# 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). Some time stepping methods for fractional diffusion problems with nonsmooth data. Computational Methods in Applied Mathematics, 18(1), 129-146. 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
Type:
Article
Language:
en
EISSN:
1609-9389
Appears in Collections:
Mathematics

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). Some time stepping methods for fractional diffusion problems with nonsmooth data. Computational Methods in Applied Mathematics, 18(1), 129-146. 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.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