On interface transmission conditions for conservation laws with discontinuous flux of general shape

Abstract

International audienceConservation laws of the form $\partial_t u+ \partial_x f(x;u)=0$ with space-discontinuous flux $f(x;\cdot)=f_l(\cdot)\mathbf{1}_{x<0}+f_r(\cdot)\mathbf{1}_{x>0}$ were deeply investigated in the last ten years, with a particular emphasis in the case where the fluxes are ''bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of $f_{l,r}$. The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers is then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications

New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux

“…A non-measurable choice of t → a(t) would lead to a family of germs G(t) = G a(t) for which existence of solution fails, indeed, (11) cannot hold because t → u l (t) = u(t, 0 − ) must be measurable while (11) leaves no other choice than u l (t) = a(t). So, existence of a sufficiently rich family of measurable in (t, x) functions u which traces u l,r on Σ fulfill (11) is an implicit measurability requirement on the considered family of germs.…”

“…In [14] existence and uniqueness of vanishing viscosity admissible solutions was proved in a multi-dimensional setting, with only one sufficiently regular interface Σ. Let us stress that the proof used only the characterization (11), where the family of germs underlying the admissibility criterion was derived from the model case (4) studied in § 2.1 7 . Therefore the issue of defining and proving measurability of this family of germs was not addressed; heuristically, one can say that the fact that it should be seen as measurable is actually contained in the existence result.…”

“…These developments are detailed in Section 2, Section 3 and finally in Section 4 that is devoted to interface coupling conditions. Details for Section 2 can be found in the work [19] of D. Mitrović and the author; for Section 3, we refer to the work [11] (see also [8,10]) of C. Cancès and the author. Section 4, which idea was already outlined in this introduction, transposes the tools of the works [20,21] of K. Sbihi and the author (recalled in § 4.1) from the case of boundary-value problems to the case of interface coupling problems.…”

“…A new characterization of limits of vanishing viscosity approximation proposed in [Andreianov and Mitrović, 2014] permits to encode admissibility in singular but intuitively appealing entropy inequalities. Transmission maps introduced in ([Andreianov and Cancès, 2015]) have applications in modeling flows in strongly heterogeneous porous media and lead to a simple algorithm for numerical approximation of the associated solutions.Moreover, in order to embed all the aforementioned results into a natural framework, we put forward the concept of interface coupling conditions (ICC) which role is analogous to the role of boundary conditions for boundary-value problems. We link this concept to known examples and techniques.…”

“…A new characterization of limits of vanishing viscosity approximation proposed in [Andreianov and Mitrović, 2014] permits to encode admissibility in singular but intuitively appealing entropy inequalities. Transmission maps introduced in ([Andreianov and Cancès, 2015]) have applications in modeling flows in strongly heterogeneous porous media and lead to a simple algorithm for numerical approximation of the associated solutions.…”