Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

Handle URI:
http://hdl.handle.net/10754/623573
Title:
Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
Authors:
Carrillo, José A.; Ranetbauer, Helene; Wolfram, Marie-Therese
Abstract:
In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on its variational formulation as a gradient flow with respect to the Wasserstein distance. Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this property with various examples in spatial dimension one and two.
Citation:
Carrillo JA, Ranetbauer H, Wolfram M-T (2016) Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms. Journal of Computational Physics 327: 186–202. Available: http://dx.doi.org/10.1016/j.jcp.2016.09.040.
Publisher:
Elsevier BV
Journal:
Journal of Computational Physics
Issue Date:
22-Sep-2016
DOI:
10.1016/j.jcp.2016.09.040
Type:
Article
ISSN:
0021-9991
Sponsors:
JAC was partially supported by the Royal Society via a Wolfson Research Merit Award. HR and MTW acknowledge financial support from the Austrian Academy of Sciences ÖAW via the New Frontiers Group NSP-001. The authors would like to thank the King Abdullah University of Science and Technology for its hospitality and partial support while preparing the manuscript.
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DC FieldValue Language
dc.contributor.authorCarrillo, José A.en
dc.contributor.authorRanetbauer, Heleneen
dc.contributor.authorWolfram, Marie-Thereseen
dc.date.accessioned2017-05-15T10:35:09Z-
dc.date.available2017-05-15T10:35:09Z-
dc.date.issued2016-09-22en
dc.identifier.citationCarrillo JA, Ranetbauer H, Wolfram M-T (2016) Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms. Journal of Computational Physics 327: 186–202. Available: http://dx.doi.org/10.1016/j.jcp.2016.09.040.en
dc.identifier.issn0021-9991en
dc.identifier.doi10.1016/j.jcp.2016.09.040en
dc.identifier.urihttp://hdl.handle.net/10754/623573-
dc.description.abstractIn this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on its variational formulation as a gradient flow with respect to the Wasserstein distance. Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this property with various examples in spatial dimension one and two.en
dc.description.sponsorshipJAC was partially supported by the Royal Society via a Wolfson Research Merit Award. HR and MTW acknowledge financial support from the Austrian Academy of Sciences ÖAW via the New Frontiers Group NSP-001. The authors would like to thank the King Abdullah University of Science and Technology for its hospitality and partial support while preparing the manuscript.en
dc.publisherElsevier BVen
dc.subjectLagrangian coordinatesen
dc.subjectVariational schemeen
dc.subjectOptimal transporten
dc.subjectImplicit in time discretizationen
dc.titleNumerical simulation of nonlinear continuity equations by evolving diffeomorphismsen
dc.typeArticleen
dc.identifier.journalJournal of Computational Physicsen
dc.contributor.institutionDepartment of Mathematics, Imperial College London, London SW7 2AZ, UKen
dc.contributor.institutionRadon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Strasse 69, 4040 Linz, Austriaen
dc.contributor.institutionMathematics Institute, University of Warwick, Coventry CV4 7AL, UKen
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