Accurate numerical solutions of conservative nonlinear oscillators
NASA Astrophysics Data System (ADS)
Khan, Najeeb Alam; Nasir Uddin, Khan; Nadeem Alam, Khan
2014-12-01
The objective of this paper is to present an investigation to analyze the vibration of a conservative nonlinear oscillator in the form u" + lambda u + u^(2n-1) + (1 + epsilon^2 u^(4m))^(1/2) = 0 for any arbitrary power of n and m. This method converts the differential equation to sets of algebraic equations and solve numerically. We have presented for three different cases: a higher order Duffing equation, an equation with irrational restoring force and a plasma physics equation. It is also found that the method is valid for any arbitrary order of n and m. Comparisons have been made with the results found in the literature the method gives accurate results.
Accurate numerical solution of compressible, linear stability equations
NASA Technical Reports Server (NTRS)
Malik, M. R.; Chuang, S.; Hussaini, M. Y.
1982-01-01
The present investigation is concerned with a fourth order accurate finite difference method and its application to the study of the temporal and spatial stability of the three-dimensional compressible boundary layer flow on a swept wing. This method belongs to the class of compact two-point difference schemes discussed by White (1974) and Keller (1974). The method was apparently first used for solving the two-dimensional boundary layer equations. Attention is given to the governing equations, the solution technique, and the search for eigenvalues. A general purpose subroutine is employed for solving a block tridiagonal system of equations. The computer time can be reduced significantly by exploiting the special structure of two matrices.
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.
NASA Astrophysics Data System (ADS)
Jiang, Shidong; Luo, Li-Shi
2016-07-01
The integral equation for the flow velocity u (x ; k) in the steady Couette flow derived from the linearized Bhatnagar-Gross-Krook-Welander kinetic equation is studied in detail both theoretically and numerically in a wide range of the Knudsen number k between 0.003 and 100.0. First, it is shown that the integral equation is a Fredholm equation of the second kind in which the norm of the compact integral operator is less than 1 on Lp for any 1 ≤ p ≤ ∞ and thus there exists a unique solution to the integral equation via the Neumann series. Second, it is shown that the solution is logarithmically singular at the endpoints. More precisely, if x = 0 is an endpoint, then the solution can be expanded as a double power series of the form ∑n=0∞∑m=0∞cn,mxn(xln x) m about x = 0 on a small interval x ∈ (0 , a) for some a > 0. And third, a high-order adaptive numerical algorithm is designed to compute the solution numerically to high precision. The solutions for the flow velocity u (x ; k), the stress Pxy (k), and the half-channel mass flow rate Q (k) are obtained in a wide range of the Knudsen number 0.003 ≤ k ≤ 100.0; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.
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].
Predict amine solution properties accurately
Cheng, S.; Meisen, A.; Chakma, A.
1996-02-01
Improved process design begins with using accurate physical property data. Especially in the preliminary design stage, physical property data such as density viscosity, thermal conductivity and specific heat can affect the overall performance of absorbers, heat exchangers, reboilers and pump. These properties can also influence temperature profiles in heat transfer equipment and thus control or affect the rate of amine breakdown. Aqueous-amine solution physical property data are available in graphical form. However, it is not convenient to use with computer-based calculations. Developed equations allow improved correlations of derived physical property estimates with published data. Expressions are given which can be used to estimate physical properties of methyldiethanolamine (MDEA), monoethanolamine (MEA) and diglycolamine (DGA) solutions.
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.
Numerical Asymptotic Solutions Of Differential Equations
NASA Technical Reports Server (NTRS)
Thurston, Gaylen A.
1992-01-01
Numerical algorithms derived and compared with classical analytical methods. In method, expansions replaced with integrals evaluated numerically. Resulting numerical solutions retain linear independence, main advantage of asymptotic solutions.
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.
Accurate deterministic solutions for the classic Boltzmann shock profile
NASA Astrophysics Data System (ADS)
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
Accurate complex scaling of three dimensional numerical potentials
Cerioni, Alessandro; Genovese, Luigi; Duchemin, Ivan; Deutsch, Thierry
2013-05-28
The complex scaling method, which consists in continuing spatial coordinates into the complex plane, is a well-established method that allows to compute resonant eigenfunctions of the time-independent Schroedinger operator. Whenever it is desirable to apply the complex scaling to investigate resonances in physical systems defined on numerical discrete grids, the most direct approach relies on the application of a similarity transformation to the original, unscaled Hamiltonian. We show that such an approach can be conveniently implemented in the Daubechies wavelet basis set, featuring a very promising level of generality, high accuracy, and no need for artificial convergence parameters. Complex scaling of three dimensional numerical potentials can be efficiently and accurately performed. By carrying out an illustrative resonant state computation in the case of a one-dimensional model potential, we then show that our wavelet-based approach may disclose new exciting opportunities in the field of computational non-Hermitian quantum mechanics.
Numerical solution of under-resolved detonations
NASA Astrophysics Data System (ADS)
Tosatto, Luca; Vigevano, Luigi
2008-02-01
A new fractional-step method is proposed for the numerical solution of high speed reacting flows, where the chemical time scales are often much smaller than the fluid dynamical time scales. When the problem is stiff, because of insufficient spatial/temporal resolution, a well-known spurious numerical phenomenon occurs in standard finite volume schemes: the incorrect calculation of the speed of propagation of discontinuities. The new method is first illustrated considering a one-dimensional scalar hyperbolic advection/reaction equation with stiff source term, which may be considered as a model problem to under-resolved detonations. During the reaction step, the proposed scheme replaces the cell average representation with a two-value reconstruction, which allows us to locate the discontinuity position inside the cell during the computation of the source term. This results in the correct propagation of discontinuities even in the stiff case. The method is proved to be second-order accurate for smooth solutions of scalar equations and is applied successfully to the solution of the one-dimensional reactive Euler equations for Chapman-Jouguet detonations.
Spurious Numerical Solutions Of Differential Equations
NASA Technical Reports Server (NTRS)
Lafon, A.; Yee, H. C.
1995-01-01
Paper presents detailed study of spurious steady-state numerical solutions of differential equations that contain nonlinear source terms. Main objectives of this study are (1) to investigate how well numerical steady-state solutions of model nonlinear reaction/convection boundary-value problem mimic true steady-state solutions and (2) to relate findings of this investigation to implications for interpretation of numerical results from computational-fluid-dynamics algorithms and computer codes used to simulate reacting flows.
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.
Vibration of clamped right triangular thin plates: Accurate simplified solutions
NASA Astrophysics Data System (ADS)
Saliba, H. T.
1994-12-01
Use of the superposition techniques in the free-vibration analyses of thin plates, as they were first introduced by Gorman, has provided simple and effective solutions to a vast number of rectangular plate problems. A modified superposition method is presented that is a noticeable improvement over existing techniques. It deals only with simple support conditions, leading to a simple, highly accurate, and very economical solution to the free-vibration problem of simply-supported right angle triangular plates. The modified method is also applicable to clamped-edge conditions.
A hybrid numerical scheme for the numerical solution of the Burgers' equation
NASA Astrophysics Data System (ADS)
Jiwari, Ram
2015-03-01
In this article, a hybrid numerical scheme based on Euler implicit method, quasilinearization and uniform Haar wavelets has been developed for the numerical solutions of Burgers' equation. Most of the numerical methods available in the literature fail to capture the physical behavior of the equations when viscosity ν → 0. In Jiwari (2012), the author presented the numerical results up to ν = 0.003 and the scheme failed for values smaller than ν = 0.003. The main aim in the development of the present scheme is to overcome the drawback of the scheme developed in Jiwari (2012). Lastly, three test problems are chosen to check the accuracy of the proposed scheme. The approximated results are compared with existing numerical and exact solutions found in literature. The use of uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and at small computation costs.
Vibration of clamped right triangular thin plates: Accurate simplified solutions
NASA Astrophysics Data System (ADS)
Saliba, H. T.
1994-12-01
Use of the superposition techniques in the free-vibration analyses of thin plates, as they were first introduced by Gorman, has provided simple and effective solutions to a vast number of rectangular plate problems. The method has also been extended to nonrectangular plates such as triangular and trapezoidal plates. However, serious difficulties were encountered in some of these analyses. These difficulties were discussed and obviated in Salibra, 1990. This reference, however, dealt only with simple support conditions, leading to a simple, highly accurate, and very economical solution to the free-vibration problem of simply supported right angle triangular plates. The purpose of this Note is to show that the modified superposition method of Salibra, 1990 is also applicable to clamped-edge conditions. This is accomplished through the application of this method to the title problem.
Fast and Accurate Learning When Making Discrete Numerical Estimates.
Sanborn, Adam N; Beierholm, Ulrik R
2016-04-01
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
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
Accurate ω-ψ Spectral Solution of the Singular Driven Cavity Problem
NASA Astrophysics Data System (ADS)
Auteri, F.; Quartapelle, L.; Vigevano, L.
2002-08-01
This article provides accurate spectral solutions of the driven cavity problem, calculated in the vorticity-stream function representation without smoothing the corner singularities—a prima facie impossible task. As in a recent benchmark spectral calculation by primitive variables of Botella and Peyret, closed-form contributions of the singular solution for both zero and finite Reynolds numbers are subtracted from the unknown of the problem tackled here numerically in biharmonic form. The method employed is based on a split approach to the vorticity and stream function equations, a Galerkin-Legendre approximation of the problem for the perturbation, and an evaluation of the nonlinear terms by Gauss-Legendre numerical integration. Results computed for Re=0, 100, and 1000 compare well with the benchmark steady solutions provided by the aforementioned collocation-Chebyshev projection method. The validity of the proposed singularity subtraction scheme for computing time-dependent solutions is also established.
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.
The development of accurate and efficient methods of numerical quadrature
NASA Technical Reports Server (NTRS)
Feagin, T.
1973-01-01
Some new methods for performing numerical quadrature of an integrable function over a finite interval are described. Each method provides a sequence of approximations of increasing order to the value of the integral. Each approximation makes use of all previously computed values of the integrand. The points at which new values of the integrand are computed are selected in such a way that the order of the approximation is maximized. The methods are compared with the quadrature methods of Clenshaw and Curtis, Gauss, Patterson, and Romberg using several examples.
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.
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.
Numerical methods for finding stationary gravitational solutions
NASA Astrophysics Data System (ADS)
Dias, Óscar J. C.; Santos, Jorge E.; Way, Benson
2016-07-01
The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations and give a practical guide for the numerical solution of gravitational boundary value problems. We present these methods by way of example: resolving asymptotically flat black rings, singly spinning lumpy black holes in anti-de Sitter (AdS), and the Gregory–Laflamme zero modes of small rotating black holes in AdS{}5× {S}5. We also include several tools and tricks that have been useful throughout the literature.
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.
Numerical solution of large Lyapunov equations
NASA Technical Reports Server (NTRS)
Saad, Youcef
1989-01-01
A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.
Numerical analysis of the asymptotic two-point boundary value solution for N-body trajectories.
NASA Technical Reports Server (NTRS)
Lancaster, J. E.; Allemann, R. A.
1972-01-01
Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical boundary value solution applicable to a broad class of trajectory problems. In addition, the earlier first-order solutions have been extended to second-order to determine if improved accuracy is possible. Comparisons between the asymptotic solution and numerical integration for several lunar and interplanetary trajectories show that the asymptotic solution is generally quite accurate. Also, since no iterations are required, a solution to the boundary value problem is obtained in a fraction of the time required for numerically integrated solutions.
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.
Multigrid techniques for the numerical solution of the diffusion equation
NASA Technical Reports Server (NTRS)
Phillips, R. E.; Schmidt, F. W.
1984-01-01
An accurate numerical solution of diffusion problems containing large local gradients can be obtained with a significant reduction in computational time by using a multigrid computational scheme. The spatial domain is covered with sets of uniform square grids of different sizes. The finer grid patterns overlap the coarse grid patterns. The finite-difference expressions for each grid pattern are solved independently by iterative techniques. Two interpolation methods were used to establish the values of the potential function on the fine grid boundaries with information obtained from the coarse grid solution. The accuracy and computational requirements for solving a test problem by a simple multigrid and a multilevel-multigrid method were compared. The multilevel-multigrid method combined with a Taylor series interpolation scheme was found to be best.
Accurate solution of the Dirac equation on Lagrange meshes.
Baye, Daniel; Filippin, Livio; Godefroid, Michel
2014-04-01
The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a 1/r singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers n+1 of mesh points, where n is the principal quantum number. Numerically exact mean values of powers -2 to 3 of the radial coordinate r can also be obtained with n+2 mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 or fewer mesh points. PMID:24827362
Accurate solution of the Dirac equation on Lagrange meshes
NASA Astrophysics Data System (ADS)
Baye, Daniel; Filippin, Livio; Godefroid, Michel
2014-04-01
The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a 1/r singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers n +1 of mesh points, where n is the principal quantum number. Numerically exact mean values of powers -2 to 3 of the radial coordinate r can also be obtained with n +2 mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 or fewer mesh points.
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.
Meek, Garrett A; Levine, Benjamin G
2014-07-01
Spikes in the time-derivative coupling (TDC) near surface crossings make the accurate integration of the time-dependent Schrödinger equation in nonadiabatic molecular dynamics simulations a challenge. To address this issue, we present an approximation to the TDC based on a norm-preserving interpolation (NPI) of the adiabatic electronic wave functions within each time step. We apply NPI and two other schemes for computing the TDC in numerical simulations of the Landau-Zener model, comparing the simulated transfer probabilities to the exact solution. Though NPI does not require the analytical calculation of nonadiabatic coupling matrix elements, it consistently yields unsigned population transfer probability errors of ∼0.001, whereas analytical calculation of the TDC yields errors of 0.0-1.0 depending on the time step, the offset of the maximum in the TDC from the beginning of the time step, and the coupling strength. The approximation of Hammes-Schiffer and Tully yields errors intermediate between NPI and the analytical scheme. PMID:26279558
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. PMID:26034665
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.
