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

Abstract

Based on the superconvergent approximation at some point (depending on the fractional order α, but not belonging to the mesh points) for Grünwald 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 nonhomogeneous boundary conditions are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders.

Algebraic construction of a third order difference approximations for fractional derivatives and applications

“…We consider the following test example for the fractional diffusion problem (24). The test problem was applied to the CN type numerical schemes developed in Section 5.…”

“…We present the CN type scheme with the third order approximation in (18) for the space fractional derivative using W 2,1 (z) with the preconditioner operator P x = 1 + h 2 a 2 (1)D 2 . For the second order approximation in (15), P x will be the unit operator I. Preconditioning (24) by P x , one gets the equivalent equation…”

“…Proof. Let e m = u m −Û m be the error vector of the solutions, where u m ,Û m are the exact and approximate solutions of the diffusion problem (24). Then the error of the internal grid valuesê m satisfy the system The convergence is then established as h, τ → 0.…”

“…Convex combinations of various shifts of the shifted Grünwald approximation were used to obtain higher order approximation in Chinese schools [20,6,25]. Zhao and Deng [24] extended the concept of super convergence to derive a series of higher order approximations. Earlier, Lubich [9] obtained higher order approximations for the fractional derivative for orders up to 6 with no shifts involved.…”

Finite difference approximation of space-fractional diffusion problems: The matrix transformation method

“…The finite difference methods for the conventional diffusion problems were extended in some sense including the development of higher-order methods for the spatial discretization [1], [2] and the time integration [3], generalization of ADI methods [4], [5], construction of appropriate iterative solvers [6] and computing on non-uniform meshes [7]. On the development of the computational efficiency we refer to [5] and [7].…”