论文标题
稳定的基于跨越的局部时间步变方法，用于波动方程
Stabilized leapfrog based local time-stepping method for the wave equation
论文作者
论文摘要
局部时间步变方法允许克服由于局部网状细化而不牺牲明确性而引起的明确方法的严重稳定性约束。在\ cite {diazgrote09}中，提出了一种基于跨越的显式局部时间步变（LF-LTS）方法，以用于二阶波方程的时间整合。最近，尽管在CFL稳定性条件下，全局时间步长$ΔT$取决于网格\ cite {grote_sauter_1}中最小的元素，但证明了最佳的收敛速率。通常，由于LF-LTS方法在$ΔT$的某些离散值下可能会变得不稳定，因此无法改善该稳定性约束。要删除$ΔT$的关键值，我们将稍作修改（如最近在LF-Chebyshev方法上进行的工作\ cite {carhocstu19}）上的原始LF-LTS方法，尽管如此，它仍然保留了其可取的属性：它是完全明确的，第二级准确的能量，可以满足三个期限的依赖（保守），并保守了三个 - 保守的（保守）。新的稳定LF-LTS方法还为标准的FE离散化产生了最佳收敛速率，但在CFL条件下，$ΔT$不再取决于本地精制区域内的网格大小。
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, $Δt$, 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 $Δt$. To remove those critical values of $Δt$, 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 $Δt$ no longer depends on the mesh size inside the locally refined region.
