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Zbl 1119.65379
Yuste, S.B.; Acedo, L.
An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. (English)
[J] SIAM J. Numer. Anal. 42, No. 5, 1862-1874 (2005). ISSN 0036-1429; ISSN 1095-7170/e

Summary: A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grünwald-Letnikov discretization of the Riemann-Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.

MSC 2000:: *65M06 Finite difference methods (IVP of PDE)
35K15 Second order parabolic equations, initial value problems
26A33 Fractional derivatives and integrals (real functions)

Keywords: fractional diffusion equation; von Neumann stability analysis; parabolic integro-differential equations; finite difference methods; numerical examples; forward time centered space method; Grünwald-Letnikov discretization

Cited in: Zbl 1094.65085

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