Exponential integrators with parallel-in-time rational approximations for the shallow-water equations on the rotating sphere

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 6× higher accuracy within the same time frame or an about 3× reduced time-to-solution for a similar error threshold. Our results motivate further explorations of REXI for operational weather/climate systems.

Semi-Lagrangian Exponential Integration with Application to the Rotating Shallow Water Equations

“…Krylov solvers, such as those presented in [28], are used in [15] and [25] for the matrix exponentiation of a dynamic linearization of the shallow water system. Furthermore, [48] adopts a rational approximation based on [27] for the rotating SWE on the plane, which is also used for the sphere in [47,49] with a global spectral spherical harmonics representation. This rational approximation approach calculates the matrix exponentials with a very high degree of parallelism, so the additional computational costs of the calculating such exponential may be absorbed by extra compute nodes to still reduce the time-to-solution.…”

Time-parallel simulation of the Schrödinger Equation

“…We want to use rational approximations to compute the matrix exponential numerically. There are various ways to compute the exponential of a matrix [32,33], however, we are interested in methods that use rational approximations, because these methods can be constructed in a way that allows for parallelizing the time integration scheme itself, increasing the parallelism of the overall solution process [19,21,20].…”

“…It can thus be classified as parallel across the method, although its approach allows to take much larger time-steps as more classical methods like Crank-Nicolson. REXI has been successfully applied to shallow-water equation on the rotating sphere [20] and to linear oscillatory problems [21], making parallel-in-time integration possible even for these type of problems.…”

High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid

“…The implementation of high-order Eulerian techniques in operational models is made difficult by the Courant-Friedrichs-Lewy (CFL) condition that limits the time step in several explicit time integration schemes [6]. This motivated the recent investigation of exponential time integrators in geophysical applications [7,8,9,10,11,12,13,14]. These approaches allow for longer time steps and yield higher accuracy than traditional algorithms.…”

“…L, u 1 + h11 g r H R (140) in the x 1 direction andα = max u 2 + h 22 g r H L , u 2 + h 22 g r H R(141)in the x 2 direction. Here, the superscripts L and R indicate quantities that are evaluated at the two sides of the element interface under consideration.…”