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, \Delta 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 \Delta t. To remove those critical values of \Delta 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 \Delta t no longer depends on the mesh size inside the locally refined region.
Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. In \mbox{%DIFAUXCMD 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 \mbox{%DIFAUXCMD 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 \mbox{%DIFAUXCMD CarHocStu19 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.
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Recently
In \mbox{%DIFAUXCMD DiazGrote09
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Optimal
Recently, optimal
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\mbox{%DIFAUXCMD grote_sauter_1
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(linearly)
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propose
apply
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of
(as in recent work on LF-Chebyshev methods \mbox{%DIFAUXCMD CarHocStu19