SM stability for time-dependent problems

TL;DR

This paper introduces the concept of SM stability for finite difference schemes applied to time-dependent problems, focusing on selecting optimal schemes based on additional criteria for parabolic equations.

Contribution

It proposes the concept of SM stable schemes and discusses their selection criteria for time-dependent finite difference methods, especially for diffusion and convection-diffusion equations.

Findings

Defined requirements for unconditionally stable schemes

Introduced SM stability concept based on Padé approximations

Analyzed stability for diffusion and convection-diffusion problems

Abstract

Various classes of stable finite difference schemes can be constructed to obtain a numerical solution. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In this study, we use a simple boundary value problem for a one-dimensional parabolic equation to discuss the selection of an approximation with respect to time. We consider the pure diffusion equation, the pure convective transport equation and combined convection-diffusion phenomena. Requirements for the unconditionally stable finite difference schemes are formulated that are related to retaining the main features of the differential problem. The concept of SM stable finite difference scheme is introduced. The starting point are difference schemes constructed on the basis of the various Pad$\acute{e}$ approximations.