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.

A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville derivative

“…Here, we consider a discretization of the Riemann-Liouville fractional derivative in an unbounded domain and prove its second order consistency. We would like to point out that during the time this work was under revision, some authors have been using the discretization of the fractional derivative introduced here in different problems [3,4,8,13,33]. At the same time, second order and higher order approximations for the fractional derivative, based in different ideas, have been appearing in literature [5,6,39].…”

Fast numerical solution for fractional diffusion equations by exponential quadrature rule

“…For steps 3 and 4 in Algorithm 1, note that A h in (6) is diagonally dominant Toeplitz-like. There are many fast algorithms for solving such a linear system in O(n log n) operations; see [22,25,30,32,36] for more discussions. In this paper, we employ the preconditioned GMRES method [38] with the generalized Strang's circulant preconditioner proposed in [22] to solve the linear system iteratively.…”

“…Therefore, the computational cost for such a matrix-vector multiplication can be carried out in O(n log n) operations using a fast algorithm based on the fast Fourier transform (FFT), and the storage requirement is reduced from O(n 2 ) to O(n), where n is the number of spatial grid points in the discretization. As the resulting coefficient matrix is still ill-conditioned [32], many fast iterative methods have been proposed to speed up the convergence rate; see [22,25,30,32,36]. The complexity by those methods for solving the resulting system at each time step is of order O(n log n).…”

Fast Numerical Contour Integral Method for Fractional Diffusion Equations

“…Recall that the resulting coefficient matrices of space-fractional diffusion equations possess Toeplitz-like structures [38,40]; see also (3.8) and (3.20). Many efficient strategies have been proposed and used for fast solving the resulting systems emerged in the time-stepping scheme, such as the fast conjugate gradient squared method [38], multigrid method [25], preconditioned Krylov subspace methods with circulant-type preconditioners [10,24,27], and band preconditioners [12], etc. In our numerical experiments, we adopt the preconditioning strategies proposed in [10] to solve the resulting shifted linear systems.…”