A {\tau} Matrix Based Approximate Inverse Preconditioning for Tempered Fractional Diffusion Equations

TL;DR

This paper introduces a novel preconditioning technique based on the ta matrix for efficiently solving discretized tempered fractional diffusion equations, improving convergence of iterative methods.

Contribution

It proposes a ta matrix-based approximate inverse preconditioner tailored for the coefficient matrices of tempered fractional diffusion equations.

Findings

Eigenvalues of the preconditioned matrix cluster around 1

Preconditioner computed efficiently via discrete sine transform

Numerical experiments confirm improved convergence

Abstract

Tempered fractional diffusion equations are a crucial class of equations widely applied in many physical fields. In this paper, the Crank-Nicolson method and the tempered weighted and shifts Gr\"unwald formula are firstly applied to discretize the tempered fractional diffusion equations. We then obtain that the coefficient matrix of the discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite(SPD) Toeplitz matrix. Based on the properties of SPD Toeplitz matrices, we use $\tau$ matrix approximate it and then propose a novel approximate inverse preconditioner to approximate the coefficient matrix. The $\tau$ matrix based approximate inverse preconditioner can be efficiently computed using the discrete sine transform(DST). In spectral analysis, the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Finally, numerical experiments demonstrate the effectiveness of the proposed preconditioner.