1999-03-01
The mutual information of a single-layer perceptron with N Gaussian inputs and P deterministic binary outputs is studied by numerical simulations. The relevant parameters of the problem are the ratio between the number of output and input units, α = P/N, and those describing the two-point correlations between inputs. The main motivation of this work refers to the comparison between the replica computation of the mutual information and an analytical solution valid up to α ~ O(1). The most relevant results are: 1) the simulation supports the validity of the analytical prediction, and 2) it also verifies a previously proposed conjecture that the replica solution interpolates well between large and small values of α.
Mustafa, Meraj; Farooq, Muhammad A; Hayat, Tasawar; Alsaedi, Ahmed
2013-01-01
This investigation is concerned with the stagnation-point flow of nanofluid past an exponentially stretching sheet. The presence of Brownian motion and thermophoretic effects yields a coupled nonlinear boundary-value problem (BVP). Similarity transformations are invoked to reduce the partial differential equations into ordinary ones. Local similarity solutions are obtained by homotopy analysis method (HAM), which enables us to investigate the effects of parameters at a fixed location above the sheet. The numerical solutions are also derived using the built-in solver bvp4c of the software MATLAB. The results indicate that temperature and the thermal boundary layer thickness appreciably increase when the Brownian motion and thermophoresis effects are strengthened. Moreover the nanoparticles volume fraction is found to increase when the thermophoretic effect intensifies.
NASA Astrophysics Data System (ADS)
Santos, Leonardo S. F.; Pires, Marcelo O. C.; Giugno, Davi
2015-03-01
We study the stationary solution of an atomic Bose-Einstein condensate coupled coherently to a molecular condensate with both repulsive and attractive interspecies interactions confined in an isotropic harmonic trap. We use the Thomas-Fermi approximation and find four kinds of analytical solution for the cases. These analytical solutions are adopted as trial function for the diffusive numerical solution of the Gross-Pitaevskii equations. For the repulsive interspecies interaction, the case in which the atomic and molecular wavefunctions are out-phase, the densities have similar profiles for both methods, however, the case where the wavefunctions are in-phase, there are considerable difference between the density profiles. For the attractive interspecies interaction, there are two cases in the Thomas-Fermi approximation where the wavefunctions are in-phase. One of them has numerical solution that agree with the approximation and the other does not have corresponding numerical solution.
Solution of the main problem of the lunar physical libration by a numerical method
NASA Astrophysics Data System (ADS)
Zagidullin, Arthur; Petrova, Natalia; Nefediev, Yurii
2016-10-01
Series of the lunar programs requires highly accurate ephemeris of the Moon at any given time. In the light of the new requirements on the accuracy the requirements to the lunar physical libration theory increase.At the Kazan University there is the experience of constructing the lunar rotation theory in the analytical approach. Analytical theory is very informative in terms of the interpretation of the observed data, but inferior to the accuracy of numerical theories. The most accurate numerical ephemeris of the Moon is by far the ephemeris DE430 / 431 built in the USA. It takes into account a large number of subtle effects both in external perturbations of the Moon, and in its internal structure. Before the Russian scientists the task is to create its own numerical theory that would be consistent with the American ephemeris. On the other hand, even the practical application of the american ephemeris requires a deep understanding of the principles of their construction and the intelligent application.As the first step, we constructed a theory in the framework of the main problem. Because we compare our theory with the analytical theory of Petrova (1996), all the constants and the theory of orbital motion are taken identical to the analytical theory. The maximum precision, which the model can provide is 0.01 seconds of arc, which is insufficient to meet the accuracy of modern observations, but this model provides the necessary basis for further development.We have constructed the system of the libration equations, for which the numerical integrator was developed. The internal accuracy of the software integrator is several nanoseconds. When compared with the data of Petrova the differences of order of 1 second are observed at the resonant frequencies. The reason, we believe, in the inaccuracy of the analytical theory. We carried out a comparison with the Eroshkin's data [2], which gave satisfactory agreement, and with Rambaux data. In the latter case, as expected
Numerical solution for viscous flow for two-dimensional domains using orthogonal coordinate systems
NASA Astrophysics Data System (ADS)
Rangwalla, A. A.; Munson, B. R.
1987-06-01
A numerical technique is developed to generate two-dimensional orthogonal maps for simply and doubly connected domains, solving the strong constraint as defined by Ryskin and Leal (1983) for the case of known domain boundaries. The approach employed is shown to avoid the use of Dirichlet boundary conditions and to perform accurately even with highly skewed initial grids. The technique is then used to solve the unsteady Navier-Stokes equations (in the vorticity/stream-function formulation of Roache, 1972) for a doubly connected two-dimensional domain (a rotating circular cylinder with a stationary rounded-end tab). The results are presented graphically and compared with analytical results and experimental data: good agreement is obtained at low Reynolds numbers, and at higher Re when boundary vorticity relaxation is applied.
Formalism of two potentials for the numerical solution of Maxwell's equations
NASA Astrophysics Data System (ADS)
Kudryavtsev, A. N.; Trashkeev, S. I.
2013-11-01
A new formulation of Maxwell's equations based on the introduction of two vector and two scalar potentials is proposed. As a result, the electromagnetic field equations are written as a hyperbolic system that contains, in contrast to the original Maxwell system, only evolution equations and does not involve equations in the form of differential constraints. This makes the new equations especially convenient for the numerical simulation of electromagnetic processes. Specifically, they can be solved by applying powerful modern shock-capturing methods based on the approximation of spatial derivatives by upwind differences. The cases of an electromagnetic field in a vacuum and an inhomogeneous material are considered. Examples are given in which electromagnetic wave propagation is simulated by solving the formulated system of equations with the help of modern high-order accurate schemes.
Numerical solution of an elastic and viscoelastic gravitational models by the finite element method
NASA Astrophysics Data System (ADS)
Arjona Almodóvar, A.; Chacón Rebollo, T.; Gómez Marmol, M.
2014-12-01
Volcanic areas present a lower effective viscosity than usually in the Earth's crust. Both the elastic-gravitational and the viscoelastic-gravitational models allow the computation of gravity, deformation, and gravitational potential changes in order to investigate crustal deformations of Earth (see for instance Battaglia & Segall, 2004; Fernández et al. 1999, 2001; Rundle 1980 and 1983). These models can be represented by a coupled system of linear parabolic (for the elastic deformations), hyperbolic (for the viscoelastic deformations) and elliptic partial differential equations (for gravitational potential changes) (see for instance Arjona et al. 2008 and 2010). The existence and uniqueness of weak solutions for both the elastic-gravitational and viscoelastic-gravitational problem was demonstrated in Arjona et al. (2008 and 2014). The stabilization to solutions of the associated stationary system was proved in Arjona and Díaz (2007). Here we consider the internal source as response to the effect of a pressurized magma reservoir into a multilayered, elastic-gravitational and viscoelastic-gravitational earth model. We introduce the numerical analysis of a simplified steady elastic-gravitational model, solved by means of the finite element method. We also present some numerical tests in realistic situations that confirm the predictions of theoretical order of convergence. Finally, we describe the methodology for both the elastic-gravitational and the viscoelastic-gravitational models using 2D and 3D test examples performed with FreeFEM++.
An efficient numerical technique for the solution of the monodomain and bidomain equations.
Whiteley, Jonathan P
2006-11-01
Most numerical schemes for solving the monodomain or bidomain equations use a forward approximation to some or all of the time derivatives. This approach, however, constrains the maximum timestep that may be used by stability considerations as well as accuracy considerations. Stability may be ensured by using a backward approximation to all time derivatives, although this approach requires the solution of a very large system of nonlinear equations at each timestep which is computationally prohibitive. In this paper we propose a semi-implicit algorithm that ensures stability. A linear system is solved on each timestep to update the transmembrane potential and, if the bidomain equations are being used, the extracellular potential. The remainder of the equations to be solved uncouple into small systems of ordinary differential equations. The backward Euler method may be used to solve these systems and guarantee numerical stability: as these systems are small, only the solution of small nonlinear systems are required. Simulations are carried out to show that the use of this algorithm allows much larger timesteps to be used with only a minimal loss of accuracy. As a result of using these longer timesteps the computation time may be reduced substantially.
NASA Astrophysics Data System (ADS)
Hrubý, Jan
2012-04-01
Mathematical modeling of the non-equilibrium condensing transonic steam flow in the complex 3D geometry of a steam turbine is a demanding problem both concerning the physical concepts and the required computational power. Available accurate formulations of steam properties IAPWS-95 and IAPWS-IF97 require much computation time. For this reason, the modelers often accept the unrealistic ideal-gas behavior. Here we present a computation scheme based on a piecewise, thermodynamically consistent representation of the IAPWS-95 formulation. Density and internal energy are chosen as independent variables to avoid variable transformations and iterations. On the contrary to the previous Tabular Taylor Series Expansion Method, the pressure and temperature are continuous functions of the independent variables, which is a desirable property for the solution of the differential equations of the mass, energy, and momentum conservation for both phases.
Numerical steady flow solutions of the lower leg venous circulation: effects of external compression
NASA Astrophysics Data System (ADS)
Fullana, J.-M.; Flaud, P.
2007-06-01
We present a numerical model used to compute steady flow solutions of the venous circulation of the leg. The network topology is based on clinical data and the flow is assumed to be steady, incompressible, and one-dimensional. We develop a non Newtonian approach to a one-dimensional flow because the blood viscosity depends on the velocity profile, and we demonstrate theoretically the pertinence of a phenomenological law of equivalent viscosity. Clinical experiments observe hemodynamical variables (i.e. venous pressure, venous area, blood velocity) only at the accessible places. In contrast the numerical model results are not limited to particular locations but can be evaluated on every point of the network. It provides important help to the definition of a clinical protocol. The model was designed to quantify a compression level of elastic compression stockings and to plan clinical studies. We validate the numerical approach using a published clinical trial, where the diameter of superficial and deep veins were measured at different compression pressures. We show also that the viscosity variations in a bed-rest position as a consequence of the application of a European Class II compression stockings. These variations could prevent the hyper-coagulability and the stasis of the blood.
Numerical solution of the quantum Lenard-Balescu equation for non-degenerate plasmas
NASA Astrophysics Data System (ADS)
Graziani, Frank; Scullard, Christian; Belt, Andrew; Fennell, Susan; Jankovic, Marija; Ng, Nathan; Serna, Susana
2016-10-01
For weakly-coupled plasmas, time-dependent non-equilibrium effects are usually studied by numerically solving the Landau equation in Fokker-Planck form. This system requires an input Coulomb logarithm, which adds a level of ambiguity to the calculation that can only be remedied by considering a more sophisticated collision operator. We have recently developed a spectral method for numerically solving the quantum Lenard-Balescu equation, which includes the effects of both quantum diffraction and dynamic screening, eliminating the divergences that require an input Coulomb logarithm. Our method allows a fast and accurate integration over the dielectric function for general non-equilibrium distributions. I will present calculations on various systems, including one- and two-component plasmas, and comparisons with the Landau equation. I will also discuss future prospects for the method. This work was performed un- der the auspices of the U.S. Department of Energy at the Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.
Numerical solution of the Hartree-Fock equation in molecular geometries
Talman, James D.
2010-11-15
Solutions of the restricted Hartree-Fock equations are obtained for small molecules using a combination of variationally optimized atomic orbitals centered on the nuclei and corrections computed on a Cartesian mesh. The problem of finding the corrections is reduced to the problem of solving the Hartree-Fock equations with inhomogeneous terms. An iterative method is developed in which the equation is treated as an inhomogeneous Helmholtz equation with the potential terms transferred to the inhomogeneous term. Terms in the equation that arise from rapid variation of the orbitals in the neighborhoods of the nuclei are treated analytically. The Helmholtz equation can then be solved using a fast Fourier transform method. Results for a number of small molecules that are accurate at the millihartree level are presented. The method for solving the inhomogeneous Hartree-Fock equation should be applicable to other problems of quantum chemistry.
Numerical solution of transonic wing flows using an Euler/Navier-Stokes zonal approach
NASA Technical Reports Server (NTRS)
Holst, T. L.; Gundy, K. L.; Thomas, S. D.; Chaderjian, N. M.; Flores, J.
1985-01-01
Transonic flow fields about wing geometries are computed using an Euler/Navier-Stokes approach in which the flow field is divided into several zones. The grid zones immediately adjacent to the wing surface are suitably clustered and solved with the Navier-Stokes equations. Grid zones removed from the wing are less finely clustered and are solved with the Euler equations. Wind tunnel wall effects are easily and accurately modeled with the new grid-zoning algorithm because the wind tunnel grid is constructed as an exact subset of the corresponding free-air grid. Solutions are obtained that are in good agreement with experiment, including cases with significant wind tunnel wall effects and shock-induced separation on the upper wing surface.
Baehr, A.L.; Corapcioglu, M.Y.
1987-01-01
In this paper we develop a numerical solution to equations developed in part 1 (M.Y. Corapcioglu and A.L. Baehr, this issue) to predict the fate of an immiscible organic contaminant such as gasoline in the unsaturated zone subsequent to plume establishment. This solution, obtained by using a finite difference scheme and a method of forward projection to evaluate nonlinear coefficients, provides estimates of the flux of solubilized hydrocarbon constituents to groundwater from the portion of a spill which remains trapped in a soil after routine remedial efforts to recover the product have ceased. The procedure was used to solve the one-dimensional (vertical) form of the system of nonlinear partial differential equations defining the transport for each constituent of the product. Additionally, a homogeneous, isothermal soil with constant water content was assumed. An equilibrium assumption partitions the constituents between air, water, adsorbed, and immiscible phases. Free oxygen transport in the soil was also simulated to provide an upper bound estimate of aerobic biodegradation rates. Results are presented for a hypothetical gasoline consisting of eight groups of hydrocarbon constituents. Rates at which hydrocarbon mass is removed from the soil, entering either the atmosphere or groundwater, or is biodegraded are presented. A significant sensitivity to model parameters, particularly the parameters characterizing diffusive vapor transport, was discovered. We conclude that hypocarbon solute composition in groundwater beneath a gasoline contaminated soil would be heavily weighted toward aromatic constituents like benzene, toluene, and xylene.In this paper we develop a numerical solution to equations developed in part 1 (M. Y. Corapcioglu and A. L. Baehr, this issue) to predict the fate of an immiscible organic contaminant such as gasoline in the unsaturated zone subsequent to plume establishment. This solution, obtained by using a finite difference scheme and a method
NASA Technical Reports Server (NTRS)
Rogers, S. E.; Kwak, D.; Chang, J. L. C.
1986-01-01
Numerically solving the incompressible Navier-Stokes equations is known to be time consuming and expensive. Testing of the INS3D computers code, which solves these equations with the use of the pseudocompressibility method, shows this method to be an efficient way to obtain the steady state solution. The effects of the waves introduced by the pseudocompressibility method are analyzed and criteria are set and tested for the choice of the pseudocompressibility parameter which governs the artificial sound speed. The code is tested using laminar flow over a two dimensional backward-facing step, and laminar flow over a two dimensional circular cylinder. The results of the computations over the backward-facing step are in excellent agreement with experimental results. The transient solution of the flow over the cylinder impulsively started from rest is in good agreement with experimental results. However, the computed frequency of periodic shedding of vortices behind the cylinder is not in agreement with the experimental value. For a three dimensional test case, computations were conducted for a cylinder end wall junction. The saddle point separation and horseshoe vortex system appear in the computed field. The solution also shows secondary vortex filaments which wrap around the cylinder and spiral up in the wake.
Numerical solutions of Navier-Stokes equations for the structure of a trailing vortex
NASA Technical Reports Server (NTRS)
Jain, A. C.
1977-01-01
The structure and decay of a trailing vortex were analyzed during the numerical solutions of the full Navier-Stokes equations. Unsteady forms of the governing equations were recast in terms of circulation, vorticity, and stream function as dependent variables, and a second upwind finite difference scheme was used to integrate them with prescribed initial and boundary conditions. The boundary conditions at the outer edge and at the outflow section of the trailing vortex were considered. Different models of the flow were postulated, and solutions were obtained describing the development of the flow as integration proceeds in time. A parametric study was undertaken with a view to understanding the various phenomena that may possibly occur in the trailing vortex. Using the Hoffman and Joubert law of circulation at the inflow section, the results of this investigation were compared with experimental data for a Convair 990 wind model and a rectangular wing. With an exponentially decaying law of circulation at the inflow section and an adverse pressure gradient at the outer edge of the trailing vortex, solutions depict vortex bursting through the sudden expansion of the core and/or through the stagnation and consequent reversal of the flow on the axis. It was found that this bursting takes place at lower values of the swirl ratio as the Reynolds number increases.
NASA Technical Reports Server (NTRS)
Lummerzheim, D.; Lilensten, J.
1994-01-01
Auroral electron transport calculations are a critical part of auroral models. We evaluate a numerical solution to the transport and energy degradation problem. The numerical solution is verified by reproducing simplified problems to which analytic solutions exist, internal self-consistency tests, comparison with laboratory experiments of electron beams penetrating a collision chamber, and by comparison with auroral observations, particularly the emission ratio of the N2 second positive to N2(+) first negative emissions. Our numerical solutions agree with range measurements in collision chambers. The calculated N(2)2P to N2(+)1N emission ratio is independent of the spectral characteristics of the incident electrons, and agrees with the value observed in aurora. Using different sets of energy loss cross sections and different functions to describe the energy distribution of secondary electrons that emerge from ionization collisions, we discuss the uncertainties of the solutions to the electron transport equation resulting from the uncertainties of these input parameters.
NASA Astrophysics Data System (ADS)
Fan, Cui-Ying; Zhao, Ming-Hao; Zhou, You-He
2009-09-01
The polarization saturation (PS) model [Gao, H., Barnett, D.M., 1996. An invariance property of local energy release rates in a strip saturation model of piezoelectric fracture. Int. J. Fract. 79, R25-R29; Gao, H., Zhang, T.Y., Tong, P., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491-510], and the dielectric breakdown (DB) model [Zhang, T.Y., Zhao, M.H., Cao, C.F., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311-327] explain very well some experimental observations of fracture of piezoelectric ceramics. In this paper, the nonlinear hybrid extended displacement discontinuity-fundamental solution method (NLHEDD-FSM) is presented for numerical analysis of both the PS and DB models of two-dimensional (2D) finite piezoelectric media under impermeable and semi-permeable electric boundary conditions. In this NLHEDD-FSM, the solution is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack and the electric yielding zone. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy certain conditions on the boundary of the domain, on the crack face and the electric yielding zone. The zero electric displacement intensity factor in the PS model or the zero electric field strength intensity factor in the DB model at the outer tips of the electric yielding zone is used as a supplementary condition to determine the size of the electric yielding zone. Iteration approaches are adopted in the NLHEDD-FSM. The electric yielding zone is determined, and the extended intensity factors and the local J-integral are calculated for
NASA Astrophysics Data System (ADS)
Chiumenti, M.; Cervera, M.; Agelet de Saracibar, C.; Dialami, N.
