On the numerical solution of hyperbolic IBVP with high-order stable finite difference schemes

Allaberen Ashyralyev,Ozgur Yildirim

doi:10.1186/1687-2770-2013-29

Allaberen Ashyralyev, Ozgur Yildirim

Open Access

https://doi.org/10.1186/1687-2770-2013-29

Copy DOI

Journal: Boundary Value Problems	Publication Date: Feb 15, 2013
Citations: 8	License type: CC BY 2.0

Affiliation: Fatih University

Abstract

The abstract Cauchy problem for the hyperbolic equation in a Hilbert space H with self-adjoint positive definite operator A is considered. The third and fourth orders of accuracy difference schemes for the approximate solution of this problem are presented. The stability estimates for the solutions of these difference schemes are established. A finite difference method and some results of numerical experiments are presented in order to support theoretical statements. MSC:65J10, 65M12, 65N12, 35L30.

Highlights

Partial differential equations of the hyperbolic type play an important role in many branches of science and engineering
Acoustics, electromagnetics, hydrodynamics, elasticity, fluid mechanics, and other areas of physics lead to partial differential equations of the hyperbolic type
A large cycle of works on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitude of the grid steps τ and h with respect to time and space variables are connected

Summary

Introduction

Partial differential equations of the hyperbolic type play an important role in many branches of science and engineering. A large cycle of works on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitude of the grid steps τ and h with respect to time and space variables are connected (see, e.g., [ – ] and the references therein). Of great interest is the study of absolute stable difference schemes of a high order of accuracy for hyperbolic partial differential equations, in which stability was established without any assumptions in respect of the grid steps τ and h. Such type stability inequalities for the solutions of the first order of accuracy difference scheme for the differential equations of hyperbolic type were established for the first time in [ ]. Note that boundary value problems for parabolic equations, elliptic equations, and equations of mixed type have been studied extensively by many scientists (see, e.g., [ – ] and the references therein)

The stability estimates

Method