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...