Weighted average finite difference methods for fractional diffusion equations

By Santos B. Yuste. Journal of Computational Physics, 2004

380 citations recorded by the source corpus.

A class of finite difference methods for solving fractional diffusion equations is considered. These methods are an extension of the weighted average methods for ordinary (non-fractional) diffusion equations. Their accuracy is of order (Δx)2 and Δt, except for the fractional version of the Crank-Nicholson method, where the accuracy with respect to the timestep is of order (Δt)2 if a second-order approximation to the fractional time-derivative is used. Their stability is analyzed by means of a recently proposed procedure akin to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, is found and checked numerically. Some examples are provided in which the new methods' numerical solutions are obtained and compared against exact solutions.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b4a1427c7f1b84493ebe4575b83c3871ca5cad59

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Citations in the corpus

The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation (2013, methodology): “A class of fractional weighted average finite difference methods were developed in [55] by Yuste, in which the stability was investigated by using the fractional von Neumann stability analysis.”
A new operational matrix for solving fractional-order differential equations (2010, methodology): “dynamic systems containing fractional derivatives, such as Adomian’s decomposition method [19–23], He’s variational iterationmethod [24–26], homotopyperturbationmethod [27,28], homotopy analysismethod [29], collocationmethod [30], Galerkin method [31] and other methods [32–34].”
An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data (2017, neutral): “1) are sufficiently smooth, for example u ∈ C2[0, T ], there are many time discretisation schemes in the literature, for example, Lubich’s fractional multistep methods [7], [30], [55], [65], [61], [62], [64], the L1 scheme and its modification [28], [26], [18], [29], [47], the spectral method [58] [”
Implicit finite difference approximation for time fractional diffusion equations (2008, neutral): “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.”
Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation (2013, methodology): “It is also worth to mention that stability of forward-Euler scheme and weighted averaged difference scheme based on GrünwaldLetnikov approximation were analyzed in [21, 22] for time fractional diffusion equations.”
Exact Solution of Some Linear Fractional Differential Equations by Laplace Transform (2013, methodology): “Keywords: Fractional-order differential equations; Laplace Transform; Inverse Laplace Transform.”