Fully implicit timestepping methods have several potential advantages for atmosphere/ocean simulation. First, being unconditionally stable, they degrade more gracefully as the Courant number increases, typically requiring more solver iterations rather than suddenly blowing up. Second, particular choices of implicit timestepping methods can extend energy conservation properties of spatial discretisations to the fully discrete method. Third, these methods avoid issues related to splitting errors that can occur in some situations, and avoid the complexities of splitting methods. Fully implicit timestepping methods have had limited application in geophysical fluid dynamics due to challenges of finding suitable iterative solvers, since the coupled treatment of advection prevents the standard elimination techniques. However, overlapping Additive Schwarz methods, provide a robust, scalable iterative approach for solving the monolithic coupled system for all fields and Runge-Kutta stages. In this study we investigate this approach applied to the rotating shallow water equations, facilitated by the Irksome package which provides automated code generation for implicit Runge-Kutta methods. We compare various schemes in terms of accuracy and efficiency using an implicit/explicit splitting method, namely the ARK2 scheme of Giraldo et al (2013), as a benchmark. This provides an initial look at whether implicit Runge-Kutta methods can be viable for atmosphere and ocean simulation.

翻译：完全隐式时间步进方法在大气/海洋模拟中具有若干潜在优势。首先，由于其无条件稳定性，这些方法在库朗数增大时性能衰减更为平缓——通常仅表现为求解器迭代次数增加，而非突然发散。其次，特定隐式时间步进方法的选择可将空间离散格式的能量守恒特性拓展至全离散方法。第三，这些方法避免了某些情况下可能出现的分裂误差问题，同时规避了分裂方法的复杂性。由于平流项的耦合处理阻碍了标准消元技术的应用，寻找合适的迭代求解器存在挑战，完全隐式时间步进方法在地球物理流体动力学中的应用一直受限。然而，重叠型加性施瓦茨方法为求解所有场变量和龙格-库塔阶段的全耦合系统提供了稳健、可扩展的迭代途径。本研究利用Irksome包（该包为隐式龙格-库塔方法提供自动化代码生成功能）探讨了该方法在旋转浅水方程中的应用。我们以隐式/显式分裂方法——即Giraldo等人（2013）提出的ARK2格式——作为基准，从精度和效率两个维度比较了多种方案。这为探究隐式龙格-库塔方法在大气与海洋模拟中的可行性提供了初步视角。