High-Order Time-Stepping Methods for Two-Dimensional Space-Fractional Reaction-Diffusion Models

In this paper, we developed novel fourth-order Runge-Kutta type exponential time differencing (ETD) A-stable and L-stable methods for space-fractional nonlinear reaction-diffusion equations with initial non-smooth or smooth data. Based on compact finite differences, a fourth-order technique is used for spatial discretization, while ETD is employed to discretize the time. Our novel numerical schemes have the benefit of explicitly handling the nonlinear term. The well-known issue of numerical instability related to computing the matrix exponential is addressed using the real single-pole rational approximation, namely the restricted Padé approximation approach. The corresponding ETD-A-stable and L-stable methods are obtained. Convergence, error estimates, and stability analysis of the suggested approaches are studied theoretically. Under a global Lipschitz continuity assumption, the unconditional L2 numerical stability is established. Moreover, the convergence order of O(k4) for the derived methods is also studied in the norm L2. Numerical experiments demonstrate the advantages of the methods in computational accuracy, efficiency, and reliability.