  With the stagnation of processor core performance, further reductions in the
time-to-solution for geophysical fluid problems are becoming increasingly
difficult with standard time integrators. Parallel-in-time exposes and exploits
additional parallelism in the time dimension which is inherently sequential in
traditional methods. The rational approximation of exponential integrators
(REXI) method allows taking arbitrarily long time steps based on a sum over a
number of decoupled complex PDEs that can be solved independently massively
parallel. Hence REXI is assumed to be well suited for modern massively parallel
super computers which are currently trending. To date the study and development
of the REXI approach has been limited to linearized problems on the periodic 2D
plane. This work extends the REXI time stepping method to the linear
shallow-water equations (SWE) on the rotating sphere, thus moving the method
one step closer to solving fully nonlinear fluid problems of geophysical
interest on the sphere. The rotating sphere poses particular challenges for
finding an efficient solver due to the zonal dependence of the Coriolis term.
Here we present an efficient REXI solver based on spherical harmonics, showing
the results of: a geostrophic balance test, a comparison with alternative time
stepping methods, an analysis of dispersion relations, indicating superior
properties of REXI, and finally a performance comparison on Cheyenne
supercomputer. Our results indicate that REXI is not only able to take larger
time steps, but that REXI can also be used to gain higher accuracy and
significantly reduced time-to-solution compared to currently existing time
stepping methods.
