An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations

By Santos B. Yuste, Luis Acedo. SIAM Journal on Numerical Analysis, 2003

579 citations recorded by the source corpus.

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 Grunwald--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.

Subfields: Mechanics of Materials, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/6882719723145ebd89d1032e2c1fbe99fd82e98a

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Citations in the corpus

On the existence of solutions of fractional integro-differential equations (2011, unknown): “[3], Pilipović and Stojanović [30], Yuste and Acedo [41], Idczak and Kamocki [16], and Kostić [22], between so many more, have investigated the existence of solutions for various types of fractional differential and integral equations.”
A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications (2019, background): “Subsequently, VO-FDEs have attracted more and more attention, ascribing to its suitability in modeling along with a large variety of phenomena, ranging many fields of science and engineering fields, including anomalous diffusion [19, 103, 139], viscoelastic mechanics [30, 36, 50, 96], control system”
A Fourier method for the fractional diffusion equation describing sub-diffusion (2007, background): “It has been suggested that the probability density function (pdf) u(x;t) that describes anomalous sub-difiusive particles follows the fractional difiusion equation [1],[13],[ 14 ]:”
Time two-mesh algorithm combined with finite element method for time fractional water wave model (2018, unknown): “In view of the difficulty for solving the analytic solutions, increasing scholars have started to study some effective numerical methods for solving complex FPDEs with time, space and space-time fractional derivatives, which include finite difference methods [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1”
A backward problem for the time-fractional diffusion equation (2010, methodology): “As for the numerical methods for the initial/boundary value problem for fractional diffusion equations, see [18,19].”
Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion (2013, methodology): “Yuste and Acedo [25] presented a numerical scheme by combining the forward time centered space method for the ordinary diffusion equation and Grunwald-Letnikov discretization of the (Riemann-Liouville type) fractional-derivative and provided a von Neumann type stability analysis.”
On Multistep Methods for Differential Equations of Fractional Order (2006, methodology): “An example of successfully application is given in [ 14 ].”
Implicit finite difference approximation for time fractional diffusion equations (2008, unknown): “Explicit methods, and hence conditionally stable, finite difference schemes are available in the literature, [9,10], for the numerical solution of the time fractional diffusion equation of the form ∂u(x, t) ∂t = ∂1−α ∂t1−α ∂2u(x, t) ∂x2 (1.”