An optimal preconditioner for a high-order scheme arising from multi-dimensional Riesz space fractional diffusion equations with variable coefficients

In this paper, we propose a method for solving multi-dimensional Riesz space fractional diffusion equations with variable coefficients. The Crank–Nicolson (CN) method is used for temporal discretization, while the fourth-order fractional centered difference (4FCD) method is employed for spatial discretization. Using a novel technique, we show that the CN-4FCD scheme for the multi-dimensional case is unconditionally stable and convergent, achieving second-order accuracy in time and fourth-order accuracy in space with respect to the discrete L2-norm. Moreover, leveraging the symmetric multilevel Toeplitz-like structure of the coefficient matrix in the discrete linear systems, we enhance the computational efficiency of the proposed scheme with a sine transform based preconditioner, ensuring a mesh-size-independent convergence rate for the conjugate gradient method. Finally, numerical examples validate the theoretical analysis and demonstrate the superior performance of the proposed preconditioner compared to existing methods.