Abstract:
Burger et al.in \cite{karlsen-1} proposed a flux TVD (FTVD) second order scheme by using a new non local limiter algorithm for conservation laws with discontinuous flux modeling clarifier thickener units. In this work we show that their idea of constructing FTVD second order schemes also can be used to construct second order schemes satisfying (A,B)-entropy condition for the scalar conservation law with discontinuous flux with proper modification at the interface. We present numerical experiments to show the superiority of the second order schemes over the monotone first order schemes. We show further from numerical experiments that solutions from these schemes are comparable with the second order schemes obtained from minimod limiter.

Abstract:
We consider multidimensional 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 ？？-measures to investigate the zero diffusion-dispersion-smoothing limit. 1. Introduction We consider the convergence of smooth solutions ？？=？？？？(？？,？？) with (？？,？？)∈？？+×？？？？ of the nonlinear partial differential equation ？？？？？？+div？？？？？？(？？,？？,？？)=？？div？？？？(？？？)+？？？？？？=1？？3？？？？？？？？？？？？？？(1.1) as ？？→0 and ？？=？？(？？),？？=？？(？？)→0. Here ？？∈？？(？？;？？？？(？？+？？×？？？？？？)) is the Caratheodory flux vector such that max|？？|≤？？||？？？？||(？？,？？,？？)？？？(？？,？？,？？)？0,？？？0,in？？？？loc？？+×？？？？,(1.2) for ？？>2 and every ？？>0. The aim is to show convergence to a weak solution of the corresponding hyperbolic conservation law: ？？？？？？+div？？？(？？,？？,？？)=0,？？=？？(？？,？？),？？∈？？？？,？？≥0.(1.3) We refer to this problem as the zero diffusion-dispersion-smoothing limit. In the case when the flux ？？ is at least Lipschitz continuous, it is well known that the Cauchy problem corresponding to (1.3) has a unique admissible entropy solution in the sense of Kru？hkov [1] (or measure valued solution in the sense of DiPerna [2]). The situation is more complicated when the flux is discontinuous and it has been the subject of intensive investigations in the recent years (see, e.g., [3] and references therein). The one-dimensional case of the problem is widely investigated using several approaches (numerical techniques [3, 4], compensated compactness [5, 6], and kinetic approach [7, 8]). In the multidimensional case there are only a few results concerning existence of a weak solution. In [9] existence is obtained by a two-dimensional variant of compensated compactness, while in [10] the approach of ？？-measures [11, 12] is used for the case of arbitrary space dimensions. Still, many open questions remain such as the uniqueness and stability of solutions. A problem that has not yet been studied in the context of conservation laws with discontinuous flux, and which is the topic of the present paper, is that of zero diffusion-dispersion limits. When the flux is independent of the spatial and temporal positions, the study of zero diffusion-dispersion limits was initiated in [13] and further addressed in numerous works by LeFloch et al. (e.g., [14–17]). The compensated compactness method is the basic tool used in the one-dimensional situation for the so-called limiting case in which the diffusion and dispersion parameters are in an appropriate balance. On the other hand, when diffusion dominates

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.

Abstract:
We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized Kato inequality for any pair of solutions. Applications to uniqueness of solutions are then given.

Abstract:
We study the BGK approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy Problem for the BGK approximation is well-posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.

Abstract:
We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux. We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under assumptions on the flux which provide the maximum principle. In particular, we allow multiple flux crossings and we do not need any kind of genuine nonlinearity conditions.

Abstract:
We propose new entropy admissibility conditions for multidimensional hyperbolic scalar conservation laws with discontinuous flux which generalize one-dimensional Karlsen-Risebro-Towers entropy conditions. These new conditions are designed, in particular, in order to characterize the limit of vanishing viscosity approximations. On the one hand, they comply quite naturally with a certain class of physical and numerical modeling assumptions; on the other hand, their mathematical assessment turns out to be intricate. \smallskip The generalization we propose is not only with respect to the space dimension, but mainly in the sense that the "crossing condition" of [K.H. Karlsen, N.H. Risebro, J. Towers, Skr.\,K.\,Nor.\,Vid.\,Selsk. (2003)] is not mandatory for proving uniqueness with the new definition. We prove uniqueness of solutions and give tools to justify their existence via the vanishing viscosity method, for the multi-dimensional spatially inhomogeneous case with a finite number of Lipschitz regular hypersurfaces of discontinuity for the flux function.

Abstract:
In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.

Abstract:
The confidence of many authors in the possibility to use superconducting loop interrupted by Josephson junctions as a basis for quantum bit, flux qubit, presumes the assumption on superposition of two macroscopically distinct quantum states with macroscopically different angular momentum. The contradiction of this assumption with macroscopic realism and the conservation law must call the numerous publications about flux qubit in question. These publications uncover misunderstanding by many modern physicists of the essence of the superposition principle. The Einstein - Podolsky - Rosen (EPR) correlation or entanglement, introduced in 1935 by opponents of the Copenhagen interpretation in order to reveal the contradiction of this principle with realism, has provided a basis of the idea of quantum computation. The problem of the EPR correlation has emerged thanks to philosophical controversy between the creators of the quantum theory about the subject of its description. Therefore it is impossible to solve correctly the problem of quantum computer creation without the insight into the essence of this philosophical controversy. The total neglect of the philosophical problems of quantum foundation results to concrete mistakes, the example of which are the publications about flux qubit. In order to prevent such mistakes in the future the philosophical questions about the essence of superposition and entanglement and about the subject of quantum description are considered.

Abstract:
This
paper is concerned with the initial-boundary value problem of scalar conservation
laws with weak discontinuous flux, whose initial data are a function with two
pieces of constant and whose boundary data are a constant function.
Under the condition that the flux function has a finite number of weak discontinuous
points, by using the structure of weak entropy solution of the corresponding
initial value problem and the boundary entropy condition developed by
Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy
solution for this initial-boundary value problem, and by investigating the
interaction of elementary waves and the boundary, we clarify the geometric
structure and the behavior of boundary for the weak entropy solution.