The Numeric Solution of Eigenvalue Problems.
ERIC Educational Resources Information Center
Bauer, H.; Roth, K.
1980-01-01
Presents the mathematical background for solving eigenvalue problems, with illustrations of the applications in computer programing. The numerical matrix treatment is presented, with a demonstration of the simple HMO theory. (CS)
NASA Astrophysics Data System (ADS)
Kafri, H. Q.; Khuri, S. A.; Sayfy, A.
2016-03-01
In this paper, a novel approach is introduced for the solution of the non-linear Troesch's boundary value problem. The underlying strategy is based on Green's functions and fixed-point iterations, including Picard's and Krasnoselskii-Mann's schemes. The resulting numerical solutions are compared with both the analytical solutions and numerical solutions that exist in the literature. Convergence of the iterative schemes is proved via manipulation of the contraction principle. It is observed that the method handles the boundary layer very efficiently, reduces lengthy calculations, provides rapid convergence, and yields accurate results particularly for large eigenvalues. Indeed, to our knowledge, this is the first time that this problem is solved successfully for very large eigenvalues, actually the rate of convergence increases as the magnitude of the eigenvalues increases.
Accurate Critical Stress Intensity Factor Griffith Crack Theory Measurements by Numerical Techniques
Petersen, Richard C.
2014-01-01
Critical stress intensity factor (KIc) has been an approximation for fracture toughness using only load-cell measurements. However, artificial man-made cracks several orders of magnitude longer and wider than natural flaws have required a correction factor term (Y) that can be up to about 3 times the recorded experimental value [1-3]. In fact, over 30 years ago a National Academy of Sciences advisory board stated that empirical KIc testing was of serious concern and further requested that an accurate bulk fracture toughness method be found [4]. Now that fracture toughness can be calculated accurately by numerical integration from the load/deflection curve as resilience, work of fracture (WOF) and strain energy release (SIc) [5, 6], KIc appears to be unnecessary. However, the large body of previous KIc experimental test results found in the literature offer the opportunity for continued meta analysis with other more practical and accurate fracture toughness results using energy methods and numerical integration. Therefore, KIc is derived from the classical Griffith Crack Theory [6] to include SIc as a more accurate term for strain energy release rate (𝒢Ic), along with crack surface energy (γ), crack length (a), modulus (E), applied stress (σ), Y, crack-tip plastic zone defect region (rp) and yield strength (σys) that can all be determined from load and deflection data. Polymer matrix discontinuous quartz fiber-reinforced composites to accentuate toughness differences were prepared for flexural mechanical testing comprising of 3 mm fibers at different volume percentages from 0-54.0 vol% and at 28.2 vol% with different fiber lengths from 0.0-6.0 mm. Results provided a new correction factor and regression analyses between several numerical integration fracture toughness test methods to support KIc results. Further, bulk KIc accurate experimental values are compared with empirical test results found in literature. Also, several fracture toughness mechanisms
Numerical error in groundwater flow and solute transport simulation
NASA Astrophysics Data System (ADS)
Woods, Juliette A.; Teubner, Michael D.; Simmons, Craig T.; Narayan, Kumar A.
2003-06-01
Models of groundwater flow and solute transport may be affected by numerical error, leading to quantitative and qualitative changes in behavior. In this paper we compare and combine three methods of assessing the extent of numerical error: grid refinement, mathematical analysis, and benchmark test problems. In particular, we assess the popular solute transport code SUTRA [Voss, 1984] as being a typical finite element code. Our numerical analysis suggests that SUTRA incorporates a numerical dispersion error and that its mass-lumped numerical scheme increases the numerical error. This is confirmed using a Gaussian test problem. A modified SUTRA code, in which the numerical dispersion is calculated and subtracted, produces better results. The much more challenging Elder problem [Elder, 1967; Voss and Souza, 1987] is then considered. Calculation of its numerical dispersion coefficients and numerical stability show that the Elder problem is prone to error. We confirm that Elder problem results are extremely sensitive to the simulation method used.
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
Explicit numerical solutions of a microbial survival model under nonisothermal conditions.
Zhu, Si; Chen, Guibing
2016-03-01
Differential equations used to describe the original and modified Geeraerd models were, respectively, simplified into an explicit equation in which the integration of the specific inactivation rate with respect to time was numerically approximated using the Simpson's rule. The explicit numerical solutions were then used to simulate microbial survival curves and fit nonisothermal survival data for identifying model parameters in Microsoft Excel. The results showed that the explicit numerical solutions provided an easy way to accurately simulate microbial survival and estimate model parameters from nonisothermal survival data using the Geeraerd models. PMID:27004117
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.
NASA Astrophysics Data System (ADS)
Blackman, Jonathan; Field, Scott E.; Galley, Chad R.; Szilágyi, Béla; Scheel, Mark A.; Tiglio, Manuel; Hemberger, Daniel A.
2015-09-01
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 -2Yℓm waveform modes resolved by the NR code up to ℓ=8 . We compare our surrogate model to effective one body waveforms from 50 M⊙ to 300 M⊙ for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases).
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.
Milne, a routine for the numerical solution of Milne's problem
NASA Astrophysics Data System (ADS)
Rawat, Ajay; Mohankumar, N.
2010-11-01
The routine Milne provides accurate numerical values for the classical Milne's problem of neutron transport for the planar one speed and isotropic scattering case. The solution is based on the Case eigen-function formalism. The relevant X functions are evaluated accurately by the Double Exponential quadrature. The calculated quantities are the extrapolation distance and the scalar and the angular fluxes. Also, the H function needed in astrophysical calculations is evaluated as a byproduct. Program summaryProgram title: Milne Catalogue identifier: AEGS_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGS_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 701 No. of bytes in distributed program, including test data, etc.: 6845 Distribution format: tar.gz Programming language: Fortran 77 Computer: PC under Linux or Windows Operating system: Ubuntu 8.04 (Kernel version 2.6.24-16-generic), Windows-XP Classification: 4.11, 21.1, 21.2 Nature of problem: The X functions are integral expressions. The convergence of these regular and Cauchy Principal Value integrals are impaired by the singularities of the integrand in the complex plane. The DE quadrature scheme tackles these singularities in a robust manner compared to the standard Gauss quadrature. Running time: The test included in the distribution takes a few seconds to run.
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.
Numerical study of the Kerr solution in rotating coordinates
NASA Astrophysics Data System (ADS)
Bai, S.; Izquierdo, G.; Klein, C.
2016-06-01
The Kerr solution in coordinates corotating with the horizon is studied as a testbed for a spacetime with a helical Killing vector in the Ernst picture. The solution is numerically constructed by solving the Ernst equation with a spectral method and a Newton iteration. We discuss convergence of the iteration for several initial iterates and different values of the Kerr parameters.
A third-order-accurate upwind scheme for Navier-Stokes solutions at high Reynolds numbers
NASA Astrophysics Data System (ADS)
Agarwal, R. K.
1981-01-01
A third-order-accurate upwind scheme is presented for solution of the steady two-dimensional Navier-Stokes equations in stream-function/vorticity form. The scheme is found to be accurate and stable at high Reynolds numbers. A series of test computations is performed on flows with large recirculating regions. In particular, highly accurate solutions are obtained for flow in a driven square cavity up to Reynolds numbers of 10,000. These computations are used to critically evaluate the accuracy of other existing first- and second-order-accurate upwind schemes. In addition, computations are carried out for flow in a channel with symmetric sudden expansion, flow in a channel with a symmetrically placed blunt base, and the flowfield of an impinging jet. Good agreement is obtained with the computations of other investigators as well as with the available experimental data.
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. PMID:7699200
Numerical Solution of Natural Convection in Eccentric Annuli
Pepper, D.W.
2001-09-18
The governing equations for transient natural convection in eccentric annular space are solved with two high-order accurate numerical algorithms. The equation set is transformed into bipolar coordinates and split into two one-dimensional equations: finite elements are used in the direction normal to the cylinder surfaces; the pseudospectral technique is used in the azimuthal direction. This report discusses those equations.
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. PMID:17510203
Recommendations for accurate numerical blood flow simulations of stented intracranial aneurysms.
Janiga, Gábor; Berg, Philipp; Beuing, Oliver; Neugebauer, Mathias; Gasteiger, Rocco; Preim, Bernhard; Rose, Georg; Skalej, Martin; Thévenin, Dominique
2013-06-01
The number of scientific publications dealing with stented intracranial aneurysms is rapidly increasing. Powerful computational facilities are now available; an accurate computational modeling of hemodynamics in patient-specific configurations is, however, still being sought. Furthermore, there is still no general agreement on the quantities that should be computed and on the most adequate analysis for intervention support. In this article, the accurate representation of patient geometry is first discussed, involving successive improvements. Concerning the second step, the mesh required for the numerical simulation is especially challenging when deploying a stent with very fine wire structures. Third, the description of the fluid properties is a major challenge. Finally, a founded quantitative analysis of the simulation results is obviously needed to support interventional decisions. In the present work, an attempt has been made to review the most important steps for a high-quality computational fluid dynamics computation of virtually stented intracranial aneurysms. In consequence, this leads to concrete recommendations, whereby the obtained results are not discussed for their medical relevance but for the evaluation of their quality. This investigation might hopefully be helpful for further studies considering stent deployment in patient-specific geometries, in particular regarding the generation of the most appropriate computational model. PMID:23729530
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). PMID:26430979
Earthquake Rupture Dynamics using Adaptive Mesh Refinement and High-Order Accurate Numerical Methods
NASA Astrophysics Data System (ADS)
Kozdon, J. E.; Wilcox, L.
2013-12-01
Our goal is to develop scalable and adaptive (spatial and temporal) numerical methods for coupled, multiphysics problems using high-order accurate numerical methods. To do so, we are developing an opensource, parallel library known as bfam (available at http://bfam.in). The first application to be developed on top of bfam is an earthquake rupture dynamics solver using high-order discontinuous Galerkin methods and summation-by-parts finite difference methods. In earthquake rupture dynamics, wave propagation in the Earth's crust is coupled to frictional sliding on fault interfaces. This coupling is two-way, required the simultaneous simulation of both processes. The use of laboratory-measured friction parameters requires near-fault resolution that is 4-5 orders of magnitude higher than that needed to resolve the frequencies of interest in the volume. This, along with earlier simulations using a low-order, finite volume based adaptive mesh refinement framework, suggest that adaptive mesh refinement is ideally suited for this problem. The use of high-order methods is motivated by the high level of resolution required off the fault in earlier the low-order finite volume simulations; we believe this need for resolution is a result of the excessive numerical dissipation of low-order methods. In bfam spatial adaptivity is handled using the p4est library and temporal adaptivity will be accomplished through local time stepping. In this presentation we will present the guiding principles behind the library as well as verification of code against the Southern California Earthquake Center dynamic rupture code validation test problems.
A numerical model of acoustic choking. II - Shocked solutions
NASA Astrophysics Data System (ADS)
Walkington, N. J.; Eversman, W.
1986-01-01
The one dimensional equations of gas dynamics are used to model subsonic acoustic choking. This model can accommodate non-linear distortion of waves and the eventual formation of shock waves. Several finite differencing schemes are adapted to obtain solutions. The results obtained with the various schemes are compared with the asymptotic results available. The results suggest that no one finite differencing scheme gives solutions significantly better than the others and that most of the difference solutions are close to the asymptotic results. If the acoustic shock wave is sufficiently strong it almost annihilates the acoustic wave; in this situation numerical errors may dominate the results. Such solutions involve very large acoustic attenuations.
Numerical solutions of telegraph equations with the Dirichlet boundary condition
NASA Astrophysics Data System (ADS)
Ashyralyev, Allaberen; Turkcan, Kadriye Tuba; Koksal, Mehmet Emir
2016-08-01
In this study, the Cauchy problem for telegraph equations in a Hilbert space is considered. Stability estimates for the solution of this problem are presented. The third order of accuracy difference scheme is constructed for approximate solutions of the problem. Stability estimates for the solution of this difference scheme are established. As a test problem to support theoretical results, one-dimensional telegraph equation with the Dirichlet boundary condition is considered. Numerical solutions of this equation are obtained by first, second and third order of accuracy difference schemes.
Some recent advances in the numerical solution of differential equations
NASA Astrophysics Data System (ADS)
D'Ambrosio, Raffaele
2016-06-01
The purpose of the talk is the presentation of some recent advances in the numerical solution of differential equations, with special emphasis to reaction-diffusion problems, Hamiltonian problems and ordinary differential equations with discontinuous right-hand side. As a special case, in this short paper we focus on the solution of reaction-diffusion problems by means of special purpose numerical methods particularly adapted to the problem: indeed, following a problem oriented approach, we propose a modified method of lines based on the employ of finite differences shaped on the qualitative behavior of the solutions. Constructive issues and a brief analysis are presented, together with some numerical experiments showing the effectiveness of the approach and a comparison with existing solvers.
Numerical solution of a microbial growth model applied to dynamic environments.
