http://www.mat.uc.pt/~ecs/2015-sousa-li.pdf

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

In this paper we intend to establish fast numerical approaches to solve a class of initial-boundary problem of time-space fractional convection–diffusion equations. We present a new unconditionally stable implicit difference method, which is derived from the weighted and shifted Grunwald formula, and converges with the second-order accuracy in both time and space variables. Then, we show that the discretizations lead to Toeplitz-like systems of linear equations that can be efficiently solved by Krylov subspace solvers with suitable circulant preconditioners. 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 $${\mathcal {O}}(N^3)$$
 to $${\mathcal {O}}(N\log N)$$
 in each iterative step, where N is the number of grid nodes. Extensive numerical examples are reported to support our theoretical findings and show the utility of these methods over traditional direct solvers of the implicit difference method, in terms of computational cost and memory requirements.

/pdf/fast-iterative-method-with-a-second-order-implicit-1icqo7j4qs.pdf

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

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 Grunwald 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.

/pdf/fast-iterative-method-with-a-second-order-implicit-4jm8tamkam.pdf

Fast iterative method with a second order implicit difference scheme for time-space fractional convection-diffusion equations

The numerical contour integral method with hyperbolic contour is exploited to solve space-fractional diffusion equations. By making use of the Toeplitz-like structure of spatial discretized matrices and the relevant properties, the regions that the spectra of resulting matrices lie in are derived. The resolvent norms of the resulting matrices are also shown to be bounded outside of the regions. Suitable parameters in the hyperbolic contour are selected based on these regions to solve the fractional diffusion equations. Numerical experiments are provided to demonstrate the efficiency of our contour integral methods.

/pdf/fast-numerical-contour-integral-method-for-fractional-51zhe4c3rm.pdf

Fast Numerical Contour Integral Method for Fractional Diffusion Equations

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

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.

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

A non-modulus linear method for solving the linear complementarity problem

The unconditional stability of a second order finite
difference scheme for space fractional diffusion equations is proved
theoretically for a class of variable diffusion coefficients. In
particular, the scheme is unconditionally stable for all one-sided
problems and problems with Riesz fractional derivative. For problems
with general smooth diffusion coefficients, numerical experiments
show that the scheme is still stable if the space step is small
enough.

https://www.aimsciences.org/article/exportPdf?id=2eb07d05-575f-47a1-9d5e-e625145701e9

A note on the stability of a second order finite difference scheme for space fractional diffusion equations

Circulant and skew-circulant splitting iteration method is employed for solving linear systems from finite difference discretisation of tempered fractional diffusion equations. The method is shown to be convergent unconditionally and the convergence rate is fast in numerical tests. In each iteration, a circulant system and a skew-circulant system are required to be solved which cost only $$O(N\log N)$$
 operations by fast Fourier transform, where N is the number of interior mesh points in space. Moreover, the induced preconditioner possesses circulant-times-skew-circulant structure so that it can be inverted in $$O(N\log N)$$
 operation. In all of our numerical experiments, the preconditioner performs very well with a very simple choice of the method parameter.

Wei Qu

Papers

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

A non-modulus linear method for solving the linear complementarity problem

A note on the stability of a second order finite difference scheme for space fractional diffusion equations

On CSCS-based iteration method for tempered fractional diffusion equations

An unconditionally convergent RSCSCS iteration method for Riesz space fractional diffusion equations with variable coefficients