An Engquist–Osher-Type Scheme for Conservation Laws with Discontinuous Flux Adapted to Flux Connections

Bürger, Raimund; Karlsen, Kenneth H.; Towers, John D.

doi:10.1137/07069314x

Cited by 85 publications

(174 citation statements)

References 57 publications

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“…The following definition (cf. [10,9,17,7]), however, avoids the explicit reference to the point (ii) of the introduction. …”

Section: Assumptions Definitions and Resultsmentioning

confidence: 99%

“…For the sake of generality we will consider R-valued bounded functions u 0 and g, although (H1) naturally appears in the case where solutions are [0, 1]-valued (such solutions represent saturations in the porous media, sedimentation or road traffic models; see, e.g., [1,17,5]). Throughout this paper, we assume that f l,r verify …”

Section: Assumptions Definitions and Resultsmentioning

confidence: 99%

“…This weaker assumption leads to a smaller number of global adapted entropy inequalities to be checked. E.g., in the situation where the fluxes f l,r are "bell-shaped", only one global adapted entropy inequality is needed in (2.3), see [17,4,5]. In the present paper, one can replace G * by G in the above definition for the stationary problem (StPb) but not for the evolution problem.…”

Section: Definition 25 (G-entropy Solution Of the Stationary Problem)mentioning

confidence: 99%

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The Semigroup Approach to Conservation Laws with Discontinuous Flux

Andreïanov

2013

Springer Proceedings in Mathematics &Amp; Statistics

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The model one-dimensional conservation law with discontinuous spatially heterogeneous flux isWe prove well-posedness for the Cauchy problem for (EvPb) in the framework of solutions satisfying the so-called adapted entropy inequalities.Exploiting the notion of integral solution that comes from the nonlinear semigroup theory, we propose a way to circumvent the use of strong interface traces for the evolution problem (EvPb) (in fact, proving existence of such traces for the case of x-dependent f l,r would be a delicate technical issue). The difficulty is shifted to the study of the associated one-dimensional stationary problem u + f(x, u)x = g, where existence of strong interface traces of entropy solutions is an easy fact. We give a direct proof of this fact, avoiding the subtle arguments of kinetic formulation [23] or of the H-measure approach [27].

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“…The following definition (cf. [10,9,17,7]), however, avoids the explicit reference to the point (ii) of the introduction. …”

Section: Assumptions Definitions and Resultsmentioning

confidence: 99%

Section: Assumptions Definitions and Resultsmentioning

confidence: 99%

Section: Definition 25 (G-entropy Solution Of the Stationary Problem)mentioning

confidence: 99%

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The Semigroup Approach to Conservation Laws with Discontinuous Flux

Andreïanov

2013

Springer Proceedings in Mathematics &Amp; Statistics

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“…In general, serious work is needed in order to identify the germ G in 1. that encodes specific modeling assumptions at interfaces. Designing conditions of kind 2. may require less work, because the knowledge of a definite subset of a germ G (which is sometimes fairly small, see, e.g., [9,30]) is enough to write down the adapted entropy inequalities. However, the latter advantage disappears if the family of germs should vary along the interface.…”

Section: Towards More Convenient Admissibility Criteriamentioning

confidence: 99%

“…In order to justify existence, one needs in addition the completeness of the maximal extension G * of G and one exploits the adapted entropy inequalities (cf. [24,30]), shown to be equivalent to the admissibility criterion (9):…”

Section: On Different Notions Of Solution and Admissibility Conditionsmentioning

confidence: 99%

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

Andreïanov

2015

ESAIM: Proc.

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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.

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Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

Adimurthi

Ghoshal

Dutta

et al. 2010

Comm Pure Appl Math

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For the scalar conservation laws with discontinuous flux, an infinite family of .A; B/-interface entropies are introduced and each one of them is shown to form an L 1 -contraction semigroup (see [2]). One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [6]. In this paper, we discuss this particular issue in detail and produce a counterexample to show that the solution, in general, has unbounded total variation near the interface. In fact, this example illustrates that smallness of the BV norm of the initial data is immaterial. We hereby settle the question of determining for which of the aforementioned .A; B/ pairs the solution will have bounded total variation in the case of strictly convex fluxes.

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An Engquist–Osher-Type Scheme for Conservation Laws with Discontinuous Flux Adapted to Flux Connections

Cited by 85 publications

References 57 publications

The Semigroup Approach to Conservation Laws with Discontinuous Flux

The Semigroup Approach to Conservation Laws with Discontinuous Flux

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

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

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