A Novel Fourth‐Order Scheme for Two‐Dimensional Riesz Space Fractional Nonlinear Reaction‐Diffusion Equations and Its Optimal Preconditioned Solver
Wei Qu, Yuanyuan Huang, Sean Hon, Siu‐Long Lei
Abstract
ABSTRACT A novel fourth‐order finite difference formula coupling the Crank–Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction‐diffusion equations in two dimensions. Theoretically, under the Lipschitz assumption on the nonlinear term, the proposed high‐order scheme is proved to be unconditionally stable and convergent in the discrete ‐norm. Moreover, a ‐matrix‐based preconditioner is developed to speed up the convergence of the conjugate gradient method with an optimal convergence rate (a convergence rate independent of mesh sizes) for solving the symmetric discrete linear system. Theoretical analysis shows that the spectra of the preconditioned matrices are uniformly bounded in the open interval . This preconditioned iterative solver, to the best of our knowledge, is a new development with a mesh‐independent convergence rate for the linearized high‐order scheme. Numerical examples are given to validate the accuracy of the scheme and the effectiveness of the proposed preconditioned solver.