A unified difference-spectral method for time–space fractional diffusion equations

In this paper we develop a unified difference-spectral method for stably solving time–space fractional sub-diffusion and super-diffusion equations. Based on the equivalence between Volterra integral equations and fractional ordinary differential equations with initial conditions, this proposed method is constructed by combining the spectral Galerkin method in space and the fractional trapezoid formula in time. Numerical experiments are carried out to verify the effectiveness of the method, and demonstrate that the unified method can achieve spectral accuracy in space and second-order accuracy in time for solving two kinds of time–space fractional diffusion equations.     

Homotopy analysis method for solving multi-term linear and nonlinear diffusion–wave equations of fractional order

In this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.     

A space–time spectral collocation method for the two-dimensional variable-order fractional percolation equations

In this article, we introduce a space–time spectral collocation method for solving the two-dimensional variable-order fractional percolation equations. The method is based on a Legendre–Gauss–Lobatto (LGL) spectral collocation method for discretizing spatial and the spectral collocation method for the time integration of the resulting linear first-order system of ordinary differential equation. Optimal priori error estimates in $L^2$ norms for the semi-discrete and full-discrete formulation are derived. The method has spectral accuracy in both space and time. Numerical results confirm the exponential convergence of the proposed method in both space and time.     

An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations

In this paper, we propose a split-step quasi-compact finite difference method to solve the nonlinear fractional Ginzburg–Landau equations both in one and two dimensions. The original equations are split into linear and nonlinear subproblems. The Riesz space fractional derivative is approximated by a fourth-order fractional quasi-compact method. Furthermore, an alternating direction implicit scheme is constructed for the two dimensional linear subproblem. The unconditional stability and convergence of the schemes are proved rigorously in the linear case. Numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method.     

A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.     

A Galerkin finite element scheme for time–space fractional diffusion equation

In this paper, a Galerkin finite element scheme to approximate the time–space fractional diffusion equation is studied. Firstly, the fractional diffusion equation is transformed into a fractional Volterra integro-differential equation. And a second-order fractional trapezoidal formula is used to approach the time fractional integral. Then a Galerkin finite element method is introduced in space direction, where the semi-discretization scheme and fully discrete scheme are given separately. The stability analysis of semi-discretization scheme is discussed in detail. Furthermore, convergence analysis of semi-discretization scheme and fully discrete scheme are given in details. Finally, two numerical examples are displayed to demonstrate the effectiveness of the proposed method.     

A fast algorithm for solving the space–time fractional diffusion equation

In this paper, we propose a fast algorithm for efficient and accurate solution of the space–time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian ${(?\Delta )}_s^{\alpha /2}$ results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix–vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method.     

Unconditionally stable high-order time integration for moving mesh finite difference solution of linear convection–diffusion equations

This paper is concerned with moving mesh finite difference solution of partial differential equations. It is known that mesh movement introduces an extra convection term and its numerical treatment has a significant impact on the stability of numerical schemes. Moreover, many implicit second and higher order schemes, such as the Crank–Nicolson scheme, will lose their unconditional stability. A strategy is presented for developing temporally high-order, unconditionally stable finite difference schemes for solving linear convection–diffusion equations using moving meshes. Numerical results are given to demonstrate the theoretical findings.     

Circulant and skew-circulant splitting iteration for fractional advection–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.     

Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

One of the ongoing issues with time fractional diffusion models is the design of efficient high-order numerical schemes for the solutions of limited regularity. We construct in this paper two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection–diffusion–reaction equations with constant and variable coefficients. The model solution is discretized in time with a spectral expansion of fractional-order Jacobi orthogonal functions. For the space discretization, the proposed schemes accommodate high-order Jacobi Galerkin spectral discretization. The numerical schemes do not require imposition of artificial smoothness assumptions in time direction as is required for most methods based on polynomial interpolation. We illustrate the flexibility of the algorithms by comparing the standard Jacobi and the fractional Jacobi spectral methods for three numerical examples. The numerical results indicate that the global character of the fractional Jacobi functions makes them well-suited to time fractional diffusion equations because they naturally take the irregular behavior of the solution into account and thus preserve the singularity of the solution.

     

Linearized Crank–Nicolson method for solving the nonlinear fractional diffusion equation with multi-delay

This paper is concerned with numerical solution of the nonlinear fractional diffusion equation with multi-delay. The studied model plays a significant role in population ecology. A linearized Crank–Nicolson method for such problem is proposed by combing the Crank–Nicolson approximation in time with the fractional centred difference formula in space. Using the discrete energy method, the suggested scheme is proved to be uniquely solvable, stable and convergent with second-order accuracy in both space and time for sufficiently small space and time increments. Several numerical experiments for solving the delay fractional Hutchinson equation and two real problems in population dynamics are provided to verify our theoretical results.     

Relaxation method for unsteady convection–diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection–diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection–diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit–explicit (IMEX) Runge–Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection–diffusion equations with both smooth structures and discontinuities.     

High-order time stepping Fourier spectral method for multi-dimensional space-fractional reaction–diffusion equations

This paper introduces a high-order time stepping scheme, that is based on using Fourier spectral in space and a fourth-order diagonal Padé approximation to the matrix exponential function for solving multi-dimensional space-fractional reaction–diffusion equations. The resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non-linear system at each time step while maintaining the stability of the scheme. The non-locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices. This scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator. It also attains spectral convergence for multiple spatial dimensions. The stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions, which provides an indication of the stability of the method. The convergence analysis is performed empirically to show that the scheme is fourth-order accurate in time, as expected. Numerical experiments on reaction–diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space-fractional reaction–diffusion equations and to validate the effectiveness of the scheme.     

The local discontinuous Galerkin method for 2D nonlinear time-fractional advection–diffusion equations

This paper presents a numerical solution of time-fractional nonlinear advection–diffusion equations (TFADEs) based on the local discontinuous Galerkin method. The trapezoidal quadrature scheme (TQS) for the fractional order part of TFADEs is investigated. In TQS, the fractional derivative is replaced by the Volterra integral equation which is computed by the trapezoidal quadrature formula. Then the local discontinuous Galerkin method has been applied for space-discretization in this scheme. Additionally, the stability and convergence analysis of the proposed method has been discussed. Finally some test problems have been investigated to confirm the validity and convergence of the proposed method.

     

A meshless method for solving the time fractional advection–diffusion equation with variable coefficients

In this paper, an efficient and accurate meshless method is proposed for solving the time fractional advection–diffusion equation with variable coefficients which is based on the moving least square (MLS) approximation. In the proposed method, firstly the time fractional derivative is approximated by a finite difference scheme of order $O\left({\left(\delta t\right)}^{2?\alpha }\right),0<\alpha \le 1$ and then the MLS approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the Caputo sense. Also, the validity of the proposed method is investigated in error analysis discussion. The main aim is to show that the meshless method based on the MLS shape functions is highly appropriate for solving fractional partial differential equations (FPDEs) with variable coefficients. The efficiency and accuracy of the proposed method are verified by solving several examples.     

A stable Gaussian radial basis function method for solving nonlinear unsteady convection–diffusion–reaction equations

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection–diffusion–reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor–corrector schemes such as Adams–Bashforth and implicit–explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.     

A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.     

On linear stability of predictor–corrector algorithms for fractional differential equations

This paper deals with the numerical approximation of differential equations of fractional order by means of predictor–corrector algorithms. A linear stability analysis is performed and the stability regions of different methods are compared. Furthermore the effects on stability of multiple corrector iterations are verified.