A compact ADI finite difference method for 2D reaction-diffusion equations with variable diffusion coefficients

Reaction-diffusion systems on a spatially heterogeneous domain have been widely used to model various biological applications. However, solving such partial differential equa-tions (PDEs) analytically is rarely possible. Therefore, efficient and accurate numerical methods for solving such PDEs are desired. In this paper, we apply the well-known Pade approximation-based operator splitting techniques and develop a fourth-order compact alternative directional implicit (ADI) scheme. The new scheme is compact and fourth-order accurate in space. Combined with the Richardson extrapolation, the method can be improved to fourth-order accuracy in time. Stability analysis shows that the method is unconditionally stable; thus, a large time step can be used to improve the overall computational efficiency. Numerical examples have also demonstrated the new scheme's high efficiency and high order accuracy.& COPY; 2023 Elsevier B.V. All rights reserved.