  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.
