Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modelling, improving their stability and convergence remains a substantial challenge for their application to operational models. In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modelling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in Time (MGRIT), when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved for better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on the choice of coarse scheme. Next, we perform numerical simulations of two standard tests to evaluate the stability, convergence and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.

翻译：尽管并行时间方法作为一种加速大气模式数值模拟的手段日益受到关注，但提升其稳定性和收敛性仍是其在业务化模式中应用面临的重大挑战。本研究结合大气模式中因稳定性特性而常用的时间步进格式（即欧拉隐式-显式（IMEX）方法和半拉格朗日半隐式方法（SL-SI-SETTLS）），系统研究了旋转球面上浅水方程的时间并行化问题。核心目标是探究当这些成熟格式应用于粗网格层时，并行时间方法（即Parareal和多重网格时间约化法，MGRIT）的性能表现，并为其性能优化提供改进思路。我们首先对基于不同粗网格格式选择下的Parareal和MGRIT方法进行了线性化常微分方程的解析稳定性分析。随后，通过两个标准测试算例的数值模拟，评估了并行时间方法相较于串行精细参考解在稳定性、收敛性和加速比方面的表现。此外，我们还深入研究了粗网格层上人工粘性和超粘性方法对时间并行化性能的影响。解析稳定性分析与数值模拟均表明：与IMEX格式相比，在粗网格层使用SL-SI-SETTLS格式时稳定性更差。采用IMEX格式时，通过合理选择参数和人工粘性，可在收敛性、稳定性与加速比之间实现优于串行模拟的平衡，这为并行时间方法在真实模式中的潜在竞争力提供了前景。