Unconditional Stability for Multistep ImEx Schemes: Practice

TL;DR

This paper develops strategies for achieving unconditional stability in multistep ImEx schemes, enabling efficient solutions of stiff nonlinear problems without implicit nonlinear solves.

Contribution

It introduces new ImEx multistep schemes with a free parameter and strategies for choosing splittings and parameters to ensure unconditional stability, surpassing SBDF limitations.

Findings

Higher order time stepping enabled by new strategies.

Efficient solution of stiff nonlinear problems demonstrated.

Unconditional stability achieved in practical nonlinear diffusion and flow problems.

Abstract

This paper focuses on the question of how unconditional stability can be achieved via multistep ImEx schemes, in practice problems where both the implicit and explicit terms are allowed to be stiff. For a class of new ImEx multistep schemes that involve a free parameter, strategies are presented on how to choose the ImEx splitting and the time stepping parameter, so that unconditional stability is achieved under the smallest approximation errors. These strategies are based on recently developed stability concepts, which also provide novel insights into the limitations of existing semi-implicit backward differentiation formulas (SBDF). For instance, the new strategies enable higher order time stepping that is not otherwise possible with SBDF. With specific applications in nonlinear diffusion problems and incompressible channel flows, it is demonstrated how the unconditional stability property can be leveraged to efficiently solve stiff nonlinear or nonlocal problems without the need to solve nonlinear or nonlocal problems implicitly.