Wave Front Tracking Research Articles

Spurious Rayleigh-wave apparent anisotropy near major structural boundaries: a numerical and theoretical investigation

SUMMARY The recent developments in array-based surface-wave tomography have made it possible to directly measure apparent phase velocities through wave front tracking. While directionally dependent measurements have been used to infer intrinsic $2\psi $ azimuthal anisotropy (with a 180° periodicity), a few studies have also demonstrated strong but spurious $1\psi $ azimuthal anisotropy (360° periodicity) near major structure boundaries particularly for long period surface waves. In such observations, Rayleigh waves propagating in the direction perpendicular to the boundary from the slow to the fast side persistently show a higher apparent velocity compared to waves propagating in the opposite direction. In this study, we conduct numerical and theoretical investigations to explore the effect of scattering on the apparent Rayleigh-wave phase velocity measurement. Using 2-D spectral-element numerical wavefield simulations, we first reproduce the observation that waves propagating in opposite directions show different apparent phase velocities when passing through a major velocity contrast. Based on mode coupling theory and the locked mode approximation, we then investigate the effect of the scattered fundamental-mode Rayleigh wave and body waves interfering with the incident Rayleigh wave separately. We show that scattered fundamental-mode Rayleigh waves, while dominating the scattered wavefield, mostly cause short wavelength apparent phase velocity variations that could only be studied if the station spacing is less than about one tenth of the surface wave wavelength. Scattered body waves, on the other hand, cause longer wavelength velocity variations that correspond to the existing real data observations. Because of the sensitivity of the $1\psi $ apparent anisotropy to velocity contrasts, incorporating such measurements in surface wave tomography could improve the resolution and sharpen the structural boundaries of the inverted model.

Hypersonic similarity for steady compressible full Euler flows over two-dimensional Lipschitz wedges

We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. Mathematically, it can be formulated as the comparison of the entropy solutions in BV∩L1 between the two initial-boundary value problems for the compressible full Euler equations with parameter τ>0 and the hypersonic small-disturbance equations (the scaled compressible full Euler equations with parameter τ=0) with curved characteristic boundaries. We establish the L1–convergence estimate of these two solutions with the optimal convergence rate, which justifies the Van Dyke's similarity theory rigorously for the compressible full Euler flows. This is the first mathematical result on the comparison of two solutions of the compressible Euler equations with characteristic boundary conditions. To achieve this, we first employ the special structures of the two systems and establish the global existence and the L1–stability of the entropy solutions via the wave-front tracking scheme under the smallness assumption on the total variation of both the initial data and the tangential slope function of the wedge boundary. Based on the L1–stability properties of the approximate solutions to the scaled equations with parameter τ>0, a uniform Lipschtiz continuous map with respect to the initial data and the wedge boundary is obtained, which is the first time for the characteristic boundary conditions. Next, we compare the solutions given by the Riemann solvers of the two systems by taking the boundary perturbations into account case by case. Then, for a given fixed hypersonic similarity parameter, as the Mach number tends to infinity, by employing the Lipschitz continuous properties of the map, we establish the desired L1–convergence estimate with the optimal convergence rate. Finally, we show the optimality of the convergence rate by investigating a special solution.

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.

Mapping structures on the core–mantle boundary using Sdiff postcursors: Part I. Method and Validation

SUMMARY Ultra-low velocity zones (ULVZs) are patches of extremely slow seismic velocities on the core–mantle boundary (CMB). Here, we target them using the postcursors to S core-diffracted phases (Sdiff) caused by ULVZs. We use traveltimes of these postcursors to make probabilistic maps using a reversible-jump Markov chain Monte Carlo inversion setup. For the forward model, we extend 2-D wave front tracking (2DWT) software, previously developed for surface wave multipathing studies, to the CMB. The 2DWT is able to model the full multipathing behaviour of Sdiff postcursors and compute arrival times for a given ULVZ input velocity structure on the order of a few CPU seconds, as opposed to 100s of CPU hours required for 3-D full waveform synthetics. We validate the method using synthetic data sets produced by the 2DWT, as well as 3-D full waveform synthetics, using a parametrisation formed from a collection of ellipses. We also test idealistic data coverage versus a case of more realistic coverage. We show ULVZ size and velocity reduction can typically be well recovered, and our maps show the inherent trade-off between these parameters around the edge of the ULVZ. Our method cannot directly constrain ULVZ height; tests show that it underestimates ULVZ velocity reductions and overestimates ellipticity for thinner ULVZs due to neglecting mantle effects.

