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

TL;DR

This paper introduces an efficient, stable, and accurate numerical scheme for multi-dimensional Riesz space fractional diffusion equations with variable coefficients, featuring a novel preconditioner for improved computational performance.

Contribution

It develops a second-order in time, fourth-order in space scheme with a sine transform-based preconditioner, enhancing efficiency for solving complex fractional diffusion equations.

Findings

The scheme is unconditionally stable and convergent.

The preconditioner ensures mesh-size-independent convergence.

Numerical examples confirm superior performance over existing methods.

Abstract

In this paper, we propose an efficient 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 multi-level 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, two numerical examples validate the theoretical analysis and demonstrate the superior performance of the proposed preconditioner compared to existing methods.