Numerical analysis and fast implementation of a fourth-order difference scheme for two-dimensional space-fractional diffusion equations

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.

TT-M FE method for a 2D nonlinear time distributed-order and space fractional diffusion equation

Small perturbations in the type of boundary conditions for an elliptic operator

A domain decomposition method for determining the diffusion coefficient of a two-dimensional linear diffusion equation in the time domain

In this paper a family of fourth-order and sixth-order compact difference schemes for the three dimensional (3D) linear Poisson equation are derived in detail. By using finite volume (FV) method for derivation, the highest-order compact schemes based on two different types of dual partitions are obtained. Moreover, a new fourth-order compact scheme is gained and numerical experiments show the new scheme is much better than other known fourth-order schemes. The outline for the nonlinear problems are also given. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference scheme.

We consider the preconditioned Krylov subspace method for linear systems arising from the finite volume discretization method of steady-state variable-coefficient conservative space-fractional diffusion equations. We propose to use a scaled-circulant preconditioner to deal with such Toeplitz-like discretization matrices. We show that the difference between the scaled-circulant preconditioner and the coefficient matrix is equal to the sum of a small-norm matrix and a low-rank matrix. Numerical tests are conducted to show the effectiveness of the proposed method for one- and two-dimensional steady-state space-fractional diffusion equations and demonstrate that the preconditioned Krylov subspace method converges very quickly.

Compact alternating direction implicit method for two-dimensional time fractional diffusion equation

Implicit Runge-Kutta and spectral Galerkin methods for the two-dimensional nonlinear Riesz space fractional diffusion equation

The combination of Sinc quadrature method and double exponential transformation (DE) is a powerful tool to approximate the singular integrals, and radial basis functions (RBFs) are useful for the higher-dimensional space problem. In this study, we develop a numerical method base on Gaussian-RBF combined with QR-factorization of arising matrix and DE-quadrature Sinc method to approximate the solution of two-dimensional space-fractional diffusion equations. When the number of central nodes increases, the ill-conditioning of resultant matrix can be eliminated by using GRBF-QR method. Two numerical examples have been presented to test the efficiency and accuracy of the method.

We develop a fast finite difference method for time-dependent variable-coefficient space-fractional diffusion equations with fractional derivative boundary-value conditions in three dimensional spaces. Fractional differential operators appear in both of the equation and the boundary conditions. Because of the nonlocal nature of the fractional Neumann boundary operator, the internal and boundary nodes are strongly coupled together in the coupled linear system. The stability and convergence of the finite difference method are discussed. For the implementation, the development of the fast method is based upon a careful analysis and delicate decomposition of the structure of the coefficient matrix. The fast method has approximately linear computational complexity per Krylov subspace iteration and an optimal-order memory requirement. Numerical results are presented to show the utility of the method.

The usual three node approximations for the first-order spatial derivatives in the radial basis functions based finite difference (RBF-FD) scheme allow relatively smaller time step size than the standard scheme (forward difference in time and central difference in space) in obtaining stable results. In this paper, to better mimic the continuous model, we use nine nodal points in RBF-FD approximation of first-order spatial derivatives. We then employ these approximations to solve a two-dimensional linear diffusion equation. The constructed numerical scheme is shown to be stable using the maximum-minimum principle. We also show that the scheme allows four times bigger time step size than the standard scheme.

ABSTRACTIn this paper, we consider a numerical technique for solving the two-dimensional fractional order diffusion equation with a time fractional derivative. The proposed technique depends basically on the fact that an expansion of the required approximate solution in a series of shifted Chebyshev polynomials of the first kind in the time and the Sinc function in the space. Then, the expansion coefficients are determined by reducing the problem with the initial conditions to a system of algebraic equations. The fractional derivatives are expressed in the Caputo sense. The numerical results of the proposed technique are compared with other published results to show the efficiency of the presented technique.

A preconditioned fast quadratic spline collocation method for two-sided space-fractional partial differential equations

Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions

Two-dimensional non-Fickian diffusion equation is solved analytically under arbitrary initial condition and two kinds of periodic boundary conditions. The concentration field distributions are analytically obtained with a form of double Fourier series, and the damped diffusion wave transport is discussed. At the same time, the numerical simulation is carried out for the problem with homogeneous boundary condition and arbitrary initial condition, which shows that the concentration field gradually changes from the initial distribution to the steady distribution and it changes faster for the smaller Vernotte number. The numerical results agree well with the experimental results.