  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.
