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