Uniform stability of a class of parabolic difference operators

Conditions on a class of parabolic difference operators are given which are sufficient for maximumnorm stability, uniform in the mesh-widths. This paper is a generalization of a previous result by Hakberg [4].     

Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces

The well-posedness of the nonlocal boundary-value problem for abstract parabolic differential equations in Bochner spaces is established. The first and second order of accuracy difference schemes for the approximate solutions of this problem are considered. The coercive inequalities for the solutions of these difference schemes are established. In applications, the almost coercive stability and coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary-value problem for parabolic equation are obtained.     

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.     

The hyperbolic–elliptic equation with the nonlocal condition

The nonlocal boundary value problem for a hyperbolic–elliptic equation in a Hilbert space is considered. The stability estimate for the solution of the given problem is obtained. The first and second orders of difference schemes approximately solving this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established. The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Properties of Difference Schemes with Oblique Stencils for Hyperbolic Equations

In this paper, various difference schemes with oblique stencils, i.e., schemes using different space grids at different time levels, are studied. Such schemes may be useful in solving boundary value problems with moving boundaries, regular grids of a non-standard structure (for example, triangular or cellular ones), and adaptive methods. To study the stability of finite difference schemes with oblique stencils, we analyze the first differential approximation and dispersion. We study stability conditions as constraints on the geometric locations of stencil elements with respect to characteristics of the equation. We compare our results with a geometric interpretation of the stability of some classical schemes. The paper also presents generalized oblique schemes for a quasilinear equation of transport and the results of numerical experiments with these schemes.     

Stable difference schemes for the hyperbolic problems subject to nonlocal boundary conditions with self-adjoint operator

In the present paper the first and second orders of accuracy difference schemes for the numerical solution of multidimensional hyperbolic equations with nonlocal boundary and Dirichlet conditions are presented. The stability estimates for the solution of difference schemes are obtained. A method is used for solving these difference schemes in the case of one dimensional hyperbolic equation.     

THE UNCONDITIONAL STABLE DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR TWO DIMENSIONAL SEMILINEAR PARABOLIC SYSTEMS

In this paper we are going to discuss the difference schemes with intrinsic parallelism for the boundary value problem of the two dimesional semilinear parabolic systems.The unconditional stability of the general finite difference schemes with intrinsic parallelism is justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problems in the discrete W2^(2,1) norms.Then the uniqueness of the discrete vector solution of this difference scheme follows as the consequence of the stability.     

广义K.d.V.方程的数值方法

本文研究一般的广义K.d.V.方程的数值方法,给出了广义K.d.V.方程的一类半离散差分格式,证明了它们的守恒性。作者还严格证明了这类格式的广义稳定性,并由此推出收敛性。文章的最后考虑了全离散情形和两步格式。     

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.     

Stability of Implicit Difference Schemes with Space and Time-Dependent Coefficients

A stability theorem is derived for implicit difference schemes approximating multidimensional initial-value problems for linear hyperbolic systems with variable coefficients, and lots of widely used difference schemes are proved to be stable under the conditions similar to those for the cases of constant coefficients. This theorem is an extension of the stability theorem due to Lax-Nirenberg. The proof is quite simple.     

A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

PARALLEL COMPUTING METHOD OF PURE ALTERNATIVE SEGMENT EXPLICIT-IMPLICIT DIFFERENCE SCHEME FOR NONLINEAR LELAND EQUATION

The research on the numerical solution of the nonlinear Leland equation has important theoretical significance and practical value. To solve nonlinear Leland equation, this paper offers a class of difference schemes with parallel nature which are pure alternative segment explicit-implicit(PASE-I) and implicit-explicit(PASI-E) schemes. It also gives the existence and uniqueness,the stability and the error estimate of numerical solutions for the parallel difference schemes. Theoretical analysis demonstrates that PASE-I and PASI-E schemes have obvious parallelism, unconditionally stability and second-order convergence in both space and time. The numerical experiments verify that the calculation accuracy of PASE-I and PASI-E schemes are better than that of the existing alternating segment Crank-Nicolson scheme, alternating segment explicit-implicit and implicit-explicit schemes. The speedup of PASE-I scheme is 9.89, compared to classical Crank-Nicolson scheme. Thus the schemes given by this paper are high efficient and practical for solving the nonlinear Leland equation.     

