A fourth-order exponential time differencing scheme with real and distinct poles rational approximation for solving non-linear reaction-diffusion systems

TL;DR

This paper introduces a fourth-order exponential time differencing scheme with rational approximation for reaction-diffusion systems, offering improved stability, efficiency, and parallelization capabilities for solving nonlinear PDEs with nonsmooth data.

Contribution

The paper presents a novel ETDRK4RDP scheme using rational functions with real and distinct poles, enhancing damping of oscillations and parallel efficiency compared to existing methods.

Findings

Achieves fourth-order accuracy for reaction-diffusion systems.

Demonstrates up to six times CPU time speedup over competing schemes.

Ensures effective damping of oscillations with nonsmooth data.

Abstract

A fourth-order, L-stable, exponential time differencing Runge-Kutta type scheme is developed to solve nonlinear systems of reaction diffusion equations with nonsmooth data. The new scheme, ETDRK4RDP, is constructed by approximating the matrix exponentials in the ETDRK4 scheme with a fourth order, L-acceptable, non-Pad\'e rational function having real and distinct poles (RDP). Using RDP rational functions to construct the scheme ensures efficient damping of spurious oscillations arising from non-smooth initial and boundary conditions and a straightforward parallelization. We verify empirically that the new ETDRK4RDP scheme is fourth-order accurate for several reaction diffusion systems with Dirichlet and Neumann boundary conditions and show it to be more efficient than competing exponential time differencing schemes, especially when implemented in parallel, with up to six times speedup in CPU time.