High-order algorithms for Riesz derivative and their applications (II)☆
Introduction
Recently, fractional differential equations have attracted increasing interests mainly because they have been proved to be successful in modeling many natural phenomena [1], [19], [20], [27], [30], [31]. However, most of fractional differential equations cannot be easily analytically solved. So, more and more researchers are interested in studying the numerical methods for the fractional differential equations [8], [11], [15], [24], [25], [38]. Meerschaert and Tadjeran [23] considered the finite difference method for the space fractional advection–dispersion equation with Riemann–Liouville derivative, where a shifted Grünwald–Letnikov scheme was proposed. In [21], [22], Murillo and Yuste constructed the difference schemes for the fractional diffusion equation by using the so-called L1 formula and for the fractional diffusion-wave equation by using the L2 formula, respectively. In [17], Liu et al. used L1 scheme to discrete the time Caputo derivative and the shifted Grünwald–Letnikov formulae to discrete the left- and right-sided Riemann–Liouville derivatives. Yang et al. [35] obtained numerical solutions for the space Riesz fractional diffusion and advection–dispersion equations on a finite domain by using the L1, L2 methods. Shen et al. [32] used a second-order numerical scheme for the space Riesz fractional advection–dispersion equation based on fractional central difference operator. In addition, Podlubny firstly proposed the matrix approach to discrete fractional calculus and developed the idea to approximate fractional partial differential equations in [28]. Li et al. [18] firstly developed a finite element method for solving the nonlinear time–space fractional subdiffusion and superdiffusion equations, respectively. They analyzed the stability and error estimates for the semi-discrete and fully discrete schemes. Jiang and Ma [13] constructed a high-order finite element method for solving the time Caputo fractional partial differential equations. A space fractional reaction–diffusion equation has been considered by Burrage et al. [2]. In [6], Deng and Hesthaven proposed discontinuous Galerkin methods for fractional diffusion equations. Wang et al. [16] proposed a difference method for the fractional diffusion equation. The Petrov–Galerkin spectral method for fractional initial-value problems and fractional final-value problems was developed by Zayernouri and Karniadakis [36]. Later on, they developed an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional partial differential equations [37]. It is worth noting that these difference schemes have high-accuracy in space and the order of accuracy in time is second-order. Very recently, Ding et al. [7] constructed two different fourth-order numerical schemes for the Riesz derivative and they applied one of them to the spatial Riesz fractional diffusion equation.
As far as we know, there are no numerical formulas more than sixth-order of the accuracy for the Riesz fractional derivative. The aim of this paper is to construct several high-order approximate schemes (the orders more than 6) for the Riesz fractional derivative and use sixth- and eighth-order schemes to solve the following space Riesz fractional reaction-dispersion equation
Suppose that the above equation (1) is subject to the initial and boundary value conditions as below where is the dispersion coefficient. denotes the Riesz fractional derivative with order α defined by [14] where , and are the left- and right-sided Riemann–Liouville derivatives represented in the following forms and Here indicates the Euler's Gamma function. In particular, when , the Riesz derivative reduces to the general second-order partial derivative .
In addition, operators and denote the left- and right-sided Riemann–Liouville integral operators and are defined as follows : and
The Dirichlet boundary value conditions (or the first kind of boundary value conditions) (1.2) and (1.3) for Eq. (1) mean that, and
If the integral operators and the limits, and , and , can be exchanged, then it follows from Eqs. (1.2) and (1.3) that and So the boundary value conditions (1.2) and (1.3) can be converted into respectively. This is why the boundary value conditions like (1.4) and (1.5) were chosen in all the past studies, where the assumption of the exchangeability of the integral operators and the limits was used. Throughout this paper, such exchangeability of the integral operators and the limits is always assumed.
Generally speaking, definite conditions of fractional differential equations with Riemann–Liouville or Riesz derivatives are quite different from those of typical differential equations. So to propose definite conditions for Riemann–Liouville type differential equations must be very careful. For more details, see Appendix A.
We remark that there exists another equivalent definition for the Riesz derivative by utilizing the Fourier transform on an infinite domain under homogeneous Dirichlet boundary conditions [35] where the fractional power of the Laplace operator is defined as follows: in which and denote the Fourier and inverse Fourier transforms, respectively.
The plan of this paper is listed as follows. In Section 2, we construct several high-order numerical formulas for the Riesz fractional derivative. In Section 3, two new finite difference schemes are proposed for solving a Riesz fractional reaction-dispersion equation. In Section 4, we discuss the stability and convergence of the proposed numerical methods in details. Some numerical examples are taken to show the effectiveness and accuracy of the numerical schemes in Section 5. At last, some remarks are concluded in Section 6.
