Compact Difference Schemes on a Three-Point Stencil for Second-Order Hyperbolic Equations

P P Matus,Hoang Thi Kieu Anh Hoang Thi Kieu Anh

doi:10.1134/s0012266121070090

P P Matus, Hoang Thi Kieu Anh Hoang Thi Kieu Anh

Open Access

https://doi.org/10.1134/s0012266121070090

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Journal: Differential Equations	Publication Date: Jul 1, 2021
Citations: 8	License type: open-access

Affiliation: Belarusian State University

Abstract

We consider compact difference schemes of approximation order 4+2 on a three-point spatial stencil for theKlein–Gordon equations with constant and variable coefficients. New compact schemes areproposed for one type of second-order quasilinear hyperbolic equations. In the case of constantcoefficients, we prove the strong stability of the difference solution under small perturbations ofthe initial conditions, the right-hand side, and the coefficients of the equation. A priori estimatesare obtained for the stability and convergence of the difference solution in strong mesh norms.

Highlights

Improving the accuracy of numerical methods for solving problems of mathematical physics on minimal stencils has always been a topical problem in numerical analysis
A special place among the methods for constructing difference schemes of higher approximation order is occupied by so-called compact schemes, which are written on a stencil that differs insignificantly from the ones traditional for the equation in question [5]
We study compact difference schemes of approximation order 4 + 2 on the usual three-point stencil for various types of the Klein–Gordon equation

Summary

INTRODUCTION

Improving the accuracy of numerical methods for solving problems of mathematical physics on minimal stencils has always been a topical problem in numerical analysis (see, e.g., [1,2,3,4]). We study compact difference schemes of approximation order 4 + 2 on the usual three-point stencil for various types of the Klein–Gordon equation. Some results on the topic were announced in [12, 13] For this equation, the differential and difference problems with variable coefficients are linear, one fails to obtain relevant a priori estimates by applying well-known results from Samarskii’s stability theory of three-level operator-difference schemes [1, Ch. VI, Sec. 3]. We apply the method of energy inequalities to compact difference schemes approximating the Klein–Gordon equations with variable coefficients to obtain a priori estimates for the stability and convergence of the difference solution in the mesh norms of L2(ωh), W21(ωh), C(ωh), or L∞(ωh).

Statement of the Problem and the Difference Scheme

Stability with Respect to the Initial Data and the Right-Hand Side

Strong Stability

Convergence Theorem

Stability

QUASILINEAR KLEIN–GORDON EQUATIONS

COMPUTATIONAL EXPERIMENT