# An efficient second-order convergent scheme for one-side space fractional diffusion equations with variable coefficients

## TL;DR

This paper introduces a second-order finite difference scheme for one-sided space fractional diffusion equations with variable coefficients, achieving higher accuracy and efficiency through stability analysis and a Toeplitz preconditioner.

## Contribution

The paper develops a second-order convergent scheme with an efficient Toeplitz preconditioner for variable coefficient fractional diffusion equations, improving upon existing methods.

## Key findings

- Unconditionally stable with second-order convergence in time and space.
- Preconditioned linear systems have bounded condition numbers, ensuring fast convergence.
- Numerical results confirm the scheme's accuracy and efficiency.

## Abstract

In this paper, a second order finite difference scheme is investigated for time-dependent one-side space fractional diffusion equations with variable coefficients. The existing schemes for the equation with variable coefficients have temporal convergence rate no better than second order and spatial convergence rate no better than first order, theoretically. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Gr\"unwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the diffusion coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme.