Wave Front Tracking Method Research Articles

Acoustic black hole designs for circular beams under axial loading

Acoustic Black Hole (ABH), as a non-reflecting wave effect, has been realised in beams and plates by locally changing their thickness. Geometrical ABH designs stem from their transverse-load bearing properties, assuming isotropic material properties associated with engineering materials (e.g., steels). The underlying physics is to manipulate stiffness locally for a transversely-loaded structure, leading to a change in the elastic wave speed. However, the same ABH would have limitations for axial loading, i.e., slender beams for which longitudinal waves dominate. Here, we present a wave propagation approach using wavefront tracking to identify a potential ABH for axially-loaded circular beams. We study wave propagation in three exponential designs of finite length, monitoring the wave-travel time. Indicative of an effective ABH in finite length sections, the wave-travel time increases compared to the case of a beam without ABH. By employing the wave front tracking method for the design of an ABH with axial loading, it is possible to verify the effectiveness of ABHs. Also, various material models, e.g., orthotropic materials such as wood, and different loading conditions can be considered, which opens a new avenue in applications of ABH phenomena beyond conventional vibro-acoustic control problems.

Stability of solutions constructed by the wave front tracking method for conservation laws

We prove the Lloc1 stability of sequence of piecewise constant solutions constructed by the wave front tracking method for conservation laws. As a result, we can see concretely the convergence rate of sequence of piecewise constant solutions constructed by the wave front tracking method for conservation laws.

Entropy solutions for a two-phase transition model for vehicular traffic with metastable phase and time depending point constraint on the density flow

We consider a macroscopic two-phase transition model for vehicular traffic flow subject to a point constraint on the density flux. The two phases correspond to light and heavy traffic and their dynamics are described respectively by the Lighthill–Whitham–Richards model and the Aw–Rascle–Zhang model. Their intersection, the so-called metastable phase, is assumed to be non empty. The discrete in time point constraint mechanism, inducing flux limitation at bottlenecks, is explored within this two-phase model. We introduce a new definition of admissible solutions for the Cauchy problem, for which we prove existence and we provide a characterization. In particular, these admissible solutions attain the maximal flux allowed by the constraint whenever it is enforced, which guarantees compatibility of the constructed solutions with the modeling assumption imposed at the level of the Riemann solver. These results rely on the wave-front tracking method and on adaptation of the specific entropies and renormalization properties introduced in Andreainov, Donadello, Rosini, M3AS (2016) while dealing with the Aw–Rascle–Zhang model with point constraint.

On entropic solutions to conservation laws coupled with moving bottlenecks

Moving bottlenecks in road traffic represent an interesting mathematical problem, which can be modeled via coupled PDE-ODE systems. We consider the case of a scalar conservation law modeling the evolution of vehicular traffic and an ODE with discontinuous right-hand side for the bottleneck. The bottleneck usually corresponds to a slow moving vehicle influencing the bulk traffic flow via a moving flux pointwise constraint. The definition of solutions requires a special entropy condition selecting nonclassical shocks and we prove existence of such solutions for initial data with bounded variation. Approximate solutions are constructed via the wave-front tracking method and their limit are solutions of the Cauchy problem PDE-ODE.

On the Cauchy problem for Aw-Rascle system with linear damping

The existence of global BV solutions for the Aw-Rascle system with linear damping is considered. In order to get approximate solutions we consider the system in Lagrangian coordinates, then by using the wave front tracking method coupling with and suitable splitting algorithm and the ideas of [1] we get a sequence of approximate solutions. Finally we show the convergence of this approximate sequence to the weak entropic solution.

Stability of approximate solutions constructed by the wave front tracking method

We prove the stability of approximate solutions constructed by the wave front tracking method for the Cauchy problem of scalar conservation laws. The ideas of the proof are based upon an application of the proof technique of the uniqueness theorem to the entropy solution given by Kružkov and a local generalization of the method by Liu and Yang.

The zero relaxation limit for the Aw–Rascle–Zhang traffic flow model

We study the behavior of the Aw–Rascle–Zhang model when the relaxation parameter converges to zero. In a Lagrangian setting, we use the wavefront tracking method with splitting technique to construct a sequence of approximate solutions. We prove that this sequence converges to a weak entropy solution of the relaxed system associated to a given initial datum with bounded variation. Besides, we also provide an estimate on the decay of positive waves. We finally prove that the solutions of the Aw–Rascle–Zhang system with relaxation converge to a weak solution of the corresponding scalar conservation law when the relaxation parameter goes to zero.

Stability of Rarefaction Wave to the 1-D Piston Problem for the Pressure-Gradient Equations

The 1-D piston problem for the pressure gradient equations arising from the flux-splitting of the compressible Euler equations is considered. When the total variations of the initial data and the velocity of the piston are both sufficiently small, the author establishes the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength by employing a modified wave front tracking method.

Static and Dynamic Permittivity Measurement of High Explosives in the W Band to Investigate Shock and Detonation Phenomena

AbstractRadio interferometry techniques are often used to investigate shock and detonation phenomena thanks to the radio‐transparency of high explosives in the gigahertz frequency band. These techniques require the knowledge of the permittivity of studied explosives. Although the permittivity has been thoroughly studied for many materials, very few data are available at high frequencies (>75 GHz) for high explosives. In this paper, we report static measurement data of the permittivity for various reactive materials using the standard line transmission method between 75 GHz and 110 GHz (W frequency Band), and we present dynamic measurement results at 94 GHz obtained from the so‐called detonation wavefront tracking method. It is shown that the measurement results provided by these two methods are in good agreement. As a consequence, this work validates the detonation wavefront tracking method for the dynamic measurement of high explosives permittivity, and shows that the static experimental results are relevant for shock wave propagation analysis from millimeter‐wave measurement techniques.

