Weighted average finite difference methods for fractional diffusion equations

Weighted averaged finite difference methods for solving fractional diffusion equations are discussed and different formulae of the discretization of the Riemann-Liouville derivative are considered. The stability analysis of the different numerical schemes is carried out by means of a procedure close to the well-known von Neumann method of ordinary diffusion equations. The stability bounds are easily found and checked in some representative examples.Comment: Communication presented at the FDA'04 Workshop (with some minor corrections and updates

An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains

In this article, an advanced differential quadrature (DQ) approach is proposed for the high-dimensional multi-term time-space-fractional partial differential equations (TSFPDEs) on convex domains. Firstly, a family of high-order difference schemes is introduced to discretize the time-fractional derivative and a semi-discrete scheme for the considered problems is presented. We strictly prove its unconditional stability and error estimate. Further, we derive a class of DQ formulas to evaluate the fractional derivatives, which employs radial basis functions (RBFs) as test functions. Using these DQ formulas in spatial discretization, a fully discrete DQ scheme is then proposed. Our approach provides a flexible and high accurate alternative to solve the high-dimensional multi-term TSFPDEs on convex domains and its actual performance is illustrated by contrast to the other methods available in the open literature. The numerical results confirm the theoretical analysis and the capability of our proposed method finally.Comment: 22 pages, 26 figure

Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation

In this paper, compact finite difference schemes for the modified anomalous fractional sub-diffusion equation and fractional diffusion-wave equation are studied. Schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Based on the idea of weighted and shifted Grunwald difference operator, we establish schemes with temporal and spatial accuracy order equal to two and four respectively.Comment: 20 pages, 1 figure

Numerical Approximations for Fractional Differential Equations

The Gr\"unwald and shifted Gr\"unwald formulas for the function $y(x)-y(b)$ are first order approximations for the Caputo fractional derivative of the function $y(x)$ with lower limit at the point $b$. We obtain second and third order approximations for the Gr\"unwald and shifted Gr\"unwald formulas with weighted averages of Caputo derivatives when sufficient number of derivatives of the function $y(x)$ are equal to zero at $b$, using the estimate for the error of the shifted Gr\"unwald formulas. We use the approximations to determine implicit difference approximations for the sub-diffusion equation which have second order accuracy with respect to the space and time variables, and second and third order numerical approximations for ordinary fractional differential equations

Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation

We propose a locally one dimensional (LOD) finite difference method for multidimensional Riesz fractional diffusion equation with variable coefficients on a finite domain. The numerical method is second-order convergent in both space and time directions, and its unconditional stability is strictly proved. Comparing with the popular first-order finite difference method for fractional operator, the form of obtained matrix algebraic equation is changed from $(I-A)u^{k+1}=u^k+b^{k+1}$ to $(I-{\widetilde A})u^{k+1}=(I+{\widetilde B})u^k+{\tilde b}^{k+1/2}$; the three matrices $A$, ${\widetilde A}$ and ${\widetilde B}$ are all Toeplitz-like, i.e., they have completely same structure and the computational count for matrix vector multiplication is $\mathcal{O}(N {log} N)$; and the computational costs for solving the two matrix algebraic equations are almost the same. The LOD-multigrid method is used to solve the resulting matrix algebraic equation, and the computational count is $\mathcal{O}(N {log} N)$ and the required storage is $\mathcal{O}(N)$, where $N$ is the number of grid points. Finally, the extensive numerical experiments are performed to show the powerfulness of the second-order scheme and the LOD-multigrid method.Comment: 26 page

Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

In this paper, fast numerical methods are established for solving a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by the weighted and shifted Gr$\ddot{\rm{u}}$nwald formula in time and the fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing the preconditioned Krylov subspace method is developed to solve the symmetric positive definite Toeplitz linear systems derived from the proposed difference scheme. In the two-dimensional case, we also design a global preconditioned conjugate gradient method with a truncated preconditioner to solve the discretized Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around one, such that the proposed numerical methods with preconditioners converge very quickly. Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference schemes and show that the performances of the proposed fast solution algorithms are better than other numerical methods.Comment: 36 pages, 7 figures, 12 table

Efficient method for fractional L\'{e}vy-Feller advection-dispersion equation using Jacobi polynomials

In this paper, a novel formula expressing explicitly the fractional-order derivatives, in the sense of Riesz-Feller operator, of Jacobi polynomials is presented. Jacobi spectral collocation method together with trapezoidal rule are used to reduce the fractional L\'{e}vy-Feller advection-dispersion equation (LFADE) to a system of algebraic equations which greatly simplifies solving like this fractional differential equation. Numerical simulations with some comparisons are introduced to confirm the effectiveness and reliability of the proposed technique for the L\'{e}vy-Feller fractional partial differential equations.Comment: 23 pages, 4 figure

Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations

This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are respectively verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.Comment: 31 pages, 5 figure

An a posteriori error analysis for an optimal control problem involving the fractional Laplacian

In a previous work, we introduced a discretization scheme for a constrained optimal control problem involving the fractional Laplacian. For such a control problem, we derived optimal a priori error estimates that demand the convexity of the domain and some compatibility conditions on the data. To relax such restrictions, in this paper, we introduce and analyze an efficient and, under certain assumptions, reliable a posteriori error estimator. We realize the fractional Laplacian as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi--infinite cylinder in one more spatial dimension. This extra dimension further motivates the design of an posteriori error indicator. The latter is defined as the sum of three contributions, which come from the discretization of the state and adjoint equations and the control variable. The indicator for the state and adjoint equations relies on an anisotropic error estimator in Muckenhoupt weighted Sobolev spaces. The analysis is valid in any dimension. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that exhibits optimal experimental rates of convergence

Finite difference/local discontinuous Galerkin method for solving the fractional diffusion-wave equation

In this paper a finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation is presented and analyzed. We first propose a new finite difference method to approximate the time fractional derivatives, and give a semidiscrete scheme in time with the truncation error $O((\Delta t)^2)$, where $\Delta t$ is the time step size. Further we develop a fully discrete scheme for the fractional diffusion-wave equation, and prove that the method is unconditionally stable and convergent with order $O(h^{k+1}+(\Delta t)^{2})$, where $k$ is the degree of piecewise polynomial. Extensive numerical examples are carried out to confirm the theoretical convergence rates.Comment: 18 pages, 2 figure