2013-05-01
In this work a novel finite element technology based on a three-field mixed formulation is presented. The Variational Multi Scale (VMS) method is used to circumvent the LBB stability condition allowing the use of linear piece-wise interpolations for displacement, stress and pressure fields, respectively. The result is an enhanced stress field approximation which enables for stress-accurate results in nonlinear computational mechanics. The use of an independent nodal variable for the pressure field allows for an adhoc treatment of the incompressibility constraint. This is a mandatory requirement due to the isochoric nature of the plastic strain in metal forming processes. The highly non-linear stress field typically encountered in the Friction Stir Welding (FSW) process is used as an example to show the performance of this new FE technology. The numerical simulation of the FSW process is tackled by means of an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The computational domain is split into three different zones: the work.piece (defined by a rigid visco-plastic behaviour in the Eulerian framework), the pin (within the Lagrangian framework) and finally the stirzone (ALE formulation). A fully coupled thermo-mechanical analysis is introduced showing the heat fluxes generated by the plastic dissipation in the stir-zone (Sheppard rigid-viscoplastic constitutive model) as well as the frictional dissipation at the contact interface (Norton frictional contact model). Finally, tracers have been implemented to show the material flow around the pin allowing a better understanding of the welding mechanism. Numerical results are compared with experimental evidence.
Nelson, B; Liu, E; Kirby, R M; Haimes, R
2012-12-01
This paper presents the Element Visualizer (ElVis), a new, open-source scientific visualization system for use with high-order finite element solutions to PDEs in three dimensions. This system is designed to minimize visualization errors of these types of fields by querying the underlying finite element basis functions (e.g., high-order polynomials) directly, leading to pixel-exact representations of solutions and geometry. The system interacts with simulation data through runtime plugins, which only require users to implement a handful of operations fundamental to finite element solvers. The data in turn can be visualized through the use of cut surfaces, contours, isosurfaces, and volume rendering. These visualization algorithms are implemented using NVIDIA's OptiX GPU-based ray-tracing engine, which provides accelerated ray traversal of the high-order geometry, and CUDA, which allows for effective parallel evaluation of the visualization algorithms. The direct interface between ElVis and the underlying data differentiates it from existing visualization tools. Current tools assume the underlying data is composed of linear primitives; high-order data must be interpolated with linear functions as a result. In this work, examples drawn from aerodynamic simulations-high-order discontinuous Galerkin finite element solutions of aerodynamic flows in particular-will demonstrate the superiority of ElVis' pixel-exact approach when compared with traditional linear-interpolation methods. Such methods can introduce a number of inaccuracies in the resulting visualization, making it unclear if visual artifacts are genuine to the solution data or if these artifacts are the result of interpolation errors. Linear methods additionally cannot properly visualize curved geometries (elements or boundaries) which can greatly inhibit developers' debugging efforts. As we will show, pixel-exact visualization exhibits none of these issues, removing the visualization scheme as a source of
Numerical solutions of the Navier-Stokes equations for transonic afterbody flows
NASA Technical Reports Server (NTRS)
Swanson, R. C., Jr.
1980-01-01
The time dependent Navier-Stokes equations in mass averaged variables are solved for transonic flow over axisymmetric boattail plume simulator configurations. Numerical solution of these equations is accomplished with the unsplit explict finite difference algorithm of MacCormack. A grid subcycling procedure and computer code vectorization are used to improve computational efficiency. The two layer algebraic turbulence models of Cebeci-Smith and Baldwin-Lomax are employed for investigating turbulence closure. Two relaxation models based on these baseline models are also considered. Results in the form of surface pressure distribution for three different circular arc boattails at two free stream Mach numbers are compared with experimental data. The pressures in the recirculating flow region for all separated cases are poorly predicted with the baseline turbulence models. Significant improvements in the predictions are usually obtained by using the relaxation models.
A comprehensive one-dimensional numerical model for solute transport in rivers
NASA Astrophysics Data System (ADS)
Barati Moghaddam, Maryam; Mazaheri, Mehdi; MohammadVali Samani, Jamal
2017-01-01
One of the mechanisms that greatly affect the pollutant transport in rivers, especially in mountain streams, is the effect of transient storage zones. The main effect of these zones is to retain pollutants temporarily and then release them gradually. Transient storage zones indirectly influence all phenomena related to mass transport in rivers. This paper presents the TOASTS (third-order accuracy simulation of transient storage) model to simulate 1-D pollutant transport in rivers with irregular cross-sections under unsteady flow and transient storage zones. The proposed model was verified versus some analytical solutions and a 2-D hydrodynamic model. In addition, in order to demonstrate the model applicability, two hypothetical examples were designed and four sets of well-established frequently cited tracer study data were used. These cases cover different processes governing transport, cross-section types and flow regimes. The results of the TOASTS model, in comparison with two common contaminant transport models, shows better accuracy and numerical stability.
Numerical solution of the Navier-Stokes equations for blunt nosed bodies in supersonic flows
NASA Technical Reports Server (NTRS)
Warsi, Z. U. A.; Devarayalu, K.; Thompson, J. F.
1978-01-01
A time dependent, two dimensional Navier-Stokes code employing the method of body fitted coordinate technique was developed for supersonic flows past blunt bodies of arbitrary shapes. The bow shock ahead of the body is obtained as part of the solution, viz., by shock capturing. A first attempt at mesh refinement in the shock region was made by using the forcing function in the coordinate generating equations as a linear function of the density gradients. The technique displaces a few lines from the neighboring region into the shock region. Numerical calculations for Mach numbers 2 and 4.6 and Reynolds numbers from 320 to 10,000 were performed for a circular cylinder with and without a fairing. Results of Mach number 4.6 and Reynolds number 10,000 for an isothermal wall temperature of 556 K are presented in detail.
Parallel solution of high-order numerical schemes for solving incompressible flows
NASA Technical Reports Server (NTRS)
Milner, Edward J.; Lin, Avi; Liou, May-Fun; Blech, Richard A.
1993-01-01
A new parallel numerical scheme for solving incompressible steady-state flows is presented. The algorithm uses a finite-difference approach to solving the Navier-Stokes equations. The algorithms are scalable and expandable. They may be used with only two processors or with as many processors as are available. The code is general and expandable. Any size grid may be used. Four processors of the NASA LeRC Hypercluster were used to solve for steady-state flow in a driven square cavity. The Hypercluster was configured in a distributed-memory, hypercube-like architecture. By using a 50-by-50 finite-difference solution grid, an efficiency of 74 percent (a speedup of 2.96) was obtained.
Numerical multistep methods for the efficient solution of quantum mechanics and related problems
NASA Astrophysics Data System (ADS)
Anastassi, Z. A.; Simos, T. E.
2009-10-01
In this paper we present the recent development in the numerical integration of the Schrödinger equation and related systems of ordinary differential equations with oscillatory solutions, such as the N-body problem. We examine several types of multistep methods (explicit, implicit, predictor-corrector, hybrid) and several properties (P-stability, trigonometric fitting of various orders, phase fitting, high phase-lag order, algebraic order). We analyze the local truncation error and the stability of the methods. The error for the Schrödinger equation is also presented, which reveals the relation of the error to the energy. The efficiency of the methods is evaluated through the integration of five problems. Figures are presented and analyzed and some general conclusions are made. Code written in Maple is given for the development of all methods analyzed in this paper. Also the subroutines written in Matlab, that concern the integration of the methods, are presented.
Formulation and numerical solution of finite-level quantum optimal control problems
NASA Astrophysics Data System (ADS)
Borzi`, A.; Salomon, J.; Volkwein, S.
2008-06-01
Optimal control of finite-level quantum systems is investigated, and iterative solution schemes for the optimization of a control representing laser pulses are developed. The purpose of this external field is to channel the system's wavefunction between given states in its most efficient way. Physically motivated constraints, such as limited laser resources or population suppression of certain states, are accounted for through an appropriately chosen cost functional. First-order necessary optimality conditions and second-order sufficient optimality conditions are investigated. For solving the optimal control problems, a cascadic non-linear conjugate gradient scheme and a monotonic scheme are discussed. Results of numerical experiments with a representative finite-level quantum system demonstrate the effectiveness of the optimal control formulation and efficiency and robustness of the proposed approaches.
NASA Astrophysics Data System (ADS)
Venturi, Daniele
2016-11-01
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics, quantum field theory and statistical physics. For example, in the context of fluid dynamics, the Hopf characteristic functional equation was deemed by Monin and Yaglom to be "the most compact formulation of the turbulence problem", which is the problem of determining the statistical properties of the velocity and pressure fields of Navier-Stokes equations given statistical information on the initial state. However, no effective numerical method has yet been developed to compute the solution to functional differential equations. In this talk I will provide a new perspective on this general problem, and discuss recent progresses in approximation theory for nonlinear functionals and functional equations. The proposed methods will be demonstrated through various examples.
Bahşı, Ayşe Kurt; Yalçınbaş, Salih
2016-01-01
In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduced to a set of linear algebraic equations. Also, an error estimation algorithm which is based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation algorithm. If the exact solution of the problem is not known, the absolute error function of the problems can be approximately computed by using the Fibonacci polynomial solution. By using this error estimation function, we can find improved solutions which are more efficient than direct numerical solutions. Numerical examples, figures, tables are comparisons have been presented to show efficiency and usable of proposed method.
NASA Astrophysics Data System (ADS)
Cremer, Clemens; Neuweiler, Insa; Bechtold, Michel; Vanderborght, Jan
2016-04-01
Quantification of flow and solute transport in the shallow subsurface adjacent to the atmosphere is decisive to prevent groundwater pollution and conserve groundwater quality, to develop successful remediation strategies and to understand nutrient cycling. In nature, due to erratic precipitation-evaporation patterns, soil moisture content and related hydraulic conductivity in the vadose zone are not only variable in space but also in time. Flow directions and flow paths locally change between precipitation and evaporation periods. This makes the identification and description of solute transport processes in the vadose zone a complex problem. Recent studies (Lehmann and Or, 2009; Bechtold et al., 2011a) focused on the investigation of upward transport of solutes during evaporation in heterogeneous soil columns, where heterogeneity was introduced by a sharp vertical material interface between two types of sand. Lateral solute transport through the interface in both (lateral) directions was observed at different depths of the investigated soil columns. Following recent approaches, we conduct two-dimensional numerical simulations in a similar setup which is composed of two sands with a sharp vertical material interface. The investigation is broadened from the sole evaporation to combined precipitation-evaporation cycles in order to quantify transport processes resulting from the combined effects of heterogeneous soil structure and dynamic flow conditions. Simulations are performed with a coupled finite volume and random walk particle tracking algorithm (Ippisch et al., 2006; Bechtold et al., 2011b). By comparing scenarios with cyclic boundary conditions and stationary counterparts with the same net flow rate, we found that duration and intensity of precipitation and evaporation periods potentially have an influence on lateral redistribution of solutes and thus leaching rates. Whether or not dynamic boundary conditions lead to significant deviations in the transport
Bu Sunyoung Huang Jingfang Boyer, Treavor H. Miller, Cass T.
2010-07-01
The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.
Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H; Miller, Cass T
2010-07-01
The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward-difference-formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.
Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H.; Miller, Cass T.
2010-01-01
The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward-difference-formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications. PMID:20577570
Zhang, Bo; Chen, Tianning; Zhao, Yuyuan; Zhang, Weiyong; Zhu, Jian
2012-09-01
On the basis of the work of Wilson et al. [J. Acoust. Soc. Am. 84, 350-359 (1988)], a more exact numerical approach was constructed for predicting the nonlinear sound propagation and absorption properties of rigid porous media at high sound pressure levels. The numerical solution was validated by the experimental results for sintered fibrous porous steel samples and its predictions were compared with the numerical solution of Wilson et al. An approximate analytical solution was further put forward for the normalized surface acoustic admittance of rigid air-saturated porous materials with infinite thickness, based on the wave perturbation method developed by Lambert and McIntosh [J. Acoust. Soc. Am. 88, 1950-1959 (1990)]. Comparisons were made with the numerical results.
Botello-Smith, Wesley M.; Luo, Ray
2016-01-01
Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membrane into existing continuum Poisson-Boltzmann solvent models to extend their applications to membrane systems. Inclusion of an implicit membrane complicates numerical solutions of the underlining Poisson-Boltzmann equation due to the dielectric inhomogeneity on the boundary surfaces of a computation grid. This can be alleviated by the use of the periodic boundary condition, a common practice in electrostatic computations in particle simulations. The conjugate gradient and successive over-relaxation methods are relatively straightforward to be adapted to periodic calculations, but their convergence rates are quite low, limiting their applications to free energy simulations that require a large number of conformations to be processed. To accelerate convergence, the Incomplete Cholesky preconditioning and the geometric multi-grid methods have been extended to incorporate periodicity for biomolecular applications. Impressive convergence behaviors were found as in the previous applications of these numerical methods to tested biomolecules and MMPBSA calculations. PMID:26389966
Linear stability analysis in the numerical solution of initial value problems
NASA Astrophysics Data System (ADS)
van Dorsselaer, J. L. M.; Kraaijevanger, J. F. B. M.; Spijker, M. N.
This article addresses the general problem of establishing upper bounds for the norms of the nth powers of square matrices. The focus is on upper bounds that grow only moderately (or stay constant) where n, or the order of the matrices, increases. The so-called resolvant condition, occuring in the famous Kreiss matrix theorem, is a classical tool for deriving such bounds.Recently the classical upper bounds known to be valid under Kreiss's resolvant condition have been improved. Moreover, generalizations of this resolvant condition have been considered so as to widen the range of applications. The main purpose of this article is to review and extend some of these new developments.The upper bounds for the powers of matrices discussed in this article are intimately connected with the stability analysis of numerical processes for solving initial(-boundary) value problems in ordinary and partial linear differential equations. The article highlights this connection.The article concludes with numerical illustrations in the solution of a simple initial-boundary value problem for a partial differential equation.
The numerical solution of the transient two-phase flow in rigid pipelines
NASA Astrophysics Data System (ADS)
Hadj-Taieb, Ezzeddine; Lili, Taieb
1999-03-01
Consideration is given in this paper to the numerical solution of the transient two-phase flow in rigid pipelines. The governing equations for such flows are two coupled, non-linear, hyperbolic, partial differential equations with pressure dependent coefficients. The fluid pressure and velocity are considered as two principle dependent variables. The fluid is a homogeneous gas-liquid mixture for which the density is defined by an expression averaging the two-component densities where a polytropic process of the gaseous phase is admitted. Instead of the void fraction, which varies with the pressure, the gas-fluid mass ratio (or the quality) is assumed to be constant, and is used in the mathematical formulation. The problem has been solved by the method of non-linear characteristics and the finite difference conservative scheme. To verify their validity, the computed results of the two numerical techniques are compared for different values of the quality, in the case where the liquid compressibility and the pipe wall elasticity are neglected. Copyright
Botello-Smith, Wesley M; Luo, Ray
2015-10-26
Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membranes into existing continuum Poisson-Boltzmann solvent models to extend their applications to membrane systems. Inclusion of an implicit membrane complicates numerical solutions of the underlining Poisson-Boltzmann equation due to the dielectric inhomogeneity on the boundary surfaces of a computation grid. This can be alleviated by the use of the periodic boundary condition, a common practice in electrostatic computations in particle simulations. The conjugate gradient and successive over-relaxation methods are relatively straightforward to be adapted to periodic calculations, but their convergence rates are quite low, limiting their applications to free energy simulations that require a large number of conformations to be processed. To accelerate convergence, the Incomplete Cholesky preconditioning and the geometric multigrid methods have been extended to incorporate periodicity for biomolecular applications. Impressive convergence behaviors were found as in the previous applications of these numerical methods to tested biomolecules and MMPBSA calculations.
Open issues in devising software for the numerical solution of implicit delay differential equations
NASA Astrophysics Data System (ADS)
Guglielmi, Nicola
2006-01-01
We consider initial value problems for systems of implicit delay differential equations of the formMy'(t)=f(t,y(t),y([alpha]1(t,y(t))),...,y([alpha]m(t,y(t)))),where M is a constant square matrix (with arbitrary rank) and [alpha]i(t,y(t))[less-than-or-equals, slant]t for all t and i.For a numerical treatment of this kind of problems, a software tool has been recently developed [6]; this code is called RADAR5 and is based on a suitable extension to delay equations of the 3-stage Radau IIA Runge-Kutta method.The aim of this work is that of illustrating some important topics which are being investigated in order to increase the efficiency of the code. They are mainly relevant to(i) the error control strategies in relation to derivative discontinuities arising in the solutions of delay equations;(ii) the integration of problems with unbounded delays (like the pantograph equation);(iii) the applications to problems with special structure (as those arising from spatial discretization of evolutions PDEs with delays).Several numerical examples will also be shown in order to illustrate some of the topics discussed in the paper.
Analytical and numerical solutions to the amplifier with incoherent pulse temporal overlap
NASA Astrophysics Data System (ADS)
Li, M.; Zhang, X. M.; Wang, Z. G.; Cui, X. D.; Yan, X. W.; Jiang, X. Y.; Zheng, J. G.; Wang, W.; Li, Mingzhong
2017-01-01
Serious pulse temporal overlap in amplifiers would result in the decrease of energy extraction efficiency and the increase of pulse-shape distortion (PSD). Precisely predicting pulse temporal overlap is of significance to an effective amplifier design. In this work, the analytical expressions with complete pulse overlap are derived and a numerical method is proposed to solve the case with partial temporal overlap for a double-pass Nd:YAG amplifier. Our studies, in which pulse temporal overlap is taken into account, can precisely predict the output energy and temporal shape, compared to the results from Hirano and other experiments. In addition, our numerical routes could provide the applicable range of analytical solutions to conventional Frantz-Nodvik equations in the case of pulse overlap, further extending the applicability and reducing computational costs. For given conditions, energy reduction and PSD are mainly determined by the overlap degree. For step-shaped pulse, we demonstrate that avoiding overlap in the peak pulse and allowing overlap in the foot pulse have small impacts on the energy extraction and PSD, which extends the range of duration of the pulse for a designed amplifier. Our investigations might provide an efficient way to carefully design a pulsed amplifier with controllable temporal overlap.
Real-time Numerical Solution for the Plasma Response Matrix for Disruption Avoidance in ITER
NASA Astrophysics Data System (ADS)
Glasser, Alexander; Kolemen, Egemen; Glasser, A. H.
2016-10-01
Real-time analysis of plasma stability is essential to any active feedback control system that performs ideal MHD disruption avoidance. Due to singularities and poor numerical conditioning endemic to ideal MHD models of tokamak plasmas, current state-of-the-art codes require serial operation, and are as yet inoperable on the sub- O (1s) timescale required by ITER's MHD evolution time. In this work, low-toroidal-n ideal MHD modes are found in near real-time as solutions to a well-posed boundary value problem. Using a modified parallel shooting technique and linear methods to subdue numerical instability, such modes are integrated with parallelization across spatial and ``temporal'' parts, via a Riccati approach. The resulting state transition matrix is shown to yield the desired plasma response matrix, which describes how magnetic perturbations may be employed to maintain plasma stability. Such an algorithm may be helpful in designing a control system to achieve ITER's high-performance operational objectives. Sponsored by US DOE under DE-SC0015878 and DE-FC02-04ER54698.
NASA Astrophysics Data System (ADS)
Ivanov, I. G.; Netov, N. C.; Bogdanova, B. C.
