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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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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creator Carrillo, José A.
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description 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.
doi_str_mv 10.1016/j.jcp.2016.09.040
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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
title Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
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