Riesz Space Fractional Diffusion Equation Research Articles

Unconditionally convergent [formula omitted] splitting iterative methods for variable coefficient Riesz space fractional diffusion equations

In this paper, we consider fast solvers for discrete linear systems generated by Riesz space fractional diffusion equations. We extract a scalar matrix, a compensation matrix, and a τ matrix from the coefficient matrix, and use their sum to construct a class of τ splitting iterative methods. Additionally, we design a preconditioner for the conjugate gradient method. Theoretical analyses show that the proposed τ splitting iterative methods are unconditionally convergent with convergence rates independent of step-sizes. Numerical results are provided to demonstrate the effectiveness of the proposed iterative methods.

Banded preconditioning with shift compensation for solving discrete Riesz space-fractional diffusion equations

Banded preconditioning with shift compensation for solving discrete Riesz space-fractional diffusion equations

A new diagonal and Toeplitz splitting preconditioning method for solving time-dependent Riesz space-fractional diffusion equations

The initial–boundary value problem of the Riesz space-fractional diffusions equation is an important class of equations arising in many application fields. In this paper, we apply the Grünwald–Letnikov type formulas to discretize the time-dependent Riesz space-fractional diffusion equations, and obtain a system of linear equations from the discretization results. A new diagonal and Toeplitz splitting (NDTS) iteration method is constructed for this linear system. Based on the NDTS iteration method, an NDTSτ preconditioner is proposed and the generalized minimal residual (GMRES) method combined with this preconditioner is applied to solve the linear system. We theoretically show that the eigenvalues of the NDTSτ preconditioned matrix are clustered. Numerical experiments illustrate the efficiency of the proposed method.

Richardson extrapolation method for solving the Riesz space fractional diffusion problem

AbstractThe Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.

On τ-preconditioner for a novel fourth-order difference scheme of two-dimensional Riesz space-fractional diffusion equations

On τ-preconditioner for a novel fourth-order difference scheme of two-dimensional Riesz space-fractional diffusion equations

Fast solution methods for Riesz space fractional diffusion equations with non-separable coefficients

In this paper, efficient preconditioned iterative methods for solving the Riesz space fractional diffusion equations with variable diffusion coefficients are considered. Crank–Nicolson and weighted and shifted Grünwald difference scheme are applied to discretize the problem. For one-dimensional problems, we transform the asymmetric discretized linear system into a symmetric one and solve the resulted linear system by preconditioned conjugate gradient method. The preconditioned conjugate gradient method with a symmetric Toeplitz preconditioner and a sine transform based preconditioner are employed to solve the resulting linear system. Theoretically, we respectively prove that the condition numbers of preconditioned matrices have uniform upper bounds, which is independent of discretization step-sizes and fractional order. For two-dimensional problems, we propose an optional and symmetric splitting iteration method, which can be embedded by conjugate gradient method although the diffusion coefficients are non-separable. We prove that the optional and symmetric splitting iteration method is unconditionally convergent. Numerical results are reported to illustrate the efficiency of the proposed methods.

A fourth‐order accurate numerical method for the distributed‐order Riesz space fractional diffusion equation

AbstractIn this article, a novel fourth‐order accurate difference method is derived for the distributed‐order Riesz space fractional diffusion equation in one‐dimensional (1D) and two‐dimensional (2D) cases, respectively. First, the distributed integral terms are discretized by using the Simpson quadrature rule into the multi‐term Riesz space fractional diffusion equations. Then, a fourth‐order accurate difference scheme is presented to approximate the multi‐term Riesz fractional diffusion equations. Moreover, the proposed difference schemes are proved to be unconditionally stable and convergent in norm for both 1D and 2D cases. Finally, numerical experiments are given to verify the efficiency of the schemes.

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

In this paper, a respectively scaled circulant and skew-circulant splitting (RSCSCS) iteration method is employed to solve the Toeplitz-like linear systems arising from time-dependent Riesz space fractional diffusion equations with variable coefficients. The RSCSCS iteration method is shown to be convergent unconditionally by a novel technique, and only requires computational costs of O(NlogN) with N denoting the number of interior mesh points in space. In theory, we obtain an upper bound for the convergence factor of the RSCSCS iteration method and discuss the optimal value of its iteration parameter that minimizes the corresponding upper bound. Meanwhile, we also design a fast induced RSCSCS preconditioner to accelerate the convergence rate of the Krylov subspace iteration method likes generalized minimal residual (GMRES) method. Numerical results are presented to show that the efficiencies of our proposed RSCSCS iteration method and the preconditioned GMRES method with the RSCSCS preconditioner.

Accurate Boundary Treatment for Riesz Space Fractional Diffusion Equations

Accurate Boundary Treatment for Riesz Space Fractional Diffusion Equations

Efficient second-order ADI difference schemes for three-dimensional Riesz space-fractional diffusion equations

In this paper, a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered, and an alternating direction implicit (ADI) difference scheme is proposed, in which a weighted and shifted Grünwald difference scheme (see Tian et al. (2015) [33]) is utilized for the discretizations of space-fractional derivatives, and a fractional-Douglas-Gunn type ADI method is utilized for the discretization of time derivative. The method is proved to be unconditionally stable and convergent with second-order accuracy both in time and space with respect to a weighted discrete energy norm. Efficient implementation of the method is carefully discussed, and then based on fast matrix-vector multiplications, a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic system is developed. Numerical experiments support the theoretical analysis and show strong effectiveness and efficiency of the method for large-scale modeling and simulations. Finally, a linearized ADI scheme based on second-order extrapolation method is developed and tested for the nonlinear Riesz space-fractional diffusion equation.

A preconditioner based on sine transform for two-dimensional semi-linear Riesz space fractional diffusion equations in convex domains

In this paper, we develop a fast numerical method for solving the time-dependent Riesz space fractional diffusion equations with a nonlinear source term in the convex domain. An implicit finite difference method is employed to discretize the Riesz space fractional diffusion equations with a penalty term in a rectangular region by the volume-penalization approach. The stability and the convergence of the proposed method are studied. As the coefficient matrix is with the Toeplitz-like structure, the generalized minimum residual method with a preconditioner based on the sine transform is exploited to solve the discretized linear system, where the preconditioner is constructed in view of the combination of two approximate inverse τ matrices, which can be diagonalized by the sine transform. The spectrum of the preconditioned matrix is also investigated. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.

Fourth-order numerical method for the Riesz space fractional diffusion equation with a nonlinear source term

‎This paper aims to propose a high-order and accurate numerical scheme for the solution of the nonlinear diffusion equation with Riesz space fractional derivative. To this end, we first discretize the Riesz fractional derivative with a fourth-order finite difference method, then we apply a boundary value method (BVM) of fourth-order for the time integration of the resulting system of ordinary differential equations. The proposed method has a fourth-order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of BVM. The numerical results are compared with analytical solutions and with those provided by other methods in the literature. Numerical experiments obtained from solving several problems including fractional Fisher and fractional parabolic-type sine-Gordon equations show that the proposed method is an efficient algorithm for solving such problems and can arrive at the high-precision.

Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations

Based on the Crank-Nicolson alternating direction implicit finite volume (CN-ADI-FV) discretization of two dimensional Riesz space-fractional diffusion equations (2 D RSFDEs) proposed in [1], a fast direct solver is proposed to obtain the numerical solutions. Linear systems result from the CN-ADI-FV discretization can be treated as two matrix equations with symmetric positive definite (SPD) Toeplitz coefficient matrices. Then numerical solutions, expressed as multiple matrix-vector multiplications involving the inverse of SPD Toeplitz matrices, can be effectively obtained by using Gohberg-Semencul formula (GSF), where the preconditioned conjugate gradient (PCG) method is applied as an acceleration technique. Moreover, under certain assumptions, the spectral of the preconditioned matrices of the linear system associated with the GSF are theoretically proved to be clustered around 1. Then the PCG method is verified to converge superlinearly and can be implemented with O(NlogN) operations, where N is the number of spatial unknowns in both x- and y- directions. Besides, the PCG method is used only twice in the whole solution process, and the complete solution process can be finished with O(MN2logN) operations and O(N) memory, where M is the number of temporal unknowns. Numerical experiments are given to demonstrate the efficiency of our approach.

Banded Preconditioners for Riesz Space Fractional Diffusion Equations

In this paper, we consider numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations. We apply the Crank–Nicolson technique to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. We transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. We prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners.

