We study the Cauchy problem for the Aw-Rascle-Zhang model for traffic flow with a flux constraint at x = 0. More precisely we consider the Riemann solver, conserving the number of cars at x = 0 but not the generalized momentum, introduced in [9] for the problem with flux constrained. For such a Riemann solver, we prove existence of a solution for the Cauchy problem. The proof is based on the wa...

Advection-reaction type partial differential equations model a wide variety of phenomena in several disciplines in physics, chemistry, environmental sciences, geometry, financial mathematics and many others. Generally, these non-linear inhomo-geneous equations must be solved in complicated multi-dimensional domains and thus analytical solutions are only available under very special circumstance...

We study the regularity of semihyperbolic patches of self-similar solutions near sonic lines to a Riemann problem for the two-dimensional (2-D) Euler system. As a result, it is verified that there exists a global solution in the semihyperbolic patch up to the sonic boundary and that the sonic boundary has C1-regularity. The study of the semihyperbolic patches of solutions for the Euler system w...

In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation ut + (−∂ 2 x) α/2u+ uux = 0 with α ∈ (1, 2) is studied. It is shown that if the nondecreasing initial datum approaches the constant states u± (u− < u+) as x → ±∞, respectively, then the corresponding solution converges toward the rarefaction wave, i.e. the unique entropy sol...

Abstract. We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross–section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L–M and R–M cu...

We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is non-strictly hyperbolic and does not admit a fully conservative form, and we establish the existence of two-parameter wave sets, rather than wave curves. The selection of admissible waves is particularly challenging. Our construction is fully explicit...

Abstract. This paper is concerned with the numerical approximation of the solutions of a macroscopic model for the description of the flow of pedestrians. Solutions of the associated Riemann problem are known to be possibly nonclassical in the sense that the underlying discontinuities may well violate the Lax inequalities, which makes their numerical approximation very sensitive. This study pro...

Abstract The present paper is devoted to the construction and comparative study of upwind methods as applied to the system of one-dimensional nonlinear elasticity equations. We derive a simple approach for building up exact solutions to the Riemann problem and construct a suite of test problems to assess numerical methods. Then we carry out the implementation and a systematic comparative study ...

We deal with the viscous approximation of a system of conservation laws in one space dimension and we focus on initial-boundary value problems. It is known that, in general, different viscous approximation provide different limits because of boundary layer phenomena. We focus on Riemann-type data and we discuss a uniqueness criterion for distributional solutions which applies to both the non ch...

In the previous ghost fluid methods (GFMs) developed, the focus is on the definition of ghost fluid states while the pressure and velocity in the real fluid sides are taken for granted, except for the correction made to the density at the real fluid nodes next to the interface to overcome the possible problems related to overheating. It has been found that such GFMs encounter many difficulties ...