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arXiv · 1605.05921

Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

José A. Carrillo·Helene Ranetbauer·Marie-Therese Wolfram
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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 properties with various examples in spatial dimension one and two.

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José A. Carrillo, Helene Ranetbauer, Marie-Therese Wolfram. 2016-05-19. Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms. https://doi.org/10.1016/j.jcp.2016.09.040

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