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.

翻译：本地时间步方法可以克服本地网格精细细细细细微的清晰方法对明确方法造成的严重稳定性限制,而不会牺牲清晰度。 在\ cite{ DiazGrote09} 中,为二阶波方程式的时间整合提出了基于明确本地时间步的飞跃方法(LF-LTS ) 。 最近, 最佳趋同率被证明符合FEM的离散性, 尽管在 CFL 稳定性条件下, 全球时间步进$\ Delta t$, 取决于网格的细细微值, 取决于网格中的最小元素。 一般来说, 无法在这种稳定性限制下改进, 因为LF- LTS 方法在某些离散值($\ Delta t) 上可能会变得不稳定。 要去除这些关键值$\ Delta t$, 我们对LF- C- C- C- C- C- C- C- C- CarhoStu19} 新的LTS- LTS 方法进行了微小的修改, 但仍保持其理想性特性: 它完全清晰的、 稳定的固定的C- LF- 标准 和稳定率, 和最精确的C- LF- d- tral- d- d- d- tral- tral- real) 率 的精确 和最接近率。