Zhu, Si; Chen, Guibing
2015-05-01
The Baranyi and Roberts model is one of the most frequently used microbial growth models. It has been successfully applied to numerous studies of various microorganisms in different food products. Under dynamic conditions, the model is implicitly formulated as a set of two coupled differential equations which could be numerically solved using the Runge-Kutta method. In this study, a simplified numerical solution of the coupled differential equations was derived and used to simulate microbial growth under dynamic conditions in Microsoft Excel. As expected, the results obtained were the same as those from solving the coupled differential equations using a MATLAB Solver. In addition, model parameters were accurately identified by fitting the numerical solution to simulated growth curves under dynamic (time-varying) temperature conditions using the Microsoft Excel Solver. PMID:25765150
An Accurate Upper Bound Solution for Plane Strain Extrusion through a Wedge-Shaped Die
Mustafa, Yusof; Lyamina, Elena
2014-01-01
An upper bound method for the process of plane strain extrusion through a wedge-shaped die is derived. A technique for constructing a kinematically admissible velocity field satisfying the exact asymptotic singular behavior of real velocity fields in the vicinity of maximum friction surfaces (the friction stress at sliding is equal to the shear yield stress on such surfaces) is described. Two specific upper bound solutions are found using the method derived. The solutions are compared to an accurate slip-line solution and it is shown that the accuracy of the new method is very high. PMID:25101311
Implicit numerical integration for periodic solutions of autonomous nonlinear systems
NASA Technical Reports Server (NTRS)
Thurston, G. A.
1982-01-01
A change of variables that stabilizes numerical computations for periodic solutions of autonomous systems is derived. Computation of the period is decoupled from the rest of the problem for conservative systems of any order and for any second-order system. Numerical results are included for a second-order conservative system under a suddenly applied constant load. Near the critical load for the system, a small increment in load amplitude results in a large increase in amplitude of the response.
Numerical solutions for heat flow in adhesive lap joints
NASA Technical Reports Server (NTRS)
Howell, P. A.; Winfree, William P.
1992-01-01
The present formulation for the modeling of heat transfer in thin, adhesively bonded lap joints precludes difficulties associated with large aspect ratio grids required by standard FEM formulations. This quasi-static formulation also reduces the problem dimensionality (by one), thereby minimizing computational requirements. The solutions obtained are found to be in good agreement with both analytical solutions and solutions from standard FEM programs. The approach is noted to yield a more accurate representation of heat-flux changes between layers due to a disbond.
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.
NASA Astrophysics Data System (ADS)
Saliba, H. T.
1995-06-01
The practical engineering problem of right angled triangular plates with combinations of clamped and simply supported boundary conditions is dealt with in this paper. A highly accurate, economical and practical solution is used for the transverse free vibration analysis of these plates. The solution is based on a modified superposition method. The accuracy of the solution is discussed. Numerical results are compared with previously published reliable data. The advantages of the solution used in the paper over previously published solutions are discussed. Eigenvalues, mode shapes and contour plots are provided for a large number of plates.
Numerical Simulation of the 2004 Indian Ocean Tsunami: Accurate Flooding and drying in Banda Aceh
NASA Astrophysics Data System (ADS)
Cui, Haiyang; Pietrzak, Julie; Stelling, Guus; Androsov, Alexey; Harig, Sven
2010-05-01
The Indian Ocean Tsunami on December 26, 2004 caused one of the largest tsunamis in recent times and led to widespread devastation and loss of life. One of the worst hit regions was Banda Aceh, which is the capital of the Aceh province, located in the northern part of Sumatra, 150km from the source of the earthquake. A German-Indonesian Tsunami Early Warning System (GITEWS) (www.gitews.de) is currently under active development. The work presented here is carried out within the GITEWS framework. One of the aims of this project is the development of accurate models with which to simulate the propagation, flooding and drying, and run-up of a tsunami. In this context, TsunAWI has been developed by the Alfred Wegener Institute; it is an explicit, () finite element model. However, the accurate numerical simulation of flooding and drying requires the conservation of mass and momentum. This is not possible in the current version of TsunAWi. The P1NC - P1element guarantees mass conservation in a global sense, yet as we show here it is important to guarantee mass conservation at the local level, that is within each individual cell. Here an unstructured grid, finite volume ocean model is presented. It is derived from the P1NC - P1 element, and is shown to be mass and momentum conserving. Then a number of simulations are presented, including dam break problems flooding over both a wet and a dry bed. Excellent agreement is found. Then we present simulations for Banda Aceh, and compare the results to on-site survey data, as well as to results from the original TsunAWI code.
Stochasticity in numerical solutions of the nonlinear Schroedinger equation
NASA Technical Reports Server (NTRS)
Shen, Mei-Mei; Nicholson, D. R.
1987-01-01
The cubically nonlinear Schroedinger equation is an important model of nonlinear phenomena in fluids and plasmas. Numerical solutions in a spatially periodic system commonly involve truncation to a finite number of Fourier modes. These solutions are found to be stochastic in the sense that the largest Liapunov exponent is positive. As the number of modes is increased, the size of this exponent appears to converge to zero, in agreement with the recent demonstration of the integrability of the spatially periodic case.
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.
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.
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.
Numerical solution-space analysis of satisfiability problems.
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 α(c)≈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 α(s)≈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. PMID:21230614
Numerical solution of boundary-integral equations for molecular electrostatics.
Bardhan, Jaydeep P
2009-03-01
Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived. PMID:19275391
Accurate solution of the proton-hydrogen three-body scattering problem
NASA Astrophysics Data System (ADS)
Abdurakhmanov, I. B.; Kadyrov, A. S.; Bray, I.
2016-02-01
An accurate solution to the fundamental three-body problem of proton-hydrogen scattering including direct scattering and ionization, electron capture and electron capture into the continuum (ECC) is presented. The problem has been addressed using a quantum-mechanical two-center convergent close-coupling approach. At each energy the internal consistency of the solution is demonstrated with the help of single-center calculations, with both approaches converging independently to the same electron-loss cross section. This is the sum of the electron capture, ECC and direct ionization cross sections, which are only obtainable separately in the solution of the problem using the two-center expansion. Agreement with experiment for the electron-capture cross section is excellent. However, for the ionization cross sections some discrepancy exists. Given the demonstrated internal consistency we remain confident in the provided theoretical solution.
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.
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.
TOPICA: an accurate and efficient numerical tool for analysis and design of ICRF antennas
NASA Astrophysics Data System (ADS)
Lancellotti, V.; Milanesio, D.; Maggiora, R.; Vecchi, G.; Kyrytsya, V.
2006-07-01
The demand for a predictive tool to help in designing ion-cyclotron radio frequency (ICRF) antenna systems for today's fusion experiments has driven the development of codes such as ICANT, RANT3D, and the early development of TOPICA (TOrino Polytechnic Ion Cyclotron Antenna) code. This paper describes the substantive evolution of TOPICA formulation and implementation that presently allow it to handle the actual geometry of ICRF antennas (with curved, solid straps, a general-shape housing, Faraday screen, etc) as well as an accurate plasma description, accounting for density and temperature profiles and finite Larmor radius effects. The antenna is assumed to be housed in a recess-like enclosure. Both goals have been attained by formally separating the problem into two parts: the vacuum region around the antenna and the plasma region inside the toroidal chamber. Field continuity and boundary conditions allow formulating of a set of two coupled integral equations for the unknown equivalent (current) sources; then the equations are reduced to a linear system by a method of moments solution scheme employing 2D finite elements defined over a 3D non-planar surface triangular-cell mesh. In the vacuum region calculations are done in the spatial (configuration) domain, whereas in the plasma region a spectral (wavenumber) representation of fields and currents is adopted, thus permitting a description of the plasma by a surface impedance matrix. Owing to this approach, any plasma model can be used in principle, and at present the FELICE code has been employed. The natural outcomes of TOPICA are the induced currents on the conductors (antenna, housing, etc) and the electric field in front of the plasma, whence the antenna circuit parameters (impedance/scattering matrices), the radiated power and the fields (at locations other than the chamber aperture) are then obtained. An accurate model of the feeding coaxial lines is also included. The theoretical model and its TOPICA
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.
Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics
Hou, Thomas Y. . E-mail: hou@ama.caltech.edu; Luo Wuan; Rozovskii, Boris; Zhou Haomin
2006-08-10
In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier-Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner. The separation principle effectively reduces a stochastic equation to its associated propagator, a system of deterministic equations for the coefficients of the Wiener Chaos expansion. Simple formulas for statistical moments of the stochastic solution are presented. These formulas only involve the solutions of the propagator. We demonstrate that for short time solutions the numerical methods based on the Wiener Chaos expansion are more efficient and accurate than those based on the Monte Carlo simulations.
Numerical time-dependent solutions of the Schrödinger equation with piecewise continuous potentials
NASA Astrophysics Data System (ADS)
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.
NASA Astrophysics Data System (ADS)
Zhang, Zhizeng; Zhao, Zhao; Li, Yongtao
2016-06-01
This paper attempts to verify the correctness of the analytical displacement solution in transversely isotropic rock mass, and to determine the scope of its application. The analytical displacement solution of a circular tunnel in transversely isotropic rock mass was derived firstly. The analytical solution was compared with the numerical solution, which was carried out by FLAC3D software. The results show that the expression of the analytical displacement solution is correct, and the allowable engineering range is that the dip angle is less than 15 degrees.
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.
Numerical solution for option pricing with stochastic volatility model
NASA Astrophysics Data System (ADS)
Mariani, Andi; Nugrahani, Endar H.; Lesmana, Donny C.
2016-01-01
The option pricing equations derived from stochatic volatility models in finance are often cast in the form of nonlinear partial differential equations. To solve the equations, we used the upwind finite difference scheme for the spatial discretisation and a fully implicit time-stepping scheme. The result of this scheme is a matrix system in the form of an M-Matrix and we proof that the approximate solution converges to the viscosity solution to the equation by showing that the scheme is monotone, consistent and stable. Numerical experiments are implemented to show that the behavior and the order of convergence of upwind finite difference method.
Free Vibration of Simply Supported General Triangular Thin Plates: AN Accurate Simplified Solution
NASA Astrophysics Data System (ADS)
Saliba, H. T.
1996-09-01
In this paper, a highly accurate, simplified, and economical solution is provided for the free vibration problem of simply supported thin general triangular plates. The method is applicable to thin plates with linear boundaries regardless of there geometrical shapes. Results are compared with previously published reliable data for both the isosceles as well as the general triangles. Excellent agreements are reported. Eigenvalues are provided for a wide range of place aspect ratios. The first five mode shapes for some isosceles triangles are also provided for illustrative purposes. The advantages of the solution presented in the paper over previously published solutions are briefly discussed. Although the paper deals only with simple support conditions, it is mentioned that any combination of classical boundary conditions, with or without complicating factors, can easily be handled.
Numerical Solutions in 5d Standing Wave Braneworld
NASA Astrophysics Data System (ADS)
Gogberashvili, Merab; Sakhelashvili, Otari; Tukhashvili, Giorgi
2013-06-01
Within the 5D standing wave braneworld model numerical solutions of the equations for matter fields with various spins are found. It is shown that corresponding action integrals are factorizable and convergent over the extra coordinate, i.e. 4D fields are localized on the brane. We find that only left massless fermions are localized on the brane, while the right fermions are localized in the bulk. We demonstrate also quantization of Kaluza-Klein excited modes in our model.
Numerical Solution of a Model Equation of Price Formation
NASA Astrophysics Data System (ADS)
Chernogorova, T.; Vulkov, L.
2009-10-01
The paper [2] is devoted to the effect of reconciling the classical Black-Sholes theory of option pricing and hedging with various phenomena observed in the markets such as the influence of trading and hedging on the dynamics of an asset. Here we will discuss the numerical solution of initial boundary-value problems to a model equation of the theory. The lack of regularity in the solution as a result from Dirac delta coefficient reduces the accuracy in the numerical computations. First, we apply the finite volume method to discretize the differential problem. Second, we implement a technique of local regularization introduced by A-K. Tornberg and B. Engquist [7] for handling this equation. We derived the numerical regularization process into two steps: the Dirac delta function is regularized and then the regularized differential equation is discretized by difference schemes. Using the discrete maximum principle a priori bounds are obtained for the difference equations that imply stability and convergence of difference schemes for the problem under consideration. Numerical experiments are discussed.
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
A spectrally accurate method for overlapping grid solution of incompressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Merrill, Brandon E.; Peet, Yulia T.; Fischer, Paul F.; Lottes, James W.
2016-02-01
An overlapping mesh methodology that is spectrally accurate in space and up to third-order accurate in time is developed for solution of unsteady incompressible flow equations in three-dimensional domains. The ability to decompose a global domain into separate, but overlapping, subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries. The methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally-accurate spatial interpolation and high-order accurate temporal extrapolation, and requires few, if any, iterations, yet maintains the global accuracy and stability of the underlying flow solver. The overlapping mesh methodology is thoroughly validated using two-dimensional and three-dimensional benchmark problems in laminar and turbulent flows. The spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver. The influence of long integration times, as well as inflow-outflow global boundary conditions on the performance of the overlapping grid solver is assessed. In a turbulent benchmark of fully-developed turbulent pipe flow, the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data. Scaling tests are presented that show near linear strong scaling, even for moderately large processor counts.
Unique solution for accurate in-situ infrared profiling in reheat furnaces
NASA Astrophysics Data System (ADS)
Primhak, David; Wileman, Ben; Drögmöller, Peter
2010-05-01
As thermal imaging becomes a more accepted technology in industrial environments it can provide exciting new solutions to applications that have been previously dominated by single point pyrometers. The new development of an uncooled focal plane array thermal imager with a narrow band 3.9μm filter and background compensation processing enables measurements in industrial furnaces to provide temperature profiling of the product. This paper will show why the use of a 3.9μm camera with a borescope optic is the most accurate noncontact method for in-furnace temperature measurement. This will be done using the example of a reheat furnace where in a controlled trial using an instrumented billet the measurement from the IR device was shown to accurately track the thermocouple temperature during a variety of furnace operating conditions.
Numerical Bianchi type I solutions in semiclassical gravitation
Vitenti, Sandro D. P.; Mueller, Daniel
2006-09-15
It is believed that soon after the Planck era, spacetime should have a semiclassical nature. In this context we consider quantum fields propagating in a classical gravitational field and study the backreaction of these fields, using the expected value of the energy-momentum tensor as source of the gravitational field. According to this theory, the escape from general relativity theory is unavoidable. Two geometric counter-term are needed to regularize the divergences which come from the expected value. There is a parameter associated to each counter-term and in this work we found numerical solutions of this theory to particular initial conditions, for general Bianchi Type I spaces. We show that even though there are spurious solutions some of them can be considered physical. These physical solutions include de Sitter and Minkowski that are obtained asymptotically.
A new algorithm for generating highly accurate benchmark solutions to transport test problems
Azmy, Y.Y.
1997-06-01
We present a new algorithm for solving the neutron transport equation in its discrete-variable form. The new algorithm is based on computing the full matrix relating the scalar flux spatial moments in all cells to the fixed neutron source spatial moments, foregoing the need to compute the angular flux spatial moments, and thereby eliminating the need for sweeping the spatial mesh in each discrete-angular direction. The matrix equation is solved exactly in test cases, producing a solution vector that is free from iteration convergence error, and subject only to truncation and roundoff errors. Our algorithm is designed to provide method developers with a quick and simple solution scheme to test their new methods on difficult test problems without the need to develop sophisticated solution techniques, e.g. acceleration, before establishing the worthiness of their innovation. We demonstrate the utility of the new algorithm by applying it to the Arbitrarily High Order Transport Nodal (AHOT-N) method, and using it to solve two of Burre`s Suite of Test Problems (BSTP). Our results provide highly accurate benchmark solutions, that can be distributed electronically and used to verify the pointwise accuracy of other solution methods and algorithms.
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.
Sensitivity of the solution of the Elder problem to density, velocity and numerical perturbations
NASA Astrophysics Data System (ADS)
Park, Chan-Hee; Aral, Mustafa M.
2007-06-01
In this paper the Elder problem is studied with the purpose of evaluating the inherent instabilities associated with the numerical solution of this problem. Our focus is first on the question of the existence of a unique numerical solution for this problem, and second on the grid density and fluid density requirements necessary for a unique numerical solution. In particular we have investigated the instability issues associated with the numerical solution of the Elder problem from the following perspectives: (i) physical instability issues associated with density differences; (ii) sensitivity of the numerical solution to idealization irregularities; and, (iii) the importance of a precise velocity field calculation and the association of this process with the grid density levels that is necessary to solve the Elder problem accurately. In the study discussed here we have used a finite element Galerkin model we have developed for solving density-dependent flow and transport problems, which will be identified as TechFlow. In our study, the numerical results of Frolkovič and de Schepper [Frolkovič, P. and H. de Schepper, 2001. Numerical modeling of convection dominated transport coupled with density-driven flow in porous media, Adv. Water Resour., 24, 63-72.] were replicated using the grid density employed in their work. We were also successful in duplicating the same result with a less dense grid but with more computational effort based on a global velocity estimation process we have adopted. Our results indicate that the global velocity estimation approach recommended by Yeh [Yeh, G.-T., 1981. On the computation of Darcian velocity and mass balance in finite element modelling of groundwater flow, Water Resour. Res., 17(5), 1529-1534.] allows the use of less dense grids while obtaining the same accuracy that can be achieved with denser grids. We have also observed that the regularity of the elements in the discretization of the solution domain does make a difference
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.
Numerical solution for a spinning, nutating, fluid-filled cylinder
Vaughn, H.R.; Oberkampf, W.L.; Wolfe, W.P.
1983-12-01
The incompressible, three-dimensional Navier-Stokes equations are solved numerically for a fluid filled cylindrical cannister that is spinning and nutating. The motion of the cannister is characteristic of that experienced by spin stabilized artillery projectiles. Equations for the internal fluid motion are derived in a noninertial, aeroballistic coordinate system. Steady state numerical solutions are obtained by an iterative finite difference procedure. Flow fields and liquid induced moments have been calculated for viscosities in the range of 0.9 x 10/sup 4/ to 1 x 10/sup 9/ centistokes. The calculated moments agree favorably with experimental measurements. These results have direct application to analyzing the flight stability of spinning flight vehicles having highly viscous liquid payloads.
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.
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 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.
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 of a semilinear elliptic equation via difference scheme
NASA Astrophysics Data System (ADS)
Beigmohammadi, Elif Ozturk; Demirel, Esra
2016-08-01
We consider the Bitsadze-Samarskii type nonlocal boundary value problem { -d/2v (t ) d t2 +B v (t ) =h (t ,v (t ) ) ,0
Numerical solution of the imprecisely defined inverse heat conduction problem
NASA Astrophysics Data System (ADS)
Smita, Tapaswini; Chakraverty, S.; Diptiranjan, Behera
2015-05-01
This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solution process, double parametric forms of fuzzy numbers are used with the variational iteration method (VIM). This problem first computes the uncertain temperature distribution in the domain. Next, when the uncertain temperature measurements in the domain are known, the functions describing the uncertain temperature and heat flux on the boundary are reconstructed. Related example problems are solved using the present procedure. We have also compared the present results with those in [Inf. Sci. (2008) 178 1917] along with homotopy perturbation method (HPM) and [Int. Commun. Heat Mass Transfer (2012) 39 30] in the special cases to demonstrate the validity and applicability.
Numerical Solutions of Supersonic and Hypersonic Laminar Compression Corner Flows
NASA Technical Reports Server (NTRS)
Hung, C. M.; MacCormack, R. W.
1976-01-01
An efficient time-splitting, second-order accurate, numerical scheme is used to solve the complete Navier-Stokes equations for supersonic and hypersonic laminar flow over a two-dimensional compression corner. A fine, exponentially stretched mesh spacing is used in the region near the wall for resolving the viscous layer. Good agreement is obtained between the present computed results and experimental measurement for a Mach number of 14.1 and a Reynolds number of 1.04 x 10(exp 5) with wedge angles of 15 deg, 18 deg, and 24 deg. The details of the pressure variation across the boundary layer are given, and a correlation between the leading edge shock and the peaks in surface pressure and heat transfer is observed.
Numerical solution of the radiation transport equation in disk geometry
NASA Technical Reports Server (NTRS)
Spagna, George F., Jr.; Leung, Chun Ming
1987-01-01
An efficient numerical method for solving the problem of radiation transport in a dusty medium with two dimensional (2-D) disk geometry is described. It is a generalization of the one-dimensional quasi-diffusion method in which the transport equation is cast in diffusion form and then solved as a boundary value problem. The method should be applicable to a variety of astronomical sources, the dynamics of which are angular-momentum dominated and hence not accurately treated by spherical geometry, e.g., protoplanetary nebulae, circumstellar disks, interstellar molecular clouds, accretion disks, and disk galaxies. The computational procedure and practical considerations for implementing the method are described in detail. To illustrate the effects of 2-D radiation transport, some model results (dust temperature distributions and IR flux spectra) for externally heated, interstellar dust clouds with spherically symmetric and disk geometry are compared.
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.
A New Solution to the Problem of Finding All Numerical Solutions to Ordered Metric Structures.
ERIC Educational Resources Information Center
Lehner, Paul E.; Norma, Elliot
1980-01-01
A new algorithm is used to test and describe the set of all possible solutions for any linear model of an empirical ordering derived from techniques such as additive conjoint measurement, unfolding theory, general Fechnerian scaling, and ordinal multiple regression. The algorithm is computationally faster and numerically superior to previous…
Accurate and molecular-size-tolerant NMR quantitation of diverse components in solution
Okamura, Hideyasu; Nishimura, Hiroshi; Nagata, Takashi; Kigawa, Takanori; Watanabe, Takashi; Katahira, Masato
2016-01-01
Determining the amount of each component of interest in a mixture is a fundamental first step in characterizing the nature of the solution and to develop possible means of utilization of its components. Similarly, determining the composition of units in complex polymers, or polymer mixtures, is crucial. Although NMR is recognized as one of the most powerful methods to achieve this and is widely used in many fields, variation in the molecular sizes or the relative mobilities of components skews quantitation due to the size-dependent decay of magnetization. Here, a method to accurately determine the amount of each component by NMR was developed. This method was validated using a solution that contains biomass-related components in which the molecular sizes greatly differ. The method is also tolerant of other factors that skew quantitation such as variation in the one-bond C–H coupling constant. The developed method is the first and only way to reliably overcome the skewed quantitation caused by several different factors to provide basic information on the correct amount of each component in a solution. PMID:26883279
NASA Astrophysics Data System (ADS)
Soffer, Avy; Zhao, Xiaofei
2016-04-01
Based on our previous work for solving the nonlinear Schrödinger equation with multichannel dynamics that is given by a localized standing wave and radiation, in this work we deal with the multichannel solution which consists of a moving soliton and radiation. We apply the modulation theory to give a system of ODEs coupled to the radiation term for describing the solution, which is valid for all times. The modulation equations are solved accurately by the proposed numerical method. The soliton and radiation are captured separately in the computation, and they are solved on the translated domain that is moving with them. Thus for a fixed finite physical domain in the lab frame, the multichannel solution can pass through the boundary naturally, which cannot be done by imposing any existing boundary conditions. We comment on the differences of this method from the collective coordinates.
Numerical solution of the three-dimensional fluid flow in a rotating heterogeneous porous channel
NASA Astrophysics Data System (ADS)
Havstad, Mark A.; Vadasz, Peter
1999-09-01
A numerical solution to the problem of the three-dimensional fluid flow in a long rotating heterogeneous porous channel is presented. A co-ordinate transformation technique is employed to obtain accurate solutions over a wide range of porous media Ekman number values and consequent boundary layer thicknesses. Comparisons with an approximate asymptotic solution (for large values of Ekman number) and with theoretical predictions on the validity of Taylor-Proudman theorem in porous media for small values of Ekman number show good qualitative agreement. An evaluation of the boundary layer thickness is presented and a power-law correlation to Ekman number is shown to well-represent the results for small values of Ekman number. The different three-dimensional fluid flow regimes are presented graphically, demonstrating the distinct variation of the flow field over the wide range of Ekman numbers used. Copyright
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)
The spatially nonuniform convergence of the numerical solution of flows
NASA Technical Reports Server (NTRS)
Panaras, Argyris G.
1987-01-01
The spatial distribution of the numerical disturbances that are generated during the numerical solution of a flow is examined. It is shown that the distribution of the disturbances is not uniform. In regions where the structure of a flow is simple, the magnitude of the generated disturbances is small and their decay is fast. However, in complex flow regions, as in separation and vortical areas, large magnitude disturbances appear and their decay may be very slow. The observed nonuniformity of the numerical disturbances makes possible the reduction of the calculation time by application of what may be called the partial-grid calculation technique, in which a major part of the calculation procedure is applied in selective subregions, where the velocity disturbances are large, and not within the whole grid. This technique is expected to prove beneficial in large-scale calculations such as the flow about complete aircraft configurations at high angle of attack. Also, it has been shown that if the Navier-Stokes equations are written in a generalized coordinate system, then in regions in which the grid is fine, such as near solid boundaries, the norms become infinitesimally small, because in these regions the Jacobian has very large values. Thus, the norms, unless they are unscaled by the Jacobians, reflect only the changes that happen at the outer boundaries of the computation domain, where the value of the Jacobian approaches unity, and not in the whole flow field.
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.
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...
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.
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.
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. PMID:20160303
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.
Numerical continuation of solution at singular points of codimension one
NASA Astrophysics Data System (ADS)
Krasnikov, S. D.; Kuznetsov, E. B.
2015-11-01
Numerical continuation of a solution through some singular points of the curve of solutions to algebraic or transcendental equations with a parameter is considered. Singular points of codimension one are investigated. An algorithm for constructing all the branches of the curve at a simple bifurcation point is proposed. A special regularization that allows one to pass simple cusp points as limit points is obtained. For the regularized simple cusp point, a bound on the norm of the inverse Jacobian matrix in a neighborhood of this point is found. Using this bound, the convergence of the continuation process in a neighborhood of the simple cusp point is proved; an algorithm for the discrete continuation of the solution at the singular point along a smooth curve is obtained and its validity is proved. Based on a unified approach, a bound on the norm of the inverse Jacobian matrix and results on the convergence of continuation process in the case of the simple bifurcation point are also obtained. The operation of computational programs is demonstrated on benchmarks, which proves their effectiveness and confirms theoretical results. The effectiveness of software is investigated by solving the applied problem of three-rod truss stability.
Danshita, Ippei; Polkovnikov, Anatoli
2010-09-01
We study the quantum dynamics of supercurrents of one-dimensional Bose gases in a ring optical lattice to verify instanton methods applied to coherent macroscopic quantum tunneling (MQT). We directly simulate the real-time quantum dynamics of supercurrents, where a coherent oscillation between two macroscopically distinct current states occurs due to MQT. The tunneling rate extracted from the coherent oscillation is compared with that given by the instanton method. We find that the instanton method is quantitatively accurate when the effective Planck's constant is sufficiently small. We also find phase slips associated with the oscillations.
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)
Chae, Kyu-Hyun
2002-04-01
Fourier series solutions to the deflection and magnification by a family of three-dimensional cusped two-power-law ellipsoidal mass distributions are presented. The cusped two-power-law ellipsoidal mass distributions are characterized by inner and outer power-law radial indices and a break (or transition) radius. The model family includes mass models mimicking Jaffe, Hernquist, and η models and dark matter halo profiles from numerical simulations. The Fourier series solutions for the cusped two-power-law mass distributions are relatively simple and allow a very fast calculation, even for a chosen small fractional calculational error (e.g., 10-5). These results will be particularly useful for studying lensed systems that provide a number of accurate lensing constraints and for systematic analyses of large numbers of lenses. Subroutines employing these results for the two-power-law model and the results by Chae, Khersonsky, & Turnshek for the generalized single-power-law mass model are made publicly available.
NASA Technical Reports Server (NTRS)
Kylling, Arve; Stamnes, Knut
1992-01-01
The present solutions to the linear transport equation pertain to monoenergetic particles' interaction with a multiple scattering/absorbing layered medium with a general anisotropic internal source term. Attention is given to a novel exponential-linear approximation to the internal source, as a function of scattering depth, which furnishes an at-once efficient and accurate solution to the linear transport equation through its reduction of the spatial mesh size. The great superiority of the proposed method is demonstrated by the numerical results obtained in the illustrative cases of (1) an embedded thermal source and (2) a rapidly varying beam pseudosource.
Differential-equation-based representation of truncation errors for accurate numerical simulation
NASA Astrophysics Data System (ADS)
MacKinnon, Robert J.; Johnson, Richard W.
1991-09-01
High-order compact finite difference schemes for 2D convection-diffusion-type differential equations with constant and variable convection coefficients are derived. The governing equations are employed to represent leading truncation terms, including cross-derivatives, making the overall O(h super 4) schemes conform to a 3 x 3 stencil. It is shown that the two-dimensional constant coefficient scheme collapses to the optimal scheme for the one-dimensional case wherein the finite difference equation yields nodally exact results. The two-dimensional schemes are tested against standard model problems, including a Navier-Stokes application. Results show that the two schemes are generally more accurate, on comparable grids, than O(h super 2) centered differencing and commonly used O(h) and O(h super 3) upwinding schemes.
Towards more accurate numerical modeling of impedance based high frequency harmonic vibration
NASA Astrophysics Data System (ADS)
Lim, Yee Yan; Kiong Soh, Chee
2014-03-01
The application of smart materials in various fields of engineering has recently become increasingly popular. For instance, the high frequency based electromechanical impedance (EMI) technique employing smart piezoelectric materials is found to be versatile in structural health monitoring (SHM). Thus far, considerable efforts have been made to study and improve the technique. Various theoretical models of the EMI technique have been proposed in an attempt to better understand its behavior. So far, the three-dimensional (3D) coupled field finite element (FE) model has proved to be the most accurate. However, large discrepancies between the results of the FE model and experimental tests, especially in terms of the slope and magnitude of the admittance signatures, continue to exist and are yet to be resolved. This paper presents a series of parametric studies using the 3D coupled field finite element method (FEM) on all properties of materials involved in the lead zirconate titanate (PZT) structure interaction of the EMI technique, to investigate their effect on the admittance signatures acquired. FE model updating is then performed by adjusting the parameters to match the experimental results. One of the main reasons for the lower accuracy, especially in terms of magnitude and slope, of previous FE models is the difficulty in determining the damping related coefficients and the stiffness of the bonding layer. In this study, using the hysteretic damping model in place of Rayleigh damping, which is used by most researchers in this field, and updated bonding stiffness, an improved and more accurate FE model is achieved. The results of this paper are expected to be useful for future study of the subject area in terms of research and application, such as modeling, design and optimization.
Assessment of an efficient numerical solution of the 1D Richards' equation on bare soil
NASA Astrophysics Data System (ADS)
Varado, N.; Braud, I.; Ross, P. J.; Haverkamp, R.
2006-05-01
A new numerical scheme has been proposed by Ross [Ross, P.J., 2003. Modeling soil water and solute transport—fast, simplified numerical solutions. Agronomy Journal 95, 1352-1361] to solve the 1D Richards' equation [Richards, L.A., 1931. Capillary conduction of liquids through porous medium. Physics 1, 318-333]. This non-iterative solution uses the description of soil properties proposed by Brooks and Corey [Brooks, R.H., Corey, A.T., 1964. Hydraulic properties of porous media. Colorado State University, Fort Collins]. It allows the derivation of an analytical expression for the Kirchhoff potential used in the calculation of water fluxes. The degree of saturation is used as the dependent variable when the soil is unsaturated and the Kirchhoff potential is used in case of saturation. A space and time discretisation scheme leads to a tridiagonal set of linear equations that is solved non-iteratively. We propose in this paper an extensive test of this numerical method, evaluated only on a single case by Ross. The tests are conducted in two steps. First, the solution is assessed against two analytical solutions. The first one [Basha, H.A., 1999. Multidimensional linearized nonsteady infiltration with prescribed boundary conditions at the soil surface. Water Resources Research 35(1), 75-93] provides the water content profile when simplified soil characteristics such as the exponential law of Gardner [Gardner, W.R., 1958. Some steady-state solutions of the unsaturated moisture flow equations with application to evaporation from a water table. Soil Science 85, 228-232] are used. The Ross solution is compared to this solution on eight different soils that were fitted to this law. Analytical solution with the Brooks and Corey models is not available at the moment for the moisture profile but some exist for cumulative infiltration. Therefore, the second analytical solution, used in this study, is the one developed by Parlange et al. [Parlange, J.-Y., Haverkamp, R., Touma, J
Numerical solution of nonlinear heat problem with moving boundary
NASA Astrophysics Data System (ADS)
AL-Mannai, Mona; Khabeev, Nail
2012-01-01
Two phase gas-liquid flow in pipes is widely spread in space applications: bubble flows appear in cryogenic components transport through fuel/oxidant supply lines. Another important application is based on the fact that in liquid flows with small bubbles a close contact between the two phases occurs resulting in high rates of transfer between them. The compactness of a system makes it ideally suited to serve as a space-based two-phase bio-reactor which forms an important unit in environmental control and life support system deployed onboard. A numerical method was developed for solving a nonlinear problem of thermal interaction between a spherical gas bubble and surrounding liquid. The system of equations for describing this interaction was formulated. It includes ordinary and nonlinear partial differential equations. The problem was solved using finite-difference technique by dividing the system into spherical layers inside the bubble and employing the new variable which "freezes" the moving boundary of the bubble. A numerical solution is obtained for the problem of radial bubble motion induced by a sudden pressure change in the liquid—a situation which corresponds to the behavior of bubbles beyond a shock wave front when the latter enters a bubble curtain.
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.
Reaction fronts in a porous medium. Approximation techniques versus numerical solution
Escobedo, F.; Viljoen, H.J.
1995-03-01
The combustion of a gas fuel inside a porous media, as found in radiant porous burners (PRB), is one of the most promising novel combustion concepts. The flame sheet approximation (FS) and a novel polynomial approximation technique (PA) are compared in terms of their capability to describe reaction fronts of highly exothermic reactions in a porous medium. A one-phase model and a two-phase model of a system with adiabatic walls and a radiant output (to approximate the case of a porous radiant burner) are included in the analysis. By matching the reaction zone solution found by either the FS or PA method with the solutions of the nonreacting zones, the temperature, conversion, and position of the reaction zone were determined. Numerical solutions for catalytic and noncatalytic oxidation reactions were used to compare the predictions of both approaches. It was found that although both techniques yielded good approximations to the solutions, the PA technique proved to be more accurate, producing results with 3.5% of the numerical results. Both methods can find useful application in the analysis of this class of problems.
A numerical solution of 3D inviscid rotational flow in turbines and ducts
NASA Astrophysics Data System (ADS)
Oktay, Erdal; Akmandor, Sinan; Üçer, Ahmet
1998-04-01
The numerical solutions of inviscid rotational (Euler) flows were obtained using an explicit hexahedral unstructured cell vertex finite volume method. A second-order-accurate, one-step Lax-Wendroff scheme was used to solve the unsteady governing equations discretized in conservative form. The transonic circular bump, in which the location and the strength of the captured shock are well predicted, was used as the first test case. The nozzle guide vanes of the VKI low-speed turbine facility were used to validate the Euler code in highly 3D environment. Despite the high turning and the secondary flows which develop, close agreements have been obtained with experimental and numerical results associated with these test cases.
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.
Numerical solution of Boltzmann equation using discrete velocity grids
NASA Astrophysics Data System (ADS)
Vedula, Prakash
2015-11-01
An importance sampling based approach for numerical solution of the (single species) Boltzmann equation using discrete velocity grids is proposed. This approach involves a stochastic method for evaluation of the collision integral based on sampling of depleting/replenishing collisions and is designed to preserve important symmetries of the collision operator, including collision invariants. The underlying particle distribution function is represented as a collection of delta functions with associated weights that are non-negative. A key feature in the construction of the proposed method is that it ensures that the weights associated with the distribution function remain non-negative during collisional relaxation, thereby satisfying an important realizability condition. Performance of the proposed approach will be studied using test problems involving spatially homogeneous collisional relaxation flow and microchannel flows. Results obtained from the proposed method will be compared with those obtained from the (deterministic) collisional Lattice Boltzmann Method (cLBM) and the traditional direct simulation Monte Carlo (DSMC) method for solution of Boltzmann equation. Extension of the proposed method using discrete velocity grids for multicomponent mixtures will also be discussed.
TOPLHA: an accurate and efficient numerical tool for analysis and design of LH antennas
NASA Astrophysics Data System (ADS)
Milanesio, D.; Lancellotti, V.; Meneghini, O.; Maggiora, R.; Vecchi, G.; Bilato, R.
2007-09-01
Auxiliary ICRF heating systems in tokamaks often involve large complex antennas, made up of several conducting straps hosted in distinct cavities that open towards the plasma. The same holds especially true in the LH regime, wherein the antennas are comprised of arrays of many phased waveguides. Upon observing that the various cavities or waveguides couple to each other only through the EM fields existing over the plasma-facing apertures, we self-consistently formulated the EM problem by a convenient set of multiple coupled integral equations. Subsequent application of the Method of Moments yields a highly sparse algebraic system; therefore formal inversion of the system matrix happens to be not so memory demanding, despite the number of unknowns may be quite large (typically 105 or so). The overall strategy has been implemented in an enhanced version of TOPICA (Torino Polytechnic Ion Cyclotron Antenna) and in a newly developed code named TOPLHA (Torino Polytechnic Lower Hybrid Antenna). Both are simulation and prediction tools for plasma facing antennas that incorporate commercial-grade 3D graphic interfaces along with an accurate description of the plasma. In this work we present the new proposed formulation along with examples of application to real life large LH antenna systems.
Kottmann, Jakob S; Höfener, Sebastian; Bischoff, Florian A
2015-12-21
In the present work, we report an efficient implementation of configuration interaction singles (CIS) excitation energies and oscillator strengths using the multi-resolution analysis (MRA) framework to address the basis-set convergence of excited state computations. In MRA (ground-state) orbitals, excited states are constructed adaptively guaranteeing an overall precision. Thus not only valence but also, in particular, low-lying Rydberg states can be computed with consistent quality at the basis set limit a priori, or without special treatments, which is demonstrated using a small test set of organic molecules, basis sets, and states. We find that the new implementation of MRA-CIS excitation energy calculations is competitive with conventional LCAO calculations when the basis-set limit of medium-sized molecules is sought, which requires large, diffuse basis sets. This becomes particularly important if accurate calculations of molecular electronic absorption spectra with respect to basis-set incompleteness are required, in which both valence as well as Rydberg excitations can contribute to the molecule's UV/VIS fingerprint. PMID:25913482
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.
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. PMID:27105653
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
NASA Technical Reports Server (NTRS)
Hasiuk, Jan; Hindman, Richard; Iversen, James
1988-01-01
The azimuthal-invariant, three-dimensional cylindrical, incompressible Navier-Stokes equations are solved to steady state for a finite-length, physically realistic model. The numerical method relies on an alternating-direction implicit scheme that is formally second-order accurate in space and first-order accurate in time. The equations are linearized and uncoupled by evaluating variable coefficients at the previous time iteration. Wall grid clustering is provided by a Roberts transformation in radial and axial directions. A vorticity-velocity formulation is found to be preferable to a vorticity-streamfunction approach. Subject to no-slip, Dirichlet boundary conditions, except for the inner cylinder rotation velocity (impulsive start-up) and zero-flow initial conditions, nonturbulent solutions are obtained for sub- and supercritical Reynolds numbers of 100 to 400 for a finite geometry where R(outer)/R(inner) = 1.5, H/R(inner) = 0.73, and H/Delta-R = 1.5. An axially-stretched model solution is shown to asymptotically approach the one-dimensional analytic Couette solution at the cylinder midheight. Flowfield change from laminar to Taylor-vortex flow is discussed as a function of Reynolds number. Three-dimensional velocities, vorticity, and streamfunction are presented via two-dimensional graphs and three-dimensional surface and contour plots.
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
Numerical solution of a flow inside a labyrinth seal
NASA Astrophysics Data System (ADS)
Šimák, Jan; Straka, Petr; Pelant, Jaroslav
2012-04-01
The aim of this study is a behaviour of a flow inside a labyrinth seal on a rotating shaft. The labyrinth seal is a type of a non-contact seal where a leakage of a fluid is prevented by a rather complicated path, which the fluid has to overcome. In the presented case the sealed medium is the air and the seal is made by a system of 20 teeth on a rotating shaft situated against a smooth static surface. Centrifugal forces present due to the rotation of the shaft create vortices in each chamber and thus dissipate the axial velocity of the escaping air.The structure of the flow field inside the seal is studied through the use of numerical methods. Three-dimensional solution of the Navier-Stokes equations for turbulent flow is very time consuming. In order to reduce the computational time we can simplify our problem and solve it as an axisymmetric problem in a two-dimensional meridian plane. For this case we use a transformation of the Navier-Stokes equations and of the standard k-omega turbulence model into a cylindrical coordinate system. A finite volume method is used for the solution of the resulting problem. A one-side modification of the Riemann problem for boundary conditions is used at the inlet and at the outlet of the axisymmetric channel. The total pressure and total density (temperature) are to be used preferably at the inlet whereas the static pressure is used at the outlet for the compatibility. This idea yields physically relevant boundary conditions. The important characteristics such as a mass flow rate and a power loss, depending on a pressure ratio (1.1 - 4) and an angular velocity (1000 - 15000 rpm) are evaluated.
