Unconditional Stability for Multistep ImEx Schemes: Theory

TL;DR

This paper introduces high-order linear ImEx multistep schemes with large unconditional stability regions, enabling stable and accurate time integration for stiff problems such as variable-coefficient PDEs and Navier-Stokes equations.

Contribution

The paper develops a new class of high-order ImEx schemes with large unconditional stability regions and provides a stability analysis framework and explicit coefficients up to fifth order.

Findings

Schemes achieve unconditional stability for stiff problems.

Introduction of an unconditional stability region concept.

Coefficients for schemes up to fifth order are provided.

Abstract

This paper presents a new class of high order linear ImEx multistep schemes with large regions of unconditional stability. Unconditional stability is a desirable property of a time stepping scheme, as it allows the choice of time step solely based on accuracy considerations. Of particular interest are problems for which both the implicit and explicit parts of the ImEx splitting are stiff. Such splittings can arise, for example, in variable-coefficient problems, or the incompressible Navier-Stokes equations. To characterize the new ImEx schemes, an unconditional stability region is introduced, which plays a role analogous to that of the stability region in conventional multistep methods. Moreover, computable quantities (such as a numerical range) are provided that guarantee an unconditionally stable scheme for a proposed implicit-explicit matrix splitting. The new approach is illustrated with several examples. Coefficients of the new schemes up to fifth order are provided.