Spatiotemporal pattern formations in stiff reaction-diffusion systems by new time marching methods

Abstract

Reaction-diffusion systems are extensively used in the modeling of developmental biology and in chemical systems to explain the Turing instability are generally highly stiff in both reaction and diffusion terms. For numerical simulations of stiff reaction-diffusion systems, this paper introduces a novel family of implicit-explicit Runge-Kutta (IMEX RK) type methods. Despite being implicit in nature, the developed methods require no numerical or analytical inversion of the coefficient matrix, that is, computationally explicit. Fourier analysis is used to assess the stability results for the developed methods with the model two-dimensional reaction diffusion equation. The efficiency and robustness of the developed methods are validated by numerical simulations of spatiotemporal patterns for reaction-diffusion systems governing phase-separation, the Schnakenberg model, and the electrodeposition process. The results of computed solutions are also compared to those reported in the literature.