Stable rotational symmetric schemes for nonlinear reaction-diffusion equations

In the simulation of biological pattern forming, it has been observed that the numerical solution is more sensitive to the spatial mesh resolution than the temporal one. Such a higher sensitivity to the spatial resolution is mainly originated from an inaccurate approximation of diffusion differential operators, which might violate the rotational symmetry to be seriously erroneous in low spatial resolutions. Also, it has been known that the second-order Crank-Nicolson time-stepping procedure may introduce spurious oscillations when the initial data or the source term is nonsmooth and the temporal step size is set relatively large. This article studies 9-point finite difference schemes for the diffusion operator to enhance the rotational symmetry, employs the variable-theta method to achieve a nonoscillatory second-order time-stepping procedure, and adopts an effective relaxation linear solver to solve the algebraic systems efficiently. The variable-theta method is proved to satisfy the maximum principle, which guarantees that the time-stepping procedure is unconditionally stable. When the successive over-relaxation method with an optimal relaxation parameter is adopted for the algebraic solver, the iteration converges in 2-4 iterations in most time steps. The overall algorithm is second-order in accuracy and scalable in efficiency. Various examples are given to show the accuracy and efficiency of the proposed algorithm for the numerical solution of the system of nonlinear reaction-diffusion equations.