Stabilized leapfrog based local time-stepping method for the wave equation

Abstract

Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. In \cite{DiazGrote09}, a leapfrog based explicit local time-stepping (LF-LTS) method was proposed for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step, $\Delta t$, depends on the smallest elements in the mesh \cite{grote_sauter_1}. In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain discrete values of $\Delta t$. To remove those critical values of $\Delta t$, we apply a slight modification (as in recent work on LF-Chebyshev methods \cite{CarHocStu19}) to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate, satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition where $\Delta t$ no longer depends on the mesh size inside the locally refined region.

arXiv commentIn this new version of our paper we have corrected the proof of Lemma A.1 in the appendix. The stabilization parameter nu now has to obey the condition nu<= 1/2 instead of nu <= 1. In practice, however, nu is chosen much smaller, e.g., nu = 0.05 for the stabilized LF-LTS method; hence, the smaller upper bound on the admissible values for nu has no bearing in practice