Relaxing the CFL condition for the wave equation on adaptive meshes

TL;DR

This paper introduces a subspace projection method that relaxes the CFL condition for the wave equation on adaptive meshes, enabling larger time steps without sacrificing stability or accuracy, especially near singularities.

Contribution

It proposes a simple subspace projection inspired by numerical homogenisation to relax the CFL condition for adaptive mesh wave simulations.

Findings

Allows larger time steps while maintaining stability

Balances CFL condition and approximation properties optimally

Effective near spatial singularities

Abstract

The Courant-Friedrichs-Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. This paper shows how a simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities.