Circulant and skew-circulant splitting iteration for fractional advection–diffusion equations

Abstract

An implicit second-order finite difference scheme, which is unconditionally stable, is employed to discretize fractional advection–diffusion equations with constant coefficients. The resulting systems are full, unsymmetric, and possess Toeplitz structure. Circulant and skew-circulant splitting iteration is employed for solving the Toeplitz system. The method is proved to be convergent unconditionally to the solution of the linear system. Numerical examples show that the convergence rate of the method is fast.

Keywords:

• Toeplitz matrix
• fractional advection–diffusion equation
• fast Fourier transform
• circulant and skew-circulant splitting iteration

2010 AMS Subject Classifications:

• 26A33
• 65L12
• 65L20
• 65F10
• 65T50

Acknowledgements

Siu-Long Lei is supported by research grant MYRG071(Y1-L2)-FST13-LSL from University of Macau. Seak-Weng Vong is supported by the Macao Science and Technology Development Fund FDCT/001/2013/A.