Exponential Integrators with Parallel-in-Time Rational Approximations for the Shallow-Water Equations on the Rotating Sphere

TL;DR

This paper explores a massively parallel rational approximation method called REXI for exponential integrators, demonstrating its efficiency and accuracy in simulating the nonlinear shallow-water equations on a rotating sphere, relevant for atmospheric modeling.

Contribution

It introduces a parallel-in-time rational approximation approach (REXI) for exponential integrators applied to nonlinear atmospheric equations, showing improved accuracy and efficiency over traditional methods.

Findings

REXI allows larger time steps with lower errors.

REXI achieves over 6x higher accuracy in the same time frame.

REXI reduces time-to-solution by about 3x for similar errors.

Abstract

High-performance computing trends towards many-core systems are expected to continue over the next decade. As a result, parallel-in-time methods, mathematical formulations which exploit additional degrees of parallelism in the time dimension, have gained increasing interest in recent years. In this work we study a massively parallel rational approximation of exponential integrators (REXI). This method replaces a time integration of stiff linear oscillatory and diffusive systems by the sum of the solutions of many decoupled systems, which can be solved in parallel. Previous numerical studies showed that this reformulation allows taking arbitrarily long time steps for the linear oscillatory parts. The present work studies the non-linear shallow-water equations on the rotating sphere, a simplified system of equations used to study properties of space and time discretization methods in the context of atmospheric simulations. After introducing time integrators, we first compare the time step sizes to the errors in the simulation, discussing pros and cons of different formulations of REXI. Here, REXI already shows superior properties compared to explicit and implicit time stepping methods. Additionally, we present wallclock-time-to-error results revealing the sweet spots of REXI obtaining either an over 6x higher accuracy within the same time frame or an about 3x reduced time-to-solution for a similar error threshold. Our results motivate further explorations of REXI for operational weather/climate systems.