Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

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 driv...

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Bibliographic Details
Published in:Journal of computational physics 2016-12, Vol.327, p.186-202
Main Authors: Carrillo, José A., Ranetbauer, Helene, Wolfram, Marie-Therese
Format: Article
Language:English
Subjects:
Computational physics
Computer simulation
Continuity equation
Energy
Gradient flow
Implicit in time discretization
Lagrangian coordinates
Nonlinear equations
Numerical analysis
Optimal transport
Studies
Variational scheme
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2016.09.040