2015-10-01
This paper addresses the problem of solving a generalized algebraic Riccati equation with an indefinite sign of its quadratic term. We extend the approach introduced by Lanzon, Feng, Anderson and Rotkowitz (2008) for solving similar Riccati equations. We numerically investigate two types of iterative methods for computing the stabilizing solution. The first type of iterative methods constructs two matrix sequences, where the sum of them converges to the stabilizing solution. The second type of methods defines one matrix sequence which converges to the stabilizing solution. Computer realizations of the presented methods are numerically tested and compared on the test of family examples. Based on the experiments some conclusions are derived.
A numerical code for the solution of the Kompaneets equation in cosmological context
NASA Astrophysics Data System (ADS)
Procopio, P.; Burigana, C.
2009-12-01
Context: After fundamental ground-based, balloon-born, and space experiments, and, in particular, after the COBE/FIRAS results, confirming that only very small deviations from a Planckian shape can be present in the CMB spectrum, current and future CMB absolute temperature experiments aim at discovering very small distortions such as those associated with the cosmological reionization process or that could be generated by different kinds of earlier processes. Aims: Interpretation of future data calls for a continuous improvement in the theoretical modeling of CMB spectrum. In this work we describe the fundamental approach and, in particular, the update to recent NAG versions of a numerical code, KYPRIX, specifically written to solve the Kompaneets equation in a cosmological context. It was first implemented in the years 1989-1991 to accurately compute the CMB spectral distortions under general assumptions. Methods: Specifically, we describe the structure and the main subdivisions of the code and discuss the most relevant aspects of its technical implementation. After a presentation of the equation formalism and of the boundary conditions added to the set of ordinary differential equations derived from the original parabolic partial differential equation, we provide details on the adopted space variable (i.e. dimensionless frequency) and space discretization, on time variables, on the output results, on the accuracy parameters, and on the used auxiliary integration routines. The problem with introducing the time dependence of the ratio between electron and photon temperatures and of the radiative Compton scattering term, both of them introducing integral terms into the Kompaneets equation, is addressed in the specific context of the recent NAG versions. We describe the introduction of the cosmological constant in the terms controlling the general expansion of the Universe in agreement with the current concordance model, of the relevant chemical abundances, and on
Nadeem, Sohail; Masood, Sadaf; Mehmood, Rashid; Sadiq, Muhammad Adil
2015-01-01
The present analysis deals with flow and heat transfer aspects of a micropolar nanofluid between two horizontal parallel plates in a rotating system. The governing partial differential equations for momentum, energy, micro rotation and nano-particles concentration are presented. Similarity transformations are utilized to convert the system of partial differential equations into system of ordinary differential equations. The reduced equations are solved analytically with the help of optimal homotopy analysis method (OHAM). Analytical solutions for velocity, temperature, micro-rotation and concentration profiles are expressed graphically against various emerging physical parameters. Physical quantities of interest such as skin friction co-efficient, local heat and local mass fluxes are also computed both analytically and numerically through mid-point integration scheme. It is found that both the solutions are in excellent agreement. Local skin friction coefficient is found to be higher for the case of strong concentration i.e. n=0, as compared to the case of weak concentration n=0.50. Influence of strong and weak concentration on Nusselt and Sherwood number appear to be similar in a quantitative sense.
Kaczmarski, Krzysztof; Guiochon, Georges A
2011-01-01
In supercritical fluid chromatography (SFC), the significant expansion of the mobile phase along the column causes the formation of axial and radial gradients of temperature. Due to these gradients, the mobile phase density, its viscosity, its velocity, its diffusion coefficients, etc. are not constant throughout the column. This results in a nonuniform flow velocity distribution, itself causing a loss of column efficiency in certain cases, even at low flow rates, as they do in HPLC. At high flow rates, an important deformation of the elution profiles of the sample components may occur. The model previously used to account satisfactorily for the retention of an unsorbed solute in SFC is applied to the modeling of the elution peak profiles of retained compounds. The numerical solution of the combined heat and mass balance equations provides the temperature and the pressure profiles inside the column and values of the retention time and the band profiles of retained compounds that are in excellent agreement with independent experimental data for large value of mobile phase reduced density. At low reduced densities, the band profiles can strongly depend on the column axial distribution of porosity.
Nadeem, Sohail; Masood, Sadaf; Mehmood, Rashid; Sadiq, Muhammad Adil
2015-01-01
The present analysis deals with flow and heat transfer aspects of a micropolar nanofluid between two horizontal parallel plates in a rotating system. The governing partial differential equations for momentum, energy, micro rotation and nano-particles concentration are presented. Similarity transformations are utilized to convert the system of partial differential equations into system of ordinary differential equations. The reduced equations are solved analytically with the help of optimal homotopy analysis method (OHAM). Analytical solutions for velocity, temperature, micro-rotation and concentration profiles are expressed graphically against various emerging physical parameters. Physical quantities of interest such as skin friction co-efficient, local heat and local mass fluxes are also computed both analytically and numerically through mid-point integration scheme. It is found that both the solutions are in excellent agreement. Local skin friction coefficient is found to be higher for the case of strong concentration i.e. n=0, as compared to the case of weak concentration n=0.50. Influence of strong and weak concentration on Nusselt and Sherwood number appear to be similar in a quantitative sense. PMID:26046637
NASA Astrophysics Data System (ADS)
Rodriguez-Nunez, Jesus; Castillo, Jesus; Molinar-Tabares, Martin
The solutions of the logistic difference equation when they are under the influence of the chaotic regime are very sensitive to initial conditions due to the butterfly effect. In this study we used arbitrary significant digits to generate solutions of the logistic difference equation under the influence of chaos, and a follow of its effects along each digit of the solutions was made. A large amount of significant digits to generate the solutions is necessary since it is the only way of naturally appreciating the implications of chaos on these solutions. We compared digit by digit the numerical solutions that were generated by several different initial conditions that contain modifications in a very far significant digit, with respect to the solution of another initial condition that was selected for a control solution. The results shown that it is possible to track the butterfly effect and easily predict the moment on which its effects will be noticeable.
Numerical solution of the asymmetric water impact of a wedge in three degrees of freedom
NASA Astrophysics Data System (ADS)
Ghazizade-Ahsaee, H.; Nikseresht, A. H.
2013-06-01
Impact problems associated with water entry have important applications in various aspects of naval architecture and ocean engineering. Estimation of hydrodynamic impact forces especially during the first instances after the impact is very important and is of interest. Since the estimation of hydrodynamic impact load plays an important role in safe design and also in evaluation of structural weight and costs, it is better to use a reliable and accurate prediction method instead of a simple estimation resulted by analyzing methods. In landing of flying boats, some phenomena such as weather conditions and strong winds can cause asymmetric instead of symmetric descent. In this paper, a numerical simulation of the asymmetric impact of a wedge, as the step of a flying boat, considering dynamic equations in two-phase flow is taken into account. The dynamic motion of the wedge in two-phase flow is solved based on finite volume method with volume of fluid (VOF) scheme considering dynamic equations. Then the effects of different angles of impact and water depth on the velocity change and slamming forces in an asymmetric impact are investigated. The comparison between the simulation results and experimental data verifies the accuracy of the method applied in the present study.
Malik, Suheel Abdullah; Qureshi, Ijaz Mansoor; Amir, Muhammad; Haq, Ihsanul
2014-01-01
We present a hybrid heuristic computing method for the numerical solution of nonlinear singular boundary value problems arising in physiology. The approximate solution is deduced as a linear combination of some log sigmoid basis functions. A fitness function representing the sum of the mean square error of the given nonlinear ordinary differential equation (ODE) and its boundary conditions is formulated. The optimization of the unknown adjustable parameters contained in the fitness function is performed by the hybrid heuristic computation algorithm based on genetic algorithm (GA), interior point algorithm (IPA), and active set algorithm (ASA). The efficiency and the viability of the proposed method are confirmed by solving three examples from physiology. The obtained approximate solutions are found in excellent agreement with the exact solutions as well as some conventional numerical solutions.
NASA Technical Reports Server (NTRS)
Thompson, J. F.; Mastin, C. W.; Thames, F. C.; Shanks, S. P.
1975-01-01
A procedure for numerical solution of the time-dependent, two-dimensional incompressible Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple airfoils, and submerged hydrofoils, as naturally as it can deal with the flow about simple bodies. The solution is based on a method of automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multiconnected region containing any number of arbitrarily shaped bodies. The curvilinear coordinates are generated as the solution of two elliptical partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other being specified along the boundaries. The solution compares excellently with the Blasius boundary layer solution for the flow past a semiinfinite flat plate.
Malik, Suheel Abdullah; Qureshi, Ijaz Mansoor; Haq, Ihsanul
2014-01-01
We present a hybrid heuristic computing method for the numerical solution of nonlinear singular boundary value problems arising in physiology. The approximate solution is deduced as a linear combination of some log sigmoid basis functions. A fitness function representing the sum of the mean square error of the given nonlinear ordinary differential equation (ODE) and its boundary conditions is formulated. The optimization of the unknown adjustable parameters contained in the fitness function is performed by the hybrid heuristic computation algorithm based on genetic algorithm (GA), interior point algorithm (IPA), and active set algorithm (ASA). The efficiency and the viability of the proposed method are confirmed by solving three examples from physiology. The obtained approximate solutions are found in excellent agreement with the exact solutions as well as some conventional numerical solutions. PMID:24672381
NASA Technical Reports Server (NTRS)
Penny, M. M.; Smith, S. D.; Anderson, P. G.; Sulyma, P. R.; Pearson, M. L.
1976-01-01
A numerical solution for chemically reacting supersonic gas-particle flows in rocket nozzles and exhaust plumes was described. The gas-particle flow solution is fully coupled in that the effects of particle drag and heat transfer between the gas and particle phases are treated. Gas and particles exchange momentum via the drag exerted on the gas by the particles. Energy is exchanged between the phases via heat transfer (convection and/or radiation). Thermochemistry calculations (chemical equilibrium, frozen or chemical kinetics) were shown to be uncoupled from the flow solution and, as such, can be solved separately. The solution to the set of governing equations is obtained by utilizing the method of characteristics. The equations cast in characteristic form are shown to be formally the same for ideal, frozen, chemical equilibrium and chemical non-equilibrium reacting gas mixtures. The particle distribution is represented in the numerical solution by a finite distribution of particle sizes.
NASA Astrophysics Data System (ADS)
Abbasbandy, S.; Van Gorder, R. A.; Hajiketabi, M.; Mesrizadeh, M.
2015-10-01
We consider traveling wave solutions to the Casimir equation for the Ito system (a two-field extension of the KdV equation). These traveling waves are governed by a nonlinear initial value problem with an interesting nonlinearity (which actually amplifies in magnitude as the size of the solution becomes small). The nonlinear problem is parameterized by two initial constant values, and we demonstrate that the existence of solutions is strongly tied to these parameter values. For our interests, we are concerned with positive, bounded, periodic wave solutions. We are able to classify parameter regimes which admit such solutions in full generality, thereby obtaining a nice existence result. Using the existence result, we are then able to numerically simulate the positive, bounded, periodic solutions. We elect to employ a group preserving scheme in order to numerically study these solutions, and an outline of this approach is provided. The numerical simulations serve to illustrate the properties of these solutions predicted analytically through the existence result. Physically, these results demonstrate the existence of a type of space-periodic structure in the Casimir equation for the Ito model, which propagates as a traveling wave.
NASA Astrophysics Data System (ADS)
Tirupathi, S.; Schiemenz, A. R.; Liang, Y.; Parmentier, E.; Hesthaven, J.
2013-12-01
The style and mode of melt migration in the mantle are important to the interpretation of basalts erupted on the surface. Both grain-scale diffuse porous flow and channelized melt migration have been proposed. To better understand the mechanisms and consequences of melt migration in a heterogeneous mantle, we have undertaken a numerical study of reactive dissolution in an upwelling and viscously deformable mantle where solubility of pyroxene increases upwards. Our setup is similar to that described in [1], except we use a larger domain size in 2D and 3D and a new numerical method. To enable efficient simulations in 3D through parallel computing, we developed a high-order accurate numerical method for the magma dynamics problem using discontinuous Galerkin methods and constructed the problem using the numerical library deal.II [2]. Linear stability analyses of the reactive dissolution problem reveal three dynamically distinct regimes [3] and the simulations reported in this study were run in the stable regime and the unstable wave regime where small perturbations in porosity grows periodically. The wave regime is more relevant to melt migration beneath the mid-ocean ridges but computationally more challenging. Extending the 2D simulations in the stable regime in [1] to 3D using various combinations of sustained perturbations in porosity at the base of the upwelling column (which may result from a viened mantle), we show the geometry and distribution of dunite channel and high-porosity melt channels are highly correlated with inflow perturbation through superposition. Strong nonlinear interactions among compaction, dissolution, and upwelling give rise to porosity waves and high-porosity melt channels in the wave regime. These compaction-dissolution waves have well organized but time-dependent structures in the lower part of the simulation domain. High-porosity melt channels nucleate along nodal lines of the porosity waves, growing downwards. The wavelength scales
Gursoy, Gamze; Terebus, Anna; Youfang Cao; Jie Liang
2016-08-01
Stochasticity plays important roles in regulation of biochemical reaction networks when the copy numbers of molecular species are small. Studies based on Stochastic Simulation Algorithm (SSA) has shown that a basic reaction system can display stochastic focusing (SF) by increasing the sensitivity of the network as a result of the signal noise. Although SSA has been widely used to study stochastic networks, it is ineffective in examining rare events and this becomes a significant issue when the tails of probability distributions are relevant as is the case of SF. Here we use the ACME method to solve the exact solution of the discrete Chemical Master Equations and to study a network where SF was reported. We showed that the level of SF depends on the degree of the fluctuations of signal molecule. We discovered that signaling noise under certain conditions in the same reaction network can lead to a decrease in the system sensitivities, thus the network can experience stochastic defocusing. These results highlight the fundamental role of stochasticity in biological reaction networks and the need for exact computation of probability landscape of the molecules in the system.
On the formation, growth, and shapes of solution pipes - insights from numerical modeling
NASA Astrophysics Data System (ADS)
Szymczak, Piotr; Tredak, Hanna; Upadhyay, Virat; Kondratiuk, Paweł; Ladd, Anthony J. C.
2015-04-01
Cylindrical, vertical structures called solution pipes are a characteristic feature of epikarst, encountered in different parts of the world, both in relatively cold areas such as England and Poland (where their formation is linked to glacial processes) [1] and in coastal areas in tropical or subtropical climate (Bermuda, Australia, South Africa, Caribbean, Mediterranean) [2,3]. They are invariably associated with weakly cemented, porous limestones and relatively high groundwater fluxes. Many of them develop under the colluvial sandy cover and contain the fill of clayey silt. Although it is widely accepted that they are solutional in origin, the exact mechanism by which the flow becomes focused is still under debate. The hypotheses include the concentration of acidified water around stems and roots of plants, or the presence of pre-existing fractures or steeply dipping bedding planes, which would determine the points of entry for the focused groundwater flows. However, there are field sites where neither of this mechanisms was apparently at play and yet the pipes are formed in large quantities [1]. In this communication we show that the systems of solution pipes can develop spontaneously in nearly uniform matrix due to the reactive-infiltration instability: a homogeneous porous matrix is unstable with respect to small variations in local permeability; regions of high permeability dissolve faster because of enhanced transport of reactants, which leads to increased rippling of the front. This leads to the formation of a system of solution pipes which then advance into the matrix. We study this process numerically, by a combination of 2d- and 3d-simulations, solving the coupled flow and transport equations at the Darcy scale. The relative simplicity of this system (pipes developing in a uniform porous matrix, without any pre-existing structure) makes it very attractive from the modeling standpoint. We quantify the factors which control the pipe diameters and the
Shear-driven particle size segregation: Models, analysis, numerical solutions, and experiments
NASA Astrophysics Data System (ADS)
May, Lindsay Bard Hilbert
, we find a layer of small particles below a layer of large particles. We also measure a velocity profile from the Couette cell experimental data, which provides parameters used to derive the solution of the initial boundary value problem. The initial condition for the partial differential equation corresponds to the one dimensional initial configuration of the experiment. We solve two initial boundary value problems, one with a piecewise linear shear rate and one with an exponential shear rate, where the parameters for both cases are derived from the experimental data. In each case, we use the method of characteristics to solve the initial boundary value problem. In both cases, almost all pieces of the solution can be explicitly calculated, and those that cannot are calculated numerically. In the piecewise linear case, there is a material interface across which the characteristic speed jumps; in the exponential case, the characteristics are curved. We compare the model with the exponential shear rate to the experimental data. The model solution is the volume fraction of small particles at time t and location z. We cannot measure the volume fraction locally in the experiment; instead, we map the volume fraction to a theoretical height which we compare to the experimental experimental height data. The height of the sample is an indirect measurement of the amount of mixing or segregation. We conclude that the model captures qualitative features of the experimental data, but there are features of the experiment that the current version of the model does not describe.
NASA Astrophysics Data System (ADS)
Holman, Benjamin R.
In recent years, revolutionary "hybrid" or "multi-physics" methods of medical imaging have emerged. By combining two or three different types of waves these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information. Thermoacoustic (and photoacoustic) tomography is the most developed multi-physics imaging modality. Thermo- and photo- acoustic tomography require reconstructing initial acoustic pressure in a body from time series of pressure measured on a surface surrounding the body. For the classical case of free space wave propagation, various reconstruction techniques are well known. However, some novel measurement schemes place the object of interest between reflecting walls that form a de facto resonant cavity. In this case, known methods cannot be used. In chapter 2 we present a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity with a constant speed of sound. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations. In chapter 3 we consider the more general problem of an arbitrarily shaped resonant cavity with a non constant speed of sound and present the gradual time reversal method for computing solutions to the inverse source problem. It consists in solving back in time on the interval [0, T] the initial/boundary value problem for the wave equation, with the Dirichlet boundary data multiplied by a smooth cutoff function. If T is sufficiently large one obtains a good approximation to the initial pressure; in the limit of large T such an approximation converges (under certain conditions) to the exact solution.
NASA Astrophysics Data System (ADS)
Zhang, Guowei
2013-01-01
SummaryMost of the coal mines of China are mining under the ground, so the artesian test which is one of the aquifer tests is conducted normally several hundred of meters below the earth surface. And the target aquifer is with very high hydraulic pressure, sometimes more than 3 MPa. Because of the high hydraulic pressure, it is most difficult to control the rate of flow out of the artesian well. Moreover, the velocity of flow out of the well cannot descend rapidly to zero, thus the analytical solution of Jacob and Lohman type curve for the artesian test will not be applicable. It is more reasonable if analyzing this test as a pumping test but with variable discharge. It is considered to rebuild that hydrogeological conceptual model in this paper. This conceptual model is similar with Theis model but with the variable discharge merely. And a general equation for any discharge variability is given. Its application for the linearly decreasing discharge is presented subsequently, and a type curve of this equation with linearly decreasing discharge will be given as well. Then a simple numerical model using FEFLOW will be built to simulate the linearly decreasing discharge from giving several different groups of the values of Transmissivity (T) and Storage coefficient (S). Both of them are much important hydrogeological parameters, and will be evaluated by using the type curve developed for this linear decreasing discharge well. The error between the given values of T and S in FEFLOW and the values of those calculated by matching point are much small. The solution gives really satisfactory values of these hydrogeological parameters.
Numerical solutions for unsteady rotating high-porosity medium channel Couette hydrodynamics
NASA Astrophysics Data System (ADS)
Zueco, Joaquin; Bég, O. Anwar; Bég, Tasveer A.
