An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems: a computational study

TL;DR

This paper introduces an efficient ADI orthogonal spline collocation method based on an extrapolated Crank-Nicolson scheme for solving nonlinear reaction-diffusion systems, demonstrating high accuracy and superconvergence through computational experiments.

Contribution

It presents a novel algebraically linear ADI OSC method with optimal accuracy and superconvergence for nonlinear reaction-diffusion systems, validated on classical models.

Findings

Efficient $O({ m N})$ computational complexity per time level.

Achieves optimal global accuracy in various norms.

Exhibits superconvergence properties in numerical solutions.

Abstract

An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only $O({\cal N})$ operations where ${\cal N}$ is the number of unknowns. Moreover,it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties.