High-order numerical methods with stability analysis for Caputo–Riesz fractional reaction-diffusion models in three dimensions

Abstract

In this paper, we propose and study a new ternary three-dimensional fractional reaction-diffusion model. The model combines Caputo fractional derivatives in time with Riesz fractional derivatives in space. It is capable of describing multi-species anomalous transport processes with memory in time and nonlocal interactions in space. To approximate the system, we design a quadratic-weighted scheme for the Caputo derivative and a meshless scheme for the Riesz derivative. This leads to a fully discrete numerical method that is accurate, flexible, and easy to implement. We establish a rigorous stability analysis using the energy method. We also prove convergence, showing second-order accuracy in both time and space. Supporting the theory, we prove auxiliary results such as spatial coercivity, discrete energy monotonicity, and boundedness of the numerical solution. Two numerical examples are presented to validate the analysis. The computed errors and convergence rates are summarized in tables and illustrated with figures. The numerical results confirm the predicted second-order accuracy in both temporal and spatial directions.