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 a 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加速大气模式数值模拟的手段日益受到关注，但在实际模式应用中提升其稳定性与收敛性仍面临重大挑战。本研究以旋转球面上浅水方程为对象，结合大气模式中因稳定性特性而广泛使用的时间步进格式——即欧拉隐式-显式（IMEX）方法与半拉格朗日半隐式方法（SL-SI-SETTLS），系统研究其时间并行化性能。核心目标在于探究当上述成熟格式应用于粗离散层时，Parareal与多重网格时间缩减（MGRIT）两种并行时间方法的性能表现，并为优化其性能提供理论支撑。我们首先对Parareal及MGRIT在粗格式选择影响下的线性化常微分方程进行解析稳定性分析；继而通过两个标准测试的数值模拟，评估并行时间方法相较串行精细参考解在稳定性、收敛性及加速比方面的表现。特别地，我们深入研究了粗离散层上引入人工粘性与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结果均表明：相较于IMEX格式，粗层采用SL-SI-SETTLS格式会呈现更弱的稳定性特性。而基于IMEX格式，在适当参数配置与人工粘性选择下，可取得较串行模拟更优的收敛性、稳定性与加速比平衡——这为并行时间方法在真实模式中的竞争力展现了潜在前景。