First order quasilinear equations with boundary conditions

Bardos, Claude; Leroux, Alain; Nédélec, Jean-Claude

doi:10.1080/03605307908820117

Cited by 577 publications

(704 citation statements)

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“…Before entering the core of the matter, we recall some basic results from [3,11,35] concerning the boundary value problem for (26):…”

Section: Some Results About Initial-boundary-value Problemsmentioning

confidence: 99%

Localization effects and measure source terms in numerical schemes for balance laws

Gosse

2001

Math. Comp.

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Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided.

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“…Before entering the core of the matter, we recall some basic results from [3,11,35] concerning the boundary value problem for (26):…”

Section: Some Results About Initial-boundary-value Problemsmentioning

confidence: 99%

Localization effects and measure source terms in numerical schemes for balance laws

Gosse

2001

Math. Comp.

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“…REMARK 1. (1) In view of previous results [1] concerning the equation (1.1), the boundary conditions (1.3.a) and (1.3.b) seem to be "natural". We consider that they are clearly justified by the semigroup property in weighted L1 -space, which will be established in §6.…”

Section: Formulationmentioning

confidence: 78%

“…We will establish that the functions r(-)f(u(-, •)) and sgn/'(u(-, •)) possess traces at the boundary in a weak sense. And, we will follow ideas of Bardos-Leroux-Nedelec [1]: if 7o(-) (respectively eo(-,-)) denotes the trace of r(-)f(u(-, ■)) (resp. sgn/'(w(-, •))) at z = 0, for example, the boundary condition at z = 0 is written as (see §1) , .…”

Section: Introductionmentioning

confidence: 99%

Explicit formula for weighted scalar nonlinear hyperbolic conservation laws

LeFloch¹,

Nédélec²

1988

Trans. Amer. Math. Soc.

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Abstract.We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form jL(ru) + |_(r/(tt))=0, dt ox with initial data and boundary condition. The scalar function u = u(x,t), x > 0, t > 0, is the unknown, the function / = f(u) is assumed to be strictly convex with inf /(•) = 0 and the weight function r = r(x), x > 0, to be positive (for example, r(x) = xa, with an arbitrary real a).We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux r(-)f(u(-, ■)) at the boundary is solved by introducing a variational inequality.The uniqueness result is obtained by extending a semigroup property due to B. L. Keyfitz. Introduction.We consider weighted scalar nonlinear hyperbolic conservation laws of the formwhere the scalar function u = u(x,t), x > 0, t > 0, is the unknown. The flux function / = f(u) is assumed to be strictly convex with inf /(•) = 0 and the weight function r = r(x), x > 0, to be positive. This paper is concerned with the mixed problem associated with the equation (0.1): we are looking for a solution u = u(x,t) of (0.1), satisfying an initial data and a boundary condition. But, it is well known that conservation laws of the type (0.1) do not possess classical solutions, even when the initial data is smooth: discontinuities appear in finite time. Hence, we consider only weak solutions of (0.1), that is solutions in the sense of distributions. And, for the sake of uniqueness, we have to add an entropy condition that selects the physical (or entropy) solution among all the solutions in the sense of distributions.For a convex flux function /(•), the entropy condition is written as We are interested (in particular) in the case where the function r = r(x) tends to zero at the point z = 0 or at the point z = +oo. For example, we can take r(x) = xa (a E R). In such a case, the equation (0.1) possesses an algebraic

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“…At point x = 0 (the entry), according to [2], the value of density r cannot be prescribed if that value is larger than r c . Why is that so?…”

Section: The Entrymentioning

confidence: 99%

“…For the rigorous mathematical derivation of BLN boundary conditions in general context one can consult [2] or [12]. It should be mentioned that the BLN condition does not work for hyperbolic systems, i. e. for the second order models of traffic flow.…”

Section: Introductionmentioning

confidence: 99%