Explicit Parallel-in-time Integration of a Linear Acoustic-Advection System

TL;DR

This paper explores a modified Parareal parallel-in-time integration scheme applied to a linear hyperbolic acoustic-advection system, demonstrating stability, accuracy, and potential for reduced computation time in numerical weather prediction models.

Contribution

It adapts a modified Parareal scheme for hyperbolic problems, enabling explicit integration without implicit schemes, and shows its effectiveness for NWP-related PDEs.

Findings

Modified Parareal is stable for hyperbolic systems.

Explicit coarse integrator achieves speedup without implicit schemes.

Reasonably accurate solutions with reduced time-to-solution.

Abstract

The applicability of the Parareal parallel-in-time integration scheme for the solution of a linear, two-dimensional hyperbolic acoustic-advection system, which is often used as a test case for integration schemes for numerical weather prediction (NWP), is addressed. Parallel-in-time schemes are a possible way to increase, on the algorithmic level, the amount of parallelism, a requirement arising from the rapidly growing number of CPUs in high performance computer systems. A recently introduced modification of the "parallel implicit time-integration algorithm" could successfully solve hyperbolic problems arising in structural dynamics. It has later been cast into the framework of Parareal. The present paper adapts this modified Parareal and employs it for the solution of a hyperbolic flow problem, where the initial value problem solved in parallel arises from the spatial discretization of a partial differential equation by a finite difference method. It is demonstrated that the modified Parareal is stable and can produce reasonably accurate solutions while allowing for a noticeable reduction of the time-to-solution. The implementation relies on integration schemes already widely used in NWP (RK-3, partially split forward Euler, forward-backward). It is demonstrated that using an explicit partially split scheme for the coarse integrator allows to avoid the use of an implicit scheme while still achieving speedup.