Numerical Investigation of the Wave-Front Tracking Algorithm for the Full Ultra-Relativistic Euler Equations

We introduce a generalized version of the front tracking algorithm for the full ultra-relativistic Euler system. The construction and analysis of this algorithm are somewhat simpler than other algorithms. Moreover, this scheme leads to a more robust and efficient result. The scheme also satisfies positivity. This scheme is compared with other two schemes by two numerical test cases. Furthermore we give another application of this scheme, namely we check the explicit formula of interaction of two generalized shocks, by further numerical test case.

BV Solutions to 1D isentropic Euler equations in the zero mach number limit

Two compressible immiscible fluids in 1D and in the isentropic approximation are considered. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, we prove the rigorous convergence for the fully nonlinear compressible to incompressible limit of the coupled dynamics of the two fluids. A key role is played by a suitably refined wave front tracking algorithm, which yields precise [Formula: see text], [Formula: see text] and weak* convergence estimates, either uniform or explicitly dependent on the Mach number.

Crowd dynamics and conservation laws with nonlocal constraints and capacity drop

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a point constraint in the spirit of [Colombo and Goatin, J. Differential Equations, 2007]. We introduce a nonlocal constraint to restrict the flux at the exit to a maximum value p(ξ), where ξ is the weighted averaged instantaneous density of the crowd in an upstream vicinity of the exit. Choosing a non-increasing constraint function p(⋅), we are able to model the capacity drop phenomenon at the exit. Existence and stability results for the Cauchy problem with Lipschitz constraint function p(⋅) are achieved by a procedure that combines the wave-front tracking algorithm with the operator splitting method. In view of the construction of explicit examples (one is provided), we discuss the Riemann problem with discretized piecewise constant constraint p(⋅). We illustrate the fact that nonlocality induces loss of self-similarity for the Riemann solver; moreover, discretization of p(⋅) may induce non-uniqueness and instability of solutions.

COUPLING OF LIGHTHILL–WHITHAM–RICHARDS AND PHASE TRANSITION MODELS

This paper deals with coupling conditions between the classical Lighthill–Whitham–Richards model and a phase transition model. We propose two different definitions of solution at the interface between the two models. The first one corresponds to maximize the flux passing through the interface, while the second one imposes an additional constraint on the flux. We prove existence of solutions to the Cauchy problem with arbitrary initial data of bounded variation, by means of the wave-front tracking technique.