# Parallel-in-Time Multi-Level Integration of the Shallow-Water Equations on the Rotating Sphere

## TL;DR

This paper introduces a multi-level parallel-in-time integration method combining PFASST with spherical harmonics for efficient simulation of shallow-water equations on the sphere, significantly reducing computation time.

## Contribution

The paper presents a novel PFASST-based multi-level scheme using spherical harmonics for spatial coarsening, enabling faster parallel-in-time integration of atmospheric PDEs on the sphere.

## Key findings

- PFASST-SH converges upon time refinement.
- The coarsening strategy effectively captures high-frequency modes.
- The scheme achieves multiple times speedup over serial methods.

## Abstract

The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field. We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel high-order fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space. The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes.