Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
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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.
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.
SponsorsJAC 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.
JournalJournal of Computational Physics