Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

José A. Carrillo; Helene Ranetbauer; Marie-Therese Wolfram

doi:10.1016/j.jcp.2016.09.040

Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

José A. Carrillo, Helene Ranetbauer, Marie-Therese Wolfram

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

34 Scopus citations

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.

Original language	English (US)
Pages (from-to)	186-202
Number of pages	17
Journal	Journal of Computational Physics
Volume	327
DOIs	https://doi.org/10.1016/j.jcp.2016.09.040
State	Published - Sep 22 2016
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1016/j.jcp.2016.09.040

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title = "Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms",

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.",

author = "Carrillo, {Jos{\'e} A.} and Helene Ranetbauer and Marie-Therese Wolfram",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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 {\"O}AW 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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AU - Carrillo, José A.

AU - Ranetbauer, Helene

AU - Wolfram, Marie-Therese

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2016/9/22

Y1 - 2016/9/22

N2 - 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.

AB - 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.

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SP - 186

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JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -

Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

Abstract

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