High-order explicit local time-stepping methods for damped wave equations

TL;DR

This paper introduces high-order explicit local time-stepping methods for damped wave equations, enabling efficient and stable simulations on locally refined meshes by using smaller time-steps only where needed.

Contribution

It develops arbitrarily high-order local time-stepping schemes based on Adams-Bashforth methods for damped wave equations, compatible with finite element discretizations.

Findings

Methods are fully explicit and parallelizable.

Numerical experiments validate stability and accuracy.

Applicable to continuous and discontinuous Galerkin discretizations.

Abstract

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations validate the theory and illustrate the usefulness of these local time-stepping methods.