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Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
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DOI：10.1016/j.jcp.2016.09.040.png)
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Abstract
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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. (C) 2016 Elsevier Inc. All rights reserved.
Keywords:
Lagrangian coordinates
Variational scheme
Optimal transport
Implicit in time discretization
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