Adaptivity and local mesh refinement are crucial for the efficient numerical simulation of wave phenomena in complex geometry. Local mesh refinement, however, can impose a tiny time-step across the entire computational domain when using explicit time integration. By taking smaller time-steps yet only inside locally refined regions, local time-stepping methods overcome the stringent CFL stability restriction imposed on the global time-step by a small fraction of the elements without sacrificing explicitness. In [21], a leapfrog based local time-stepping method was proposed for the inhomogeneous wave equation, which applies standard leapfrog time-marching with a smaller time-step inside the refined region. Here, to remove potential instability at certain time-steps, a stabilized version is proposed which leads to optimal L2-error estimates under a CFL condition independent of the coarse-to-fine mesh ratio. Moreover, a weighted transition is introduced to restore optimal H1-convergence when the source is nonzero across the coarse-to-fine mesh interface. Numerical experiments corroborate the theoretical error estimates and illustrate the usefulness of these improvements.

翻译：自适应性与局部网格细化对于复杂几何中波动现象的高效数值模拟至关重要。然而，在使用显式时间积分时，局部网格细化可能迫使整个计算域采用极小的时间步长。局部时间步进方法通过在局部细化区域内采用更小的时间步长，同时保持显式格式，从而克服了由少数单元对全局时间步长施加的严格CFL稳定性限制。文献[21]针对非均匀波动方程提出了一种基于蛙跳格式的局部时间步进方法，该方法在细化区域内采用更小步长的标准蛙跳时间推进。本文为消除特定时间步可能的不稳定性，提出了一种稳定化版本，该版本可在与粗细网格比无关的CFL条件下获得最优L2误差估计。此外，当源项在粗细网格界面处非零时，引入加权过渡以恢复最优H1收敛性。数值实验验证了理论误差估计，并说明了这些改进的有效性。