Efficient adaptive step size control for exponential integrators

TL;DR

This paper introduces an adaptive step size controller for exponential Rosenbrock methods that minimizes computational cost, significantly improving efficiency over traditional controllers in solving nonlinear PDEs.

Contribution

It proposes a novel adaptive step size control strategy tailored for exponential integrators that accounts for variable iterative costs, enhancing computational efficiency.

Findings

Up to 4x reduction in computational cost

Effective across various nonlinear PDEs

Outperforms traditional step size controllers

Abstract

Traditional step size controllers make the tacit assumption that the cost of a time step is independent of the step size. This is reasonable for explicit and implicit integrators that use direct solvers. In the context of exponential integrators, however, an iterative approach, such as Krylov methods or polynomial interpolation, to compute the action of the required matrix functions is usually employed. In this case, the assumption of constant cost is not valid. This is, in particular, a problem for higher-order exponential integrators, which are able to take relatively large time steps based on accuracy considerations. In this paper, we consider an adaptive step size controller for exponential Rosenbrock methods that determines the step size based on the premise of minimizing computational cost. The largest allowed step size, given by accuracy considerations, merely acts as a constraint. We test this approach on a range of nonlinear partial differential equations. Our results show significant improvements (up to a factor of 4 reduction in the computational cost) over the traditional step size controller for a wide range of tolerances.