We consider the Riemann problem for non-genuinely nonlinear conservation laws where the flux function admits two inflection points. This is a simplification of van der Waals fluid pressure, which can be seen as a function of the specific volume for a specific entropy at which the system lacks the non-genuine nonlinearity. Corresponding to each inflection point, A nonclassical Riemann solver can...

An extension of the Colombo phase transition model is proposed. The congestion phase is described by a two-dimensional zone defined around an equilibrium flux known as the classical fundamental diagram. General criteria to build such a set-valued fundamental diagram are enumerated, and instantiated on several equilibrium fluxes with different concavity properties. The solution of the Riemann pr...

Abstract. In this paper, we investigate a traffic model similar to the Lighthill–Whitham– Richards model, consisting of a hyperbolic conservation law with a discontinuous, non-convex, piecewise-linear flux. Using Dias and Figueira’s mollification framework, analytical solutions to the corresponding Riemann problem are derived. For certain initial data, these Riemann problems can generate zero w...

We present a five-wave Riemann solver for the equations of ideal relativistic magnetohydrodynamics. Our solver can be regarded as a relativistic extension of the five-wave HLLD Riemann solver initially developed by Miyoshi and Kusano for the equations of ideal MHD. The solution to the Riemann problem is approximated by a five wave pattern, comprised of two outermost fast shocks, two rotational ...

An approximate Riemann solver for the equations of ideal relativistic magnetohydrodynamics is presented. The solver belongs to the so-called Harten-Laxvan Leer (HLL, [1]) family of solvers where an initial guess to the characteristic wave speeds is given without any knowledge a priori of the solution. Our proposed method of solution generalizes to the relativistic case the classical five-wave H...

In this paper we present the exact solution of the Riemann Problem for the non-linear shallow water equations with a step-like bottom. The solution has been obtained by solving an enlarged system obtained by adding an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. The resulting solution is unique and satis es the p...

The aim of the paper is to study qualitative behavior of solutions to the equation du df(u) , . m + ~oT9(o)' where (x,t) £lx R+,u = u(x,t) € R. The main new feature with respect to previous works is that the flux function / may have finitely many inflection points, intervals in which it is afEne, and corner points. The function g is supposed to be zero at 0 and 1, and positive in between. We pr...

Abstract. In this article, we study the global solution of the elementary waves interaction problem for the nonlinear chromatography equations. We constructively obtain the solutions when the initial data are three piecewise constant states. The global structures and large time-asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, it is easy to ...