Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation

Abstract

We propose a locally one dimensional (LOD) finite difference method for multidimensional Riesz fractional diffusion equation with variable coefficients on a finite domain. The numerical method is second-order convergent in both space and time directions, and its unconditional stability is strictly proved. Comparing with the popular first-order finite difference method for fractional operator, the form of obtained matrix algebraic equation is changed from $(I-A)u^{k+1}=u^k+b^{k+1}$ to $(I-\tilde{A})u^{k+1}=(I+\tilde{B})u^k+{\tilde{b}}^{k+1/2}$; the three matrices $A$, $\tilde{A}$ and $\tilde{B}$ are all Toeplitz-like, i.e., they have completely same structure and the computational count for matrix vector multiplication is $\mathcal{O}(NlogN)$; and the computational costs for solving the two matrix algebraic equations are almost the same. The LOD-multigrid method is used to solve the resulting matrix algebraic equation, and the computational count is $\mathcal{O}(NlogN)$ and the required storage is $\mathcal{O}(N)$, where $N$ is the number of grid points. Finally, the extensive numerical experiments are performed to show the powerfulness of the second-order scheme and the LOD-multigrid method.

26 pages

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• Compact Finite Difference Approximations for Space Fractional Diffusion Equations
• A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation
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• WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative
• A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations
• A simple solver for the fractional Laplacian in multiple dimensions
• A Multigrid method for nonlocal problems: non-diagonally dominant Toeplitz-plus-tridiagonal systems
• Fast and High-order Accuracy Numerical Methods for Time-Dependent Nonlocal Problems in $\mathbb{R}^2
• Central local discontinuous Galerkin method for the space fractional diffusion equation
• A second-order accurate scheme for two-dimensional space fractional diffusion equations with time Caputo-Fabrizio fractional derivative