Spectral Stability Conditions for an Explicit Three-Level Finite-Difference Scheme for a Multidimensional Transport Equation with Perturbations

A A Zlotnik,B N Chetverushkin

doi:10.1134/s0012266121070065

Abstract

We study difference schemes associated with a simplified linearized multidimensionalhyperbolic quasi-gasdynamic system of differential equations. It is shown that an explicit two-levelvector difference scheme with flux relaxation for a second-order hyperbolic equation with variablecoefficients that is a perturbation of the transport equation with a parameter multiplying thehighest derivatives can be reduced to an explicit three-level difference scheme. In the case ofconstant coefficients, the spectral condition for the time-uniform stability of this explicitthree-level difference scheme is analyzed, and both sufficient and necessary conditions for thiscondition to hold are derived, in particular, in the form of Courant type conditions on the ratio oftemporal and spatial steps.

Highlights

The hyperbolic quasi-gasdynamic system is a specially perturbed system of gasdynamic equations in which the terms containing second derivatives with respect to the space and time variables are multiplied by a small parameter τ > 0
This system is used for constructing a family of three-level and two-level vector difference schemes for the numerical solution of various problems in gas dynamics [1]
Instead of the quasi-gasdynamic system, we consider the simplified case of one second-order hyperbolic equation with variable coefficients that is a perturbation of the transport equation with the parameter τ multiplying the second derivatives

Summary

INTRODUCTION

The hyperbolic quasi-gasdynamic system is a specially perturbed system of gasdynamic equations in which the terms containing second derivatives with respect to the space and time variables are multiplied by a small parameter τ > 0. The present paper is intended to make some progress in this direction To this end, instead of the quasi-gasdynamic system, we consider the simplified case of one second-order hyperbolic equation with variable coefficients that is a perturbation of the transport equation with the parameter τ multiplying the second derivatives. Most importantly, we analyze the spectral condition for the time-uniform stability of this explicit three-level scheme in the case of constant coefficients Both sufficient and close-to-them necessary conditions for the validity of the spectral condition are obtained, in particular, in the form of Courant type conditions on the ratio of temporal and spatial steps.

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