Explicit Local Time-Stepping for the Inhomogeneous Wave Equation with Optimal Convergence

TL;DR

This paper introduces a stabilized local time-stepping method for the inhomogeneous wave equation that achieves optimal convergence rates and stability, even with complex mesh refinements and nonzero sources.

Contribution

It proposes a stabilized leapfrog-based local time-stepping scheme with optimal error estimates and a weighted transition to handle nonzero sources across mesh interfaces.

Findings

The method achieves optimal L2-error estimates under a CFL condition independent of mesh ratio.

Numerical experiments confirm the theoretical error bounds.

The approach improves stability and convergence in locally refined mesh simulations.

Abstract

Adaptivity and local mesh refinement are crucial for the efficient numerical simulation of wave phenomena in complex geometry. Local mesh refinement, however, can impose a tiny time-step across the entire computational domain when using explicit time integration. By taking smaller time-steps yet only inside locally refined regions, local time-stepping methods overcome the stringent CFL stability restriction imposed on the global time-step by a small fraction of the elements without sacrificing explicitness. In [21], a leapfrog based local time-stepping method was proposed for the inhomogeneous wave equation, which applies standard leapfrog time-marching with a smaller time-step inside the refined region. Here, to remove potential instability at certain time-steps, a stabilized version is proposed which leads to optimal L2-error estimates under a CFL condition independent of the coarse-to-fine mesh ratio. Moreover, a weighted transition is introduced to restore optimal H1-convergence when the source is nonzero across the coarse-to-fine mesh interface. Numerical experiments corroborate the theoretical error estimates and illustrate the usefulness of these improvements.