2012
DOI: 10.1007/s10915-012-9661-0
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Abstract: Based on the weighted and shifted Grünwald difference (WSGD) operators [24], we further construct the compact finite difference discretizations for the fractional operators. Then the discretization schemes are used to approximate the one and two dimensional space fractional diffusion equations. The detailed numerical stability and error analysis are theoretically performed. We theoretically prove and numerically verify that the provided numerical schemes have the convergent orders 3 in space and 2 in time.

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Cited by 137 publications
(98 citation statements)
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“…In the semidiscretization, we should then choose a higher order spatial approximation, e.g., the one in (12) and the accuracy of the time stepping can also be increased, e.g., using a Crank-Nicolson scheme. For homogeneous Dirichlet boundary conditions, such a study is performed (even for the multidimensional case) in [8].…”
Section: Complexity and Extension To Higher-order Methodsmentioning
confidence: 99%
See 1 more Smart Citation
“…In the semidiscretization, we should then choose a higher order spatial approximation, e.g., the one in (12) and the accuracy of the time stepping can also be increased, e.g., using a Crank-Nicolson scheme. For homogeneous Dirichlet boundary conditions, such a study is performed (even for the multidimensional case) in [8].…”
Section: Complexity and Extension To Higher-order Methodsmentioning
confidence: 99%
“…In order to have sufficient accuracy in the finite difference approximation near to the boundary and at the boundary of a nonlocal differential operator, it is necessary to have (virtual) gridpoints outside of the original computational domain D .a; b/. This is clearly shown in (6), (7) and (12). To summarize, the sketch of our approach is the following: -we extend the problem to R to get rid of the boundary conditions -we solve the corresponding Cauchy problem (3) (we will approximate this with finite differences) -we verify that the desired homogeneous Neumann (or homogeneous Dirichlet) boundary conditions are satisfied for the restriction of the solution.…”
Section: Extensionsmentioning
confidence: 99%
“…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].…”
Section: Introductionmentioning
confidence: 99%
“…The finite difference approach based on the Grünwald-Letnikov formula has been considerably developed in the last decade starting with the paper [12]. Nowadays, higher order approximations [13], [14], ADI methods [15] are available for the corresponding numerical simulations. Moreover, recently, several kinds of linear [16], [17] and non-linear problems are studied containing fractional order Laplacian operators [18], [19].…”
Section: Introductionmentioning
confidence: 99%