Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat–Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 × 2 Riemann solver. 1.

Abstract. We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux in rather general form (no convexity or genuine linearity is needed). We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under the assumption that initial data belong to the BVclass. Such initial data enable us to prove that the sequence of solutions to a special vanishing viscosity approximation of the considered equation is, at the same time, the sequence of quasisolutions to a non-degenerate scalar conservation law. This provides existence of the solution admitting strong traces at the interface. The admissibility conditions are chosen so that a kind of crossing condition is satisfied which, together with existence of traces, provides uniqueness of the solution. In the current contribution, we consider the following problem ∂tu + ∂x (H(x)f(u) + H(−x)g(u)) = 0, (t, x) ∈ IR + × IR u|t=0 = u0(x) ∈ BV (IR), x ∈ IR (1) where u is the scalar unknown function; u0 is an integrable initial function of bounded variation such that a ≤ u0 ≤ b, a, b ∈ IR; H is the Heaviside function; and

Abstract. We prove that a family of solutions to a Cauchy problem for a two dimensional scalar conservation law with a discontinuous smoothed flux and the vanishing viscosity is strongly L1 loc – precompact under a new genuine nonlinearity condition, weaker than in previous works on the subject. 1.

Abstract. We consider a multi-dimensional triangular system of conservation laws. This system arises as a model of three-phase flow in porous media and includes multi-dimensional conservation laws with discontinuous coefficients as a special case. The system is neither strictly hyperbolic nor symmetric. We propose an Engquist-Osher type scheme for this system and show that the approximate solutions generated by the scheme converge to a weak solution. Numerical examples are also presented. 1.

Abstract. We consider multi-dimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach of H-measures [17] to investigate the zero diffusion-dispersion-smoothing limit. Content

Abstract. By discovering that solutions of the vanishing viscosity approximation (but without flux regularization) to a scalar conservation law with discontinuous flux are equal to a flux crossing point at the interface, we derive entropy conditions which provide well-posedness to a corresponding Cauchy problem. We assume that the flux is such that the maximum principle holds, but we allow multiple flux crossings and we do not need any kind of genuine nonlinearity conditions. Proposed concept is a proper generalization to the standard Kruzhkov entropy conditions and it does not involve transformation of the equation or use of adapted entropies. The subject of the paper is the following Cauchy problem ∂tu + ∂x (H(x)f(u) + H(−x)g(u)) = 0, (t, x) ∈ IR + × IR u|t=0 = u0(x) ∈ L ∞ (IR), x ∈ IR where u is the scalar unknown function; u0 is a function such that a ≤ u0 ≤ b, a, b ∈ IR; H is the Heaviside function; and f, g ∈ C 1 (R) are such that f(a) = g(a) = c1,

Abstract. We prove existence and uniqueness of an entropy solution to a multidimensional scalar conservation law with discontinuous flux. The proof is based on the corresponding kinetic formulation of the considered equation and a ”smart ” change of an unknown function. 1.

Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux 1 E.Yu. Panov Sequences of entropy solutions of a non-degenerate first-order quasilinear equation are shown to be strongly pre-compact in the general case of a Caratheodory flux vector. Existence of the weak and entropy solution to Cauchy problem for such equation is also established. The proofs are based on general localization principle for H-measures corresponding to sequences of measure-valued functions.

We study a system of conservation laws that describes multispecies kinematic flows with an emphasis on models of multiclass traffic flow and of the creaming of oil-in-water dispersions. The flux can have a spatial discontinuity which models abrupt changes of road surface conditions or of the cross-sectional area in a settling vessel. For this system, an entropy inequality is proposed that singles out a relevant solution at the interface. It is shown that “piecewise smooth” limit solutions generated by the semi-discrete version of a numerical scheme the authors recently proposed [R. Bürger, A. García, K.H. Karlsen and J.D. Towers, J. Engrg. Math. 60:387–425, 2008] satisfy this entropy inequality. We present an improvement to this scheme by means of a special interface flux that is activated only at a few grid points where the discontinuity is located. While an entropy inequality is established for the semi-discrete versions of the scheme only, numerical experiments support that the fully discrete scheme are equally entropy-admissible.