Time-stepping error bounds for fractional diffusion problems with non-smooth initial data

William McLean; Kassem Mustapha

doi:10.1016/j.jcp.2014.08.050

Time-stepping error bounds for fractional diffusion problems with non-smooth initial data

William McLean^*
, Kassem Mustapha

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

75 Scopus citations

Abstract

We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the nth time level t_n, but the error bound includes a factor tn-1 if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as t_n decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.

Original language	English
Pages (from-to)	201-217
Number of pages	17
Journal	Journal of Computational Physics
Volume	293
DOIs	https://doi.org/10.1016/j.jcp.2014.08.050
State	Published - 5 Jul 2015

Bibliographical note

Publisher Copyright:
© 2014 Elsevier Inc.

Keywords

Discontinuous Galerkin method
Implicit Euler method
Laplace transform
Polylogarithm

ASJC Scopus subject areas

Numerical Analysis
Modeling and Simulation
Physics and Astronomy (miscellaneous)
General Physics and Astronomy
Computer Science Applications
Computational Mathematics
Applied Mathematics

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10.1016/j.jcp.2014.08.050

Cite this

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abstract = "We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the nth time level tn, but the error bound includes a factor tn-1 if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as tn decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.",

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AU - Mustapha, Kassem

PY - 2015/7/5

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N2 - We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the nth time level tn, but the error bound includes a factor tn-1 if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as tn decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.

AB - We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the nth time level tn, but the error bound includes a factor tn-1 if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as tn decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.

KW - Discontinuous Galerkin method

KW - Implicit Euler method

KW - Laplace transform

KW - Polylogarithm

UR - https://www.scopus.com/pages/publications/84946160836

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M3 - Article

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SP - 201

EP - 217

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -

Time-stepping error bounds for fractional diffusion problems with non-smooth initial data

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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