Weihua Deng

Numerical algorithm for the time fractional Fokker-Planck equation

Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker-Planck equation. In this paper, firstly the time fractional, the sense of Riemann-Liouville derivative, Fokker-Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann-Liouville derivative and Caputo derivative. Then combining the predictor-corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O(k^m^i^n^{^1^+^2^@a^,^2^})+O(h^2), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for @a=1.0 with the ones of directly discretizing classical Fokker-Planck equation, some numerical results for time fractional Fokker-Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for @a=0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown.

Fourth Order Accurate Scheme for the Space Fractional Diffusion Equations

Because of the nonlocal properties of fractional operators, higher order schemes play a more important role in discretizing fractional derivatives than classical ones. The striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first order schemes but greatly improve the accuracy. Nowadays, there are already two types of second order discretization schemes for space fractional derivatives: the first type is given and discussed in [Sousa and Li, arXiv:1109.2345v1, 2011; Chen and Deng, Appl. Math. Model., 38 (2014), pp. 3244--3259; Chen, Deng, and Wu, Appl. Numer. Math., 70 (2013), pp. 22--41]; and the second type is a class of schemes presented in [Tian, Zhou, and Deng, Math. Comp., to appear; also available online from arXiv:1201.5949, 2012]. The core object of this paper is to derive a class of fourth order approximations, called the weighted and shifted Lubich difference operators, for space fractional derivatives. Then we use the derived schemes t...Because of the nonlocal properties of fractional operators , higher order schemes play more important role in discretizing fractional derivatives than classical one s. The striking feature is that higher order schemes of fractional derivatives can keep the same computation cos t with first-order schemes but greatly improve the accuracy. Nowadays, there are already two types of secon d order discretization schemes for space fractional derivatives: the first type is given and discusse d in [Sousa & Li, arXiv:1109.2345; Chen & Deng, arXiv:1304.3788; Chen et al., Appl. Numer. Math., 70, 22-41]; and the second type is a class of schemes presented in [Tian et al., arXiv:1201.5949]. The co r object of this paper is to derive a class of fourth order approximations, called the weighted and shi fted Lubich difference (WSLD) operators, for space fractional derivatives. Then we use the derived sc hemes to solve the space fractional diffusion equation with variable coefficients in one-dimensional and two-dimensional cases. And the unconditional stability and the convergence with the global truncation er ror O(τ2 + h4) are theoretically proved and numerically verified.

A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation

Abstract Space fractional convection diffusion equation describes physical phenomena where particles or energy (or other physical quantities) are transferred inside a physical system due to two processes: convection and superdiffusion. In this paper, we discuss the practical alternating directions implicit method to solve the two-dimensional two-sided space fractional convection diffusion equation on a finite domain. We theoretically prove and numerically verify that the presented finite difference scheme is unconditionally von Neumann stable and second order convergent in both space and time directions.

Superlinearly convergent algorithms for the two-dimensional space-time Caputo-Riesz fractional diffusion equation

In this paper, we discuss the time-space Caputo-Riesz fractional diffusion equation with variable coefficients on a finite domain. The finite difference schemes for this equation are provided. We theoretically prove and numerically verify that the implicit finite difference scheme is unconditionally stable (the explicit scheme is conditionally stable with the stability condition τ γ (∆x)α + τ γ (∆y)β < C) and 2nd order convergent in space direction, and (2− γ)-th order convergent in time direction, where γ ∈ (0, 1].

A series of high‐order quasi‐compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

Based on the superconvergent approximation at some point (depending on the fractional order , but not belonging to the mesh points) for Grunwald discretization to fractional derivative, we develop a series of high-order quasi-compact schemes for space fractional diffusion equations. Because of the quasi-compactness of the derived schemes, no points beyond the domain are used for all the high-order schemes including second-order, third-order, fourth-order, and even higher-order schemes; moreover, the algebraic equations for all the high-order schemes have the completely same matrix structure. The stability and convergence analysis for some typical schemes are made; the techniques of treating the fractional derivatives with nonhomogeneous boundaries are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders. (c) 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1345-1381, 2015

Discretized fractional substantial calculus

This paper discusses the properties and the numerical discretizations of the fractional substantial integral I-s(v) f(x) = 1/Gamma(v) integral(x)(a) (x-tau)(v-1)e(-sigma(x-tau)) f(tau)d tau, v>0, and the fractional substantial derivative D-s(mu) f(x) = D-s(m) [I-s(v) f(x)], v = m - mu, where D-s = partial derivative/partial derivative x + sigma, sigma can be a constant or a function not related to x, say sigma(y); and m is the smallest integer that exceeds mu. The Fourier transform method and fractional linear multistep method are used to analyze the properties or derive the discretized schemes. And the convergences of the presented discretized schemes with the global truncation error O(h(p)) (p = 1, 2, 3, 4, 5) are theoretically proved and numerically verified.

High Order Algorithms for the Fractional Substantial Diffusion Equation with Truncated Lévy Flights

The equation with the time fractional substantial derivative and space fractional derivative describes the distribution of the functionals of the Levy flights, and the equation is derived as the macroscopic limit of the continuous time random walk in unbounded domain and the Levy flights have divergent second order moments. However, in more practical problems, the physical domain is bounded and the involved observables have finite moments. Then the modified equation can be derived by tempering the Levy measure of the Levy flights and the corresponding tempered space fractional derivative is introduced. This paper focuses on providing the high order algorithms for the modified equation, i.e., the equation with the time fractional substantial derivative and space tempered fractional derivative. More concretely, the contributions of this paper are as follows: (1) Detailed numerical stability analysis and error estimates of the schemes with first order accuracy in time and second order in space are given in c...

Numerical Algorithms for the Forward and Backward Fractional Feynman---Kac Equations

The Feynman–Kac equations are a type of partial differential equations describing the distribution of functionals of diffusive motion. The probability density function (PDF) of Brownian functionals satisfies the Feynman–Kac formula, being a Schrödinger equation in imaginary time. The functionals of non-Brownian motion, or anomalous diffusion, follow the fractional Feynman–Kac equation (Carmi et al. in J Stat Phys 141:1071–1092, 2010), where the fractional substantial derivative is involved. Based on recently developed discretized schemes for fractional substantial derivatives (Chen and Deng arXiv:1310.3086), this paper focuses on providing algorithms for numerically solving the forward and backward fractional Feynman–Kac equations; since the fractional substantial derivative is non-local time-space coupled operator, new challenges are introduced compared with the ordinary fractional derivative. Two ways (finite difference and finite element) of discretizing the space derivative are considered. For the backward fractional Feynman–Kac equation, the numerical stability and convergence of the algorithms with first order accuracy are theoretically discussed; and the optimal estimates are obtained. For all the provided schemes, including the first order and high order ones, of both forward and backward Feynman–Kac equations, extensive numerical experiments are performed to show their effectiveness.

Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes

This paper, as the sequel to previous work, develops numerical schemes for fractional diffusion equations on a two-dimensional finite domain with triangular meshes. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization by the use of high-order nodal basis, employing multivariate Lagrange polynomials defined on the triangles. Stability analysis and error estimates are provided, which shows that if polynomials of degree N are used, the methods are ( N + 1 ) -th order accurate for general triangulations. Finally, the performed numerical experiments confirm the optimal order of convergence.

Local discontinuous Galerkin methods for fractional ordinary differential equations

This paper discusses the upwinded local discontinuous Galerkin methods for the one-term/multi-term fractional ordinary differential equations (FODEs). The natural upwind choice of the numerical fluxes for the initial value problem for FODEs ensures stability of the methods. The solution can be computed element by element with optimal order of convergence