Particle Simulation of Fractional Diffusion Equations

Handle URI:
http://hdl.handle.net/10754/626535
Title:
Particle Simulation of Fractional Diffusion Equations
Authors:
Allouch, Samer; Lucchesi, Marco; Maître, O. P. Le; Mustapha, K. A.; Knio, Omar
Abstract:
This work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green's function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Applied Mathematics and Computational Science Program
Publisher:
arXiv
Issue Date:
12-Jul-2017
ARXIV:
arXiv:1707.03871
Type:
Preprint
Additional Links:
http://arxiv.org/abs/1707.03871v1; http://arxiv.org/pdf/1707.03871v1
Appears in Collections:
Other/General Submission; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorAllouch, Sameren
dc.contributor.authorLucchesi, Marcoen
dc.contributor.authorMaître, O. P. Leen
dc.contributor.authorMustapha, K. A.en
dc.contributor.authorKnio, Omaren
dc.date.accessioned2017-12-28T07:32:15Z-
dc.date.available2017-12-28T07:32:15Z-
dc.date.issued2017-07-12en
dc.identifier.urihttp://hdl.handle.net/10754/626535-
dc.description.abstractThis work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green's function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.en
dc.publisherarXiven
dc.relation.urlhttp://arxiv.org/abs/1707.03871v1en
dc.relation.urlhttp://arxiv.org/pdf/1707.03871v1en
dc.rightsArchived with thanks to arXiven
dc.titleParticle Simulation of Fractional Diffusion Equationsen
dc.typePreprinten
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.eprint.versionPre-printen
dc.contributor.institutionCNRS, LIMSI, Universit´e de Paris Saclay, Orsay, Franceen
dc.contributor.institutionKing Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabiaen
dc.identifier.arxividarXiv:1707.03871en
kaust.authorAllouch, Sameren
kaust.authorLucchesi, Marcoen
kaust.authorKnio, Omaren
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