Some time stepping methods for fractional diffusion problems with nonsmooth data

Hdl Handle:
http://hdl.handle.net/10034/620655
Title:
Some time stepping methods for fractional diffusion problems with nonsmooth data
Authors:
Yang, Yan; Yan, Yubin; Ford, Neville J.
Abstract:
We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293(2015), 201-217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the Riemann-Liouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
Affiliation:
Lvliang University; University of Chester
Citation:
Yang, Y., Yan, Y., & Ford, N. (2017 - in press). Some time stepping methods for fractional diffusion problems with nonsmooth data. Computational Methods in Applied Mathematics. https://doi.org/10.1515/cmam-2017-0037
Publisher:
De Gruyter
Journal:
Computational Methods in Applied Mathematics
Publication Date:
2-Sep-2017
URI:
http://hdl.handle.net/10034/620655
DOI:
10.1515/cmam-2017-0037
Additional Links:
https://www.degruyter.com/view/j/cmam.ahead-of-print/cmam-2017-0037/cmam-2017-0037.xml
Type:
Article
Language:
en
EISSN:
1609-9389
Appears in Collections:
Mathematics

Full metadata record

DC FieldValue Language
dc.contributor.authorYang, Yanen
dc.contributor.authorYan, Yubinen
dc.contributor.authorFord, Neville J.en
dc.date.accessioned2017-10-16T08:37:18Z-
dc.date.available2017-10-16T08:37:18Z-
dc.date.issued2017-09-02-
dc.identifier.citationYang, Y., Yan, Y., & Ford, N. (2017 - in press). Some time stepping methods for fractional diffusion problems with nonsmooth data. Computational Methods in Applied Mathematics. https://doi.org/10.1515/cmam-2017-0037en
dc.identifier.doi10.1515/cmam-2017-0037-
dc.identifier.urihttp://hdl.handle.net/10034/620655-
dc.description.abstractWe consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293(2015), 201-217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the Riemann-Liouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.en
dc.language.isoenen
dc.publisherDe Gruyteren
dc.relation.urlhttps://www.degruyter.com/view/j/cmam.ahead-of-print/cmam-2017-0037/cmam-2017-0037.xmlen
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectFractional diffusion problemen
dc.subjectNonsmooth dataen
dc.subjectError estimatesen
dc.subjectLaplace transformen
dc.titleSome time stepping methods for fractional diffusion problems with nonsmooth dataen
dc.typeArticleen
dc.identifier.eissn1609-9389-
dc.contributor.departmentLvliang University; University of Chesteren
dc.identifier.journalComputational Methods in Applied Mathematicsen
dc.date.accepted2017-08-18-
or.grant.openaccessYesen
rioxxterms.funderUnfundeden
rioxxterms.identifier.projectUnfundeden
rioxxterms.versionAMen
rioxxterms.licenseref.startdate2018-09-02-
This item is licensed under a Creative Commons License
Creative Commons
All Items in ChesterRep are protected by copyright, with all rights reserved, unless otherwise indicated.