Kinetic Formulation and Uniqueness for Scalar Conservation Laws with Discontinuous Flux

Crasta, Graziano; Cicco, Virginia De; Philippis, Guido De

doi:10.1080/03605302.2014.979998

Cited by 19 publications

(37 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years there have been many works on hyperbolic conservation laws with a spatially discontinuous flux function, providing a great number of results relating to existence, uniqueness, stability, and numerical approximations of entropy solutions [1,2,3,4,5,3,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,34,36,37,38,39,41,42,43,44,45,46,47]. Herein we are interested in numerical methods for the initial value problem u t + F(x, u) x = 0 for (x, t) ∈ Π T := R × (0, T ), u(x, 0) = u 0 (x) for x ∈ R, (1.1) F(x, u) := H(−x)g(u) + H(x)f (u), where H(x) is the Heaviside function.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition

Karlsen

Towers

2017

J. Hyper. Differential Equations

View full text Add to dashboard Cite

Abstract. We study a scalar conservation law whose flux has a single spatial discontinuity. There are many notions of (entropy) solution, the relevant concept being determined by the application. We focus on the so-called vanishing viscosity solution. We utilize a Kružkov-type entropy inequality which generalizes the one in [38], singles out the vanishing viscosity solution whether or not the crossing condition is satisfied, and has a discrete version satisfied by the Godunov variant of the finite difference scheme of [26,38]. We show that the solutions produced by that scheme converge to the unique vanishing viscosity solution. The scheme does not require a Riemann solver for the discontinuous flux problem. This makes its implementation simple even when the flux is multimodal, and there are multiple flux crossings.

show abstract

Section: Introductionmentioning

confidence: 99%

Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition

Karlsen

Towers

2017

J. Hyper. Differential Equations

View full text Add to dashboard Cite

show abstract

“…and the first statement of Lemma 2 is proved combining (15) and (16). The result for non classical shocks holds because of the flux conservation through the discontinuities z i .…”

Section: Proposition 2 Under Assumptions (H-0) (H-1) (H-2)mentioning

confidence: 73%

“…Other formulation can also be found in the litterature, for instance in [16] [17] where the authors introduce an entropy criterion based on the two criteria mentioned here and obtain uniqueness (but not existence) of BV entropy solutions. We provide here some examples to compare the two formulations.…”

Section: Comparison Between the Two Formulationsmentioning

confidence: 99%

A General BV Existence Result for Conservation Laws with Spatial Heterogeneities

Piccoli¹,

Tournus²

2018

SIAM J. Math. Anal.

View full text Add to dashboard Cite

We consider a scalar conservation law with a flux containing spatial heterogeneities of bounded variation, where the number of discontinuities may be infinite. We prove existence of an adapted entropy solution in the sense of Audusse-Perthame in the BV framework, under the assumptions that guarantee uniqueness of the solution, plus one additional technical assumption. We prove that existence in the BV framework fails withouth this additional technical assumption, by providing a counter-example.

show abstract

“…To sum up, in [19] we have established another intrinsic characterization of vanishing viscosity limits which, in contrast to (VV.c), is applicable to fluxes which crossing is not restricted: (VV.c ) the characterization by singular Kruzhkov entropy inequalities of the kind (19), see [19] (see also [42]). …”

Section: Singular Values and Singular Kruzhkov Entropy Inequalitiesmentioning

confidence: 99%

“…In [42], it was proved that BV solutions satisfying inequalities of the kind (20) are unique for a quite general flux function f with SBV dependence in (t, x), SBV being the special class of functions of bounded variation lacking the Cantor part. Existence of such solution is an open question.…”

Section: Existence For the Multi-dimensional Case: Adapted Viscositiesmentioning

confidence: 99%

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

Andreïanov

2015

ESAIM: Proc.

View full text Add to dashboard Cite

Abstract. Hyperbolic conservation laws of the form ut + div f(t, x; u) = 0 with discontinuous in (t, x) flux function f attracted much attention in last 20 years, because of the difficulties of adaptation of the classical Kruzhkov approach developed for the smooth case. In the discontinuous-flux case, non-uniqueness of mathematically consistent admissibility criteria results in infinitely many different notions of solution. A way to describe all the resulting L 1 -contractive solvers within a unified approach was proposed in the work [Andreianov, Karlsen, Risebro, 2011]. We briefly recall the ideas and results developed there for the model one-dimensional case with f(t, x; u) = f l (u)1 1x<0 + fr(u)1 1x>0 and highlight the main hints needed to address the multi-dimensional situation with curved interfaces.Then we discuss two recent developments in the subject which permit to better understand the issue of admissibility of solutions in relation with specific modeling assumptions; they also bring useful numerical approximation strategies. 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.

show abstract

Kinetic Formulation and Uniqueness for Scalar Conservation Laws with Discontinuous Flux

Abstract: Abstract. We prove a uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.

Cited by 19 publications

References 25 publications

Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition

Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition

A General BV Existence Result for Conservation Laws with Spatial Heterogeneities

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

Contact Info

Product

Resources

About