Yong-Liang Zhao

Yong-Liang Zhao, Ting-Zhu Huang, Xian-Ming Gu et al.

In this paper, we consider a fast and second-order implicit difference method for approximation of a class of time-space fractional variable coefficients advection-diffusion equation. To begin with, we construct an implicit difference scheme, based on $L2-1_σ$ formula [A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, \emph{J. Comput. Phys.}, 280 (2015)] for the temporal discretization and weighted and shifted Grünwald method for the spatial discretization. Then, unconditional stability of the implicit difference scheme is proved, and we theoretically and numerically show that it converges in the $L_2$-norm with the optimal order $\mathcal{O}(τ^2 + h^2)$ with time step $τ$ and mesh size $h$. Secondly, three fast Krylov subspace solvers with suitable circulant preconditioners are designed to solve the discretized linear systems with the Toeplitz matrix. In each iterative step, these methods reduce the memory requirement of the resulting linear equations from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $\mathcal{O}(N^3)$ to $\mathcal{O}(N \log N)$, where $N$ is the number of grid nodes. Finally, numerical experiments are carried out to demonstrate that these methods are more practical than the traditional direct solvers of the implicit difference methods, in terms of memory requirement and computational cost.

Yong-Liang Zhao, Pei-Yong Zhu, Xian-Ming Gu et al.

A block lower triangular Toeplitz system arising from time-space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is developed for the block lower triangular Toeplitz system, whose storage is of $\mathcal{O}(N)$ with $N$ being the spatial grid number; (ii) A new skew-circulant preconditioner is designed to fast calculate the inverse of the block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical experiments are given to demonstrate the efficiency of our preconditioners.

Huan-Yan Jian, Ting-Zhu Huang, Xi-Le Zhao et al.

The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional problems as follows: we first transform the time distributed-order fractional diffusion problem into the multi-term time-space fractional diffusion problem with the composite trapezoid formula. Then, we propose a second-order accurate difference scheme based on the interpolation approximation on a special point to solve the resultant problem. Meanwhile, the unconditional stability and convergence of the new difference scheme in $L_2$-norm are proved. Furthermore, we find that the discretizations lead to a series of Toeplitz systems which can be efficiently solved by Krylov subspace methods with suitable circulant preconditioners. Finally, numerical results are presented to show the effectiveness of the proposed difference methods and demonstrate the fast convergence of our preconditioned Krylov subspace methods.

Yong-Liang Zhao, Pei-Yong Zhu, Xian-Ming Gu et al.

An implicit finite difference scheme based on the $L2$-$1_σ$ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability and convergence of this scheme are proved rigorously by the discrete energy method, and the optimal convergence order in the $L_2$-norm is $\mathcal{O}(τ^2 + h^2)$ with time step $τ$ and mesh size $h$. Then, the same measure is exploited to solve the two-dimensional case of this problem and a rigorous theoretical analysis of the stability and convergence is carried out. Several numerical simulations are provided to show the efficiency and accuracy of our proposed schemes and in the last numerical experiment of this work, three preconditioned iterative methods are employed for solving the linear system of the two-dimensional case.

Yong-Liang Zhao, Pei-Yong Zhu, Xian-Ming Gu et al.

A kind of spatial fractional diffusion equations in this paper are studied. Firstly, an L1 formula is employed for the spatial discretization of the equations. Then, a second order scheme is derived based on the resulting semi-discrete ordinary differential system by using the implicit integration factor method, which is a class of efficient semi-implicit temporal scheme. Numerical results show that the proposed scheme is accurate even for the discontinuous coefficients.