Efficient numerical algorithms for Riesz-space fractional partial differential equations based on finite difference/operational matrix

In this paper, we construct two efficient numerical schemes by combining the finite difference method and operational matrix method (OMM) to solve Riesz-space fractional diffusion equation (RFDE) and Riesz-space fractional advection-dispersion equation (RFADE) with initial and Dirichlet boundary conditions. We applied matrix transform method (MTM) for discretization of Riesz-space fractional derivative and OMM based on shifted Legendre polynomials (SLP) and shifted Chebyshev polynomial (SCP) of second kind for approximating the time derivatives. The proposed schemes transform the RFDE and RFADE into the system of linear algebraic equations. For a better understanding of the methods, numerical algorithms are also provided for the considered problems. Furthermore, optimal error bound for the numerical solution is derived, and theoretical unconditional stability has been proved with respect to L2-norm. The stability of the schemes is also verified numerically. The schemes are observed to be of second-order accurate in space. The effectiveness and accuracy of the schemes are tested by taking two numerical examples of RFDE and RFADE and found to be in good agreement with the exact solutions. It is observed that the numerical schemes are simple, easy to implement, yield high accurate results with both the basis functions. Moreover, the CPU time taken by the schemes with SLP basis is very less as compared to schemes with SCP basis.

Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equations

We study the sine-transform-based splitting preconditioning technique for the linear systems arising in the numerical discretization of time-dependent one dimensional and two dimensional Riesz space fractional diffusion equations. Those linear systems are Toeplitz-like. By making use of diagonal-plus-Toeplitz splitting iteration technique, a sine-transform-based splitting preconditioner is proposed to accelerate the convergence rate efficiently when the Krylov subspace method is implemented. Theoretically, we prove that the spectrum of the preconditioned matrix of the proposed method is clustering around 1. In practical computations, by the fast sine transform the computational complexity at each time level can be done in $${{\mathcal {O}}}(n\log n)$$ operations where n is the matrix size. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.

Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation

This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm’s accuracy.

A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation

This paper introduces a high-order numerical procedure to solve the two-dimensional distributed-order Riesz space-fractional diffusion equation. In the proposed technique, first, a second-order numerical integration rule is employed to estimate the integral of the distributed-order Riesz space-fractional derivative. Then, the time derivative is discretized by a second-order difference scheme. Finally, the spatial direction is approximated by a difference formulation with fourth-order accuracy. The stability of the semi-discrete scheme is analyzed. We conclude that the difference between two consecutive time steps i.e. Ui,jn−Ui,jn−1 is nearly zero when n→∞. So, a suitable term is added to the main difference scheme as by adding this term we could derive the main ADI scheme. Furthermore, to reduce the used CPU time, we combine the fourth-order ADI formulation with the proper orthogonal decomposition method and then we gain a POD based reduced-order compact ADI finite difference plane. In the next, the convergence order of the fully discrete formulation has been investigated. The numerical results show the efficiency of new technique. It must be noted that the finite difference method is an effective and robust numerical technique for solving nonlinear equations that the ADI approach can be combined with it to improve the numerical simulations.

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

A numerical method with high accuracy both in time and in space is proposed for the two-dimensional nonlinear Riesz space fractional diffusion equation. The main idea is based on a spectral Galerkin method in spatial direction and an s-stage implicit Runge-Kutta method in temporal direction. A rigorous stability and error analysis is performed for the proposed method. It is shown that the proposed method is stable and convergent. The optimal spatial error estimate is also derived. Numerical experiments are provided to illustrate the theoretical results.

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

To discretize the distributed-order term of two-dimensional nonlinear Riesz space fractional diffusion equation, we consider the high accuracy Gauss-Legendre quadrature formula. By combining an s-stage implicit Runge-Kutta method in temporal direction with a spectral Galerkin method in spatial direction, we construct a numerical method with high global accuracy. If the nonlinear function satisfies the local Lipschitz condition, the s-stage implicit Runge-Kutta method with order p (p≥s+1) is coercive and algebraically stable, then we can prove that the proposed method is stable and convergent of order s+1 in time. In addition, we also derive the optimal error estimate for the discretization of distributed-order term and spatial term. Finally, numerical experiments are presented to verify the theoretical results.