2009-09-01
We investigate theoretically and numerically the unsteady, viscous, incompressible, hydrodynamic, Newtonian Couette flow in a Darcy-Forchheimer porous medium parallel-plate channel rotating with uniform angular velocity about an axis normal to the plates. The upper plate is translating at uniform velocity with the lower plate stationary. The two-dimensional reduced Navier-Stokes equations are transformed to a pair of nonlinear dimensionless momentum equations, neglecting convective inertial terms. The network simulation method, based on a thermoelectric analogy, is employed to solve the transformed dimensionless partial differential equations under prescribed boundary conditions. We examine here graphically the effect of Ekman number, Forchheimer number and Darcy number on the shear stresses at the plates over time. Excellent agreement is also obtained for the infinite permeability i.e. purely fluid (vanishing porous medium) case (Da→∞) with the analytical solutions of Guria et al (2006 Int. J. Nonlinear Mechanics 41 838-43). Backflow is observed in certain cases. Increasing Ekman number, Ek (corresponding to decreasing Coriolis force) is found to accentuate the primary shear stress component (τx) considerably but to reduce magnitudes of the secondary shear stress component (τy). The flow is also found to be accelerated generally with increasing Darcy number and decelerated with increasing Forchheimer number. The present model has applications in geophysical flows, chemical engineering systems and also fundamental studies in fluid dynamics.
NASA Technical Reports Server (NTRS)
Pinsky, P. M.; Ortiz, M.; Taylor, R. L.
1982-01-01
The spatial formulation of the elastoplastic dynamic problem for finite deformations is considered. A thermodynamic argument leads to an additive decomposition of the spatial rate of deformation tensor and allows an operator split of the evolutionary equations of the problem into elastic and plastic parts. This operator split is taken as the basis for the definition of a global product algorithm. In the context of finite element discretization the product algorithm entails, for every time step, the solution of a nonlinear elastodynamic problem followed by the application of plastic algorithms that operate on the stresses and internal variables at the integration points and bring in the plastic constitutive equations. Suitable plastic algorithms are discussed for the cases of perfect and hardening plasticity and viscoplasticity. The proposed formalism does not depend on any notion of smoothness of the yield surface and is applicable to arbitrary convex elastic regions, with or without corners. The stabiity properties of the global product algorithm are shown to be identical to those of the algorithm used for the integration of the nonlinear elastodynamic problem. Numerical examples illustrate the accuracy of the method.
Numerical solution of an inverse obstacle scattering problem with near-field data
NASA Astrophysics Data System (ADS)
Li, Peijun; Wang, Yuliang
2015-06-01
Consider the scattering of an arbitrary time-harmonic incident wave by a sound soft obstacle. In this paper, a novel method is presented for solving the inverse obstacle scattering problem of the two-dimensional Helmholtz equation, which is to reconstruct the obstacle surface by using the near-field data. The obstacle is assumed to be a small and smooth perturbation of a disc. The method uses the transformed field expansion to reduce the boundary value problem into a successive sequence of one-dimensional problems which are solved in closed forms. By dropping the higher order terms in the power series expansion and truncating the infinite linear system for the first order term, the inverse problem is linearized and an approximate but explicit formula is obtained between the Fourier coefficients of the solution and data. A nonlinear correction algorithm is introduced to improve the accuracy of the reconstructions for large deformations. Numerical examples show that the method is simple, efficient, and stable to reconstruct the obstacle with subwavelength resolution.
NASA Astrophysics Data System (ADS)
Sjöberg, L. E.
2012-11-01
We derive computational formulas for determining the Clairaut constant, i.e. the cosine of the maximum latitude of the geodesic arc, from two given points on the oblate ellipsoid of revolution. In all cases the Clairaut constant is unique. The inverse geodetic problem on the ellipsoid is to determine the geodesic arc between and the azimuths of the arc at the given points. We present the solution for the fixed Clairaut constant. If the given points are not(nearly) antipodal, each azimuth and location of the geodesic is unique, while for the fixed points in the ”antipodal region”, roughly within 36”.2 from the antipode, there are two geodesics mirrored in the equator and with complementary azimuths at each point. In the special case with the given points located at the poles of the ellipsoid, all meridians are geodesics. The special role played by the Clairaut constant and the numerical integration make this method different from others available in the literature.
Surface boundary conditions for the numerical solution of the Euler equations
NASA Technical Reports Server (NTRS)
Dadone, A.; Grossman, B.
1993-01-01
We consider the implementation of boundary conditions at solid walls in inviscid Euler solutions by upwind, finite-volume methods. We review some current methods for the implementation of surface boundary conditions and examine their behavior for the problem of an oblique shock reflecting off a planar surface. We show the importance of characteristic boundary conditions for this problem and introduce a method of applying the classical flux-difference splitting of Roe as a characteristic boundary condition. Consideration of the equivalent problem of the intersection of two (equal and opposite) oblique shocks was very illuminating on the role of surface boundary conditions for an inviscid flow and led to the introduction of two new boundary-condition procedures, denoted as the symmetry technique and the curvature-corrected symmetry technique. Examples of the effects of the various surface boundary conditions considered are presented for the supersonic blunt body problem and the subcritical compressible flow over a circular cylinder. Dramatic advantages of the curvature-corrected symmetry technique over the other methods are shown, with regard to numerical entropy generation, total pressure loss, drag and grid convergence.
NASA Technical Reports Server (NTRS)
Gnoffo, Peter A.; Inger, George R.
1999-01-01
The local viscous-inviscid interaction field generated by a wall temperature jump on a flat plate in supersonic flow and on the windside of a Reusable Launch Vehicle in hypersonic flow is studied in detail by both a Navier-Stokes numerical code and an analytical triple-deck model. Treatment of the rapid heat transfer changes both upstream and downstream of the jump is included. Closed form relationships derived from the triple-deck theory are presented. The analytically predicted pressure and heating variations including upstream influence are found to be in generally good agreement with the Computational Fluid Dynamic (CFD) predictions. These analyses not only clarify the interactive physics involved but also are useful in preliminary design of thermal protection systems and as an insertable module to improve CFD code efficiency when applied to such small-scale interaction problems. The analyses only require conditions at the wall and boundary-layer edge which are easily extracted from a baseline, constant wall temperature, CFD solution.
NASA Astrophysics Data System (ADS)
Kröger, Tim
2011-11-01
A mathematical model of radiofrequency ablation (a form of therapy for tumors) consists essentially of the thermistor problem with an additional Helmholtz term. Thereby, the computational domain is neither convex nor has smooth boundary, and the electric conductivity σ(u) experiences a drastic drop for high temperatures u. The paper shows that these properties make the system very difficult to solve numerically. A literature survey shows that no efficient numerical solution for the problem has yet been published, whereas a number of theoretical existence results for solutions exist.
Anastassi, Z. A.; Simos, T. E.
2010-09-30
We develop a new family of explicit symmetric linear multistep methods for the efficient numerical solution of the Schroedinger equation and related problems with oscillatory solution. The new methods are trigonometrically fitted and have improved intervals of periodicity as compared to the corresponding classical method with constant coefficients and other methods from the literature. We also apply the methods along with other known methods to real periodic problems, in order to measure their efficiency.
NASA Technical Reports Server (NTRS)
Stalos, S.
1990-01-01
The double-lunar swingby trajectory is a method for maintaining alignment of an Earth satellite's line of apsides with the Sun-Earth line. From a Keplerian point of view, successive close encounters with the Moon cause discrete, instantaneous changes in the satellite's eccentricity and semimajor axis. Numerical solutions to the planar, restricted problem of three bodies as double-lunar swingby trajectories are identified. The method of solution is described and the results compared to the Keplerian formulation.
The scattering of Lyα radiation in the intergalactic medium: numerical methods and solutions
NASA Astrophysics Data System (ADS)
Higgins, Jonathan; Meiksin, Avery
2012-11-01
Two methods are developed for solving the steady-state spherically symmetric radiative transfer equation for resonance line radiation emitted by a point source in the intergalactic medium, in the context of the Wouthuysen-Field mechanism for coupling the hyperfine structure spin temperature of hydrogen to the gas temperature. One method is based on solving the ray and moment equations using finite differences. The second uses a Monte Carlo approach incorporating methods that greatly improve the accuracy compared with previous approaches in this context. Several applications are presented serving as test problems for both a static medium and an expanding medium, including inhomogeneities in the density and velocity fields. Solutions are obtained in the coherent scattering limit and for Doppler RII redistribution with and without recoils. We find generally that the radiation intensity is linear in the cosine of the azimuthal angle with respect to radius to high accuracy over a broad frequency region across the line centre for both linear and perturbed velocity fields, yielding the Eddington factors fν ≃ 1/3 and gν ≃ 3/5. The radiation field produced by a point source divides into three spatial regimes for a uniformly expanding homogeneous medium. The regimes are governed by the fraction of the distance r from the source in terms of the distance r* required for a photon to redshift from line centre to the frequency needed to escape from the expanding gas. For a standard cosmology, before the Universe was reionized r* takes on the universal value independent of redshift of 1.1 Mpc, depending only on the ratio of the baryon to dark matter density. At r/r* < 1, the radiation field is accurately described in the diffusion approximation, with the scattering rate declining with the distance from the source as r-7/3, except at r/r* ≪ 1 where frequency redistribution nearly doubles the mean intensity around line centre. At r/r* > 1, the diffusion approximation breaks
Analytical-numerical solution of a nonlinear integrodifferential equation in econometrics
NASA Astrophysics Data System (ADS)
Kakhktsyan, V. M.; Khachatryan, A. Kh.
2013-07-01
A mixed problem for a nonlinear integrodifferential equation arising in econometrics is considered. An analytical-numerical method is proposed for solving the problem. Some numerical results are presented.
NASA Astrophysics Data System (ADS)
Wang, Y.-S.; Chien, C.-S.
2014-01-01
We describe a novel two-parameter continuation method combined with a spectral-collocation method (SCM) for computing the ground state and excited-state solutions of spin-1 Bose-Einstein condensates (BEC), where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. To compute the ground state solution of spin-1 BEC, we implement the single parameter continuation algorithm with the chemical potential μ as the continuation parameter, and trace the first solution branch of the Gross-Pitaevskii equations (GPEs). When the curve-tracing is close enough to the target point, where the normalization condition of the wave function is going to be satisfied, we add the magnetic potential λ as the second continuation parameter with the magnetization M as the additional constraint condition. Then we implement the two-parameter continuation algorithm until the target point is reached, and the ground state solution of the GPEs is obtained. The excited state solutions of the GPEs can be treated in a similar way. Some numerical experiments on Na23 and Rb87 are reported. The numerical results on the spin-1 BEC are the same as those reported in [10]. Further numerical experiments on excited-state solutions of spin-1 BEC suffice to show the robustness and efficiency of the proposed two-parameter continuation algorithm.
NASA Astrophysics Data System (ADS)
Przybylska, Maria; Rauch-Wojciechowski, Stefan
2016-03-01
We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straight-line rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoop.
NASA Astrophysics Data System (ADS)
Hooshmandasl, M. R.; Heydari, M. H.; Cattani, C.
2016-08-01
Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.
Ertan, H.B.
1999-09-01
For prediction of static and dynamic performance of doubly-salient motors, it is essential to know their flux linkage-position-excitation characteristics and also the static torque characteristics. At the design stage determination of these characteristics presents difficulties because of highly nonlinear behavior of the magnetic circuit. It is possible to use numerical field solution of the complete motor to obtain this information. This, however, requires expertise on a professional program and may be expensive if used to search for the best design. This paper shows that a reduced model can be used to obtain the desired information accurately. It is also shown that in fact obtaining field solutions just for a pair of teeth is enough for accurately predicting the flux linkage and torque characteristics of a motor. The approach introduced here makes possible searching for an optimum design (even on a PC) for maximizing average torque or reducing noise and vibration problems, since the effort for producing the model and computation time are greatly reduced.
Deb, Kalyanmoy; Sinha, Ankur
2010-01-01
Bilevel optimization problems involve two optimization tasks (upper and lower level), in which every feasible upper level solution must correspond to an optimal solution to a lower level optimization problem. These problems commonly appear in many practical problem solving tasks including optimal control, process optimization, game-playing strategy developments, transportation problems, and others. However, they are commonly converted into a single level optimization problem by using an approximate solution procedure to replace the lower level optimization task. Although there exist a number of theoretical, numerical, and evolutionary optimization studies involving single-objective bilevel programming problems, not many studies look at the context of multiple conflicting objectives in each level of a bilevel programming problem. In this paper, we address certain intricate issues related to solving multi-objective bilevel programming problems, present challenging test problems, and propose a viable and hybrid evolutionary-cum-local-search based algorithm as a solution methodology. The hybrid approach performs better than a number of existing methodologies and scales well up to 40-variable difficult test problems used in this study. The population sizing and termination criteria are made self-adaptive, so that no additional parameters need to be supplied by the user. The study indicates a clear niche of evolutionary algorithms in solving such difficult problems of practical importance compared to their usual solution by a computationally expensive nested procedure. The study opens up many issues related to multi-objective bilevel programming and hopefully this study will motivate EMO and other researchers to pay more attention to this important and difficult problem solving activity.
Numerical modelling of qualitative behaviour of solutions to convolution integral equations
NASA Astrophysics Data System (ADS)
Ford, Neville J.; Diogo, Teresa; Ford, Judith M.; Lima, Pedro
2007-08-01
We consider the qualitative behaviour of solutions to linear integral equations of the formwhere the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley-Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.
Meng, Da; Zheng, Bin; Lin, Guang; Sushko, Maria L.
2014-08-29
We have developed efficient numerical algorithms for the solution of 3D steady-state Poisson-Nernst-Planck equations (PNP) with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by finite difference scheme and solved iteratively by Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Algebraic multigrid method is then applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations. A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed which reduces computational complexity from O(N2) to O(NlogN) where N is the number of grid points. Integrals involving Dirac delta function are evaluated directly by coordinate transformation which yields more accurate result compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for Li ion batteries are shown to be in good agreement with the experimental data and the results from previous studies.
NASA Astrophysics Data System (ADS)
Saha Ray, S.
2013-12-01
In this paper, the modified fractional reduced differential transform method (MFRDTM) has been proposed and it is implemented for solving fractional KdV (Korteweg-de Vries) equations. The fractional derivatives are described in the Caputo sense. In this paper, the reduced differential transform method is modified to be easily employed to solve wide kinds of nonlinear fractional differential equations. In this new approach, the nonlinear term is replaced by its Adomian polynomials. Thus the nonlinear initial-value problem can be easily solved with less computational effort. In order to show the power and effectiveness of the present modified method and to illustrate the pertinent features of the solutions, several fractional KdV equations with different types of nonlinearities are considered. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional KdV equations.
Numerical solutions of a control problem governed by functional differential equations
NASA Technical Reports Server (NTRS)
Banks, H. T.; Thrift, P. R.; Burns, J. A.; Cliff, E. M.
1978-01-01
A numerical procedure is proposed for solving optimal control problems governed by linear retarded functional differential equations. The procedure is based on the idea of 'averaging approximations', due to Banks and Burns (1975). For illustration, numerical results generated on an IBM 370/158 computer, which demonstrate the rapid convergence of the method are presented.
NASA Astrophysics Data System (ADS)
Kahnert, Michael
2016-07-01
Numerical solution methods for electromagnetic scattering by non-spherical particles comprise a variety of different techniques, which can be traced back to different assumptions and solution strategies applied to the macroscopic Maxwell equations. One can distinguish between time- and frequency-domain methods; further, one can divide numerical techniques into finite-difference methods (which are based on approximating the differential operators), separation-of-variables methods (which are based on expanding the solution in a complete set of functions, thus approximating the fields), and volume integral-equation methods (which are usually solved by discretisation of the target volume and invoking the long-wave approximation in each volume cell). While existing reviews of the topic often tend to have a target audience of program developers and expert users, this tutorial review is intended to accommodate the needs of practitioners as well as novices to the field. The required conciseness is achieved by limiting the presentation to a selection of illustrative methods, and by omitting many technical details that are not essential at a first exposure to the subject. On the other hand, the theoretical basis of numerical methods is explained with little compromises in mathematical rigour; the rationale is that a good grasp of numerical light scattering methods is best achieved by understanding their foundation in Maxwell's theory.
NASA Astrophysics Data System (ADS)
Chen, Wen-Jun; Ma, Hong; Yu, De; Zeng, Xiao-Hu
2016-08-01
A novel nuclear magnetic resonance (NMR) experimental scheme, called wideband continuous wave NMR (WB-CW-NMR), is presented in this article. This experimental scheme has promising applications in pulsed magnetic fields, and can dramatically improve the utilization of the pulsed field. The feasibility of WB-CW-NMR scheme is verified by numerically solving modified Bloch equations. In the numerical simulation, the applied magnetic field is a pulsed magnetic field up to 80 T, and the wideband continuous radio frequency (RF) excitation is a band-limited (0.68-3.40 GHz) white noise. Furthermore, the influences of some experimental parameters, such as relaxation time, applied magnetic field strength and wideband continuous RF power, on the WB-CW-NMR signal are analyzed briefly. Finally, a multi-channel system framework for transmitting and receiving ultra wideband signals is proposed, and the basic requirements of this experimental system are discussed. Meanwhile, the amplitude of the NMR signal, the level of noise and RF interference in WB-CW-NMR experiments are estimated, and a preliminary adaptive cancellation plan is given for detecting WB-CW-NMR signal from large background interference. Supported by National Natural Science Foundation of China (11475067), the Innovative Research Foundation of Huazhong University of Science and Technology (2015 ZDTD017) and the Experimental Apparatus Research Project of Wuhan Pulsed High Magnetic Field Center (2015KF17)
NASA Technical Reports Server (NTRS)
Thompson, J. F.; Turner, L., III; Long, W. S.; Bearden, J. H.
1979-01-01
The development of a numerical simulation of time dependent, turbulent, compressible flow about two dimensional multi-element airfoils of arbitrary shape is described. The basis of this simulation is a technique of automatic numerical generation of coordinate systems fitted to the multiple bodies regardless of their number or shape. Procedures developed whereby the coordinate lines are automatically concentrated in the boundary layer at any Reynolds number are discussed. The compressible turbulent solution involves an algebraic eddy viscosity turbulence model. The laminar version was run for transonic flow at free stream Mach numbers up to 0.9.
NASA Astrophysics Data System (ADS)
Vaneeva, O. O.; Papanicolaou, N. C.; Christou, M. A.; Sophocleous, C.
2014-09-01
The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.
Robertson, John B.
1976-01-01
Aqueous chemical and low-level radioactive effluents have been disposed to seepage ponds since 1952 at the Idaho National Engineering Laboratory. The solutions percolate toward the Snake River Plain aquifer (135 m below) through interlayered basalts and unconsolidated sediments and an extensive zone of ground water perched on a sedimentary layer about 40 m beneath the ponds. A three-segment numerical model was developed to simulate the system, including effects of convection, hydrodynamic dispersion, radioactive decay, and adsorption. Simulated hydraulics and solute migration patterns for all segments agree adequately with the available field data. The model can be used to project subsurface distributions of waste solutes under a variety of assumed conditions for the future. Although chloride and tritium reached the aquifer several years ago, the model analysis suggests that the more easily sorbed solutes, such as cesium-137 and strontium-90, would not reach the aquifer in detectable concentrations within 150 years for the conditions assumed. (Woodard-USGS)
Numerical solution of the two-dimensional time-dependent incompressible Euler equations
NASA Technical Reports Server (NTRS)
Whitfield, David L.; Taylor, Lafayette K.
