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

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.