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.
Bangalore, Sai Santosh; Wang, Jelai; Allison, David B.
2009-01-01
In the fields of genomics and high dimensional biology (HDB), massive multiple testing prompts the use of extremely small significance levels. Because tail areas of statistical distributions are needed for hypothesis testing, the accuracy of these areas is important to confidently make scientific judgments. Previous work on accuracy was primarily focused on evaluating professionally written statistical software, like SAS, on the Statistical Reference Datasets (StRD) provided by National Institute of Standards and Technology (NIST) and on the accuracy of tail areas in statistical distributions. The goal of this paper is to provide guidance to investigators, who are developing their own custom scientific software built upon numerical libraries written by others. In specific, we evaluate the accuracy of small tail areas from cumulative distribution functions (CDF) of the Chi-square and t-distribution by comparing several open-source, free, or commercially licensed numerical libraries in Java, C, and R to widely accepted standards of comparison like ELV and DCDFLIB. In our evaluation, the C libraries and R functions are consistently accurate up to six significant digits. Amongst the evaluated Java libraries, Colt is most accurate. These languages and libraries are popular choices among programmers developing scientific software, so the results herein can be useful to programmers in choosing libraries for CDF accuracy. PMID:20161126
NASA Astrophysics Data System (ADS)
Yovanovich, M. M.; Culham, J. R.; Lemczyk, T. F.
1986-01-01
One and two-dimensional solutions are obtained for annular fins of constant cross-section having uniform base, end and side conductances. The solutions are dependent upon one geometric parameter and three fin parameters which relate the internal conductive resistance to the three boundary resistances. The two and one-dimensional solutions are compared by means of the heat flow rate or fin efficiency ratios. Simple polynomials are developed for fast, accurate numerical computation of the modified Bessel functions which appear in the solutions. For annular fins used in typical microelectronic applications the analytical expressions are also reduced to alternate expressions which are shown to be expressible by means of simple polynomials which converge to unity for large values of the arguments. Numerical computations were performed on an IBM-PC and some typical results are reported in graphical form. These plots give the heat loss ratio as a function of the dimensionless geometric and fin parameters.
NASA Astrophysics Data System (ADS)
Liu, Nansheng; Khomami, Bamin
2011-11-01
Despite tremendous progress in development of numerical techniques and constitutive theories for polymeric fluids in the past decade, Direct Numerical Simulation (DNS) of elastic turbulence has posed tremendous challenges to researchers engaged in developing first principles models and simulations that can accurately and robustly predict the dynamical behavior of polymeric flows. In this presentation, we report the first DNS of elastic turbulence in the Taylor-Couette (TC) flow. Specifically, our computations with prototypical constitutive equations for dilute polymeric solutions, such as the FENE-P model are capable of reproducing the essential features of the experimentally observed elastic turbulence in TC flow of this class of fluids, namely, randomly fluctuating fluid motion excited in a broad range of spatial and temporal scales, and a significant increase of the flow resistance. Moreover, the experimentally measured Power Spectral Density of radial velocity fluctuations, i.e., two contiguous regions of power-law decay, -1.1 at lower frequencies and -2.2 at high-frequencies is accurately computed. We would like to thank NSF through grant CBET-0755269 and NSFC through grant NO. 10972211 for supporting of this work.
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)). PMID:26679617
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...
Numerical strategies for the solution of inverse problems
NASA Astrophysics Data System (ADS)
Hebber (Haber), Eldad
This thesis deals with the numerical solutions of linear and nonlinear inverse problems. The goal of this thesis is to review and develop new techniques for solving such problems. In so doing, the computations tools for solving inverse problems are comprehensively studied. The thesis can be divided into two parts. In the first part, linear inverse theory is dealt with. Methods to estimate noise and efficiently invert large and full matrixes are reviewed and developed. Emphasis is given to Generalized Cross Validation (GCV) for noise estimation, and to Krylov space methods for efficient methods to invert large systems. This part is summarized by applying and comparing the methods developed on linear inverse problems which arise in gravity and tomography. In the second part of this thesis, extensive use of the linear algebra and the noise estimation methods which were developed in the first part of the thesis is made. A review of the current methods to carry out nonlinear inverse problems is given. A test example is constructed to demonstrate that these methods may fail. Next, a new algorithm for solving nonlinear inverse problems is developed. The algorithm is based on the ability to differentiate between correlated errors which comes from the linearization, and non-correlated noise which comes from the measurement. Based on these two types of noise, a regularization procedure which has two parts is developed. The first part is made of global regulation, to deal with the measurement noise, and the second part is made from a local regularization, to deal with the nonlinearity. The thesis demonstrates that GCV can be used in order to determine the measurement noise, and the Damped Gauss-Newton method can be used in order to deal with the local nonlinear terms. Another aspect of nonlinear inverse theory which is developed in this work concerns approximate sensitivities. A new formulation is suggested for the approximate sensitivities and bounds are calculated using
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
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.
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.
Numerical solution of initial-value problems in plasma physics
Killeen, J.
1980-10-01
The numerical models used in fusion research are briefly reviewed. The application of implicit difference techniques to problems in resistive magnetohydrodynamics, transport and the Fokker-Planck equation is discussed.
NASA Technical Reports Server (NTRS)
Zingg, D. W.; Lomax, H.
1993-01-01
A six-stage low-storage Runge-Kutta time-marching method is presented and shown to be an efficient method for use with high-accuracy spatial difference operators for wave propagation problems. The accuracy of the method for inhomogeneous ordinary differential equations is demonstrated through numerical solutions of the linear convection equation with forced boundary conditions. Numerical experiments are presented simulating a sine wave and a Gaussian pulse propagating into and through the domain. For practical levels of mesh refinement corresponding to roughly ten points per wavelength, the six-stage Runge-Kutta method is more accurate than the popular fourth-order Runge-Kutta method. Further numerical experiments are presented which show that the numerical boundary scheme at an inflow boundary can be a significant source of error when high-accuracy spatial discretizations are used.
Are 0. 1%-accurate gamma-ray assays possible for /sup 235/U solutions
Parker, J.L.
1983-01-01
The factors influencing the accuracy of passive gamma-ray assay of uniform, homogeneous solution samples have been studied in some detail, particularly for the assay of /sup 235/U in uranium solutions. Factors considered are the overall long-term electronic stability, the information losses caused by the rate-related electronic processes of pulse pileup and dead-time, and the self-attenuation of gamma rays within the samples. Both experimental and computational studies indicate that gamma-ray assay procedures for solution samples of moderate size (from approx. 10 to perhaps a few hundred milliliters) are now capable of accuracies approaching 0.1% in many practical cases.
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
von Neumann Stability Analysis of Numerical Solution Schemes for 1D and 2D Euler Equations
NASA Astrophysics Data System (ADS)
Konangi, Santosh; Palakurthi, Nikhil Kumar; Ghia, Urmila
2014-11-01
A von Neumann stability analysis is conducted for numerical schemes for the full system of coupled, density-based 1D and 2D Euler equations, closed by an isentropic equation of state. The governing equations are discretized on a staggered grid, which permits equivalence to finite-volume discretization. Presently, first-order accurate spatial and temporal finite-difference techniques are analyzed. The momentum convection term is treated as explicit, semi-implicit or implicit. Density upwind bias is included in the spatial operator of the continuity equation. By combining the discretization techniques, ten solution schemes are formulated. For each scheme, unstable and stable regimes are identified through the stability analysis, and the maximum allowable CFL number is predicted. The predictions are verified for selected schemes, using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and ``experimentally'' observed CFL values for all cases, thereby validating the analysis. The demonstrated analysis provides an accurate indication of stability conditions for the Euler equations, in contrast to the simplistic conditions arising from model equations, such as the wave equation.
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 Astrophysics Data System (ADS)
Dwyer, G. S.; Vengosh, A.
2008-12-01
The negative thermal ionization mass spectrometry technique has become the major tool for investigating boron isotopes in the environment. The high sensitivity of BO2- ionization enables measurements of ng levels of boron. However, B isotope measurement by this technique suffers from two fundamental problems (1) fractionation induced by selective ionization of B isotopes in the mass spectrometer; and (2) CNO- interference on mass 42 that is often present in some load solutions (such as B-free seawater processed through ion-exchange resin). Here we report a potentially improved methodology using an alternative filament loading solution with a recently-installed Thermo Scientific TRITON thermal ionization mass spectrometer. Our initial results suggest that this solution -- prepared by combining high-purity single- element standard solutions of Ca, Mg, Na, and K in proportions similar to those in seawater in a 5% HCl matrix -- may offer significant improvement over some other commonly used load solutions. Total loading blank is around 15pg as determined by isotope dilution (NIST952). Replicate analyses of NIST SRM951 and modern seawater thus far have yielded 11B/10B ratios of 4.0057 (±0.0008, n=14) and 4.1645 (±0.0017, n=7; δ11B=39.6 permil), respectively. Replicate analyses of samples and SRM951 yield an average standard deviation (1 σ) of approximately 0.001 (0.25 permil). Fractionation during analysis (60-90 minutes) has thus far typically been less than 0.002 (0.5 permil). The load solution delivers ionization efficiency similar to directly-loaded seawater and has negligible signal at mass 26 (CN-), a proxy for the common interfering molecular ion (CNO-) on mass 42. Standards and samples loaded with the solution behave fairly predictably during filament heating and analysis, thus allowing for the possibility of fully automated data collection.
Collocation Method for Numerical Solution of Coupled Nonlinear Schroedinger Equation
Ismail, M. S.
2010-09-30
The coupled nonlinear Schroedinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we use collocation method to solve this equation, we test this method for stability and accuracy. Numerical tests using single soliton and interaction of three solitons are used to test the resulting scheme.
NASA Astrophysics Data System (ADS)
Ngo-Cong, D.; Mohammed, F. J.; Strunin, D. V.; Skvortsov, A. T.; Mai-Duy, N.; Tran-Cong, T.
2015-06-01
The contaminant transport process governed by the advection-diffusion equation plays an important role in modelling industrial and environmental flows. In this article, our aim is to accurately reduce the 2-D advection-diffusion equation governing the dispersion of a contaminant in a turbulent open channel flow to its 1-D approximation. The 1-D model helps to quickly estimate the horizontal size of contaminant clouds based on the values of the model coefficients. We derive these coefficients analytically and investigate numerically the model convergence. The derivation is based on the centre manifold theory to obtain successively more accurate approximations in a consistent manner. Two types of the average velocity profile are considered: the classical logarithmic profile and the power profile. We further develop the one-dimensional integrated radial basis function network method as a numerical approach to obtain the numerical solutions to both the original 2-D equation and the approximate 1-D equations. We compare the solutions of the original models with their centre-manifold approximations at very large Reynolds numbers. The numerical results obtained from the approximate 1-D models are in good agreement with those of the original 2-D model for both the logarithmic and power velocity profiles.
On the numerical solution of hyperbolic equations with singular source terms
NASA Astrophysics Data System (ADS)
Turk, Irfan; Ashyraliyev, Maksat
2014-08-01
A numerical study for hyperbolic equations having singular source terms is presented. Singular in the sense that within the spatial domain the source is defined by a Dirac delta function. Solutions of such problems will have discontinuities which forms an obstacle for standard numerical methods. In this paper, a fifth order flux implicit WENO method with non-uniform meshes is studied for approximate solutions of hyperbolic equations having singular source terms. Numerical examples are provided.
Accurate and inexpensive prediction of the color optical properties of anthocyanins in solution.
Ge, Xiaochuan; Timrov, Iurii; Binnie, Simon; Biancardi, Alessandro; Calzolari, Arrigo; Baroni, Stefano
2015-04-23
The simulation of the color optical properties of molecular dyes in liquid solution requires the calculation of time evolution of the solute absorption spectra fluctuating in the solvent at finite temperature. Time-averaged spectra can be directly evaluated by combining ab initio Car-Parrinello molecular dynamics and time-dependent density functional theory calculations. The inclusion of hybrid exchange-correlation functionals, necessary for the prediction of the correct transition frequencies, prevents one from using these techniques for the simulation of the optical properties of large realistic systems. Here we present an alternative approach for the prediction of the color of natural dyes in solution with a low computational cost. We applied this approach to representative anthocyanin dyes: the excellent agreement between the simulated and the experimental colors makes this method a straightforward and inexpensive tool for the high-throughput prediction of colors of molecules in liquid solvents. PMID:25830823
Toward Accurate Modeling of the Effect of Ion-Pair Formation on Solute Redox Potential.
Qu, Xiaohui; Persson, Kristin A
2016-09-13
A scheme to model the dependence of a solute redox potential on the supporting electrolyte is proposed, and the results are compared to experimental observations and other reported theoretical models. An improved agreement with experiment is exhibited if the effect of the supporting electrolyte on the redox potential is modeled through a concentration change induced via ion pair formation with the salt, rather than by only considering the direct impact on the redox potential of the solute itself. To exemplify the approach, the scheme is applied to the concentration-dependent redox potential of select molecules proposed for nonaqueous flow batteries. However, the methodology is general and enables rational computational electrolyte design through tuning of the operating window of electrochemical systems by shifting the redox potential of its solutes; including potentially both salts as well as redox active molecules. PMID:27500744
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.
Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions
Fibich, G. . E-mail: fibich@math.tau.ac.il; Tsynkov, S. . E-mail: tsynkov@math.ncsu.edu
2005-11-20
In [J. Comput. Phys. 171 (2001) 632-677] we developed a fourth-order numerical method for solving the nonlinear Helmholtz equation which governs the propagation of time-harmonic laser beams in media with a Kerr-type nonlinearity. A key element of the algorithm was a new nonlocal two-way artificial boundary condition (ABC), set in the direction of beam propagation. This two-way ABC provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain. Altogether, the algorithm of [J. Comput. Phys. 171 (2001) 632-677] has allowed for a direct simulation of nonlinear self-focusing without neglecting nonparaxial effects and backscattering. To the best of our knowledge, this capacity has never been achieved previously in nonlinear optics. In the current paper, we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation, we now introduce Sommerfeld-type local radiation boundary conditions, which are constructed directly in the discrete framework. Numerically, implementation of the Sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non-Hermitian matrix. Subsequently, the separation of variables, which is a key building block of the aforementioned nonlocal ABC, is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors. Numerical simulations show that the new algorithm offers a considerable improvement in its numerical performance, as well as in the range of physical phenomena that it is capable of simulating.
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.
Taylor, Z A; Comas, O; Cheng, M; Passenger, J; Hawkes, D J; Atkinson, D; Ourselin, S
2009-04-01
Efficient and accurate techniques for simulation of soft tissue deformation are an increasingly valuable tool in many areas of medical image computing, such as biomechanically-driven image registration and interactive surgical simulation. For reasons of efficiency most analyses are based on simplified linear formulations, and previously almost all have ignored well established features of tissue mechanical response such as anisotropy and time-dependence. We address these latter issues by firstly presenting a generalised anisotropic viscoelastic constitutive framework for soft tissues, particular cases of which have previously been used to model a wide range of tissues. We then develop an efficient solution procedure for the accompanying viscoelastic hereditary integrals which allows use of such models in explicit dynamic finite element algorithms. We show that the procedure allows incorporation of both anisotropy and viscoelasticity for as little as 5.1% additional cost compared with the usual isotropic elastic models. Finally we describe the implementation of a new GPU-based finite element scheme for soft tissue simulation using the CUDA API. Even with the inclusion of more elaborate constitutive models as described the new implementation affords speed improvements compared with our recent graphics API-based implementation, and compared with CPU execution a speed up of 56.3 x is achieved. The validity of the viscoelastic solution procedure and performance of the GPU implementation are demonstrated with a series of numerical examples. PMID:19019721
Numerical solutions of Navier-Stokes equations for a Butler wing
NASA Technical Reports Server (NTRS)
Abolhassani, J. S.; Tiwari, S. N.
1985-01-01
The flow field is simulated on the surface of a given delta wing (Butler wing) at zero incident in a uniform stream. The simulation is done by integrating a set of flow field equations. This set of equations governs the unsteady, viscous, compressible, heat conducting flow of an ideal gas. The equations are written in curvilinear coordinates so that the wing surface is represented accurately. These equations are solved by the finite difference method, and results obtained for high-speed freestream conditions are compared with theoretical and experimental results. In this study, the Navier-Stokes equations are solved numerically. These equations are unsteady, compressible, viscous, and three-dimensional without neglecting any terms. The time dependency of the governing equations allows the solution to progress naturally for an arbitrary initial initial guess to an asymptotic steady state, if one exists. The equations are transformed from physical coordinates to the computational coordinates, allowing the solution of the governing equations in a rectangular parallel-piped domain. The equations are solved by the MacCormack time-split technique which is vectorized and programmed to run on the CDC VPS 32 computer.
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.
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 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 solution of compressible viscous flows at high Reynolds numbers
NASA Technical Reports Server (NTRS)
Maccormack, R. W.
1981-01-01
A new numerical method which was used to reduce the computation time required in fluid dynamics to solve the Navier-Stokes equations at flight Reynolds numbers is described. The method is the implicit analogue of the explicit finite different method. It uses this as its first stage, while the second stage removes the restrictive stability condition by recasting the difference equations in an implicit form. The resulting matrix equations to be solved are either upper or lower block bidiagonal equations. The new method makes it possible and practical to calculate many important three dimensional, high Reynolds number flow fields on computers.
NASA Astrophysics Data System (ADS)
Bhrawy, A. H.; Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S.
2015-07-01
In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.
Numerical solutions of the γ-index in two and three dimensions
Clasie, Benjamin M; Sharp, Gregory C; Seco, Joao; Flanz, Jacob B; Kooy, Hanne M
2012-01-01
The γ-index is used routinely to establish correspondence between two dose distributions. The definition of the γ-index can be written with a single equation but solving this equation at millions of points is computationally expensive, especially in three dimensions. Our goal is to extend the vector-equation method in Bakai et al [1] to higher order for better accuracy and, as important, to determine the magnitude of accuracy in a higher order solution. We construct a numerical framework for calculating the γ-index in two and three dimensions and present an efficient method for calculating the γ-index with 0th, 1st, and 2nd-order methods using tricubic spline interpolation. For an intensity modulated radiation therapy example with 1.78 × 106 voxels the 0th-order, 1st-order, 1st-order iterations and semi 2nd-order methods calculate the 3-dimensional γ-index in 1.5, 4.7, 34.7, and 35.6 s with 36.7%, 1.1%, 0.2% and 0.8% accuracy, respectively. The accuracy of linear interpolation with this example is 1.0%. We present efficient numerical methods for calculating the 3-dimensional γ-index with tricubic spline interpolation. The 1st-order method with iterations is the most accurate and fastest choice of the numerical methods if the dose distributions may have large second-order gradients. Furthermore, the difference between iterations can be used to determine the accuracy of the method. PMID:23044713
Numerical solutions of Maxwell's equations for nonlinear-optical pulse propagation
NASA Astrophysics Data System (ADS)
Hile, Cheryl V.; Kath, William L.
1996-06-01
A model and numerical solutions of Maxwell's equations describing the propagation of short, solitonlike pulses in nonlinear dispersive optical media are presented. The model includes linear dispersion expressed in the time domain, a Kerr nonlinearity, and a coordinate system moving with the group velocity of the pulse. Numerical solutions of Maxwell's equations are presented for circularly polarized and linearly polarized electromagnetic fields. When the electromagnetic fields are assumed to be circularly polarized, numerical solutions are compared directly with solutions of the nonlinear Schrodinger (NLS) equation. These comparisons show good agreement and indicate that the NLS equation provides an excellent model for short-pulse propagation. When the electromagnetic fields are assumed to be linearly polarized, the propagation of daughter pulses, small-amplitude pulses at three times the frequency of the solitonlike pulse, are observed in the numerical solution. These daughter pulses are shown to be the direct result of third harmonics generated by the main, solitonlike, pulse.
NASA Technical Reports Server (NTRS)
Smutek, C.; Bontoux, P.; Roux, B.; Schiroky, G. H.; Hurford, A. C.
1985-01-01
The results of a three-dimensional numerical simulation of Boussinesq free convection in a horizontal differentially heated cylinder are presented. The computation was based on a Samarskii-Andreyev scheme (described by Leong, 1981) and a false-transient advancement in time, with vorticity, velocity, and temperature as dependent variables. Solutions for velocity and temperature distributions were obtained for Rayleigh numbers (based on the radius) Ra = 74-18,700, thus covering the core- and boundary-layer-driven regimes. Numerical solutions are compared with asymptotic analytical solutions and experimental data. The numerical results well represent the complex three-dimensional flows found experimentally.
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.
Analysis of bacterial migration; 1: Numerical solution of balance equation
Frymier, P.D.; Ford, R.M.; Cummings, P.T. . Dept. of Chemical Engineering)
1994-04-01
Chemotaxis describes the ability of motile bacteria to bias their motion in the direction of increasing gradients of chemicals, usually energy sources, known as attractants. In experimental studies of the migration of chemotactic bacteria, 1-D phenomenological cell balance equations have been used to quantitatively analyze experimental observations. While attractive for their simplicity and the ease of solution, they are limited in the strict mathematical sense to the situation in which individual bacteria are confined to motion in one dimension and respond to attractant gradients in one dimension only. Recently, Ford and Cummings (1992) reduced the general 3-D cell balance equation of Alt (1980) to obtain an equation describing the migration of a bacterial population in response to a 1-D attractant gradient. Solutions of this equation for single gradients of attractants are compared to those of 1-D balance equations, results from cellular dynamics simulations, and experimental data from the authors' laboratory for E. coli responding to [alpha]-methylaspartate. The authors also investigate two aspects of the experimentally derived expression for the tumbling probability: the effect of different models for the down-gradient swimming behavior of the bacteria and the validity of ignoring the temporal derivative of the attractant concentration.
High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities
NASA Astrophysics Data System (ADS)
Britt, Darrell Steven, Jr.
Problems of time-harmonic wave propagation arise in important fields of study such as geological surveying, radar detection/evasion, and aircraft design. These often involve highfrequency waves, which demand high-order methods to mitigate the dispersion error. We propose a high-order method for computing solutions to the variable-coefficient inhomogeneous Helmholtz equation in two dimensions on domains bounded by piecewise smooth curves of arbitrary shape with a finite number of boundary singularities at known locations. We utilize compact finite difference (FD) schemes on regular structured grids to achieve highorder accuracy due to their efficiency and simplicity, as well as the capability to approximate variable-coefficient differential operators. In this work, a 4th-order compact FD scheme for the variable-coefficient Helmholtz equation on a Cartesian grid in 2D is derived and tested. The well known limitation of finite differences is that they lose accuracy when the boundary curve does not coincide with the discretization grid, which is a severe restriction on the geometry of the computational domain. Therefore, the algorithm presented in this work combines high-order FD schemes with the method of difference potentials (DP), which retains the efficiency of FD while allowing for boundary shapes that are not aligned with the grid without sacrificing the accuracy of the FD scheme. Additionally, the theory of DP allows for the universal treatment of the boundary conditions. One of the significant contributions of this work is the development of an implementation that accommodates general boundary conditions (BCs). In particular, Robin BCs with discontinuous coefficients are studied, for which we introduce a piecewise parameterization of the boundary curve. Problems with discontinuities in the boundary data itself are also studied. We observe that the design convergence rate suffers whenever the solution loses regularity due to the boundary conditions. This is
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.
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. PMID:16310203
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.
An implicit semianalytic numerical method for the solution of nonequilibrium chemistry problems
NASA Technical Reports Server (NTRS)
Graves, R. A., Jr.; Gnoffo, P. A.; Boughner, R. E.
1974-01-01
The first order differential equation form systems of equations. They are solved by a simple and relatively accurate implicit semianalytic technique which is derived from a quadrature solution of the governing equation. This method is mathematically simpler than most implicit methods and has the exponential nature of the problem embedded in the solution.
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 solutions of acoustic wave propagation problems using Euler computations
NASA Technical Reports Server (NTRS)
Hariharan, S. I.
1984-01-01
This paper reports solution procedures for problems arising from the study of engine inlet wave propagation. The first problem is the study of sound waves radiated from cylindrical inlets. The second one is a quasi-one-dimensional problem to study the effect of nonlinearities and the third one is the study of nonlinearities in two dimensions. In all three problems Euler computations are done with a fourth-order explicit scheme. For the first problem results are shown in agreement with experimental data and for the second problem comparisons are made with an existing asymptotic theory. The third problem is part of an ongoing work and preliminary results are presented for this case.
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.
Analytical Solutions Involving Shock Waves for Testing Debris Avalanche Numerical Models
NASA Astrophysics Data System (ADS)
Mungkasi, Sudi; Roberts, Stephen Gwyn
2012-10-01
Analytical solutions to debris avalanche problems involving shock waves are derived. The debris avalanche problems are described in two different coordinate systems, namely, the standard Cartesian and topography-linked coordinate systems. The analytical solutions can then be used to test debris avalanche numerical models. In this article, finite volume methods are applied as the numerical models. We compare the performance of the finite volume method with reconstruction of the conserved quantities based on stage, height, and velocity to that of the conserved quantities based on stage, height, and momentum for solving the debris avalanche problems involving shock waves. The numerical solutions agree with the analytical solution. In addition, both reconstructions lead to similar numerical results. This article is an extension of the work of Mangeney et al. (Pure Appl Geophys 157(6-8):1081-1096, 2000).
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.
NASA Astrophysics Data System (ADS)
Lee, Dongwook
2013-06-01
In this paper, we extend the unsplit staggered mesh scheme (USM) for 2D magnetohydrodynamics (MHD) [D. Lee, A.E. Deane, An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics, J. Comput. Phys. 228 (2009) 952-975] to a full 3D MHD scheme. The scheme is a finite-volume Godunov method consisting of a constrained transport (CT) method and an efficient and accurate single-step, directionally unsplit multidimensional data reconstruction-evolution algorithm, which extends Colella's original 2D corner transport upwind (CTU) method [P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys. 87 (1990) 446-466]. We present two types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTU scheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of a simple 3D extension of Collela's 2D CTU method and is considered as a direct extension from the 2D USM scheme. The full 3D CTU scheme is our primary 3D solver which includes all multidimensional cross-derivative terms for stability. The latter method is logically analogous to the 3D unsplit CTU method by Saltzman [J. Saltzman, An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys. 115 (1994) 153-168]. The major novelties in our algorithms are twofold. First, we extend the reduced CTU scheme to the full CTU scheme which is able to run with CFL numbers close to unity. Both methods utilize the transverse update technique developed in the 2D USM algorithm to account for transverse fluxes without solving intermediate Riemann problems, which in turn gives cost-effective 3D methods by reducing the total number of Riemann solves. The proposed algorithms are simple and efficient especially when including multidimensional MHD terms that maintain in-plane magnetic field dynamics. Second, we introduce a new CT scheme that makes use of proper upwind information in taking averages of electric fields. Our 3D USM schemes can be easily
Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations
NASA Technical Reports Server (NTRS)
Anderson, W. Kyle; Batina, John T.
1988-01-01
Parameter studies are conducted using the Euler and potential flow equation models for steady and unsteady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux vector splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flow results are made, as well as with experimental data where available.
Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations
NASA Technical Reports Server (NTRS)
Anderson, W. Kyle; Batina, John T.