1994-01-01
A numerical method is presented for solving the artificial compressibility form of the 2D time-dependent incompressible Euler equations. The approach is based on using an approximate Riemann solver for the cell face numerical flux of a finite volume discretization. Characteristic variable boundary conditions are developed and presented for all boundaries and in-flow out-flow situations. The system of algebraic equations is solved using the discretized Newton-relaxation (DNR) implicit method. Numerical results are presented for both steady and unsteady flow.
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.
NASA Technical Reports Server (NTRS)
Gramoll, K. C.; Dillard, D. A.; Brinson, H. F.
1989-01-01
In response to the tremendous growth in the development of advanced materials, such as fiber-reinforced plastic (FRP) composite materials, a new numerical method is developed to analyze and predict the time-dependent properties of these materials. Basic concepts in viscoelasticity, laminated composites, and previous viscoelastic numerical methods are presented. A stable numerical method, called the nonlinear differential equation method (NDEM), is developed to calculate the in-plane stresses and strains over any time period for a general laminate constructed from nonlinear viscoelastic orthotropic plies. The method is implemented in an in-plane stress analysis computer program, called VCAP, to demonstrate its usefulness and to verify its accuracy. A number of actual experimental test results performed on Kevlar/epoxy composite laminates are compared to predictions calculated from the numerical method.
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.
NASA Astrophysics Data System (ADS)
Petrov, D. I.; Petrov, I. B.; Favorskaya, A. V.; Khokhlov, N. I.
2016-06-01
The goal of this paper is the numerical solution of direct problems concerning hydrocarbon seismic exploration on the Arctic shelf. The task is addressed by solving a complete system of linear elasticity equations and a system of acoustic field equations. Both systems are solved by applying the grid-characteristic method, which takes into account all wave processes in a detailed and physically correct manner and produces a solution near the boundaries and interfaces of the integration domain, including the interface between the acoustic and linear elastic media involved. The seismograms and wave patterns obtained by numerically solving these systems are compared. The effect of ice structures on the resulting wave patterns is examined.
NASA Astrophysics Data System (ADS)
Liu, Yong-Qing; Cheng, Rong-Jun; Ge, Hong-Xia
2013-10-01
The present paper deals with the numerical solution of the coupled Schrödinger-KdV equations using the element-free Galerkin (EFG) method which is based on the moving least-square approximation. Instead of traditional mesh oriented methods such as the finite difference method (FDM) and the finite element method (FEM), this method needs only scattered nodes in the domain. For this scheme, a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method. In numerical experiments, the results are presented and compared with the findings of the finite element method, the radial basis functions method, and an analytical solution to confirm the good accuracy of the presented scheme.
NASA Technical Reports Server (NTRS)
Yon, Steven; Katz, Joseph; Plotkin, Allen
1992-01-01
The practical limit of airfoil thickness ratio for which acceptable engineering results are obtainable with the Dirichlet boundary-condition-based numerical methods is investigated. This is done by studying the effect of thickness on the calculated pressure distribution near the trailing edge and by comparing the aerodynamic coefficients with available exact solutions. The first objective of this study, owing to the wide use of such computational methods, is to demonstrate the numerical symptoms that occur when the body or wing thickness approaches zero and to increase the awareness of potential users of these methods. Additionally, an effort is made to obtain the practical limits of the trailing-edge thickness where such problems will appear in the flow solution, and to propose some possible cures for very thin airfoils or those with cusped trailing edges.
NASA Technical Reports Server (NTRS)
Lyusternik, L. A.
1980-01-01
The mathematics involved in numerically solving for the plane boundary value of the Laplace equation by the grid method is developed. The approximate solution of a boundary value problem for the domain of the Laplace equation by the grid method consists of finding u at the grid corner which satisfies the equation at the internal corners (u=Du) and certain boundary value conditions at the boundary corners.
NASA Astrophysics Data System (ADS)
Abd Elazem, Nader Y.
2016-06-01
The flow of nanofluids past a stretching sheet has attracted much attention owing to its wide applications in industry and engineering. Numerical solution has been discussed in this article for studying the effect of suction (or injection) on flow of nanofluids past a stretching sheet. The numerical results carried out using Chebyshev collocation method (ChCM). Useful results for temperature profile, concentration profile, reduced Nusselt number, and reduced Sherwood number are discussed in tabular and graphical forms. It was also demonstrated that both temperature and concentration profiles decrease by an increase from injection to suction. Moreover, the numerical results show that the temperature profiles decrease at high values of Prandtl number Pr. Finally, the present results showed that the reduced Nusselt number is a decreasing function, whereas the reduced Sherwood number is an increasing function at fixed values of Prandtl number Pr, Lewis number Le and suction (or injection) parameter s for variation of Brownian motion parameter Nb, and thermophoresis parameter Nt.
Effects of the computational time step on numerical solutions for turbulent flow
NASA Technical Reports Server (NTRS)
Choi, Haecheon; Moin, Parviz
1994-01-01
Effects of large computational time steps on the computed turbulence were investigated using a fully implicit method. In turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined. Turbulence fluctuations could not be sustained if the computational time step was near or larger than the Kolmogorov time scale.
Numerical modeling of coupled pressure solution and fluid flow in quartz sandstones
NASA Astrophysics Data System (ADS)
Sheldon, H. A.; Wheeler, J.; Worden, R.
2001-12-01
Pressure solution in quartz sandstones can be envisaged as a 3-stage process, involving dissolution along grain contacts, diffusion of the solute along the grain contact to the pore space, and removal of the solute from the pore fluid by a combination of diffusive and/or advective transport and chemical reactions (e.g. precipitation of dissolved silica on free grain surfaces). A number of authors have developed mathematical models of pressure solution in order to assess the impact of this process on porosity and permeability in sandstones. However, such models have always been based on a simplified subset of the governing equations, in order to reduce the computation time to an acceptable level. For example, some models assume diffusion through the grain contact zone to be the rate-limiting step, with all the dissolved material precipitating locally in the pore space. Other models assume that the rate of removal of solute from the pore fluid, by diffusion and precipitation, is rate-limiting. It is now possible to solve the full coupled system of equations on a PC, without such simplifications. This enables us to investigate the coupling and interactions between pressure solution, chemical reactions in the pore spaces and macroscale advective/diffusive transport in the pore fluid. Preliminary results of such modeling will be presented, highlighting the importance of modeling pressure solution in an open system, where there is a strong coupling between macroscale transport in the pore fluid and the rate of porosity loss due to compaction and cementation.
NASA Astrophysics Data System (ADS)
Cruz, Pedro A.; Tomé, Murilo F.; Stewart, Iain W.; McKee, Sean
2013-08-01
A finite difference method for solving nematic liquid crystal flows under the effect of a magnetic field is developed. The dynamic equations of nematic liquid crystals, given by the Ericksen-Leslie dynamic theory, are employed. These are expressed in terms of primitive variables and solved employing the ideas behind the GENSMAC methodology (Tomé and McKee, 1994; Tomé et al., 2002) [38,41]. These equations are nonlinear partial differential equations consisting of the mass conservation equation and the balance laws of linear and angular momentum. By employing fully developed flow assumptions an analytic solution for steady 2D-channel flow is found. The resulting numerical technique was then, in part, validated by comparing numerical solutions against this analytic solution. Convergence results are presented. To demonstrate the capabilities of the numerical method, the flow of a nematic liquid crystal through various complex geometries are then simulated. Results are obtained for L-shaped channels and planar 4:1 contraction for several values of Reynolds and Ericksen numbers.
NASA Astrophysics Data System (ADS)
Simmel, Martin; Trautmann, Thomas; Tetzlaff, Gerd
The Linear Discrete Method is used to solve the Stochastic Collection Equation (SCE) numerically. Comparisons are made with the Method of Moments, the Berry-Reinhardt model and the Linear Flux Method. Simulations for all numerical methods are shown for the kernel after Golovin [Bull. Acad. Sci. USSR, Geophys. Ser. 5 (1963) 783] and are compared with the analytical solution for two different initial distributions. BRM seems to give the best results and LDM gives good results, too. LFM overestimates the drop growth for the right tail of the distribution and MOM does the same but over the entire drop spectrum. For the hydrodynamic kernel after Long [J. Atmos. Sci. 31 (1974) 1040], simulations are presented using the four numerical methods (LDM, MOM, BRM, LFM). Especially for high resolutions, the solutions of LDM and LFM approach each other very closely. In addition, LDM simulations using the hydrodynamic kernel after Böhm [Atmos. Res. 52 (1999) 167] are presented, which show good correspondence between low- and high-resolution results. Computation efficiency is especially important when numerical schemes are to be included in larger models. Therefore, the computation times of the four methods were compared for the cases with the Golovin kernel. The result is that LDM is the fastest method by far, needing less time than other methods by a factor of 2-7, depending on the case and the bin resolution. For high resolutions, MOM is the slowest. For the lowest resolution, this holds for LFM.
NASA Astrophysics Data System (ADS)
Ying, Teh Yuan; Yaacob, Nazeeruddin
2013-04-01
In this paper, a new implicit Runge-Kutta method which based on a 7-point Gauss-Kronrod-Lobatto quadrature formula is developed. The resulting implicit method is a 7-stage tenth order Gauss-Kronrod-Lobatto IIIA method, or in brief as GKLM(7,10)-IIIA. GKLM(7,10)-IIIA requires seven function of evaluations at each integration step and it gives accuracy of order ten. In addition, GKLM(7,10)-IIIA has stage order seven and being A-stable. Numerical experiments compare the accuracy between GKLM(7,10)-IIIA and the classical 5-stage tenth order Gauss-Legendre method in solving some test problems. Numerical results reveal that GKLM(7,10)-IIIA is more accurate than the 5-stage tenth order Gauss-Legendre method because GKLM(7,10)-IIIA has higher stage order.
NASA Astrophysics Data System (ADS)
Ying, Teh Yuan; Yaacob, Nazeeruddin
2013-04-01
In this paper, a new implicit Runge-Kutta method which based on a 4-point Gauss-Kronrod-Radau II quadrature formula is developed. The resulting implicit method is a 4-stage sixth order Gauss-Kronrod-Radau IIA method, or in brief as GKRM(4,6)-IIA. GKRM(4,6)-IIA requires four function of evaluations at each integration step and it gives accuracy of order six. In addition, GKRM(4,6)-IIA has stage order four and being L-stable. Numerical experiments compare the accuracy between GKRM(4,6)-IIA and the classical 3-stage sixth order Gauss-Legendre method in solving some test problems. Numerical results reveal that GKRM(4,6)-IIA is more accurate than the 3-stage sixth order Gauss-Legendre method because GKRM(4,6)-IIA has higher stage order.
Numerical solution of the Boltzmann equation for the collective modes of trapped Fermi gases
Lepers, Thomas; Davesne, Dany; Chiacchiera, Silvia; Urban, Michael
2010-08-15
We numerically solve the Boltzmann equation for trapped fermions in the normal phase by using the test-particle method. After discussing a couple of tests in order to estimate the reliability of the method, we apply it to the description of collective modes in a spherical harmonic trap. The numerical results are compared with those obtained previously by taking moments of the Boltzmann equation. We find that the general shape of the response function is very similar in both methods, but the relaxation time obtained from the simulation is significantly longer than that predicted by the method of moments. It is shown that the result of the method of moments can be corrected by including fourth-order moments in addition to the usual second-order ones and that this method agrees very well with our numerical simulations.
NASA Astrophysics Data System (ADS)
Talamo, Alberto
2013-05-01
This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps.
Solution of AntiSeepage for Mengxi River Based on Numerical Simulation of Unsaturated Seepage
Ji, Youjun; Zhang, Linzhi; Yue, Jiannan
2014-01-01
Lessening the leakage of surface water can reduce the waste of water resources and ground water pollution. To solve the problem that Mengxi River could not store water enduringly, geology investigation, theoretical analysis, experiment research, and numerical simulation analysis were carried out. Firstly, the seepage mathematical model was established based on unsaturated seepage theory; secondly, the experimental equipment for testing hydraulic conductivity of unsaturated soil was developed to obtain the curve of two-phase flow. The numerical simulation of leakage in natural conditions proves the previous inference and leakage mechanism of river. At last, the seepage control capacities of different impervious materials were compared by numerical simulations. According to the engineering actuality, the impervious material was selected. The impervious measure in this paper has been proved to be effectible by hydrogeological research today. PMID:24707199
Study compares methods for the numerical solution of ordinary differential equations
NASA Technical Reports Server (NTRS)
1966-01-01
Study compares the use of five different methods for the computer solution of the restricted three-body problem. It describes the implementation of each method on a burroughs B-5000 computer and in terms of speed and accuracy.
NASA Astrophysics Data System (ADS)
Russo, David
2016-05-01
The aim of the present numerical study was to extend the data-driven protocol for the control of soil salinity, to control chloride and nitrate concentrations and mass fluxes below agricultural fields irrigated with treated waste water (TWW). The protocol is based on alternating irrigation water quality between TWW and desalinized water (DSW), guided by solute concentrations at soil depth, zs. Two different schemes, the first requires measurements of soil solution concentrations of chloride and nitrate at zs, while, the second scheme requires only measurements of soil solution EC at zs, were investigated. For this purpose, 3-D numerical simulations of flow and transport were performed for variably saturated, spatially heterogeneous, flow domains located at two different field sites. The sites differ in crop type, irrigation method, and in their lithology; these differences, in turn, considerably affect the performance of the proposed schemes, expressed in terms of their ability to reduce solute concentrations that drained below the root zone. Results of the analyses suggest that the proposed data-driven schemes allow the use of low-quality water for irrigation, while minimizing the consumption of high-quality water to a level, which, for given climate, soil, crop, irrigation method, and water quality, may be determined by the allowable nitrate and chloride concentrations in the groundwater. The results of the present study indicate that with respect to the diminution of groundwater contamination by chloride and nitrate, the more data demanding, first scheme is superior the second scheme.
NASA Astrophysics Data System (ADS)
Krasnikov, S. D.; Kuznetsov, E. B.
2016-09-01
Numerical continuation of solution through certain singular points of the curve of the set of solutions to a system of nonlinear algebraic or transcendental equations with a parameter is considered. Bifurcation points of codimension two and three are investigated. Algorithms and computer programs are developed that implement the procedure of discrete parametric continuation of the solution and find all branches at simple bifurcation points of codimension two and three. Corresponding theorems are proved, and each algorithm is rigorously justified. A novel algorithm for the estimation of errors of tangential vectors at simple bifurcation points of a finite codimension m is proposed. The operation of the computer programs is demonstrated by test examples, which allows one to estimate their efficiency and confirm the theoretical results.
Zeron, Eduardo S; Santillán, Moisés
2010-05-21
In this work we introduce a novel approach to study biochemical noise. It comprises a simplification of the master equation of complex reaction schemes (via an adiabatic approximation) and the numerical solution of the reduced master equation. The accuracy of this procedure is tested by comparing its results with analytic solutions (when available) and with Gillespie stochastic simulations. We further employ our approach to study the stochastic expression of a simple gene network, which is subject to negative feedback regulation at the transcriptional level. Special attention is paid to the influence of negative feedback on the amplitude of intrinsic noise, as well as on the relaxation rate of the system probability distribution function to the steady solution. Our results suggest the existence of an optimal feedback strength that maximizes this relaxation rate.
NASA Technical Reports Server (NTRS)
Kleinman, D. L.
1976-01-01
A numerical technique is given for solving the matrix quadratic equation that arises in the optimal stationary control of linear systems with state (and/or control) dependent noise. The technique exploits fully existing, efficient algorithms for the matrix Lyapunov and Ricatti equations. The computational requirements are discussed, with an associated example.
Return Trajectory of the SpaceShipTwo Spacecraft--Numerical Solution
ERIC Educational Resources Information Center
Slegr, J.; Kraus, I.
2012-01-01
SpaceShipTwo is a private spaceplane project which is intended for space tourism. Very few details about its construction and flight characteristics are available for the public, but with proper numerical methods some interesting results can be obtained using secondary school mathematics. An exercise about SpaceShipTwo can be used as a…
Exact vs. Gauss-Seidel numerical solutions of the non-LTE radiation transfer problem
NASA Astrophysics Data System (ADS)
Quang, Carine; Paletou, Frédéric; Chevallier, Loïc
2004-12-01
Although published in 1995 (Trujillo Bueno & Fabiani Bendicho, ApJ 455, 646), the Gauss-Seidel method for solving the non-LTE radiative transfer problem has deserved too little attention in the astrophysical community yet. Further tests of the performances and of the accuracy of the numerical scheme are provided.
1986-01-01
1985), 1-44. [19] V. Majer, Numerical solution of boundary value problems for ordinary differential equations of nonlinear elasticity, Ph.D. Thesis, Univ...based on the ffactoriza- tion method. 1 INTRODUCTION 1.1 Numerical methods for linear boundary value problems for ordinary differential equations The...numerical solution of linear boundary value problems for ordinary differential eqIuations are presented. The methods are optimal with respect to certain
Majer, G.; Zick, K.
2015-04-28
A pulsed field gradient spin-echo nuclear magnetic resonance (NMR) sequence with solvent suppression (PGSE-WATERGATE) was applied to accurately measure the diffusion coefficients of Rhodamine 6G (Rh6G) in low-concentration aqueous solutions. Three samples with Rh6G concentrations of C{sub Rh6G} = 1, 4.5, and 25 μM were investigated. The precise determination of the diffusion coefficients in this low-concentration range was made possible by using a cryogenically cooled NMR probe and by the effective solvent suppression of the PGSE-WATERGATE sequence. The present results bridge the gap between diffusion data measured by fluorescence correlation spectroscopy in the single molecule limit and diffusivities obtained by pulsed field gradient NMR (PFG-NMR) without solvent suppression at higher concentrations. To further extend the concentration range, the diffusion coefficient of Rh6G was also measured on a sample with C{sub Rh6G} = 410 μM by PFG-NMR. The overall concentration dependence of the Rh6G diffusion at 25 °C is discussed in terms of dimerization of the Rh6G molecules. The concentration-dependent monomer/dimer proportion is deduced from the diffusion data.
Heinz, Hendrik
2014-06-18
Adsorption of biomolecules and polymers to inorganic nanostructures plays a major role in the design of novel materials and therapeutics. The behavior of flexible molecules on solid surfaces at a scale of 1-1000 nm remains difficult and expensive to monitor using current laboratory techniques, while playing a critical role in energy conversion and composite materials as well as in understanding the origin of diseases. Approaches to implement key surface features and pH in molecular models of solids are explained, and distinct mechanisms of peptide recognition on metal nanostructures, silica and apatite surfaces in solution are described as illustrative examples. The influence of surface energies, specific surface features and protonation states on the structure of aqueous interfaces and selective biomolecular adsorption is found to be critical, comparable to the well-known influence of the charge state and pH of proteins and surfactants on their conformations and assembly. The representation of such details in molecular models according to experimental data and available chemical knowledge enables accurate simulations of unknown complex interfaces in atomic resolution in quantitative agreement with independent experimental measurements. In this context, the benefits of a uniform force field for all material classes and of a mineral surface structure database are discussed.
Recommendations for numerical solution of reinforced-panel and fuselage-ring problems
NASA Technical Reports Server (NTRS)
Hoff, N J; Libby, Paul A
1949-01-01
Procedures are recommended for solving the equations of equilibrium of reinforced panels and isolated fuselage rings as represented by the external loads and the operations table established according to Southwell's method. From the solution of these equations the stress distribution can be easily determined. The method of systematic relaxations, the matrix-calculus method, and several other methods applicable in special cases are discussed. Definite recommendations are made for obtaining the solution of reinforced-panel problems which are generally designated as shear lag problems. The procedures recommended are demonstrated in the analysis of a number of panels. In the case of fuselage rings it is not possible to make definite recommendations for the solution of the equilibrium equations for all rings and loadings. However, suggestions based on the latest experience are made and demonstrated on several rings.
NASA Technical Reports Server (NTRS)
Lim, Sang G.; Brewe, David E.; Prahl, Joseph M.
1990-01-01
The transient analysis of hydrodynamic lubrication of a point-contact is presented. A body-fitted coordinate system is introduced to transform the physical domain to a rectangular computational domain, enabling the use of the Newton-Raphson method for determining pressures and locating the cavitation boundary, where the Reynolds boundary condition is specified. In order to obtain the transient solution, an explicit Euler method is used to effect a time march. The transient dynamic load is a sinusoidal function of time with frequency, fractional loading, and mean load as parameters. Results include the variation of the minimum film thickness and phase-lag with time as functions of excitation frequency. The results are compared with the analytic solution to the transient step bearing problem with the same dynamic loading function. The similarities of the results suggest an approximate model of the point contact minimum film thickness solution.
NASA Astrophysics Data System (ADS)
Hayek, M.; Kosakowski, G.; Jakob, A.; Churakov, S.
2012-04-01
Numerical computer codes dealing with precipitation-dissolution reactions and porosity changes in multidimensional reactive transport problems are important tools in geoscience. Recent typical applications are related to CO2 sequestration, shallow and deep geothermal energy, remediation of contaminated sites or the safe underground storage of chemotoxic and radioactive waste. Although the agreement between codes using the same models and similar numerical algorithms is satisfactory, it is known that the numerical methods used in solving the transport equation, as well as different coupling schemes between transport and chemistry, may lead to systematic discrepancies. Moreover, due to their inability to describe subgrid pore space changes correctly, the numerical approaches predict discretization-dependent values of porosity changes and clogging times. In this context, analytical solutions become an essential tool to verify numerical simulations. We present a benchmark study where we compare a two-dimensional analytical solution for diffusive transport of two solutes coupled with a precipitation-dissolution reaction causing porosity changes with numerical solutions obtained with the COMSOL Multiphysics code and with the reactive transport code OpenGeoSys-GEMS. The analytical solution describes the spatio-temporal evolution of solutes and solid concentrations and porosity. We show that both numerical codes reproduce the analytical solution very well, although distinct differences in accuracy can be traced back to specific numerical implementations.
Application of multiquadric method for numerical solution of elliptic partial differential equations
Sharan, M.; Kansa, E.J.; Gupta, S.
1994-01-01
We have used the multiquadric (MQ) approximation scheme for the solution of elliptic partial differential equations with Dirichlet and/or Neumann boundary conditions. The scheme has the advantage to use the data points in arbitrary locations with an arbitrary ordering. Two dimensional Laplace, Poisson and Biharmonic equations describing the various physical processes, have been taken as the test examples. The agreement is found to be very good between the computed and exact solutions. The method also provides an excellent approximation with curve boundary.
Numerical-solution package for transient two-phase-flow equations. [PWR
Mahaffy, J.H.; Liles, D.
1982-01-01
The methods presented have proven to be extremely reliable tools for reactor safety analysis. They can handle a wide range of fluid conditions and time scales with minimal failures, while maintaining time step sizes well above those possible with most other techniques. This robustness is due not only to the finite-difference equations themselves, but also to the choice of solution technique, because poorly chosen iterative solution procedures often require a limit on the time-step size for proper convergence of the iteration.
Nonlinear grid error effects on numerical solution of partial differential equations
NASA Technical Reports Server (NTRS)
Dey, S. K.
1980-01-01
Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.
Numerical solutions and laser-Doppler measurements of spin-up
NASA Technical Reports Server (NTRS)
Warn-Varnas, A.; Piacsek, S.; Fowlis, W. W.; Lee, S. M.
1978-01-01
The spin-up flow in a cylinder of homogeneous fluid has been examined both experimentally and numerically. A series of laser-Doppler measurements was made of the zonal flow over a range of Ekman numbers and Rossby numbers at various locations in the interior of the flow. These measurements exceed previous ones in accuracy. The weak inertial modes excited by the impulsive start are detectable. The numerical simulations used the primitive equations in axisymmetric form and employed finite-difference techniques on both constant and variable grids. The number of grid points necessary to resolve the Ekman layers was determined. A thorough comparison of the simulations and the experimental measurements is made which includes the details of the amplitude and frequency of the inertial modes. Agreement to within the experimental tolerance is achieved. Analytical results for conditions identical to those in the experiments are not available but some similar linear and nonlinear theories are also compared with the experiments.
Numerical solution of the Navier-Stokes equations by a multigrid method
NASA Astrophysics Data System (ADS)
Cambier, L.; Couaillier, V.; Veuillot, J. P.
This article describes the use of a multigrid method to compute compressible two-dimensional turbulent flows by solving the averaged Navier-Stokes equations, complemented by a turbulence model. The numerical method is described in detail. It is based on an explicit, centered scheme of the Lax-Wendroff type, the convergence of which is accelerated by a multigrid phase similar to the one proposed by Ni. The effect of the parameters introduced in the multigrid acceleration phase is studied in numerical simulations to increase their effectiveness. The applications covered relate to high-Reynolds flows around a wing profile and in a two-dimensional cascade. Comparisons with experimental data are given for these two types of application.
Numerical solution of the Polanyi-DR isotherm in linear driving force models.
Lorenzetti, David M; Sohn, Michael D
2011-12-01
The Polanyi-Dubinin-Radushkevich isotherm has proven useful for modeling the adsorption of volatile organic compounds on microporous materials such as activated carbon. When embedded in a larger dynamic simulation--e.g., of whole-building pollutant transport--it is important to solve the sorption relations as quickly as possible. This work compares numerical methods for solving the Polanyi-DR model, in cases where transport to the surface is assumed linear in the bulk-to-surface concentration differences. We focus on developing numerically stable algorithms that converge across a wide range of inputs, including zero concentrations, where the isotherm is undefined. We identify several methods, including a modified Newton-Raphson search, that solve the system 3-4 times faster than simple bisection. Finally, we present a rule of thumb for identifying when boundary-layer diffusion limits the transport rate enough to justify reducing the model complexity.
Numerical solution of the thermal influence of oil well cluster on permafrost
NASA Astrophysics Data System (ADS)
Afanaseva, N. M.; Kolesov, A. E.
2016-10-01
In this work, we study the thermal effects around the oil well cluster on permafrost using numerical modeling. We use the mathematical model of heat transfer with phase transitions. To take into account the arrangement of wells in a cluster, three-dimensional domains with complex geometry are employed, which leads to the use of finite element approximation in space. For time approximation we use fully implicit scheme with linearization of nonlinear coefficients. Numerical implementations are performed using open-source libraries and programs for scientific and engineering computations. To predict the temperature field and formation of thawing area around wells with different sets of input parameters we conduct large-scale computational experiments on the supercomputer of the North-Eastern Federal University.
Study and numerical solution of a generalized mathematical model of isothermal adsorption
Komissarov, Yu.A.; Vetokhin, V.N.; Tsenev, V.A.; Gordeeva, E.L.
1995-06-01
A generalized mathematical model of isothermal adsorption that takes into account mass transfer on the surface of a particle, diffusion in micro- and macropores, and dispersion along the length of the apparatus is considered The parameters {lambda} and {var_phi}{sup 2} determine the dominating effect of any of the mass transfer mechanisms of the adsorption process. A numerical algorithm for solving the generalized adsorption model is suggested.
2010-02-20
x’ integration takes place within a single grid cell . For these energies, the finite difference 4 approximation is not even qualitatively correct...temperature variation within a few grid cells . However numerical instability occurs on the very shortest spatial scale length, so the background appears...proportional to /5-2 varies over 8 orders of magnitude, and since a laser plasma simulation typically has fewer than 1000 spatial cells , flux cannot be treated
A one-parameter family of difference schemes for the numerical solution of the Keplerian problem
NASA Astrophysics Data System (ADS)
Elenin, G. G.; Elenina, T. G.
2015-08-01
A family of numerical methods for solving the Keplerian problem is proposed. All the methods in this family are symplectic. They preserve the angular momentum, the total energy, the components of the Laplace-Runge-Lenz vector, and the phase volume. The underlying idea is an exact linearization of the problem based on the Levi-Civita transformation and two-stage symmetricsymplectic Runge-Kutta methods.
Numerical Solution of the Problem of the Computational Time Reversal in the Quadrant
2005-09-21
condition with a finite support in a hyperboilc equation, given the Cauchy data at the lateral surface. A stability estimate for this ill-posed problem...implies refocusing of the time reversed wave field. Two such two-dimensional inverse problems are solved numerically in the case when the domain is a ...inverse problem for a hyperbolic equation with the Cauchy data at a lateral surface. Consider the standard Cauchy problem for the hyperbolic equation utt
Practical aspects of spatially high accurate methods
NASA Technical Reports Server (NTRS)
Godfrey, Andrew G.; Mitchell, Curtis R.; Walters, Robert W.
1992-01-01
The computational qualities of high order spatially accurate methods for the finite volume solution of the Euler equations are presented. Two dimensional essentially non-oscillatory (ENO), k-exact, and 'dimension by dimension' ENO reconstruction operators are discussed and compared in terms of reconstruction and solution accuracy, computational cost and oscillatory behavior in supersonic flows with shocks. Inherent steady state convergence difficulties are demonstrated for adaptive stencil algorithms. An exact solution to the heat equation is used to determine reconstruction error, and the computational intensity is reflected in operation counts. Standard MUSCL differencing is included for comparison. Numerical experiments presented include the Ringleb flow for numerical accuracy and a shock reflection problem. A vortex-shock interaction demonstrates the ability of the ENO scheme to excel in simulating unsteady high-frequency flow physics.
Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation
NASA Astrophysics Data System (ADS)
Khurana, Saheba; Thachuk, Mark
2016-03-01
A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation.
Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation.
Khurana, Saheba; Thachuk, Mark
2016-03-14
A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation.
NASA Astrophysics Data System (ADS)
Sandin, Patrik; Ögren, Magnus; Gulliksson, Mârten
2016-03-01
We formulate a damped oscillating particle method to solve the stationary nonlinear Schrödinger equation (NLSE). The ground-state solutions are found by a converging damped oscillating evolution equation that can be discretized with symplectic numerical techniques. The method is demonstrated for three different cases: for the single-component NLSE with an attractive self-interaction, for the single-component NLSE with a repulsive self-interaction and a constraint on the angular momentum, and for the two-component NLSE with a constraint on the total angular momentum. We reproduce the so-called yrast curve for the single-component case, described in [A. D. Jackson et al., Europhys. Lett. 95, 30002 (2011), 10.1209/0295-5075/95/30002], and produce for the first time an analogous curve for the two-component NLSE. The numerical results are compared with analytic solutions and competing numerical methods. Our method is well suited to handle a large class of equations and can easily be adapted to further constraints and components.
NASA Astrophysics Data System (ADS)
Calini, A.; Schober, C. M.
2014-06-01
In this article we conduct a broad numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger (NLS) equation, a widely used model of rogue wave generation and dynamics in deep water. NLS breathers rising over an unstable background state are frequently used to model rogue waves. However, the issue of whether these solutions are robust with respect to the kind of random perturbations occurring in physical settings and laboratory experiments has just recently begun to be addressed. Numerical experiments for spatially periodic breathers with one or two modes involving large ensembles of perturbed initial data for six typical random perturbations suggest interesting conclusions. Breathers over an unstable background with N unstable modes are generally unstable to small perturbations in the initial data unless they are "maximal breathers" (i.e., they have N spatial modes). Additionally, among the maximal breathers with two spatial modes, the one of highest amplitude due to coalescence of the modes appears to be the most robust. The numerical observations support and extend to more realistic settings the results of our previous stability analysis, which we hope will provide a useful tool for identifying physically realizable wave forms in experimental and observational studies of rogue waves.
Sandin, Patrik; Ögren, Magnus; Gulliksson, Mårten
2016-03-01
We formulate a damped oscillating particle method to solve the stationary nonlinear Schrödinger equation (NLSE). The ground-state solutions are found by a converging damped oscillating evolution equation that can be discretized with symplectic numerical techniques. The method is demonstrated for three different cases: for the single-component NLSE with an attractive self-interaction, for the single-component NLSE with a repulsive self-interaction and a constraint on the angular momentum, and for the two-component NLSE with a constraint on the total angular momentum. We reproduce the so-called yrast curve for the single-component case, described in [A. D. Jackson et al., Europhys. Lett. 95, 30002 (2011)], and produce for the first time an analogous curve for the two-component NLSE. The numerical results are compared with analytic solutions and competing numerical methods. Our method is well suited to handle a large class of equations and can easily be adapted to further constraints and components.
Using Predictor-Corrector Methods in Numerical Solutions to Mathematical Problems of Motion
ERIC Educational Resources Information Center
Lewis, Jerome
2005-01-01
In this paper, the author looks at some classic problems in mathematics that involve motion in the plane. Many case problems like these are difficult and beyond the mathematical skills of most undergraduates, but computational approaches often require less insight into the subtleties of the problems and can be used to obtain reliable solutions.…
Kmonodium, a Program for the Numerical Solution of the One-Dimensional Schrodinger Equation
ERIC Educational Resources Information Center
Angeli, Celestino; Borini, Stefano; Cimiraglia, Renzo
2005-01-01
A very simple strategy for the solution of the Schrodinger equation of a particle moving in one dimension subjected to a generic potential is presented. This strategy is implemented in a computer program called Kmonodium, which is free and distributed under the General Public License (GPL).
vom Saal, Frederick S; Welshons, Wade V
2014-12-01
There is extensive evidence that bisphenol A (BPA) is related to a wide range of adverse health effects based on both human and experimental animal studies. However, a number of regulatory agencies have ignored all hazard findings. Reports of high levels of unconjugated (bioactive) serum BPA in dozens of human biomonitoring studies have also been rejected based on the prediction that the findings are due to assay contamination and that virtually all ingested BPA is rapidly converted to inactive metabolites. NIH and industry-sponsored round robin studies have demonstrated that serum BPA can be accurately assayed without contamination, while the FDA lab has acknowledged uncontrolled assay contamination. In reviewing the published BPA biomonitoring data, we find that assay contamination is, in fact, well controlled in most labs, and cannot be used as the basis for discounting evidence that significant and virtually continuous exposure to BPA must be occurring from multiple sources.
vom Saal, Frederick S.; Welshons, Wade V.
2016-01-01
There is extensive evidence that bisphenol A (BPA) is related to a wide range of adverse health effects based on both human and experimental animal studies. However, a number of regulatory agencies have ignored all hazard findings. Reports of high levels of unconjugated (bioactive) serum BPA in dozens of human biomonitoring studies have also been rejected based on the prediction that the findings are due to assay contamination and that virtually all ingested BPA is rapidly converted to inactive metabolites. NIH and industry-sponsored round robin studies have demonstrated that serum BPA can be accurately assayed without contamination, while the FDA lab has acknowledged uncontrolled assay contamination. In reviewing the published BPA biomonitoring data, we find that assay contamination is, in fact, well controlled in most labs, and cannot be used as the basis for discounting evidence that significant and virtually continuous exposure to BPA must be occurring from multiple sources. PMID:25304273
NASA Technical Reports Server (NTRS)
Carter, J. E.
1972-01-01
Numerical solutions have been obtained for the supersonic, laminar flow over a two-dimensional compression corner. These solutions were obtained as steady-state solutions to the unsteady Navier-Stokes equations using the finite difference method of Brailovskaya, which has second-order accuracy in the spatial coordinates. Good agreement was obtained between the computed results and wall pressure distributions measured experimentally for Mach numbers of 4 and 6.06, and respective Reynolds numbers, based on free-stream conditions and the distance from the leading edge to the corner. In those calculations, as well as in others, sufficient resolution was obtained to show the streamline pattern in the separation bubble. Upstream boundary conditions to the compression corner flow were provided by numerically solving the unsteady Navier-Stokes equations for the flat plate flow field, beginning at the leading edge. The compression corner flow field was enclosed by a computational boundary with the unknown boundary conditions supplied by extrapolation from internally computed points.
NASA Technical Reports Server (NTRS)
Wang, Xiao-Yen; Chow, Chuen-Yen; Chang, Sin-Chung
1998-01-01
Without resorting to special treatment for each individual test case, the 1D and 2D CE/SE shock-capturing schemes described previously (in Part I) are used to simulate flows involving phenomena such as shock waves, contact discontinuities, expansion waves and their interactions. Five 1D and six 2D problems are considered to examine the capability and robustness of these schemes. Despite their simple logical structures and low computational cost (for the 2D CE/SE shock-capturing scheme, the CPU time is about 2 micro-secs per mesh point per marching step on a Cray C90 machine), the numerical results, when compared with experimental data, exact solutions or numerical solutions by other methods, indicate that these schemes can accurately resolve shock and contact discontinuities consistently.
Advanced Numerical Methods for Computing Statistical Quantities of Interest from Solutions of SPDES
2012-01-19
and tested our method. Figure 1 presents a comparison of the solution of (5) obtained by a Monte- Carlo method and that of (6) using a stochastic...2 0 .3 0 .4 0.4 0.4 0.5 0 .5 0 .6 0 .6 0.6 0.6 0 .7 0.70 .7 0 .8 0.8 0 .9 x y Isotherms, Monte Carlo method, number of samples=200 0 0.2 0.4 0.6 0.8 0...of the solution of (5) obtained by a Monte- Carlo method (top) and that of (6) using a stochastic collocation method with fewer samples (bottom
Hahn, Marc Benjamin; Uhlig, Frank; Solomun, Tihomir; Smiatek, Jens; Sturm, Heinz
2016-10-19
Ectoine is an important osmolyte, which allows microorganisms to survive in extreme environmental salinity. The hygroscopic effects of ectoine in pure water can be explained by a strong water binding behavior whereas a study on the effects of ectoine in salty solution is yet missing. We provide Raman spectroscopic evidence that the influence of ectoine and NaCl are opposing and completely independent of each other. The effect can be explained by the formation of strongly hydrogen-bonded water molecules around ectoine which compensate the influence of the salt on the water dynamics. The mechanism is corroborated by first principles calculations and broadens our understanding of zwitterionic osmolytes in aqueous solution. Our findings allow us to provide a possible explanation for the relatively high osmolyte concentrations in halotolerant bacteria.
Numerical solution of singular ODE eigenvalue problems in electronic structure computations
NASA Astrophysics Data System (ADS)
Hammerling, Robert; Koch, Othmar; Simon, Christa; Weinmüller, Ewa B.
