An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs

TL;DR

This paper introduces a novel rational exponential integrator (REXI) scheme that significantly enhances accuracy and efficiency for solving hyperbolic and oscillatory PDEs, leveraging time parallelization and GPU implementation.

Contribution

The paper presents a new REXI scheme that improves accuracy and efficiency, with an easy method to determine the necessary number of terms for precise results.

Findings

Demonstrated improved accuracy and efficiency through numerical simulations.

Showed effective parallelization on GPU hardware.

Validated the method on shallow water equations.

Abstract

Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we propose a novel REXI scheme that drastically improves accuracy and efficiency. The chosen approach will also allow us to easily determine how many terms are required in the approximation in order to obtain accurate results. We provide comparative numerical simulations for a shallow water equation that highlight the efficiency of our approach and demonstrate that REXI schemes can be efficiently implemented on graphic processing units.