A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations

Wang, Hong; Basu, Treena

doi:10.1137/12086491x

Cited by 213 publications

(148 citation statements)

References 19 publications

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“…More refined analyses which take into account the singular behavior of FPDEs have been established recently in [10,9]. Moreover, some fast solvers for FD/FE approximations to FPDEs have been developed in [27,28,29] and [11,20] by exploiting their Toepliz structures. On the other hand, spectral methods for some fractional PDEs have been proposed in [12,13] where the wellposedness of some FPDEs and their spectral approximations have been established.…”

Section: Introductionmentioning

confidence: 99%

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Mao

Shen

2016

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 99%

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Mao

Shen

2016

Journal of Computational Physics

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“…Hajipour et al [17] provided a new nonstandard finite difference scheme to study the dynamic treatments of a class of fractional chaotic systems and stability analysis of fractional order systems [17]. By combining the implicit difference scheme and the preconditioned conjugate gradient method, Wang Hong et al [18,19] gave a fast implicit difference method for the two/threedimensional fractional diffusion equation. The method has a computational work count of O(N log N) per iteration and a memory requirement of O(N) [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…By combining the implicit difference scheme and the preconditioned conjugate gradient method, Wang Hong et al [18,19] gave a fast implicit difference method for the two/threedimensional fractional diffusion equation. The method has a computational work count of O(N log N) per iteration and a memory requirement of O(N) [18,19]. Gao Guanghua et al [20] derived a compact finite difference scheme for the sub-diffusion equation, which is fourth order accuracy compact approximate for the second order space derivative [20].…”

Section: Introductionmentioning

confidence: 99%

An alternating segment Crank–Nicolson parallel difference scheme for the time fractional sub-diffusion equation

Yang

Cao

2018

Adv Differ Equ

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In this paper, an alternating segment Crank-Nicolson (ASC-N) parallel difference scheme is proposed for the time fractional sub-diffusion equation, which consists of the classical Crank-Nicolson scheme, four kinds of Saul'yev asymmetric schemes, and alternating segment technique. Theoretical analysis reveals that the ASC-N scheme is unconditionally stable and convergent by mathematical induction method. Finally, the theoretical analysis is verified by numerical experiments, which show that the ASC-N scheme is efficient for solving the time fractional sub-diffusion equation.

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“…[35,32,40,34,15,16,42,22] and the references therein), which are expensive to compute and invert. It is therefore of importance to construct fast solvers by carefully analysing the structures of the matrices (see, e.g., [44,31]). This should be in marked contrast with the situations when they are applied to differential equations of integer order derivatives.…”

Section: Introductionmentioning

confidence: 99%

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

Jiao

Wang

Huang

2016

Journal of Computational Physics

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Abstract. The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order µ ∈ (0, 1) to compute that of any order k + µ with integer k ≥ 0, while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order µ ∈ (0, 1). Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at "interior" JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, are essential for our algorithm development.

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A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations

Cited by 213 publications

References 19 publications

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

An alternating segment Crank–Nicolson parallel difference scheme for the time fractional sub-diffusion equation

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

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