Explicit local time-stepping methods for time-dependent wave propagation

TL;DR

This paper introduces explicit local time-stepping methods for wave equations that allow larger time-steps without implicit schemes, improving efficiency and parallelizability in finite element discretizations.

Contribution

It develops high-order explicit LTS schemes for wave equations with energy conservation in undamped cases and high accuracy in damped cases, applicable to conforming and discontinuous Galerkin methods.

Findings

Numerical experiments validate the high-order accuracy of LTS schemes.

LTS methods enable larger time-steps while maintaining stability.

Schemes are fully explicit and highly parallelizable.

Abstract

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leap-frog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. 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.