Temple system on networks

This paper deals with the well-posedness on a network of a Temple system of nonlinear hyperbolic balance laws. Temple systems are characterized by the fact that shock and rarefaction curves coincide. This study is motivated by a model for traffic, recently proposed, inspired by kinetic considerations. The proof of the well-posedness is based on the wave-front tracking procedure, on the pseudo-polygonal technique and on the operator splitting method.

Global Well-Posedness of Shock Front Solutions to One-Dimensional Piston Problem for Combustion Euler Flows

This paper is devoted to the well-posedness theory of piston problem for compressible combustion Euler flows with physical ignition condition. A significant combustion phenomenon called detonation will occur provided that the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In the present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of space. The main difficulties here stem from the discontinuous source term without uniform dissipation structure and from the characteristic boundary associated with degenerate characteristic field. In dealing with the obstacles caused by ignition temperature, we develop a modified Glimm-type functional to control the oscillation growth of combustion waves, even if the exothermic source fails to uniformly decay. As to the characteristic boundary, the degeneracy of contact discontinuity is fully employed to get elegant stability estimates near the piston boundary. Meanwhile, we devise a weighted Lyapunov functional to balance the nonlinear effects arising from large shock, characteristic boundary, and exothermic reaction and then obtain the -stability of combustion wave solutions. Our results reveal that one-dimensional Zeldovich–von Neumann–Döring detonation waves supported by a forward piston are indeed nonlinearly stable under small perturbation in sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.

The initial boundary value problem for second order traffic flow models with vacuum: Existence of entropy weak solutions

We study the IBVP for a non-strictly hyperbolic 2 × 2 system of conservation laws on an interval with characteristic boundaries, modelling traffic dynamics including vacuum states on a road stretch. After giving a detailed characterization of the admissible states at the boundary in terms of Riemann solver and entropy conditions, we prove existence of entropy weak solutions for data of bounded variation in the Riemann invariant coordinates by convergence of wave-front tracking approximations.

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.

Initial Data Identification for the One-Dimensional Burgers Equation

In this article, we study the problem of identification for the 1-D Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of nonbackward uniqueness of the Burgers equation, there may exist multiple initial data leading to the same given target. In articles “Initial data identification in conservation laws and Hamilton-Jacobi equations” ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arXiv:1903.06448</i> , 2019) and “The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes” ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SIAM Journal on Mathematical Analysis</i> , vol. 52, the authors fully characterize the set of initial data leading to a given target using the classical Lax–Hopf formula. In this article, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigating systems of conservation laws in 1-D, where the classical Lax–Hopf formula no more holds. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wavefront tracking algorithm.

Non-relativistic limits and stability of composite wave patterns to the relativistic Euler equations

We are concerned with the structural stability of composite wave patterns to the Cauchy problem for the relativistic full Euler equations consisting of a large 1-shock, a large 2-contact discontinuity, and a large 3-shock. When the initial data are bounded but possibly large total variations, approximate solutions have been constructed via the wave front tracking scheme, a weighted Glimm functional has been introduced and its monotonicity has been proved on the basis of the local wave interaction estimates, and then the global stability of these wave patterns has been established. Moreover, the non-relativistic limits of such solutions can be obtained as the light speed c → +∞.

B-YIA2-03 A GOVERNING EQUATION TO EXPLAIN THE NUMBER OF WAVELETS AND ROTORS OBSERVED IN HUMAN VENTRICULAR FIBRILLATION

VF is commonly characterised by two mechanisms, multi wavelet reentry and rotors, but no governing equation exists to explain and predict their population dynamics. We hypothesized a single equation derived from an M/G/∞ renewal process could explain rotor and wavelet numbers in VF. Phase singularity (PS) and wavefront (WF) tracking was used to identify wavelets and rotors in epicardial recordings of induced VF during cardiac surgery (n=13 patients). Autocorrelation and distributions of PS and WF lifetimes and inter-formation times were assessed to verify an underlying renewal process. Distributions were fitted using maximum likelihood to calculate formation (λf) and destruction (λd) rates, and combined in an M/G/∞ process to develop a potential governing equation of VF dynamics. PS and WF inter-event-time distributions were consistent with the Weibull in all 210 epochs (PS: mean Χ2 P=0.23(95%CI,0.18,0.28); WF: mean Χ2 P=0.19(95%CI,0.13,0.27)), with zero autocorrelation at non-zero lags, indicative of an underlying renewal process over all stages (perfusion, ischemia and reflow). The M/G/∞ equation accurately predicted average PS and WF number (R>0.90) and population distribution (Χ2 P>0.05) in all epochs. Differences in λf (term: 0.015/ms (95%CI,0.010,0.020), non-term 0.023/ms (95%CI,0.019,0.027)), and average PS number (term: 1.68 (95%CI, 1.36, 2.00), non-term: (2.00 (95%CI, 1.86, 2.15)) was observed in spontaneous VF termination. M/G/∞ renewal process provides a governing equation to explain the number of wavelets and rotors in VF, which could be used in mechanistic studies to guide development of new therapies.

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.

A Macroscopic Traffic Flow Model Accounting for Bounded Acceleration

This paper details a new macroscopic traffic flow model accounting for the boundedness of traffic acceleration, which is required for physical realism. Our approach relies on the coupling between a scalar conservation law, which refers to the seminal LWR model, and a system of Ordinary Differential Equations describing the trajectory of accelerating vehicles, which we treat as moving constraints. We propose a Wave Front Tracking Algorithm to construct approximate solutions. We use this algorithm to prove the existence of solutions to the associated Cauchy Problem, and provide some numerical examples illustrating the solution behaviour.

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.

Shadow Wave Tracking Procedure and Initial Data Problem for Pressureless Gas Model

In this paper the new procedure for a construction of an approximated solution to initial data problem for one-dimensional pressureless gas dynamics system is introduced. The procedure is based on solving the Riemann problems and tracking singular wave interactions. For that system the new problem with initial data containing Dirac delta function is solved whenever two waves interact. Use of the shadow waves as singular solutions to such problems enables us to easily solve the interaction problems. That permits us to make a simple extension of the well known Wave Front Tracking algorithm. A non-standard part of the new algorithm is dealing with delta functions as a part of a solution. In the final part of the paper we show that the approximated solution has a subsequence converging to a signed Radon measure.

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.

Global Existence and Stability of Shock Front Solution to 1-D Piston Problem for Exothermically Reacting Euler Equations

In this paper, we are concerned with the global existence and stability of strong shock front for one-dimensional piston problem. We use the exothermically reacting Euler equations as a mathematical model to describe the gas motion. Under the assumptions that the total variations of initial data and the perturbation of piston velocity are sufficiently small, we establish the global existence and asymptotic behavior of entropy solutions including a strong shock front without restriction on the strength. A modified fractional wave front tracking scheme is developed and a modified Glimm-type functional is carefully designed.

An existence result for conservation laws having BV spatial flux heterogeneities - without concavity

We prove existence for an initial value problem featuring a conservation law whose flux has a discontinuous spatial dependence. For the type of flux considered here, where the spatial dependence occurs in a very general form, a uniqueness result is known, but the existence question was open until the recent work of Piccoli and Tournus (2018) [22]. Piccoli and Tournus proved existence using approximate solutions generated by the wave front tracking algorithm. A concavity assumption plays a simplifying role in their analysis. The main contribution of the present paper is an extension of this existence theorem in the absence of the concavity assumption. We accomplish this via finite difference approximations.

On the structure of BV entropy solutions for hyperbolic systems of balance laws with general flux function

The paper describes the qualitative structure of BV entropy solutions of a general strictly hyperbolic system of balance laws with characteristic field either piecewise genuinely nonlinear or linearly degenerate. In particular, we provide an accurate description of the local and global wave-front structure of a BV solution generated by a fractional step scheme combined with a wave-front tracking algorithm. This extends the corresponding results in [S. Bianchini and L. Yu, Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension, Comm. Partial Differential Equations 39(2) (2014) 244–273] for strictly hyperbolic system of conservation laws.