Fast iterative method with a second order implicit difference scheme for time-space fractional convection-diffusion equations
For researchers solving fractional convection-diffusion equations, this provides a more efficient numerical method, though it is an incremental improvement over existing implicit schemes.
The paper proposes an implicit difference scheme with second-order accuracy for time-space fractional convection-diffusion equations and develops fast Krylov subspace solvers with circulant preconditioners, reducing memory from O(N^2) to O(N) and computational complexity from O(N^3) to O(N log N) per iteration.
In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Grünwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $O(N^3)$ to $O(N\log N)$ in each iterative step, where $N$ is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.