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

Local time-stepping methods permit to overcome the severe stability constraint on explicit caused by local mesh refinement without sacrificing explicitness. Diaz and Grote [SIAM J. Sci. Comput. 31 (2009), pp. 1985â??2014] proposed a leapfrog based (LF-LTS) method for time integration of second-order wave equations. Recently, optimal convergence rates were proved conforming FEM discretization, albeit under CFL condition where global time-step, $\Delta t$, depends smallest elements in (see M. Grote, Mehlin, S. A. Sauter Numer. Anal. 56 (2018), 994â??1021]). In general one cannot improve upon that constraint, as LF-LTS may become unstable at certain discrete values t$. To remove those critical we apply slight modification (as recent work LF-Chebyshev Carle, Hochbruck, Sturm 58 (2020), 2404â??2433]) original which nonetheless preserves its desirable properties: it is fully explicit, accurate, satisfies three-term (leapfrog like) recurrence relation, conserves energy. The new stabilized also yields standard FE yet t$ no longer size inside locally refined region.