2010-09-01
We put forward a new method for the solution of eigenvalue problems for (systems of) ordinary differential equations, where our main focus is on eigenvalue problems for singular Schrödinger equations arising for example in electronic structure computations. In most established standard methods, the generation of the starting values for the computation of eigenvalues of higher index is a critical issue. Our approach comprises two stages: First we generate rough approximations by a matrix method, which yields several eigenvalues and associated eigenfunctions simultaneously, albeit with moderate accuracy. In a second stage, these approximations are used as starting values for a collocation method which yields approximations of high accuracy efficiently due to an adaptive mesh selection strategy, and additionally provides reliable error estimates. We successfully apply our method to the solution of the quantum mechanical Kepler, Yukawa and the coupled ODE Stark problems.
Mat Zin, Shazalina; Abbas, Muhammad; Majid, Ahmad Abd; Ismail, Ahmad Izani Md
2014-01-01
The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Several examples are discussed to exhibit the feasibility and capability of the approach. The absolute errors and L∞ error norms are also computed at different times to assess the performance of the proposed approach and the results were found to be in good agreement with known solutions and with existing schemes in literature.
NASA Technical Reports Server (NTRS)
Hanson, R. K.; Presley, L. L.; Williams, E. V.
1972-01-01
The method of characteristics for a chemically reacting gas is used in the construction of the time-dependent, one-dimensional flow field resulting from the normal reflection of an incident shock wave at the end wall of a shock tube. Nonequilibrium chemical reactions are allowed behind both the incident and reflected shock waves. All the solutions are evaluated for oxygen, but the results are generally representative of any inviscid, nonconducting, and nonradiating diatomic gas. The solutions clearly show that: (1) both the incident- and reflected-shock chemical relaxation times are important in governing the time to attain steady state thermodynamic properties; and (2) adjacent to the end wall, an excess-entropy layer develops wherein the steady state values of all the thermodynamic variables except pressure differ significantly from their corresponding Rankine-Hugoniot equilibrium values.
A fully vectorized numerical solution of the incompressible Navier-Stokes equations. Ph.D. Thesis
NASA Technical Reports Server (NTRS)
Patel, N.
1983-01-01
A vectorizable algorithm is presented for the implicit finite difference solution of the incompressible Navier-Stokes equations in general curvilinear coordinates. The unsteady Reynolds averaged Navier-Stokes equations solved are in two dimension and non-conservative primitive variable form. A two-layer algebraic eddy viscosity turbulence model is used to incorporate the effects of turbulence. Two momentum equations and a Poisson pressure equation, which is obtained by taking the divergence of the momentum equations and satisfying the continuity equation, are solved simultaneously at each time step. An elliptic grid generation approach is used to generate a boundary conforming coordinate system about an airfoil. The governing equations are expressed in terms of the curvilinear coordinates and are solved on a uniform rectangular computational domain. A checkerboard SOR, which can effectively utilize the computer architectural concept of vector processing, is used for iterative solution of the governing equations.
Recent advances in methods for numerical solution of O.D.E. initial value problems
NASA Technical Reports Server (NTRS)
Bui, T. D.; Oppenheim, A. K.; Pratt, D. T.
1984-01-01
In the mathematical modeling of physical systems, it is often necessary to solve an initial value problem (IVP), consisting of a system of ordinary differential equations (ODE). A typical program produces approximate solutions at certain mesh points. Almost all existing codes try to control the local truncation error, while the user is really interested in controlling the true or global error. The present investigation provides a review of recent advances regarding the solution of the IVP, giving particular attention to stiff systems. Stiff phenomena are customarily defined in terms of the eigenvalues of the Jacobian. There are, however, some difficulties connected with this approach. It is pointed out that an estimate of the Lipschitz constant proves to be a very practical way to determine the stiffness of a problem.
Vojta, Matthias; Mitchell, Andrew K; Zschocke, Fabian
2016-07-15
Kitaev's honeycomb-lattice compass model describes a spin liquid with emergent fractionalized excitations. Here, we study the physics of isolated magnetic impurities coupled to the Kitaev spin-liquid host. We reformulate this Kondo-type problem in terms of a many-state quantum impurity coupled to a multichannel bath of Majorana fermions and present the numerically exact solution using Wilson's numerical renormalization group technique. Quantum phase transitions occur as a function of Kondo coupling and locally applied field. At zero field, the impurity moment is partially screened only when it binds an emergent gauge flux, while otherwise it becomes free at low temperatures. We show how Majorana degrees of freedom determine the fixed-point properties, make contact with Kondo screening in pseudogap Fermi systems, and discuss effects away from the dilute limit.
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.
Numerical solutions for the spin-up of a stratified fluid
NASA Technical Reports Server (NTRS)
Hyun, J. M.; Fowlis, W. W.; Warn-Varnas, A.
1982-01-01
The model of Warn-Varnas et al. (1978) is used to numerically examine the spin-up flow of a thermally stratified fluid in a cylinder with an insulating side wall, and comparison of the results with the laser-Doppler measurements of Lee (1975) shows excellent agreement. It is shown that flow gradients are created in the interior of the fluid during the meridional circulation spin-up phase, and that the azimuthal flow decayed faster than has been predicted by Wallin (1969). It is established that viscous diffusion in the interior, arising from the interior-flow gradients, is the cause of the discrepancy with Wallin's theory.
NASA Technical Reports Server (NTRS)
Tuccillo, J. J.
1984-01-01
Numerical Weather Prediction (NWP), for both operational and research purposes, requires only fast computational speed but also large memory. A technique for solving the Primitive Equations for atmospheric motion on the CYBER 205, as implemented in the Mesoscale Atmospheric Simulation System, which is fully vectorized and requires substantially less memory than other techniques such as the Leapfrog or Adams-Bashforth Schemes is discussed. The technique presented uses the Euler-Backard time marching scheme. Also discussed are several techniques for reducing computational time of the model by replacing slow intrinsic routines by faster algorithms which use only hardware vector instructions.
Numerical solution for weight reduction model due to health campaigns in Spain
NASA Astrophysics Data System (ADS)
Mohammed, Maha A.; Noor, Noor Fadiya Mohd; Siri, Zailan; Ibrahim, Adriana Irawati Nur
2015-10-01
Transition model between three subpopulations based on Body Mass Index of Valencia community in Spain is considered. No changes in population nutritional habits and public health strategies on weight reduction until 2030 are assumed. The system of ordinary differential equations is solved using Runge-Kutta method of higher order. The numerical results obtained are compared with the predicted values of subpopulation proportion based on statistical estimation in 2013, 2015 and 2030. Relative approximate error is calculated. The consistency of the Runge-Kutta method in solving the model is discussed.
Numerical solution of two-dimensional turbulent blunt body flows with an impinging shock
NASA Technical Reports Server (NTRS)
Tannehill, J. C.; Vigneron, Y. C.; Rakich, J. V.
1978-01-01
An implicit finite-difference method has been developed to compute two-dimensional, turbulent, blunt body flows with an impinging shock wave. The full time-averaged Navier-Stokes equations are solved with algebraic eddy viscosity and turbulent Prandtl number models employed for shear stress and heat flux. The irregular-shaped bow shock is treated as a discontinuity across which the Rankine-Hugoniot equations are applied. A Type III turbulent shock interference flow field has been computed and the numerical results compare favorably with existing experimental data. In addition, comparisons are made between the present implicit code and a previous explicit code.
NASA Technical Reports Server (NTRS)
Rudy, D. H.; Morris, D. J.; Blanchard, D. K.; Cooke, C. H.; Rubin, S. G.
1975-01-01
The status of an investigation of four numerical techniques for the time-dependent compressible Navier-Stokes equations is presented. Results for free shear layer calculations in the Reynolds number range from 1000 to 81000 indicate that a sequential alternating-direction implicit (ADI) finite-difference procedure requires longer computing times to reach steady state than a low-storage hopscotch finite-difference procedure. A finite-element method with cubic approximating functions was found to require excessive computer storage and computation times. A fourth method, an alternating-direction cubic spline technique which is still being tested, is also described.
A numerical method for the solution of the bidomain equations in cardiac tissue.
Keener, J. P.; Bogar, K.
1998-03-01
A numerical scheme for efficient integration of the bidomain model of action potential propagation in cardiac tissue is presented. The scheme is a mixed implicit-explicit scheme with no stability time step restrictions and requires that only linear systems of equations be solved at each time step. The method is faster than a fully explicit scheme and there is no increase in algorithmic complexity to use this method instead of a fully explicit method. The speedup factor depends on the timestep size, which can be set solely on the basis of the demands for accuracy. (c) 1998 American Institute of Physics.
Numerical solution of transonic full stream function equations in conservation form
NASA Technical Reports Server (NTRS)
Hafez, M. M.
1979-01-01
The stream function equation in conservation form is solved iteratively based on the artificial compressibility method. The density is not a unique function of the mass flux. In order to avoid the ambiguity near the sonic line, the density is updated in terms of the velocity, which is obtained through a simple integration of a first order equation step by step in the flow field. Iteration algorithms and finite difference approximations are discussed and numerical results of both conservative and nonconservative calculations are presented.
Parmar, Payal; Samuels, Alex; Clark, Aurora E
2015-01-13
Contributing factors to the solution-phase correction to the free energy of the molecular clusters U(H2O)n(3+/4+) and UO2(H2O)m(1+/2+) (n = 8, 9, 30, 41, 77; m = 4, 5, 30, 41, 77) have been examined as a function of cavity type in the integrated-equation-formalism-protocol (IEF) and SMD polarizable continuum models (PCMs). It is observed that the free energy correction, Gcorr, does not smoothly converge to zero as the number of explicitly solvating water molecules approaches the bulk limit, and the convergence behavior varies significantly with cavity and model. The rates of convergence of the gas-phase hydration energy, ΔGhyd, wherein the bare metal ion is inserted into a molecular water cluster and ΔGcorr for the reaction exhibit wide variations as a function of ion charge, cavity, and model. This is the likely source of previously reported discrepancies in predicted free energies of solvation for metal ions when using different PCM cavities and/or models. The cancellation of errors in ΔGhyd and ΔGcorr is optimal for clusters consisting of only a second solvation shell of explicit water molecules (n = m = 30). The UFF cavity within IEF, in particular, exhibits the most consistent cancellation of errors when using a molecular cluster consisting of a second shell of solvating water for all oxidation states of uranium, leading to accurate free energies of solvation ΔGsolv for these species.
Asako, Y. . Dept. of Mechanical Engineering); Faghri, M. . Dept. of Mechanical Engineering); Charmchi, M. . Dept. of Mechanical Engineering); Bahrami, P.A. )
1994-02-01
An enthalpy method is employed to solve transport processes associated with melting of an unfixed rectangular phase change material (PCM) in a low-gravitational environment. This method permits the phase-change problems to be solved within fixed numerical grids, hence eliminating the need for coordinate transformation. The PCM, initially at its melting temperature, is placed inside a rectangular enclosure. The lower surface of the container is then exposed to a uniform temperature higher than the PCM melting temperature. The difference in densities of solid and liquid causes a force imbalance on the solid phase exceeds that of the liquid, the solid continually moves downward as melting progresses and hence generates a flow field within the liquid. The problem is formulated as a one-domain problem with the possibility of melting from all the PCM surfaces, and no approximation is made about the liquid film thickness under the melt. The governing equations are discretized by using a control-volume-based finite difference scheme with a new iterative method to correct for the downward solid-phase velocity. This will also speed up the convergence of the numerical procedure. The results are presented in the form of a parametric study of the effects of Archimedes number, Stefan number, Prandtl number, and the geometric parameters on the melt thickness, the downward solid velocity, the elevation of the top surface, and the volume of the solid PCM. They show that in a low-gravitational environment, the melting rate is very slow.
NASA Technical Reports Server (NTRS)
Ingram, H. L.
1973-01-01
Recently the determination of the best technique for numerically solving systems of ordinary differential equations on a digital computer has received much attention. The use of these formulas in conjunction with a stepsize control developed is explained, and one of the formulas is chosen for comparison with other integration techniques. This comparison of one of the best of Fehlberg's formulas with the different numerical techniques described in previous studies on a variety of test problems clearly shows the superiority of Fehlberg's formula. That is, on each of the test problems, the chosen Fehlberg formula is able to achieve a given accuracy in less computer time than any of the other techniques tested. Also, the computer program for the chosen Fehlberg formula is less complex and easier to use than the computer programs for most of the other techniques. To illustrate the use of the chosen Fehlberg formula, a computer listing of its application to several example problems is included along with a sample of the computer output from these applications.
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
Perturbation approach to resonator state solution using the Fourier-Bessel numerical solver
NASA Astrophysics Data System (ADS)
Gauthier, Robert C.
2017-02-01
The Fourier-Bessel (FFB) numerical solver is a useful tool for obtaining the steady states of resonator structures that conform to a cylindrical symmetry. Recently the FFB solver has been greatly simplified by reconfiguring the matrix generating expressions using Maxwell's curl expressions rather than the standard wave equations. This presentation provides a numerical framework suitable for the application on non-degenerate perturbation theory within the theoretical structure of the reconfigured FFB computation environment. It is shown that the resonator structure's perturbation contribution can be isolated as a separate matrix which dictates the shift in resonator state properties. Two distinct application examples are provide; the first has the perturbation possess the same rotational symmetry as the original structure and preserves azimuthal mode order families; the second perturbation has a symmetry different than the original structure and promotes a mixing between azimuthal mode order families. The perturbation extension promises to amplify the potential usefulness of the FFB technique when theoretically considering photonic sensors such as whispering-gallery mode, photonic crystal hole infiltration and a host of others in which the measurand undergoes small changes in its optical properties.
NASA Technical Reports Server (NTRS)
Kwon, J. H.
1977-01-01
Numerical solution of two dimensional, time dependent, compressible viscous Navier-Stokes equations about arbitrary bodies was treated using density gradients as additional dependent variables. Thus, six dependent variables were computed with the SOR iteration method. Besides formulation for pressure gradient terms, a formulation for computing the body density was presented. To approximate the governing equations, an implicit finite difference method was employed. In computing the solution for the flow about a circular cylinder, a problem arose near the wall at both stagnation points. Thus, computations with various conditions were tried to examine the problem. Also, computations with and without formulations are compared. The flow variables were computed on 37 by 40 field first, then on an 81 by 40 field.
On numerical solutions to the QCD ’t Hooft equation in the limit of large quark mass
Zubov, Roman; Prokhvatilov, Evgeni
2016-01-22
First we give a short informal introduction to the theory behind the ’t Hooft equation. Then we consider numerical solutions to this equation in the limit of large fermion masses. It turns out that the spectrum of eigenvalues coincides with that of the Airy differential equation. Moreover when we take the Fourier transform of eigenfunctions, they look like the corresponding Airy functions with appropriate symmetry. It is known that these functions correspond to solutions of a one dimensional Schrodinger equation for a particle in a triangular potential well. So we find the analogy between this problem and the ’t Hooft equation. We also present a simple intuition behind these results.
NASA Astrophysics Data System (ADS)
Mvogo, Alain; Tambue, Antoine; Ben-Bolie, Germain H.; Kofané, Timoléon C.
2016-10-01
We investigate localized wave solutions in a network of Hindmarsh-Rose neural model taking into account the long-range diffusive couplings. We show by a specific analytical technique that the model equations in the infrared limit (wave number k → 0) can be governed by the complex fractional Ginzburg-Landau (CFGL) equation. According to the stiffness of the system, we propose both the semi and the linearly implicit Riesz fractional finite-difference schemes to solve efficiently the CFGL equation. The obtained fractional numerical solutions for the nerve impulse reveal localized short impulse properties. We also show the equivalence between the continuous CFGL and the discrete Hindmarsh-Rose models for relatively large network.
Ahmed, Mahmoud; Eslamian, Morteza
2015-12-01
Laminar natural convection in differentially heated (β = 0°, where β is the inclination angle), inclined (β = 30° and 60°), and bottom-heated (β = 90°) square enclosures filled with a nanofluid is investigated, using a two-phase lattice Boltzmann simulation approach. The effects of the inclination angle on Nu number and convection heat transfer coefficient are studied. The effects of thermophoresis and Brownian forces which create a relative drift or slip velocity between the particles and the base fluid are included in the simulation. The effect of thermophoresis is considered using an accurate and quantitative formula proposed by the authors. Some of the existing results on natural convection are erroneous due to using wrong thermophoresis models or simply ignoring the effect. Here we show that thermophoresis has a considerable effect on heat transfer augmentation in laminar natural convection. Our non-homogenous modeling approach shows that heat transfer in nanofluids is a function of the inclination angle and Ra number. It also reveals some details of flow behavior which cannot be captured by single-phase models. The minimum heat transfer rate is associated with β = 90° (bottom-heated) and the maximum heat transfer rate occurs in an inclination angle which varies with the Ra number.
NASA Astrophysics Data System (ADS)
Ahmed, Mahmoud; Eslamian, Morteza
2015-07-01
Laminar natural convection in differentially heated ( β = 0°, where β is the inclination angle), inclined ( β = 30° and 60°), and bottom-heated ( β = 90°) square enclosures filled with a nanofluid is investigated, using a two-phase lattice Boltzmann simulation approach. The effects of the inclination angle on Nu number and convection heat transfer coefficient are studied. The effects of thermophoresis and Brownian forces which create a relative drift or slip velocity between the particles and the base fluid are included in the simulation. The effect of thermophoresis is considered using an accurate and quantitative formula proposed by the authors. Some of the existing results on natural convection are erroneous due to using wrong thermophoresis models or simply ignoring the effect. Here we show that thermophoresis has a considerable effect on heat transfer augmentation in laminar natural convection. Our non-homogenous modeling approach shows that heat transfer in nanofluids is a function of the inclination angle and Ra number. It also reveals some details of flow behavior which cannot be captured by single-phase models. The minimum heat transfer rate is associated with β = 90° (bottom-heated) and the maximum heat transfer rate occurs in an inclination angle which varies with the Ra number.
Laminar supersonic flow over a backstep - A numerical solution at higher Reynolds numbers
NASA Technical Reports Server (NTRS)
Kronzon, Y.; Rom, J.; Seginer, A.
1976-01-01
The Allen-Cheng solution of the flow over a backward facing step is extended to Reynolds numbers up to 16,000 and to inflow boundary-layer height ratios as low as 0.1 by moving the downstream boundary into the recompression region and by smoothing the resulting errors. The boundary conditions in the supersonic outer flow and the downstream boundary conditions in the wake are determined by an extrapolation procedure. Computational results are compared with relevant experimental data. Fair agreement is found between the calculated base pressures and the experimental values, whereas agreement between heat transfer rates appears to be qualitative only.
Numerical solution of a mixed singularly perturbed parabolic-elliptic problem
NASA Astrophysics Data System (ADS)
Brayanov, Iliya A.
2006-08-01
A one-dimensional singularly perturbed problem of mixed type is considered. The domain under consideration is partitioned into two subdomains. In the first subdomain a parabolic reaction-diffusion problem is given and in the second one an elliptic convection-diffusion-reaction problem. The solution is decomposed into regular and singular components. The problem is discretized using an inverse-monotone finite volume method on condensed Shishkin meshes. We establish an almost second-order global pointwise convergence in the space variable, that is uniform with respect to the perturbation parameter.