Section snippets
High-order numerical schemes for the Riesz fractional derivative
We take the mesh points , , and , , where , , i.e., h and τ are the uniform spatial and temporal step sizes, respectively.
It is obvious that the space Riesz fractional derivative (1) contains a left Riemann–Liouville derivative and a right Riemann–Liouville derivative, so numerical methods for Riemann–Liouville derivative can be applied to numerically approximate the Riesz fractional derivative, such as the standard Grünwald–Letnikov formula
Numerical methods for the space Riesz fractional reaction-dispersion equation
We study Eq. (1) at points (, ) and use the Crank–Nicolson method For smooth function , we use the second order central difference scheme Combining (13) and (14) leads to
Let be the approximation solution of .
Stability and convergence analysis
In this section, we firstly study the local truncation errors of the numerical schemes then we present the stability analysis. Finally, we give the convergence results. For convenience, we introduce some lemmas and definition for the following discussion.
Definition 1 (See [5].) Let Toeplitz matrix have the following form,
Numerical experiment
In the following, we give several examples.
Example 1 Consider the following equation subject to the initial and boundary value conditions where Its exact solution
Conclusions
In this paper, four different effective high-order numerical formulas for the Riesz fractional derivative have been constructed. Next, we use a sixth- and eighth-order numerical formulas to the space Riesz fractional reaction-dispersion equation and obtain two finite difference schemes which are both unconditionally stable and convergent. Finally, some numerical examples have been carried out to support the theoretical analysis. In addition, these numerical formulas and techniques can also be
References (38)
- et al.
Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative
J. Comput. Phys.
(2012) - et al.
New numerical methods for the Riesz space fractional partial differential equations
Comput. Math. Appl.
(2012) - et al.
Stability and convergence of a finite volume method for the space fractional advection–dispersion equation
J. Comput. Appl. Math.
(2014) - et al.
High-order finite element methods for time-fractional partial differential equations
J. Comput. Appl. Math.
(2011) - et al.
The accuracy and stability of an implicit solution method for the fractional diffusion equation
J. Comput. Phys.
(2005) - et al.
A direct finite difference method for fractional diffusion equations
J. Comput. Phys.
(2010) - et al.
Numerical methods and analysis for a class of fractional advection–dispersion models
Comput. Math. Appl.
(2012) - et al.
Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion
Comput. Math. Appl.
(2011) - et al.
Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation
J. Magn. Res.
(2008) - et al.
Finite difference approximations for fractional advection–dispersion flow equations
J. Comput. Appl. Math.
(2004)
Finite difference methods for two-dimensional fractional dispersion equation
J. Comput. Phys.
Finite difference approximations for two-sided space-fractional partial differential equations
Appl. Numer. Math.
Matrix approach to discrete fractional calculus II: partial fractional differential equations
J. Comput. Phys.
An improved dual porosity model for chemical transport in macroporous soils
J. Hydrol.
Fractional calculus and continuous time finance
Physica A
A second order explicit finite difference method for the fractional advection diffusion equation
Comput. Math. Appl.
Numerical methods for fractional partial differential equations with Riesz space fractional derivatives
Appl. Math. Model.
Exponentially accurate spectral and spectral element methods for fractional ODEs
J. Comput. Phys.
Application of a fractional advection–dispersion equation
Water Resour. Res.
Cited by (126)
A Comparative performance evaluation of a complex-order PI controller for DC–DC converters
2024, Results in Control and OptimizationA hybrid fourth order time stepping method for space distributed order nonlinear reaction-diffusion equations
2023, Computers and Mathematics with ApplicationsThe upwind PPM scheme and analysis for solving two-sided space-fractional advection-diffusion equations in three dimension
2023, Computers and Mathematics with ApplicationsOn τ-preconditioner for a novel fourth-order difference scheme of two-dimensional Riesz space-fractional diffusion equations
2023, Computers and Mathematics with ApplicationsThe construction of an optimal fourth-order fractional-compact-type numerical differential formula of the Riesz derivative and its application
2023, Communications in Nonlinear Science and Numerical SimulationFast TT-M fourth-order compact difference schemes for a two-dimensional space fractional Gray-Scott model
2023, Computers and Mathematics with Applications
- ☆
The work was partially supported by the National Natural Science Foundation of China under Grant No. 11372130, the Key Program of Shanghai Municipal Education Commission under Grant No. 12ZZ084, the grant of “The First-Class Discipline of Universities in Shanghai”, and the Tianshui Normal University “QingLan” Talent Engineering Funds under Grant No. TSQL-10-01.