Should exponential integrators be used for advection-dominated problems?

TL;DR

This paper evaluates the effectiveness of exponential integrators, specifically Leja and Krylov methods, for advection-dominated problems, finding they perform comparably or better than explicit schemes depending on domain characteristics.

Contribution

It provides a performance comparison of Leja and Krylov exponential integrators for advection-dominated problems, including a hardware-independent performance model.

Findings

Exponential integrators perform similarly to explicit Runge-Kutta methods for fully advection-dominated problems.

They outperform explicit methods when small diffusion-dominated regions are present.

Leja methods generally outperform Krylov methods, especially when inner product computations are costly.

Abstract

In this paper, we consider the application of exponential integrators to problems that are advection dominated, either on the entire or on a subset of the domain. In this context, we compare Leja and Krylov based methods to compute the action of exponential and related matrix functions. We set up a performance model by counting the different operations needed to implement the considered algorithms. This model assumes that the evaluation of the right-hand side is memory bound and allows us to evaluate performance in a hardware independent way. We find that exponential integrators perform comparably to explicit Runge-Kutta schemes for problems that are advection dominated in the entire domain. Moreover, they are able to outperform explicit methods in situations where small parts of the domain are diffusion dominated. We generally observe that Leja based methods outperform Krylov iterations in the problems considered. This is in particular true if computing inner products is expensive.