Two algorithms for a fully coupled and consistently macroscopic PDE-ODEsystem modeling a moving bottleneck on a road

In this paper we propose two numerical algorithms to solve a coupled PDE-ODE system which models a slow vehicle (bottleneck) moving on a road together with other cars. The resulting system is fully coupled because the dynamics of the slow vehicle depends on the density of cars and, at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm, based on the Wave Front Tracking method, is suitable for theoretical investigations and convergence results. The second one, based on the Godunov scheme, is used for numerical simulations. The case of multiple bottlenecks is also investigated.

Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system

We study supersonic flow past a convex corner which is surrounded by quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a strong rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to separate the supersonic flow behind the rarefaction wave from the static gas. In this paper, we employ a wave front tracking method to establish structural stability of such a flow pattern under non-smooth perturbations of the upcoming supersonic flow. It is an initial-value/free-boundary problem for the two-dimensional steady non-isentropic compressible Euler system. The main ingredients are careful analysis of wave interactions and construction of suitable Glimm functional, to overcome the difficulty that the strong rarefaction wave has a large total variation.

Stability of contact discontinuities to 1-D piston problem for the compressible Euler equations

We consider 1-D piston problem for the compressible Euler equations when the piston is static relatively to the gas in the tube. By a modified wave front tracking method, we prove that a contact discontinuity is structurally stable under the assumptions that the total variation of the initial data and the perturbation of the piston velocity are both sufficiently small. Meanwhile, we study the asymptotic behavior of the solutions by the generalized characteristic method and approximate conservation law theory as t→+∞.

The Cauchy problem for the Aw–Rascle–Zhang traffic model with locally constrained flow

We study the Cauchy problem for the Aw–Rascle–Zhang model for traffic flow with a flux constraint located at [Formula: see text]. The purpose of such a traffic model is to describe an obstruction in a road, such as a toll gate or a construction site. We consider here the concept of solutions introduced by Garavello and Goatin in 2011, which conserves the total number of cars flowing through the obstruction, but does not conserve the generalized momentum. We prove the existence of such solutions to the Cauchy problem by using the wavefront tracking method. We also compare our results with another concept of solutions introduced also by Garavello and Goatin in 2011 for the same model with a flux constraint. For this second formulation, Andreianov, Donadello, and Rosini had also proved the existence of solutions to the Cauchy problem.

A ray-based IPDG method for high-frequency time-domain acoustic wave propagation in inhomogeneous media

The numerical approximation of high-frequency wave propagation in inhomogeneous media is a challenging problem. In particular, computing high-frequency solutions by direct simulations requires several points per wavelength for stability and usually requires many points per wavelength for a satisfactory accuracy. In this paper, we propose a new method for the acoustic wave equation in inhomogeneous media in the time domain to achieve superior accuracy and stability without using a large number of unknowns. The method is based on a discontinuous Galerkin discretization together with carefully chosen basis functions. To obtain the basis functions, we use the idea from geometrical optics and construct the basis functions by using the leading order term in the asymptotic expansion. Also, we use a wavefront tracking method and a dimension reduction procedure to obtain dominant rays in each cell. We show numerically that the accuracy of the numerical solutions computed by our method is significantly higher than that computed by the IPDG method using polynomials. Moreover, the relative errors of our method grow only moderately as the frequency increases.

Non-relativistic limits of rarefaction wave to the 1-D piston problem for the isentropic relativistic Euler equations

We consider the 1-D piston problem for the isentropic relativistic Euler equations when the total variations of the initial data and the speed of the piston are both sufficiently small. By a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Meanwhile, we study the convergence of the entropy solutions to the corresponding entropy solutions of the classical non-relativistic isentropic Euler equations as the light speed c→+∞.

Boundary coupling of microscopic and first order macroscopic traffic models

We couple, at a fixed interface, the microscopic Follow the Leader model and the macroscopic Lighthill–Whitham–Richards model, both used for describing vehicular traffic. The coupling is obtained by suitable boundary conditions. We prove existence of solutions for the corresponding Cauchy problem, by using the wave-front tracking method and classical results on ordinary differential equations.

Existence and stability of rarefaction wave to 1-D piston problem for the relativistic full Euler equations

We consider the 1-D piston problem for the compressible relativistic full Euler equations when the total variations of the initial data and the piston velocity are both sufficiently small. By using a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Moreover, we study the asymptotic behavior of the solutions as t→+∞.

Stability and non-relativistic limits of rarefaction wave to the 1-D piston problem for the relativistic Euler equations

We study the 1-D piston problem for the relativistic Euler equations under the assumption that the total variations of both the initial data and the velocity of the piston are sufficiently small. By a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Meanwhile, we consider the convergence of the entropy solutions to the corresponding entropy solutions of the classical non-relativistic Euler equations as the light speed \(c\rightarrow +\infty \).

Global stability of rarefaction wave to the 1-D piston problem for the Compressible Full Euler Equations

In this paper, we study the 1-D piston problem for compressible full Euler equations under the assumptions that the total variation of the initial data and the perturbation of the piston velocity are sufficiently small. By a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Moreover, we establish the asymptotic behavior of the solutions as t→∞.

Entropy solutions for a traffic model with phase transitions

We introduce an appropriate notion of entropy solution for a generalization of the two phase macroscopic traffic model proposed in Goatin (2006). We first apply the wave-front tracking method to prove existence and a priori bounds for weak solutions. Then, in the case the characteristic field of the free phase is linearly degenerate, we prove that the obtained weak solutions are in fact entropy solutions. The case of solutions attaining values at the vacuum is considered. We also present an explicit numerical example to describe some qualitative features of the solutions.