Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part

TL;DR

This paper develops and analyzes implicit-explicit DIMSIMs with strong stability preserving explicit parts and stable implicit parts, suitable for stiff and non-stiff differential systems, providing methods up to order four with favorable stability.

Contribution

It introduces new SSP IMEX DIMSIMs with strong stability and stability analysis, advancing numerical methods for stiff and non-stiff differential equations.

Findings

Constructed SSP IMEX DIMSIMs up to order four

Demonstrated strong stability preserving explicit parts

Analyzed stability regions for implicit-explicit interactions

Abstract

For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems can be efficiently treated by a class of implicit-explicit (IMEX) diagonally implicit multistage integration methods (DIMSIMs), where the stiff part is integrated by implicit formula, and the non-stiff part is integrated by an explicit formula. We will construct methods where the explicit part has strong stability preserving (SSP) property, and the implicit part of the method is $A$-, or $L$-stable. We will also investigate stability of these methods when the implicit and explicit parts interact with each other. To be more precise, we will monitor the size of the region of absolute stability of the IMEX scheme, assuming that the implicit part of the method is $A$-, or $L$-stable. Finally we furnish examples of SSP IMEX DIMSIMs up to the order four with good stability properties.