A class of second order difference approximations for solving space fractional diffusion equations

Abstract

A class of second order approximations, called the weighted and shifted Grünwald difference (WSGD) operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions. The stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established. Several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order, and the numerical results for variable coefficients problem are also presented.

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

“…Remark 2 It should be noted that the operator (18) can also be derived by the way of weighting and shifting Grünwald difference operator [33].…”

“…More ADI schemes can be see in [33] and [37]. Letting τ = h and τ = h/20, respectively, in second and third order (in terms of spatial direction) stable schemes can make sure that the numerical errors caused by the Crank-Nicolson method in time direction is small enough, so that errors in spatial direction are dominant and the convergence rates can be testified.…”

“…Using the idea of second order central difference formula, a second order discretization for Riemann-Liouville fractional derivative is established in [31], and the scheme is extended to two dimensional two-sided space fractional convection diffusion equation in finite domain in [12]. By assembling the Grünwald difference operators with different weights and shifts, a class of stable second order discretizations for Riemann-Liouville space fractional derivative is developed in [33,17], and an abstract way of discussing the stability and convergence of the discretizations can be seen in [1]; its corresponding third order quasi-compact scheme is given in [37]. Allowing to use the points outside of the domain, a class of second, third and fourth order approximations for Riemann-Liouville space fractional derivatives are derived in [13] by using the weighted and shifted Lubich difference operators.…”

Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations

“…In [16], the authors use the superconvergent point to get the second order scheme for the Riemann-Liouville fractional derivative. The second and third order WSGD operators are provided in [17], and a third order CWSGD operator is given in [18]. The related more high order schemes can be seen, e.g., [19,20].…”

ADI Finite Element Method for 2D Nonlinear Time Fractional Reaction-Subdiffusion Equation

“…Fast iterative ADI FD schemes [24] [25] are designed for 2D/3D linear space fractional diffusion equations, which are first order accuracy in both time and space and have the advantage of low computational work and low memory storage. High order accurate ADI FD schemes are proposed for 2D linear space fractional diffusion equations [26] [27] and two-sided space fractional convection-diffusion equations [28], which are based on weighted and shifted Grünwald operators or Lubich operators approximating Riemann-Liouville fractional derivatives respectively. Spectral direction splitting methods [29] are derived for 2D space fractional differential equations.…”