A second-order accurate numerical method for the two-dimensional fractional diffusion equation
type: Item
instance of: scholarly article
title: A second-order accurate numerical method for the two-dimensional fractional diffusion equation
zbMATH Open document ID: 1113.65124
author: Charles Tadjeran
author: Mark M. Meerschaert
published in: Journal of Computational Physics
publication date: 2007-02-19T00:00:00Z
review text: The paper is devoted to a study of a numerical method for space-fractional diffusion equations in two space dimensions, with the differential operators being of Riemann-Liouville type. In contrast to the common practice, the authors impose classical Dirichlet boundary conditions. Moreover they always assume that the unknown solutions are sufficiently well behaved for the analysis of the numerical scheme to go through, without discussing concrete conditions under which this assumption is satisfied. The algorithm itself is based on a shifted Grünwald approximation of the fractional space derivatives, combined in an alternating direction implicit fashion with a Crank-Nicolson approach for the first-order time derivative. The result is (under the unspecified conditions) an unconditionally stable method with second order accuracy in time and first order accuracy in space. By adding a Richardson extrapolation step, the accuracy with respect to space can be improved to second order as well.
reviewed by: Kai Diethelm
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 5128246
zbMATH Keywords: space-fractional diffusion equation
zbMATH Keywords: extrapolation
zbMATH Keywords: Crank-Nicolson method
zbMATH Keywords: alternating directions implicit method
zbMATH Keywords: ADI method
zbMATH Keywords: Grünwald formula
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: j.jcp.2006.05.030
OpenAlex ID: W2127684492
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DOI: 10.1016/J.JCP.2006.05.030
description: scientific article; zbMATH DE number 5128246