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Finite volume approximation and well-posedness of conservation laws with moving interfaces under abstract coupling conditions

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Abstract

Scalar conservation law \(\displaystyle {\partial _t \rho (t, x) + \partial _x({\textbf{f}}(t, x, \rho )) = 0}\) with a flux \({\textbf{C}}^{1}\) in the state variable \(\rho \), piecewise \({\textbf{C}}^{1}\) in the (t, x)-plane admits infinitely many consistent notions of solution which differ by the choice of interface coupling. Only the case of the so-called vanishing viscosity solutions received full attention, while different choice of coupling is relevant in modeling situations that appear, e.g., in road traffic and in porous medium applications. In this paper, existence of solutions for a wide set of coupling conditions is established under some restrictions on \({\textbf{f}}\), via a finite volume approximation strategy adapted to slanted interfaces and to the presence of interface crossings. The notion of solution, restated under the form of an adapted entropy formulation which is consistently approximated by the numerical scheme, implies uniqueness and stability of solutions. Numerical simulations are presented to illustrate the reliability of the scheme.

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Acknowledgements

This paper has been supported by the RUDN University Strategic Academic Leadership Program. The work on this paper was supported by l’Agence Nationale de la Recherche (ANR), project ANR-22-CE40-0010 (ANR CoSS).

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Authors and Affiliations

  1. Institut Denis Poisson CNRS UMR 7013, Université de Tours, Université d’Orléans, Parc Grandmont, 37200, Tours, France

    Boris Andreianov

  2. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russian Federation, 117198

    Boris Andreianov

  3. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi 53, 20125, Milan, Italy

    Abraham Sylla

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  1. Boris Andreianov
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  2. Abraham Sylla
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All authors equally contributed to elaboration of statements and proofs, to writing and re-reading of the manuscript. AS implemented the scheme and carried out the numerical tests.

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Andreianov, B., Sylla, A. Finite volume approximation and well-posedness of conservation laws with moving interfaces under abstract coupling conditions. Nonlinear Differ. Equ. Appl. 30, 53 (2023). https://doi.org/10.1007/s00030-023-00857-9

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