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Despite the growing interest in parallel-in-time methods as an approach toaccelerate numerical simulations in atmospheric modelling, improving theirstability and convergence remains a substantial challenge for their applicationto operational models. In this work, we study the temporal parallelization ofthe shallow water equations on the rotating sphere combined with time-steppingschemes commonly used in atmospheric modelling due to their stabilityproperties, namely an Eulerian implicit-explicit (IMEX) method and asemi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is toinvestigate the performance of parallel-in-time methods, namely Parareal andMultigrid Reduction in Time (MGRIT), when these well-established schemes areused on the coarse discretization levels and provide insights on how they canbe improved for better performance. We begin by performing an analyticalstability study of Parareal and MGRIT applied to a linearized ordinarydifferential equation depending on the choice of a coarse scheme. Next, weperform numerical simulations of two standard tests to evaluate the stability,convergence and speedup provided by the parallel-in-time methods compared to afine reference solution computed serially. We also conduct a detailedinvestigation on the influence of artificial viscosity and hyperviscosityapproaches, applied on the coarse discretization levels, on the performance ofthe temporal parallelization. Both the analytical stability study and thenumerical simulations indicate a poorer stability behaviour when SL-SI-SETTLSis used on the coarse levels, compared to the IMEX scheme. With the IMEXscheme, a better trade-off between convergence, stability and speedup comparedto serial simulations can be obtained under proper parameters and artificialviscosity choices, opening the perspective of the potential competitiveness forrealistic models.

Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT