The Courant-Friedrichs-Lewy (CFL) condition is a well known, necessary condition for the stability of explicit time-stepping schemes that effectively places a limit on the size of the largest admittable time-step for a given problem. We formulate and present a new local time-stepping (LTS) scheme optimized, in the CFL sense, for the shallow water equations (SWEs). This new scheme, called FB-LTS, is based on the CFL optimized forward-backward Runge-Kutta schemes from Lilly et al. (2023). We show that FB-LTS maintains exact conservation of mass and absolute vorticity when applied to the TRiSK spatial discretization (Ringler et al., 2010), and provide numerical experiments showing that it retains the temporal order of the scheme on which it is based (second order). In terms of computational performance, we show that when applied to a real-world test case on a highly-variable resolution mesh, the MPAS-Ocean implementation of FB-LTS is up to 10 times faster than the classical four-stage, fourth-order Runge-Kutta method (RK4), and 2.3 times faster than an existing strong stability preserving Runge-Kutta based LTS scheme (LTS3). Despite this significant increase in efficiency, the solutions produced by FB-LTS are qualitatively equivalent to those produced by both RK4 and LTS3.

翻译：库朗-弗里德里希-列维（CFL）条件是显式时间步进格式稳定性的著名必要条件，它有效限制了给定问题中最大允许时间步长的取值。本文针对浅水方程（SWEs）提出并建立了一种基于CFL优化的新型局部时间步进（LTS）格式。该新格式命名为FB-LTS，其基础是Lilly等人（2023）提出的CFL优化前向-后向龙格-库塔格式。我们证明：当应用于TRiSK空间离散格式（Ringler等人，2010）时，FB-LTS能严格保持质量和绝对涡度的守恒性；数值实验表明，该格式保留了其所基于格式的时间精度阶数（二阶）。在计算性能方面，当应用于高变分辨率网格上的实际测试案例时，MPAS-Ocean实现的FB-LTS比经典四阶段四阶龙格-库塔方法（RK4）快10倍，比现有基于强稳定性保持龙格-库塔的LTS格式（LTS3）快2.3倍。尽管效率显著提升，FB-LTS生成的解在定性上与RK4和LTS3产生的解等价。