Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

Abstract

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 O(N 2 ) to 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.

Outer synchronization of a class of mixed delayed complex networks based on pinning control

“…Then by resorting to LMI in Matlab Toolbox, Theorem 3.1 can guarantee the outer synchronization of the dynamical networks, and we can derive the feasible solution to LMIs in (8) and (9) From these simulations, one can conclude that using the proposed method in this paper, outer synchronization of systems (1) and (4) can be achieved.…”

A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time–space fractional diffusion equation

“…Actually, the closed-form analytical solutions of FPDEs can be obtained in a few special cases [5], but such solutions are usually impractical. It thus becomes imperative to study the numerical solutions of FPDEs, and numerous reliable numerical methods have been developed [6][7][8][9][10][11][12][13][14][15][16][17]. Due to the nonlocality of the fractional operators, using the finite difference method to solve space/time-space fractional differential equations leads to a time-stepping scheme with dense coefficient matrices.…”

A Second-Order Accurate Implicit Difference Scheme for Time Fractional Reaction-Diffusion Equation with Variable Coefficients and Time Drift Term

“…[6][7][8]. In these models, the fractional diffusion equation (FDE) has been studied by many researchers, see [9][10][11][12][13][14][15][16][17][18] and references therein.…”

“…This scheme efficiently reduces the computational storage and cost for solving the time FDEs. Although there are many studies on the space/time FDEs, numerical studies on space-time FDEs are still not extensive, the readers are suggested to see [11,[38][39][40] and references therein.…”