Should Exponential Integrators be Used for Advection-Dominated Problems?

Abstract

In this paper, we consider the application of exponential integrators to problems that are advection dominated. 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, depending on the specific setting, either outperform or perform similarly to explicit Runge–Kutta schemes. 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.