An asymptotic parallel-in-time method for highly oscillatory PDEs

TL;DR

This paper introduces a novel parallel-in-time algorithm that combines asymptotic methods with iterative refinement to efficiently solve highly oscillatory nonlinear PDEs, achieving significant speedup and high accuracy.

Contribution

The paper proposes a new hybrid time-stepping scheme that integrates asymptotic techniques with parallel refinement for efficient, accurate solutions of oscillatory PDEs.

Findings

Achieves significant parallel speedup in solving oscillatory PDEs.

Demonstrates high accuracy with the combined method.

Effective on rotating shallow water equations.

Abstract

We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which, alone, can be inefficient for equations that exhibit rapid temporal oscillations). In particular, we use an asymptotic numerical method for computing, in serial, a solution with low accuracy, and a more expensive fine solver for iteratively refining the solutions in parallel. We present examples on the rotating shallow water equations that demonstrate that significant parallel speedup and high accuracy are achievable.