1988-01-01
Parameter studies are conducted using the Euler and potential flow equation models for unsteady and steady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux-vector-splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flows results are made as well as with experimental data where available.
NASA Astrophysics Data System (ADS)
Zhang, Na; Yao, Jun; Huang, Zhaoqin; Wang, Yueying
2013-06-01
Numerical simulation in naturally fractured media is challenging because of the coexistence of porous media and fractures on multiple scales that need to be coupled. We present a new approach to reservoir simulation that gives accurate resolution of both large-scale and fine-scale flow patterns. Multiscale methods are suitable for this type of modeling, because it enables capturing the large scale behavior of the solution without solving all the small features. Dual-porosity models in view of their strength and simplicity can be mainly used for sugar-cube representation of fractured media. In such a representation, the transfer function between the fracture and the matrix block can be readily calculated for water-wet media. For a mixed-wet system, the evaluation of the transfer function becomes complicated due to the effect of gravity. In this work, we use a multiscale finite element method (MsFEM) for two-phase flow in fractured media using the discrete-fracture model. By combining MsFEM with the discrete-fracture model, we aim towards a numerical scheme that facilitates fractured reservoir simulation without upscaling. MsFEM uses a standard Darcy model to approximate the pressure and saturation on a coarse grid, whereas fine scale effects are captured through basis functions constructed by solving local flow problems using the discrete-fracture model. The accuracy and the robustness of MsFEM are shown through several examples. In the first example, we consider several small fractures in a matrix and then compare the results solved by the finite element method. Then, we use the MsFEM in more complex models. The results indicate that the MsFEM is a promising path toward direct simulation of highly resolution geomodels.
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.
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
Some notes on the numerical solution of shear-lag and mathematically related problems
NASA Technical Reports Server (NTRS)
Kuhn, Paul
1939-01-01
The analysis of box beams with shear deformation of the flanges can be reduced to the solution of a differential equation. The same equation is met in other problems of stress analysis. No analytical solutions of this equation can be given for practical cases, and numerical methods of evaluation must be used. Available methods are briefly discussed. Two numerical examples show the application of the step-by-step method of integration to shear-lag problems.
Numerical solution of a coupled pair of elliptic equations from solid state electronics
NASA Technical Reports Server (NTRS)
Phillips, T. N.
1983-01-01
Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways, by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem.
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.
Numerical solutions of reaction-diffusion equations: Application to neural and cardiac models
NASA Astrophysics Data System (ADS)
Ji, Yanyan Claire; Fenton, Flavio H.
2016-08-01
We describe the implementation of the explicit Euler, Crank-Nicolson, and implicit alternating direction methods for solving partial differential equations and apply these methods to obtain numerical solutions of three excitable-media models used to study neurons and cardiomyocyte dynamics. We discuss the implementation, accuracy, speed, and stability of these numerical methods.
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 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.
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
Lim, S.G.; Prahl, J.M.; Brewe, D.E. Case Western Reserve University, Cleveland, OH U.S. Army, Propulsion Directorate, Cleveland, OH )
1991-04-01
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. 17 refs.
A Plane-Parallel Wind Solution for Testing Numerical Simulations of Photoevaporation
NASA Astrophysics Data System (ADS)
Hutchison, Mark A.; Laibe, Guillaume
2016-04-01
Here, we derive a Parker-wind-like solution for a stratified, plane-parallel atmosphere undergoing photoionisation. The difference compared to the standard Parker solar wind is that the sonic point is crossed only at infinity. The simplicity of the analytic solution makes it a convenient test problem for numerical simulations of photoevaporation in protoplanetary discs.
Numerical solutions to ill-posed and well-posed impedance boundary condition integral equations
NASA Astrophysics Data System (ADS)
Rogers, J. R.
1983-11-01
Exterior scattering from a three-dimensional impedance body can be formulated in terms of various integral equations derived from the Leontovich impedance boundary condition (IBC). The electric and magnetic field integral equations are ill-posed because they theoretically admit spurious solutions at the frequencies of interior perfect conductor cavity resonances. A combined field formulation is well-posed because it does not allow the spurious solutions. This report outlines the derivation of IBC integral equations and describes a procedure for constructing moment-method solutions for bodies of revolution. Numerical results for scattering from impedance spheres are presented which contrast the stability and accuracy of solutions to the ill-posed equations with those of the well-posed equation. The results show that numerical solutions for exterior scattering to the electric and magnetic field integral equations can be severely contaminated by spurious resonant solutions regardless of whether the surface impedance of the body is lossy or lossless.
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.
A numerical guide to the solution of the bi-domain equations of cardiac electrophysiology.
Pathmanathan, Pras; Bernabeu, Miguel O; Bordas, Rafel; Cooper, Jonathan; Garny, Alan; Pitt-Francis, Joe M; Whiteley, Jonathan P; Gavaghan, David J
2010-01-01
Simulation of cardiac electrical activity using the bi-domain equations can be a massively computationally demanding problem. This study provides a comprehensive guide to numerical bi-domain modelling. Each component of bi-domain simulations--discretization, ODE-solution, linear system solution, and parallelization--is discussed, and previously-used methods are reviewed, new methods are proposed, and issues which cause particular difficulty are highlighted. Particular attention is paid to the choice of stimulus currents, compatibility conditions for the equations, the solution of singular linear systems, and convergence of the numerical scheme. PMID:20553747
A numerical technique for obaining the complete bifurcated equilibrium solution space for a Tokamak
Helton, F.J.; Greene, J.M.
1992-01-01
This paper describes a new numerical method for finding solutions to the ideal MHD equilibrium problem The space of solutions thus found can have more than one bifurcation branch, depending on the tokamak being modeled. We suggest that the solutions which are difficult to obtain without the use of this technique correspond to equilibria which are difficult to maintain in the tokamak being modeled. First, this paper investigates, for a tokamak design with large poloidal field-shaping coil (PFC) to plasma distance, the bifurcated numerical solution curve as a function of flux loop position and relates this curve to the practical existence of ideal magnetohydrodynamic equilibria and practical tokamak design. In previous papers, some of the problems which could arise for large PFC-plasma distance were discussed. Then, using a regularization technique, it was shown that, for large PFC-plasma distance, the flux loops should be close to the PFCs for stable control if the full information form the flux loops is used. Here it is shown that, for large PFC-plasma distance, the structure of the equilibrium solution space becomes increasingly complex and desirable solutions become more difficult to attain as the flux loops are moved farther from the plasma. In order to explore this solution space numerically, it is necessary to obtain solutions for which the usual Picard iteration method is unstable. Here an extension of this method is given. The solution space is enlarged by adding additional variable and constraints, so that the iteration to the desired solution is stable in the extended space. A modified version of this numerical technique has been used to obtain equilibrium fits to highly elongated DIII-D plasmas. The numerical equilibria are very difficult to obtain without the use of this technique and the plasmas are difficult to maintain in the tokamak. 10 refs., 6 figs., 2 tabs.
NASA Astrophysics Data System (ADS)
Mishchenko, Michael I.; Dlugach, Janna M.; Chowdhary, Jacek; Zakharova, Nadezhda T.
2015-05-01
We describe a simple yet efficient numerical algorithm for computing polarized bidirectional reflectance of an optically thick (semi-infinite), macroscopically flat layer composed of statistically isotropic and mirror symmetric random particles. The spatial distribution of the particles is assumed to be sparse, random, and statistically uniform. The 4×4 Stokes reflection matrix is calculated by iterating the Ambartsumian's vector nonlinear integral equation. The result is a numerically exact solution of the vector radiative transfer equation and as such fully satisfies the energy conservation law and the fundamental reciprocity relation. Since this technique bypasses the computation of the internal radiation field, it is very fast and highly accurate. The FORTRAN implementation of the technique is publicly available on the World Wide Web at
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.
Analytical and numerical Gubser solutions of the second-order hydrodynamics
NASA Astrophysics Data System (ADS)
Pang, Long-Gang; Hatta, Yoshitaka; Wang, Xin-Nian; Xiao, Bo-Wen
2015-04-01
Evolution of quark-gluon plasma near equilibrium can be described by the second-order relativistic viscous hydrodynamic equations. Consistent and analytically verifiable numerical solutions are critical for phenomenological studies of the collective behavior of quark-gluon plasma in high-energy heavy-ion collisions. A novel analytical solution based on the conformal Gubser flow that is a boost-invariant solution with transverse fluid velocity is presented. Because of the nonlinear nature of the equation, the analytical solution is nonperturbative and exhibits features that are rather distinct from solutions to usual linear hydrodynamic equations. It is used to verify with high precision the numerical solution with a newly developed state-of-the-art (3 +1 ) -dimensional second-order viscous hydro code (CLVisc). The perfect agreement between the analytical and numerical solutions demonstrates the reliability of the numerical simulations with the second-order viscous corrections. This lays the foundation for future phenomenological studies that allow one to gain access to the second-order transport coefficients.
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
NASA Astrophysics Data System (ADS)
Berczyński, Paweł; Kravtsov, Yury A.; Żeglinski, Grzegorz
2008-09-01
The method of paraxial complex geometrical optics (CGO) is presented, which describes Gaussian beam diffraction in arbitrary smoothly inhomogeneous media, including lens-like waveguides. By way of an example, the known analytical solution for Gaussian beam diffraction in free space is presented. Paraxial CGO reduces the problem of Gaussian beam diffraction in inhomogeneous media to the system of the first order ordinary differential equations, which can be readily solved numerically. As a result, CGO radically simplifies the description of Gaussian beam diffraction in inhomogeneous media as compared to the numerical methods of wave optics. For the paraxial on-axis Gaussian beam propagation in lens-like waveguide, we compare CGO solutions with numerical results for finite differences beam propagation method (FD-BPM). The CGO method is shown to provide 50-times higher rate of calculation then FD-BPM at comparable accuracy. Besides, paraxial eikonal-based complex geometrical optics is generalized for nonlinear Kerr type medium. This paper presents CGO analytical solutions for cylindrically symmetric Gaussian beam in Kerr type nonlinear medium and effective numerical solutions for the self-focusing effect of Gaussian beam with elliptic cross section. Both analytical and numerical solutions are shown to be in a good agreement with previous results, obtained by other methods.
Complete numerical solution of the diffusion equation of random genetic drift.
Zhao, Lei; Yue, Xingye; Waxman, David
2013-08-01
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
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. PMID:23387826
NASA Astrophysics Data System (ADS)
Tirani, M. Dadkhah; Sohrabi, F.; Almasieh, H.; Kajani, M. Tavassoli
2015-10-01
In this paper, a collocation method based on Taylor polynomials is developed for solving systems linear differential-difference equations with variable coefficients defined in large intervals. By using Taylor polynomials and their properties in obtaining operational matrices, the solution of the differential-difference equation system with given conditions is reduced to the solution of a system of linear algebraic equations. We first divide the large interval into M equal subintervals and then Taylor polynomials solutions are obtained in each interval, separately. Some numerical examples are given and results are compared with analytical solutions and other techniques in the literature to demonstrate the validity and applicability of the proposed method.
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.
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.
Geometric invariants for initial data sets: analysis, exact solutions, computer algebra, numerics
NASA Astrophysics Data System (ADS)
Valiente Kroon, Juan A.
2011-09-01
A personal perspective on the interaction of analytical, numerical and computer algebra methods in classical Relativity is given. This discussion is inspired by the problem of the construction of invariants that characterise key solutions to the Einstein field equations. It is claimed that this kind of ideas will be or importance in the analysis of dynamical black hole spacetimes by either analytical or numerical methods.
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.
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.
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.
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.
NASA Technical Reports Server (NTRS)
Nacozy, P. E.
1984-01-01
The equations of motion are developed for a perfectly flexible, inelastic tether with a satellite at its extremity. The tether is attached to a space vehicle in orbit. The tether is allowed to possess electrical conductivity. A numerical solution algorithm to provide the motion of the tether and satellite system is presented. The resulting differential equations can be solved by various existing standard numerical integration computer programs. The resulting differential equations allow the introduction of approximations that can lead to analytical, approximate general solutions. The differential equations allow more dynamical insight of the motion.
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.-
Numerical solution of a coupled pair of elliptic equations from solid state electronics
NASA Technical Reports Server (NTRS)
Phillips, T. N.
1984-01-01
Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways, by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem. Previously announced in STAR as N83-30109
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.
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. PMID:27504247
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.
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.
Numerical precision of the solution to the running-coupling Balitsky-Kovchegov equation
NASA Astrophysics Data System (ADS)
Matas, Marek; Cepila, Jan; Guillermo Contreras Nuno, Jesus
2016-03-01
We use the running coupling Balitsky-Kovchegov (rcBK) equation to study the rapidity dependence of saturation in inclusive HERA data and we discuss the behaviour of its numerical solution. The rcBK equation has been solved using Runge-Kutta methods. The influence of the parameters implicit in the numerical evolution has been studied. They include, among others, the order of the Runge-Kutta evolution, the size of the different grids and the step in the numerical evolution. Some suggestions on the minimum value of these parameters are put forward.
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. PMID:25162064
Large-Scale Numerical Modeling of Melt and Solution Crystal Growth
NASA Astrophysics Data System (ADS)
Derby, Jeffrey J.; Chelikowsky, James R.; Sinno, Talid; Dai, Bing; Kwon, Yong-Il; Lun, Lisa; Pandy, Arun; Yeckel, Andrew
2007-06-01
We present an overview of mathematical models and their large-scale numerical solution for simulating different phenomena and scales in melt and solution crystal growth. Samples of both classical analyses and state-of-the-art computations are presented. It is argued that the fundamental multi-scale nature of crystal growth precludes any one approach for modeling, rather successful crystal growth modeling relies on an artful blend of rigor and practicality.
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.
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
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.
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.
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...