Fast permutation preconditioning for fractional diffusion equations.

Sheng-Feng Wang,Wei-Hua Luo,Xian-Ming Gu,Ting-Zhu Huang

doi:10.1186/s40064-016-2766-4

Abstract

In this paper, an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is used to discretize the fractional diffusion equations with constant diffusion coefficients. The coefficient matrix possesses the Toeplitz structure and the fast Toeplitz matrix-vector product can be utilized to reduce the computational complexity from {mathcal {O}}{(N^{3})} to {mathcal {O}}{(N log N)}, where N is the number of grid points. Two preconditioned iterative methods, named bi-conjugate gradient method for Toeplitz matrix and bi-conjugate residual method for Toeplitz matrix, are proposed to solve the relevant discretized systems. Finally, numerical experiments are reported to show the effectiveness of our preconditioners.

Highlights

In the last few decades, many anomalous diffusion phenomena have been found in the real world, which lead to the generation of the fractional diffusion equations (FDEs)
Even if the discretized approach of the FDEs is implicit, it still can result in unconditionally unstable (Meerschaert and Tadjeran 2004, 2006) because of the nonlocality of the fractional differential operators
We extend Conjugate Gradient (CG) and Conjugate Residual (CR) to BiCGT and BiCRT, respectively, which are proposed to solve the equivalent equation

Summary

Introduction

In the last few decades, many anomalous diffusion phenomena have been found in the real world, which lead to the generation of the fractional diffusion equations (FDEs). Many numerical approaches for the FDEs have been proposed and developed intensively in the last decade, for instance Zhang et al (2010), Ervin et al (2007), Langlands and Henry (2005), Liu et al (2004), Meerschaert and Tadjeran (2004, 2006), Tian et al (2015), Gu et al (2015). Even if the discretized approach of the FDEs is implicit, it still can result in unconditionally unstable (Meerschaert and Tadjeran 2004, 2006) because of the nonlocality of the fractional differential operators. In order to overcome the difficulty of the stability, (Meerschaert and Tadjeran 2004, 2006) put forward a shifted Grünwald discretization to approximate FDEs with a left-sided

Methods

Conclusion