STABILITY ANALYSIS OF YEE SCHEMES TO PML AND UPML FOR MAXWELL EQUATIONS

We utilize Fourier methods to analyze the stability of the Yee difference schemes for Berenger PML （perfectly matched layer） as well as the UPML （uniaxial perfectly matched layer） systems of two-dimensional Maxwell equations. Using a practical spectrum stability concept, we find that the two schemes are spectrum stable under the same conditions for mesh sizes. Besides, we prove that the UPML schemes with the same damping in both directions are stable. Numerical examples are given to confirm the stability analysis for the PML method.     

Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models

In this paper nonstandard finite difference (NSFD) schemes of two metapopulation models are constructed. The stability properties of the discrete models are investigated by the use of the Lyapunov stability theorem. As a result of this we have proved that the NSFD schemes preserve essential properties of the metapopulation models (positivity, boundedness and monotone convergence of the solutions, equilibria and their stability properties). Especially, the basic reproduction number of the continuous models is also preserved. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes. The method of Lyapunov functions proves to be much simpler than the standard method for studying stability of the discrete metapopulation model in our very recent paper.     

带吸收边界条件的声波方程显式差分格式的稳定性分析

１．引言 在进行无界或半无界区域上各种波动方程的数值求解时,需引进入工边界以限制计算范围,在这些边界上应加相应的人工边界条件．这种边界条件应保证所求得的有界区域上的解很好地逼近原来无界区域上的解．对波动方程来说,就是在边界上人工反射应尽可能地小,使之对区域内部解的影响在允许的误差范围以内.因而它们被称为无反射边界条件或吸收边界条件．这种边界条件还应保证所形成的有界区域上的微分方程定解问题是适定的．这也是各种数值方法稳定的必要条件。 近二十多年来,已发展了声波方程的各种类型的吸收边界条件,其中以Ｃｌ…     

Additive Difference Schemes and Iteration Methods for Problems of Mathematical Physics

We construct additive difference schemes for first-order differential–operator equations. The exposition is based on the general theory of stability for operator–difference schemes in lattice Hilbert spaces. The main focus is on the case of additive decomposition with an arbitrary number of mutually noncommuting operator terms. Additive schemes for second-order evolution equations are considered in the same way.     

Parameterized families of finite difference schemes for the wave equation

This article is devoted to an analysis of simple families of finite difference schemes for the wave equation. These families are dependent on several free parameters, and methods for obtaining stability bounds as a function of these parameters are discussed in detail. Access to explicit stability bounds such as those derived here may, it is hoped, lead to optimization techniques for so‐called spectral‐like methods, which are difference schemes dependent on many free parameters (and for which maximizing the order of accuracy may not be the defining criterion). Though the focus is on schemes for the wave equation in one dimension, the analysis techniques are extended to two dimensions; implicit schemes such as ADI methods are examined in detail. Numerical results are presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 463–480, 2004.     

Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.     

Unconditional stability of parallel alternating difference schemes for semilinear parabolic systems

The general alternating schemes with intrinsic parallelism for semilinear parabolic systems are studied. First we prove the a priori estimates in the discrete H¹ space of the difference solution for these schemes. Then the existence of the difference solution for these schemes follows from the fixed point principle. Finally the unconditional stability of the general alternating schemes is proved. The alternating group explicit scheme, the alternating segment explicit–implicit scheme and the alternating segment Crank–Nicolson scheme are the special cases of the general alternating schemes.     

Discrete shock profiles for systems of conservation laws

The existence of discrete shock profiles for difference schemes approximating a system of conservation laws is the major topic studied in this paper. The basic theorem established here applies to first-order accurate difference schemes; for weak shocks, this theorem provides necessary and sufficient conditions involving the truncation error of the linearized scheme which guarantee entropy satisfying or entropy violating discrete shock profiles. Several explicit difference schemes are used as examples illustrating the interplay between the entropy condition, monotonicity, and linearized stability. Entropy violating stationary shocks for second-order accurate Lax-Wendroff schemes approximating systems are also constructed. The only tools used in the proofs are local analysis and the center manifold theorem.