Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

TL;DR

This paper compares various exponential integrators for solving high-accuracy, stiff periodic semilinear PDEs in multiple dimensions, concluding that the simple ETDRK4 scheme performs nearly as well as more complex methods.

Contribution

The paper provides extensive MATLAB and Chebfun comparisons of exponential integrators for 1D, 2D, and 3D stiff PDEs, highlighting the effectiveness of the ETDRK4 scheme.

Findings

ETDRK4 performs nearly as well as more complex formulas

Fourth and higher order methods evaluated in multiple dimensions

Simple exponential integrator is highly effective for stiff PDEs

Abstract

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and higher order methods, and periodic semilinear stiff PDEs with constant coefficients. Our conclusion is that it is hard to do much better than one of the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews.