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 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 error $\mathcal{O}(\tau^2+h^4)$ are theoretically proved and numerically verified.

The equation with the time fractional substantial derivative and space fractional derivative describes the distribution of the functionals of the Lévy flights, and the equation is derived as the macroscopic limit of the continuous time random walk in unbounded domain and the Lévy 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 Lévy measure of the Lévy 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 complex} space, which is necessary since the inverse Fourier transform needs to be made for getting the distribution of the functionals after solving the equation. (2) We further propose the schemes with high order accuracy in both time and space, and the techniques of treating the issue of keeping the high order accuracy of the schemes for nonhomogeneous} boundary/initial conditions are introduced. (3) Multigrid methods are effectively used to solve the obtained algebraic equations which still have the Toeplitz structure. (4) We perform extensive numerical experiments, including verifying the high convergence orders and simulating the physical system which needs to numerically make the inverse Fourier transform to the numerical solutions of the equation.

Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators - Volume 16 Issue 2 - Minghua Chen, Weihua Deng