Generalized Jacobi-Gauss-Lobatto interpolation

We introduce the generalized Jacobi-Gauss-Lobatto interpolation involving the values of functions and their derivatives at the endpoints, which play important roles in the Jacobi pseudospectral methods for high order problems. We establish some results on these interpolations in non-uniformly weighted Sobolev spaces, which serve as the basic tools in analysis of numerical quadratures and various numerical methods of differential and integral equations.     

Non-Isotropic Jacobi Spectral and Pseudospectral Methods in Three Dimensions

Non-isotropic Jacobi orthogonal approximation and Jacobi-Gauss type interpolation in three dimensions are investigated. The basic approximation results are established, which serve as the mathematical foundation of spectral and pseudospectral methods for singular problems, as well as problems defined on axisymmetric domains and some unbounded domains. The spectral and pseudospectral schemes are provided for two model problems. Their spectral accuracy is proved. Numerical results demonstrate the high efficiency of suggested algorithms.     

High‐order energy‐preserving schemes for the improved Boussinesq equation

This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.     

Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well.     

A new high order compact off-step discretization for the system of 3D quasi-linear elliptic partial differential equations

We present a new fourth order compact finite difference scheme based on off-step discretization for the solution of the system of 3D quasi-linear elliptic partial differential equations subject to appropriate Dirichlet boundary conditions. We also develop new fourth order methods to obtain the numerical solution of first order normal derivatives of the solution. In all the cases, we use only 19-grid points of a single computational cell to compute the problem. The proposed methods are directly applicable to singular problems and the problems in polar coordinates, without any modification required unlike the previously developed high order schemes of [14] and [30]. We discuss the convergence analysis of the proposed method in details. Many physical problems are solved and comparative results are given to illustrate the usefulness of the proposed methods.     

A new domain decomposition algorithm for generalized Burger’s–Huxley equation based on Chebyshev polynomials and preconditioning

In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge–Kutta method for time integration to solve the generalized Burger’s–Huxley equation (GBHE). To reduce round-off error in spectral collocation (pseudospectral) method we use preconditioning. Firstly, theory of application of Chebyshev spectral collocation method with preconditioning (CSCMP) and domain decomposition on the generalized Burger’s–Huxley equation presented. This method yields a system of ordinary differential algebric equations (DAEs). Secondly, we use fourth order Runge–Kutta formula for the numerical integration of the system of DAEs. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.     

一类非线性伪抛物型方程的伪谱解法

本文考虑了一类非线性伪抛物型方程的Fourier伪谱方法,建立了该方程的Fourier伪谱方法的半离散格式和全离散格式．并利用Sobolev空间的正交映射理论,给出了这两种格式的误差估计．最后针对全离散格式给出了数值算例,数值结果表明Fourier伪谱格式能正确加解密,且计算误差较小,效率较高,具有较好的稳定性,可用于提高热流密码体制的加解密效率．     

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

We propose a 9‐point fourth‐order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two‐level high‐order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth‐order accurate in space and second‐ or lower‐order accurate in time depending on the choice of a weighted average parameter μ. Unconditional stability is proved for 0.5 ≤ μ ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high‐order accuracy of the schemes are presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 366–378, 2007     

One-step Taylor–Galerkin methods for convection–diffusion problems

Third and fourth order Taylor–Galerkin schemes have shown to be efficient finite element schemes for the numerical simulation of time-dependent convective transport problems. By contrast, the application of higher-order Taylor–Galerkin schemes to mixed problems describing transient transport by both convection and diffusion appears to be much more difficult. In this paper we develop two new Taylor–Galerkin schemes maintaining the accuracy properties and improving the stability restrictions in convection–diffusion. We also present an efficient algorithm for solving the resulting system of the finite element method. Finally we present two numerical simulations that confirm the properties of the methods.     

Monotone Finite Difference Schemes for Nonlinear Systems with Mixed Quasi-Monotonicity

Monotone finite difference schemes are proposed for nonlinear systems with mixed quasi-monotonicity. Two monotone iteration processes for the corresponding discrete problems are presented, which converge monotonically to the quasi-solutions of the discrete problems. The limits are the exact solutions under some conditions. A monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed. The numerical results coincide with theoretical analysis.     

Spectral and Pseudospectral Approximations Using Hermite Functions: Application to the Dirac Equation

We consider in this paper spectral and pseudospectral approximations using Hermite functions for PDEs on the whole line. We first develop some basic approximation results associated with the projections and interpolations in the spaces spanned by Hermite functions. These results play important roles in the analysis of the related spectral and pseudospectral methods. We then consider, as an example of applications, spectral and pseudospectral approximations of the Dirac equation using Hermite functions. In particular, these schemes preserve the essential conservation property of the Dirac equation. We also present some numerical results which illustrate the effectiveness of these methods.     

Discretization effects on approximate deconvolution modeling for LES

We study the applicability of low‐order schemes with the approximate deconvolution model (ADM) for large‐eddy simulation. As a test case compressible decaying isotropic turbulence is considered. Results obtained with low‐order finite difference schemes and a pseudospectral scheme are compared with filtered well‐resolved direct numerical simulation (DNS) data. It is found that even for low‐order schemes very good results can be obtained if the cutoff wavenumber of the filter is adjusted to the modified wavenumber of the differentiation scheme.     

A CLASS OF EXPLICIT RATIONAL SYMPLECTIC INTEGRATORS

In this paper, a class of rational explicit symplectic integrators for one-dimensional oscillatory Hamiltonian problems are presented. These methods are zero-dissipative, and of first algebraic order and high phase-lag order. By means of composition technique, we construct second and fourth order methods with high phase-lag order of this type. Based on our ideas, three applicable explicit symplectic schemes with algebraic order one, two and four are derived, respectively. We report some numerical results to illustrate the good performance of our methods.     

A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$ -Superlinear Convergence for Stochastic Games on Domains

In this work, we propose a class of numerical schemes for solving semilinear Hamilton–Jacobi–Bellman–Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the \(H^2\)-norm and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

     

WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes with a Lax–Wendroff time discretization procedure, termed WENO-LW schemes, for solving Hamilton–Jacobi equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with the original WENO with Runge–Kutta time discretizations schemes (WENO-RK) of Jiang and Peng [G. Jiang, D. Peng, Weighted ENO schemes for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2126–2143] for Hamilton–Jacobi equations, the major advantages of WENO-LW schemes are more cost effective for certain problems and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.     

Construction of a Mindlin pseudospectral plate element and evaluating efficiency of the element

In this paper, a Mindlin pseudospectral plate element is constructed to perform static, dynamic, and wave propagation analyses of plate-like structures. Chebyshev polynomials are used as basis functions and Chebyshev–Gauss–Lobatto points are used as grid points. Two integration schemes, i.e., Gauss–Legendre quadrature (GLEQ) and Chebyshev points quadrature (CPQ), are employed independently to form the elemental stiffness matrix of the present element. A lumped elemental mass matrix is generated by only using CPQ due to the discrete orthogonality of Chebyshev polynomials and overlapping of the quadrature points with the grid points. This results in a remarkable reduction of numerical operations in solving the equation of motion for being able to use explicit time integration schemes. Numerical calculations are carried out to investigate the influence of the above two numerical integration schemes in the elemental stiffness formation on the accuracy of static and dynamic response analyses. By comparing with the results of ABAQUS, this study shows that CPQ performs slightly better than GLEQ in various plates with different thicknesses, especially in thick plates. Finally, a one dimensional (1D) and a 2D wave propagation problems are used to demonstrate the efficiency of the present Mindlin pseudospectral plate element.     

Conformal structure-preserving schemes for damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise

Since many physical phenomena are often influenced by dispersive medium, energy compensation and random perturbation, exploring the dynamic behaviors of the damped-driven stochastic system has becoming a hot topic in mathematical physics in recent years. In this paper, inspired by the stochastic conformal structure, we investigate the geometric numerical integrators for the damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise. To preserve the conformal structures of the system, by using symplectic Euler method, implicit midpoint method and Fourier pseudospectral method, we propose three kinds of stochastic conformal schemes satisfying corresponding discrete stochastic multiconformal-symplectic conservation laws and discrete global/local charge conservation laws. Numerical experiments illustrate the structure-preserving properties of the proposed schemes, as well as favorable results over traditional nonconformal schemes, which are consistent with our theoretical analysis.     

A LEGENDRE PSEUDOSPECTRAL METHOD FOR SOLVINGNONLINEAR KLEIN-GORDON EQUATION

1.IntroductionAsweknow,theKlein-Gordonequationisanimportantmathematicalmodelinquantummechanics.Itisoftheformwherefi=(--1,1)",x=(xl,x25'jx.),bisarealnumber.AssumethatU000=UI(x)=0onOnandAsin[1],itcanbeshownthatifUOEHI(n)nH(n),UIEL'(fl)andfEL'(o,T;L'(n)),then(1.1)hasuniquesolutionUELoo(o,T;Hi(n)nH(fl)).oftheotherhand,somefinitedifferenceschemeswereproposedin12,31withstrictproo]ofgeneralizedstabilityandconvergence.TheirnumericalsolutionskeepthediscretEconservations.Oneofspecialcases(…     

A compact adaptive approach for degenerate singular reaction‐diffusion equations

This article concerns a compact adaptive method for the numerical solution of nonlinear degenerate singular reaction‐diffusion equations. The partial differential equation problems exhibit strong quenching blow‐up type singularities, and are critical in numerous applications ranging from optimized internal combustion designs to oil pipeline decay predictions. Adaptive schemes of fourth order in space and second order in time are acquired and discussed. Nonuniform spatial and temporal grids are utilized through suitable adaptations. Rigorous analysis is given for the numerical stability, and computational experiments are performed to illustrate our conclusions.     

A Fourier pseudospectral method for a generalized improved Boussinesq equation

In this article, we propose a Fourier pseudospectral method for solving the generalized improved Boussinesq equation. We prove the convergence of the semi‐discrete scheme in the energy space. For various power nonlinearities, we consider three test problems concerning the propagation of a single solitary wave, the interaction of two solitary waves and a solution that blows up in finite time. We compare our numerical results with those given in the literature in terms of numerical accuracy. The numerical comparisons show that the Fourier pseudospectral method provides highly accurate results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 995–1008, 2015