Fast Higher-order Numerical schemes for Two-dimensional Nonlinear Riesz Space Fractional Diffusion Equations

Key words:  linearized scheme, nonlinear Riesz space-fractional diffusion equations, preconditioner, preconditioned conjugated gradient method

Abstract:

Fast fourth-order linearized alternating direction implicit (ADI) and locally one-dimensional (LOD) schemes are presented to solve two-dimensional nonlinear Riesz space fractional diffusion equations (RSFDEs). The proposed schemes employ the Crank-Nicolson method for temporal discretization, a quartic approximation for the Riesz derivative, and an explicit treatment for linearizing the nonlinear term. Through a rigorous discrete energy analysis, we derive explicit error estimates, proving that both schemes achieve second-order accuracy in time and fourth-order accuracy in space. The ADI and LOD methods are employed to decompose two-dimensional problems into a collection of symmetric positive definite (SPD) Toeplitz systems, for which the fast sine transform can be utilized to mitigate computational complexity. Computational treatment of these subsystems is further conducted using a preconditioned conjugate gradient (PCG) approach leveraging sine transforms. Theoretical analysis demonstrates that the preconditioned matrix can be decomposed into the identity matrix, a matrix of small norm, and a low-rank matrix. The effectiveness and efficiency of the scheme are validated through numerical examples.