A fourth-order exponential time differencing scheme with dimensional splitting for non-linear reaction-diffusion systems

TL;DR

This paper introduces a fourth-order exponential time differencing scheme with dimensional splitting for multidimensional reaction-diffusion systems, achieving high accuracy and efficiency, especially for non-smooth initial conditions.

Contribution

A novel fourth-order ETD scheme using Padé approximation and dimensional splitting, offering improved efficiency over existing methods for reaction-diffusion equations.

Findings

Achieves fourth-order accuracy in multiple RDE tests.

Up to 20 times faster CPU performance compared to other schemes.

Effectively damps oscillations with pre-smoothing steps.

Abstract

A fourth-order exponential time differencing (ETD) Runge-Kutta scheme with dimensional splitting is developed to solve multidimensional non-linear systems of reaction-diffusion equations (RDE). By approximating the matrix exponential in the scheme with the A-acceptable Pad\'e (2,2) rational function, the resulting scheme (ETDRK4P22-IF) is verified empirically to be fourth-order accurate for several RDE. The scheme is shown to be more efficient than competing fourth-order ETD and IMEX schemes, achieving up to 20 times speed in CPU time. Inclusion of up to three pre-smoothing steps of a lower order L-stable scheme facilitates efficient damping of spurious oscillations arising from problems with non-smooth initial/boundary conditions.