Higher-order time-stepping methods for time-dependent reaction–diffusion equations arising in biology

摘要

This paper demonstrates the use of higher order methods to solve some time-dependent stiff PDEs. In the past, the most popular numerical methods for solving system of reaction–diffusion equations was based on the combination of low order finite difference method with low order time-stepping method. We extend in this report the compatibility of fourth-order finite difference scheme (in space) coupled with fourth-order time-stepping methods such as IMEXLM4, IMEXPC4, IMEXRK4 and ETDRK4-B (in time), for direct integration of reaction–diffusion equations in one space dimension. Some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries. Our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio-temporal dynamical structures.