High-order time-stepping methods for two-dimensional Riesz fractional nonlinear reaction–diffusion equations

Anomalous diffusion and long-range spatial interactions in anisotropic media could be captured by considering the Riesz fractional derivatives rather than the classical Laplacian. While analytical solutions of the resulting fractional reaction–diffusion models may not be available, the numerical methods for approximating them are challenging. In this paper, fourth-order L-stable (ETDRK04) and A-stable (ETDRK22) linearly implicit predictor-corrector type time-stepping methods are presented. These methods are implemented on two-dimensional Riesz fractional nonlinear reaction–diffusion equations with smooth and non-smooth initial data. Three types of nonlinear reaction–diffusion models are considered: Allen–Cahn equation with cubic nonlinearity, Fisher's equation with quadratic nonlinearity, and Enzyme Kinetics equation with rational nonlinearity. Fourth-order temporal convergence rate of the methods is proved analytically and computed numerically. Profiles of the numerical solutions corresponding to different orders and rates of diffusion are included. Computational efficiency of the ETDRK04 and ETDRK22 methods over the well known Cox–Matthews ETDRK4 is presented. The superiority of the provided methods, in terms of computational accuracy, efficiency, and reliability, is demonstrated through the numerical experiments.