Strong stability preserving explicit linear multistep methods with variable step size

TL;DR

This paper introduces the first SSP explicit linear multistep methods with variable step size, proving their optimality and stability, and demonstrating their effectiveness through numerical tests.

Contribution

It develops and analyzes the first variable step size SSP linear multistep methods of order two and three, including optimal step-size strategies and stability proofs.

Findings

Proved sharp upper bounds on step size for explicit SSP methods

Established existence of high-order SSP methods with variable step size

Numerical examples confirm the methods' effectiveness

Abstract

Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods (of order two and three) with variable step size, and prove their optimality, stability, and convergence. The choice of step size for multistep SSP methods is an interesting problem because the allowable step size depends on the SSP coefficient, which in turn depends on the chosen step sizes. The description of the methods includes an optimal step-size strategy. We prove sharp upper bounds on the allowable step size for explicit SSP linear multistep methods and show the existence of methods with arbitrarily high order of accuracy. The effectiveness of the methods is demonstrated through numerical examples.