From Memory Model to CPU Time: Exponential Integrators for Advection-Dominated Problems

TL;DR

This paper evaluates exponential integrators, specifically Krylov and Leja methods, for advection-dominated problems, demonstrating their efficiency relative to explicit Runge--Kutta schemes across different regimes.

Contribution

It extends previous work by analyzing higher-order Krylov approximations and new regimes, providing a comprehensive CPU-time efficiency comparison.

Findings

Leja methods outperform Krylov for large time steps

Krylov methods are more efficient for small time steps

Exponential integrators can match or outperform Runge--Kutta schemes

Abstract

In this paper, we investigate the application of exponential integrators to advection-dominated problems. We focus on Krylov subspace and Leja interpolation methods to compute the action of exponential and related matrix functions. Complementing our earlier paper, arXiv:2410.12765 (to appear in Advances in Applied Mathematics and Mechanics, 2025) based on a performance model, we extend the numerical investigation to higher-order Krylov approximations and new numerical regime, and assess their CPU-time efficiency relative to explicit Runge--Kutta schemes. We show that, depending on the problem setting, exponential integrators can either outperform or match explicit Runge--Kutta schemes. We also observe that Leja-based methods outperform Krylov iterations for large time steps, whereas for small time steps, Krylov-based methods provide better results than Leja-based methods.