Additive schemes (splitting schemes) for some systems of evolutionary equations

TL;DR

This paper develops additive splitting schemes for systems of evolutionary PDEs, including cases with coupled derivatives in time, enabling efficient numerical solutions for complex initial-boundary value problems.

Contribution

It introduces a novel approach to splitting schemes that handle time-derivative coupling in systems of PDEs, extending existing methods for space-coupled operators.

Findings

Effective algorithms for coupled time-derivative systems

Splitting schemes based on additive operator representations

Applicable to a broader class of evolutionary equations

Abstract

On the basis of additive schemes (splitting schemes) we construct efficient numerical algorithms to solve approximately the initial-boundary value problems for systems of time-dependent partial differential equations (PDEs). In many applied problems the individual components of the vector of unknowns are coupled together and then splitting schemes are applied in order to get a simple problem for evaluating components at a new time level. Typically, the additive operator-difference schemes for systems of evolutionary equations are constructed for operators coupled in space. In this paper we investigate more general problems where coupling of derivatives in time for components of the solution vector takes place. Splitting schemes are developed using an additive representation for both the primary operator of the problem and the operator at the time derivative. Splitting schemes are based on a triangular two-component representation of the operators.