Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

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

Research output: Contribution to journalArticlepeer-review

33 Scopus citations


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 languageEnglish (US)
Pages (from-to)186-202
Number of pages17
JournalJournal of Computational Physics
StatePublished - Sep 22 2016
Externally publishedYes

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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