# A Parallel Time-Integrator for Solving the Linearized Shallow Water Equations on the Rotating Sphere

## TL;DR

This paper extends the REXI parallel-in-time integrator to the linearized shallow water equations on a rotating sphere, demonstrating improved efficiency, larger time steps, and higher accuracy for geophysical fluid simulations on supercomputers.

## Contribution

It develops an efficient REXI solver for the linear shallow water equations on the rotating sphere, advancing the method towards fully nonlinear geophysical fluid problems.

## Key findings

- REXI enables larger time steps than traditional methods
- REXI achieves higher accuracy in simulations
- Performance tests show reduced time-to-solution on supercomputers

## Abstract

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.