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

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.

Authors
Andreianov Boris 1, 2 , Sylla Abraham3
Publisher
Birkhauser Verlag AG
Number of issue
4
Language
English
Pages
53
Status
Published
Volume
30
Year
2023
Organizations
  • 1 Université de Tours, Université d’Orléans
  • 2 Peoples’ Friendship University of Russia (RUDN University)
  • 3 Università di Milano-Bicocca
Keywords
conservation laws; discontinuous flux; Moving interface; Interface coupling conditions; finite volume scheme; Godunov flux; existence of solutions; well-posedness; 35L65; 65M08; 65M12; 35A01; 35A02; analysis

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