Finite volume numerical solution to a blood flow problem in human artery
NASA Astrophysics Data System (ADS)
Wijayanti Budiawan, Inge; Mungkasi, Sudi
2017-01-01
In this paper, we solve a one dimensional blood flow model in human artery. This model is of a non-linear hyperbolic partial differential equation system which can generate either continuous or discontinuous solution. We use the Lax–Friedrichs finite volume method to solve this model. Particularly, we investigate how a pulse propagates in human artery. For this simulation, we give a single sine wave with a small time period as an impluse input on the left boundary. The finite volume method is successful in simulating how the pulse propagates in the artery. It detects the positions of the pulse for the whole time period.
Numerical solution of 3D Navier-Stokes equations with upwind implicit schemes
NASA Technical Reports Server (NTRS)
Marx, Yves P.
1990-01-01
An upwind MUSCL type implicit scheme for the three-dimensional Navier-Stokes equations is presented. Comparison between different approximate Riemann solvers (Roe and Osher) are performed and the influence of the reconstructions schemes on the accuracy of the solution as well as on the convergence of the method is studied. A new limiter is introduced in order to remove the problems usually associated with non-linear upwind schemes. The implementation of a diagonal upwind implicit operator for the three-dimensional Navier-Stokes equations is also discussed. Finally the turbulence modeling is assessed. Good prediction of separated flows are demonstrated if a non-equilibrium turbulence model is used.
NASA Technical Reports Server (NTRS)
Devarayalu, K.
1978-01-01
The numerical solution of the full Navier-Stokes Equations for viscous flows with high Mach numbers and a strong detached bow shock was obtained. Two dimensional flows around a circular cylinder, and a circular cylinder with an aft-body in the form of a fairing, were considered. The solution of the compressible N.S. equations was accomplished by the method of finite differences. An implicit scheme of solution, the S.O.R., was used with the optimum acceleration parameters determined by trial and error. The tensor notation was used in writing the N-S Equations transformed into general curvilinear coordinates. The equations for the generation of the coordinate system were solved, followed by the solution of the N.S. equations, at the end of a set of given number of time steps. "Wiggles", constituted the one major problem that needed to be overcome. These oscillations give rise to quantities such as negative temperatures, which ultimately caused the computational program to break down. Certain dissipative finite-difference schemes damped these oscillations.
Numerical Solution of the Flow of a Perfect Gas Over A Circular Cylinder at Infinite Mach Number
NASA Technical Reports Server (NTRS)
Hamaker, Frank M.
1959-01-01
A solution for the two-dimensional flow of an inviscid perfect gas over a circular cylinder at infinite Mach number is obtained by numerical methods of analysis. Nonisentropic conditions of curved shock waves and vorticity are included in the solution. The analysis is divided into two distinct regions, the subsonic region which is analyzed by the relaxation method of Southwell and the supersonic region which was treated by the method of characteristics. Both these methods of analysis are inapplicable on the sonic line which is therefore considered separately. The shapes of the sonic line and the shock wave are obtained by iteration techniques. The striking result of the solution is the strong curvature of the sonic line and of the other lines of constant Mach number. Because of this the influence of the supersonic flow on the sonic line is negligible. On comparison with Newtonian flow methods, it is found that the approximate methods show a larger variation of surface pressure than is given by the present solution.
Hayat, Tasawar; Ali, Shafqat; Farooq, Muhammad Asif; Alsaedi, Ahmad
2015-01-01
In this paper, we have investigated the combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface. The governing non-linear analysis of partial differential equations is reduced into the ordinary differential equations using similarity transformations. The resulting problems are computed for both series and numerical solutions. Series solution is constructed using homotopy analysis method (HAM) whereas numerical solution is presented by two different techniques namely shooting method and bvp4c. A comparison of homotopy solution with numerical solution is also tabulated. Both solutions are found in an excellent agreement. Dimensionless velocity and temperature profiles are plotted and discussed for various emerging physical parameters.
Hayat, Tasawar; Ali, Shafqat; Farooq, Muhammad Asif; Alsaedi, Ahmad
2015-01-01
In this paper, we have investigated the combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface. The governing non-linear analysis of partial differential equations is reduced into the ordinary differential equations using similarity transformations. The resulting problems are computed for both series and numerical solutions. Series solution is constructed using homotopy analysis method (HAM) whereas numerical solution is presented by two different techniques namely shooting method and bvp4c. A comparison of homotopy solution with numerical solution is also tabulated. Both solutions are found in an excellent agreement. Dimensionless velocity and temperature profiles are plotted and discussed for various emerging physical parameters. PMID:26402366
Numerical solution of the Navier-Stokes equations using orthogonal boundary-fitted coordinates
NASA Astrophysics Data System (ADS)
Bramley, J. S.; Sloan, D. M.
The Navier-Stokes equations for laminar viscous flow in a bifurcating channel are solved numerically, applying the boundary-fitted coordinate method of Ryskin and Leal (1983) to generate orthogonal grids with grid lines clustered in the boundary layers of the flow. The theoretical basis of the method is outlined, and results for Re = 50, 100, and 500 are presented in graphs and briefly characterized. It is shown that the present approach reduces the number of grid points required to calculate the separation and recirculation regions by half (as compared with the method of Bramley and Sloan, 1986), with significant savings in computation time. The need for an improved vorticity boundary condition to permit smaller grids at the solid boundaries is indicated.
Numerical solution of the Black-Scholes equation using cubic spline wavelets
NASA Astrophysics Data System (ADS)
Černá, Dana
2016-12-01
The Black-Scholes equation is used in financial mathematics for computation of market values of options at a given time. We use the θ-scheme for time discretization and an adaptive scheme based on wavelets for discretization on the given time level. Advantages of the proposed method are small number of degrees of freedom, high-order accuracy with respect to variables representing prices and relatively small number of iterations needed to resolve the problem with a desired accuracy. We use several cubic spline wavelet and multi-wavelet bases and discuss their advantages and disadvantages. We also compare an isotropic and anisotropic approach. Numerical experiments are presented for the two-dimensional Black-Scholes equation.
Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential
Sharipov, Felix Bertoldo, Guilherme
2009-05-20
A numerical procedure to solve the linearized Boltzmann equation with an arbitrary intermolecular potential by the discrete velocity method is elaborated. The equation is written in terms of the kernel, which contains the differential cross section and represents a singularity. As an example, the Lennard-Jones potential is used and the corresponding differential cross section is calculated and tabulated. Then, the kernel is calculated so that to overcome its singularity. Once, the kernel is known and stored it can be used for many kinds of gas flows. In order to test the method, the transport coefficients, i.e. thermal conductivity and viscosity for all noble gases, are calculated and compared with those obtained by the variational method using the Sonine polynomials expansion. The fine agreement between the results obtained by the two different methods shows the feasibility of application of the proposed technique to calculate rarefied gas flows over the whole range of the Knudsen number.
Light-opals interaction modeling by direct numerical solution of Maxwell's equations.
Vaccari, Alessandro; Lesina, Antonino Calà; Cristoforetti, Luca; Chiappini, Andrea; Crema, Luigi; Calliari, Lucia; Ramunno, Lora; Berini, Pierre; Ferrari, Maurizio
2014-11-03
This work describes a 3-D Finite-Difference Time-Domain (FDTD) computational approach for the optical characterization of an opal photonic crystal. To fully validate the approach we compare the computed transmittance of a crystal model with the transmittance of an actual crystal sample, as measured over the 400 ÷ 750 nm wavelength range. The opal photonic crystal considered has a face-centered cubic (FCC) lattice structure of spherical particles made of polystyrene (a non-absorptive material with constant relative dielectric permittivity). Light-matter interaction is described by numerically solving Maxwell's equations via a parallelized FDTD code. Periodic boundary conditions (PBCs) at the outer edges of the crystal are used to effectively enforce an infinite lateral extension of the sample. A method to study the propagating Bloch modes inside the crystal bulk is also proposed, which allows the reconstruction of the ω-k dispersion curve for k sweeping discretely the Brillouin zone of the crystal.
Fast numerical solution for fractional diffusion equations by exponential quadrature rule
NASA Astrophysics Data System (ADS)
Zhang, Lu; Sun, Hai-Wei; Pang, Hong-Kui
2015-10-01
After spatial discretization to the fractional diffusion equation by the shifted Grünwald formula, it leads to a system of ordinary differential equations, where the resulting coefficient matrix possesses the Toeplitz-like structure. An exponential quadrature rule is employed to solve such a system of ordinary differential equations. The convergence by the proposed method is theoretically studied. In practical computation, the product of a Toeplitz-like matrix exponential and a vector is calculated by the shift-invert Arnoldi method. Meanwhile, the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift-invert Arnoldi method. Numerical results are given to demonstrate the efficiency of the proposed method.
On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs
NASA Astrophysics Data System (ADS)
Bayona, Victor; Flyer, Natasha; Fornberg, Bengt; Barnett, Gregory A.
2017-03-01
RBF-generated finite differences (RBF-FD) have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of PDEs. We find in the present study that combining polyharmonic splines (PHS) with multivariate polynomials offers an outstanding combination of simplicity, accuracy, and geometric flexibility when solving elliptic equations in irregular (or regular) regions. In particular, the drawbacks on accuracy and stability due to Runge's phenomenon are overcome once the RBF stencils exceed a certain size due to an underlying minimization property. Test problems include the classical 2-D driven cavity, and also a 3-D global electric circuit problem with the earth's irregular topography as its bottom boundary. The results we find are fully consistent with previous results for data interpolation.
Numerical solutions for spin-up from rest in a cylinder
NASA Technical Reports Server (NTRS)
Hyun, J. M.; Leslie, F.; Fowlis, W. W.; Warn-Varnas, A.
1983-01-01
A set of three-dimensional flow-field data for the region around a cylinder impulsively spun-up from rest was derived with a numerical model based on the Navier-Stokes equations. Laser-Doppler anemometer data in the azimuthal direction was employed to test the model predictions, and data was developed for a flowfield with Ekman numbers from 9.18/1,000,000 to 9.18/10,000. The contributions of inviscid and viscous terms were determined as functions of radius and time. It was found that immediately after start-up viscous diffusion is the dominant factor, which is replaced by nonlinear radial advection. The Coriolis force dominates in the later stages of spin-up. The inward radial flow is a maximum near the front, where the vertical velocity is small, but features strong radial gradients, as it does at the edge of the Ekman layer.
Designing illumination lenses and mirrors by the numerical solution of Monge-Ampère equations.
Brix, Kolja; Hafizogullari, Yasemin; Platen, Andreas
2015-11-01
We consider the inverse refractor and the inverse reflector problem. The task is to design a free-form lens or a free-form mirror that, when illuminated by a point light source, produces a given illumination pattern on a target. Both problems can be modeled by strongly nonlinear second-order partial differential equations of Monge-Ampère type. In [Math. Models Methods Appl. Sci.25, 803 (2015)MMMSEU0218-202510.1142/S0218202515500190], the authors have proposed a B-spline collocation method, which has been applied to the inverse reflector problem. Now this approach is extended to the inverse refractor problem. We explain in depth the collocation method and how to handle boundary conditions and constraints. The paper concludes with numerical results of refracting and reflecting optical surfaces and their verification via ray tracing.
The numerical solution of ordinary differential equations by the Taylor series method
NASA Technical Reports Server (NTRS)
Silver, A. H.; Sullivan, E.
1973-01-01
A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.
NASA Astrophysics Data System (ADS)
Morgan, J. P.; de Monserrat, A.; Hall, R.; Taramon, J. M.; Perez-Gussinye, M.
2015-12-01
This work focuses on improving current 2D numerical approaches to modeling the boundary conditions associated with computing accurate deformation and melting associated with continental rifting. Recent models primarily use far-field boundary conditions that have been used for decades with little assessment of their effects on asthenospheric flow beneath the rifting region. All are clearly extremely oversimplified — Huismans and Buiter assume there is no vertical flow into the rifting region, with the asthenosphere flowing uniformly into the rifting region from the sides beneath lithosphere moving in the opposing direction, Armitage et al. and van Wijk use divergent velocities on the upper boundary to impose break-up within a Cartesian box, while other studies generally assume there is uniform horizontal flow away from the center of rifting, with uniform vertical flow replenishing the material pulled out of the sides of the computational region. All are likely to significantly shape the pattern of asthenospheric flow beneath the stretching lithosphere that is associated with pressure-release melting and rift volcanism. Thus while ALL may lead to similar predictions of the effects of crustal stretching and thinning, NONE may lead to accurate determination of the the asthenospheric flow and melting associated with lithospheric stretching and breakup. Here we discuss a suite of numerical experiments that compare these choices to likely more realistic boundary condition choices like the analytical solution for flow associated with two diverging plates stretching over a finite-width region, and a high-resolution 2-D region embedded within a cylindrical annulus 'whole mantle cross-section' at 5% extra numerical problem size. Our initial results imply that the choice of far-field boundary conditions does indeed significantly influence predicted melting distributions and melt volumes associated with continental breakup. For calculations including asthenospheric melting
NASA Astrophysics Data System (ADS)
Klein, C.; Peter, R.
2015-06-01
We present a detailed numerical study of solutions to general Korteweg-de Vries equations with critical and supercritical nonlinearity, both in the context of dispersive shocks and blow-up. We study the stability of solitons and show that they are unstable against being radiated away and blow-up. In the L2 critical case, the blow-up mechanism by Martel, Merle and Raphaël can be numerically identified. In the limit of small dispersion, it is shown that a dispersive shock always appears before an eventual blow-up. In the latter case, always the first soliton to appear will blow up. It is shown that the same type of blow-up as for the perturbations of the soliton can be observed which indicates that the theory by Martel, Merle and Raphaël is also applicable to initial data with a mass much larger than the soliton mass. We study the scaling of the blow-up time t∗ in dependence of the small dispersion parameter ɛ and find an exponential dependence t∗(ɛ) and that there is a minimal blow-up time t0∗ greater than the critical time of the corresponding Hopf solution for ɛ → 0. To study the cases with blow-up in detail, we apply the first dynamic rescaling for generalized Korteweg-de Vries equations. This allows to identify the type of the singularity.
NASA Astrophysics Data System (ADS)
Smirnov, E. M.; Smirnovsky, A. A.; Schur, N. A.; Zaitsev, D. K.; Smirnov, P. E.
2016-09-01
The contribution covers results of numerical study of air flow and heat transfer past a backward-facing step at the Reynolds number of 28,000. The numerical simulation was carried out under conditions of the experiments of Vogel&Eaton (1985), where nominally 2D fluid dynamics and heat transfer in a channel with expansion ratio of 1.25 was investigated. Two approaches were used for turbulence modelling. First, the Menter SST turbulence model was used to perform refined 2D and 3D RANS steady-state computations. The 3D analysis was undertaken to evaluate effects of boundary layers developing on the sidewalls of the experimental channel. Then, 3D time-dependent computations were carried out using the vortex-resolving IDDES method and applying the spanwise-periodicity conditions. Comparative computations were performed using an in-house finite-volume code SINF/Flag-S and the ANSYS Fluent. The codes produced practically identical RANS solutions, showing in particular a difference of 4% in the central-line peak Stanton number calculated in 2D and 3D cases. The IDDES results obtained with two codes are in a satisfactory agreement. Comparing with the experimental data, the IDDES produces the best agreement for the wall friction, whereas the RANS solutions show superiority in predictions of the local Stanton number distribution.
Numerical solution of the Dirac equation in Schwarzschild de Sitter spacetime
NASA Astrophysics Data System (ADS)
Lyu, Y.; Gui, Y. X.
2007-02-01
The radial parts of the Dirac equation between the inner and the outer horizon in Schwarzschild-de Sitter geometry are solved. Two limiting cases are concerned. The first case is when the two horizons are far apart and the second case is when the horizons are close to each other. In each case, a 'tangent' approximation is used to replace the modified 'tortoise' coordinate r*, which leads to a simple analytically invertible relation between r* and the radius r. The potential V(r*) is replaced by a collection of step functions in sequence. Then the solutions of the wave equation as well as the reflection and transmission coefficients are computed by a quantum mechanical method.
NASA Technical Reports Server (NTRS)
Collier, Richard S.; Mckenna, Paul M.; Perala, Rodney A.
1991-01-01
The objective here is to describe the lightning hazards to buildings and their internal environments using advanced formulations of Maxwell's Equations. The method described is the Three Dimensional Finite Difference Time Domain Solution. It can be used to solve for the lightning interaction with such structures in three dimensions with the inclusion of a considerable amount of detail. Special techniques were developed for including wire, plumbing, and rebar into the model. Some buildings have provisions for lightning protection in the form of air terminals connected to a ground counterpoise system. It is shown that fields and currents within these structures can be significantly high during a lightning strike. Time lapse video presentations were made showing the electric and magnetic field distributions on selected cross sections of the buildings during a simulated lightning strike.
NASA Technical Reports Server (NTRS)
Peltier, Leonard Joel; Biringen, Sedat; Chait, Arnon
1990-01-01
Implicit techniques for calculating three-dimensional, time-dependent heat diffusion in a cube are tested with emphasis on storage efficiency, accuracy, and speed of calculation. For this purpose, a tensor product technique with both Chebyshev collocation and finite differences and a generalized conjugate gradient technique with finite differences are used in conjunction with Crank-Nicolson discretization. An Euler explicit finite difference calculation is performed for use as a benchmark. The implicit techniques are found to be competitive with the Euler explicit method in terms of storage efficiency and speed of calculation and offer advantages both in accuracy and stability. Mesh stretching in the finite difference calculations is shown to markedly improve the accuracy of the solution.
Numerical solution for laminar film condensation of pure refrigerant on a vertical finned surface
Yu, Jian; Matsumoto, Tatsuya; Koyama, Shigeru
1999-07-01
Plate-fin heat exchangers are widely used in chemical plants for their high heat transfer performance, and have attracted special interests recently in heat pump and refrigeration systems. Many researchers have studied this kind of heat exchanger in single-phase region in detail, but most of these studies could not be extended to two-phase flow region. In the present study, a numerical analysis for the laminar film condensation on a finned vertical surface is carried out to clarify the heat transfer characteristics of plate-fin condensers. In the analysis the following assumptions are employed. (1) The bulk vapor is pure and saturated, and the effect of viscous shear of vapor on the liquid film is negligible. (2) The condensed liquid flows not only in vertical direction by gravitational force, but in horizontal direction by surface tension. (3) The heat conduction in the fin is one-dimensional, and the base surface temperature is a constant. (4) The effect of curvature of liquid film surface in z direction is not considered on the distribution of liquid film thickness and heat transfer characteristics. The governing equation of the liquid film thickness and one-dimensional heat conduction equation in the fin are numerically solved using the finite difference method. Three-dimensional distribution of the condensed liquid film thickness, the pressure and the radius of liquid film in horizontal direction, the distributions of average heat flux and the Nusselt number along the vertical direction are obtained. From a series of calculation results, the effects of fin pitch, fin height, fin thickness, fin length and the radius of concave joint region of base plate and fin on the liquid film shape are shown, and the effects of the fin shape parameters on heat transfer enhancement ratio are examined. The average Nusselt number on a vertical finned surface, Nu{sub m}, are correlated by the Bond number Bo, the Galileo number Ga{sub L}, the phase change number Ph, and the