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 (LTS) overcome the stringent Courant-Friedrichs-Lewy (CFL) stability restriction imposed on the global time step by a small fraction of the elements without sacrificing explicitness. In (Grote, M. J. & Mitkova, T. (2010) Explicit local time-stepping methods for Maxwell’s equations. J. Comput. Appl. Math., 234, 3283–3302.), a leapfrog (LF)-based LTS was proposed for the inhomogeneous wave equation, which applies standard LF 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 $L^{2}$-error estimates under a CFL condition independent of the coarse-to-fine mesh ratio. Moreover, a weighted transition is introduced to restore optimal $H^{1}$-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.