Convergence of a Lagrangian Discretization for Barotropic Fluids and Porous Media Flow
Abstract
When expressed in Lagrangian variables, the equations of motion for compressible (barotropic) fluids have the structure of a classical Hamiltonian system in which the potential energy is given by the internal energy of the fluid. The dissipative counterpart of such a system coincides with the porous medium equation, which can be cast in the form of a gradient flow for the same internal energy. Motivated by these related variational structures, we propose a particle method for both problems in which the internal energy is replaced by its Moreau--Yosida regularization in the $L^2$ sense, which can be efficiently computed as a semidiscrete optimal transport problem. Using a modulated energy argument which exploits the convexity of the problem in Eulerian variables, we prove quantitative convergence estimates towards smooth solutions. We verify such estimates by means of several numerical tests.
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Published In
SIAM Journal on Mathematical Analysis
Pages: 2990 - 3018
DOI: 10.1137/21M1422756
ISSN (online): 1095-7154
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© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 28 May 2021
Accepted: 16 December 2021
Published online: 16 May 2022
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LabEx LMH
Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-16-CE40-0014
Commissariat Général à l'Investissement https://doi.org/10.13039/501100006072 : ANR-11-LABX-0056-LMH
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