Stabilized leapfrog based local time-stepping method for the wave equation
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Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. Recently, a leapfrog based explicit local time-stepping (LF-LTS) method was proposed for the time integration of second-order wave equations. Optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step, Δ t, depends on the smallest elements in the mesh. In general one cannot improve upon that stability constraint, as the LF-LTS method may become (linearly) unstable at certain discrete values of Δ t. To remove those critical values of Δ t, we propose a slight modification of 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 Δ t no longer depends on the mesh size inside the locally refined region.
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