Quasilinear Hyperbolic Systems Research Articles

Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review

Hyperbolic systems with stiff relaxation constitute a wide class of evolutionary partial differential equations which describe several physical phenomena, ranging from gas dynamics to kinetic theory, from semiconductor modeling to traffic flow. Peculiar features of such systems is the presence of a small parameter that determines the smallest time scale of the system. As such parameter vanishes, the system relaxes to a different one with a smaller number of equations, and sometime of a different mathematical nature. The numerical solution of such systems may present some challenges, in particular if one is interested in capturing all regimes with the same numerical method, including the one in which the small parameter vanishes (relaxed system). The design, analysis and application of numerical schemes which are robust enough to solve this class of systems for arbitrary value of the small parameter is the subject of the current paper. We start presenting different classes of hyperbolic systems with relaxation, illustrate the properties of implicit–explicit (IMEX) Runge–Kutta schemes which are adopted for the construction of efficient methods for the numerical solution of the systems, and then illustrate how to apply IMEX schemes for the construction of asymptotic preserving schemes, i.e. scheme which correctly capture the behavior of the systems even when the relaxation parameter vanishes.

Exact Boundary Controllability of Nodal Profile for Nonautonomous Quasilinear Hyperbolic Systems

Exact Boundary Controllability of Nodal Profile for Nonautonomous Quasilinear Hyperbolic Systems

Time-Periodic Solutions to Quasilinear Hyperbolic Systems on General Networks

Time-Periodic Solutions to Quasilinear Hyperbolic Systems on General Networks

Averaging Method for Quasi-Linear Hyperbolic Systems

Averaging Method for Quasi-Linear Hyperbolic Systems

Global existence and decay of small solutions for quasi-linear second-order uniformly dissipative hyperbolic-hyperbolic systems

This paper is concerned with quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively. Its main theorem establishes global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states in d≥3 space dimensions. Notably, the operators are not required to be symmetric hyperbolic, instead merely the existence of symbolic symmetrizers is assumed. The dissipation is characterized by algebraic conditions implying the uniform decay of all Fourier modes at the reference state. On a technical level, the theory developed herein uses para-differential operators as its main tool. Apparently being the first to apply such operators in the context of global-in-time existence of small solutions for quasi-linear hyperbolic systems, the present work contains a new version of the strong Gårding inequality. In the context of theoretical physics, the theorem applies to recent formulations for the relativistic dynamics of viscous, heat-conductive fluids such as notably that of Bemfica, Disconzi and Noronha (2018) [2].

Finite Time Stabilization to Zero and Exponential Stability of Quasilinear Hyperbolic Systems

Finite Time Stabilization to Zero and Exponential Stability of Quasilinear Hyperbolic Systems

The Exact Boundary Controllability of Nodal Profile for Entropy Solutions to 1-D Quasilinear Hyperbolic Systems of Conservation Laws with Linearly Degenerate Negative (Resp., Positive) Characteristic Fields

The Exact Boundary Controllability of Nodal Profile for Entropy Solutions to 1-D Quasilinear Hyperbolic Systems of Conservation Laws with Linearly Degenerate Negative (Resp., Positive) Characteristic Fields

Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems

Abstract We study the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction. In both L 1 {L^{1}} and L ∞ {L^{\infty}} frameworks, we will first show the global existence of classical solutions with small initial data, and then prove that the global solution will scatter to free linear waves and study the rigidity aspect of the scattering problem. We also show the inverse scattering result: The scattering field can determine the global solution uniquely, in the L 1 {L^{1}} framework.

An analysis of fluid–structure interaction coupling mechanisms in liquid-filled viscoelastic pipes subject to fast transients

An extension of a recently developed quasi-2D flow model for fluid transients in viscoelastic pipes to handle fluid–structure interaction mechanisms is presented. In a context in which the fluid flow is devised as a structured pseudo-mixture and the pipe’s viscoelasticity is rooted in an internal variable theory, the axial movement of the pipe wall is allowed to occur, giving rise to friction, Poisson, and junction coupling mechanisms. The resulting governing equations of the model form a quasi-linear hyperbolic system of partial differential equations, which approximated solution is achieved by means of the method of characteristics. The proposed approach is validated against pressure traces acquired from a reservoir–pipe–valve experimental setup found in the literature. In the course of the validating process, different pipe anchoring conditions are employed to study the system responses. Focus is given to pipe–fluid interface interactions, energy dissipation, and transfer of energy between both media.

A causal formulation of dissipative relativistic fluid dynamics with or without diffusion

The article proposes a causal five-field formulation of dissipative relativistic fluid dynamics as a quasilinear symmetric hyperbolic system of second order. The system is determined by four dissipation coefficients η , ζ , κ , μ \eta ,\zeta ,\kappa ,\mu , free functions of the fields, which quantify shear viscosity, bulk viscosity, heat conductivity, and diffusion.

Partially dissipative systems in the critical regularity setting, and strong relaxation limit

Many physical phenomena may be modeled by first order hyperbolic equations with degenerate dissipative or diffusive terms. This is the case for example in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term, or, in numerical simulations, of conservation laws by relaxation schemes. Such so-called partially dissipative systems have been first pointed out by S. K. Godunov in a short note in 1961. Much later, in 1984, S. Kawashima highlighted in his PhD thesis a simple criterion ensuring the existence of global strong solutions in the vicinity of a linearly stable constant state. This criterion has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. These notes originate essentially from the PhD thesis of T. Crin-Barat that was initially motivated by an earlier observation of the author in a chapter of the handbook edited by Y. Giga and A. Novotný. Our main aim is to adapt the method of Beauchard and Zuazua to a class of symmetrizable quasilinear hyperbolic systems (containing the compressible Euler equations), in a critical regularity setting that allows to keep track of the dependence with respect to e.g. the relaxation parameter. Compared to Beauchard and Zuazua’s work, we exhibit a ‘damped mode’ that will have a key role in the construction of global solutions with critical regularity, in the proof of optimal time-decay estimates and, last but not least, in the study of the strong relaxation limit. For simplicity, we here focus on a simple class of partially dissipative systems, but the overall strategy is rather flexible, and adaptable to much more general situations.

Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems and Its Asymptotic Stability

Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems and Its Asymptotic Stability

Local Solution of Three-Dimensional Axisymmetric Supersonic Flow in a Nozzle

In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.

Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems

The objective of this paper is to construct Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations geometrically. To this end, we adapt and combine elements of two approaches to the construction of Riemann $k$-waves, namely, the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the $k$-wave problem and discuss in detail the conditions for the existence of $k$-wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the $k$-wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.

Saturated boundary feedback control of quasi-linear hyperbolic balance laws with application to LWR traffic flow stabilization

The saturated boundary stabilization problem for quasi-linear hyperbolic systems of balance laws is considered under H2-norm in this paper, where the boundary conditions of the system are subject to actuator saturations. The resulting closed-loop system is proven to be locally exponentially stable with respect to the steady states in the presence of saturations. To this end, the sector nonlinearity model is introduced to deal with the saturation term, and then sufficient conditions for ensuring the locally exponential stability are established in terms of a set of matrix inequalities by employing the Lyapunov function method along with a sector condition. Furthermore, these results are applied to the stabilization of the two-lane traffic flow dynamic represented by Lighthill–Whitham–Richards (LWR) model. By utilizing variable speed limit (VSL) devices, a saturated boundary feedback controller is designed to stabilize the two-lane traffic flow, and the exponential convergence of the quasi-linear traffic flow system in H2 sense is validated by numerical simulations.

Singularity for the one-dimensional rotating Euler equations of Chaplygin gases

This paper is focused on the singularity formation of smooth solutions for the one-dimensional rotating Euler equations of Chaplygin gases, which is a nonhomogeneous quasilinear hyperbolic system with linearly degenerate characteristic fields. We overcome the influence of the rotation terms and show that the density itself of the smooth solution tends to infinity in finite time for a kind of initial data.

Numerical study of multiphase hyperbolic models

Baer–Nunziato (BN) type models introduce fundamental and widely used class of multi-fluid and multi-velocity models to describe evolution of multi-phase flows. The original BN model is supposed to describe multi-phase flows at spatial scales substantially larger than the characteristic scale of elements of the dispersed phase. Recently a number of attempts have been performed to apply it to the description of multi-phase flows with resolved, at the large spatial scale, inter-phase boundaries. A number of still not well understood issues arise in this situation, related to the complex interplay of the topics related to (i) justification of the hierarchy of BN-type models for diffuse interface type description of multi-phase flows with resolved inter-phase boundaries; (ii) the questions of “correct” definition of generalized solutions of first order quasi-linear hyperbolic systems and (iii) development of “really” path-conservative and asymptotic preserving schemes for quasilinear hyperbolic systems with stiff relaxation terms. The goal of the paper is to study numerically such interplay when BN7 model is applied to describe 1D two-phase flows. Our main target is the diffuse interface type formulations. The obtained numerical solutions are compared to analytical solutions of the “more equilibrated” BN5 equations and heterogeneous formulations for the standard Euler equation. The goal is to provide factual information for the performed tests. We also try to summarize the obtained results in a systematic way from a practical point of view. To avoid ambiguity in interpretation of the numerical results, we provide detailed description of the algorithms and perform a number of auxiliary numerical tests.

Converging strong shock wave from a cylindrical piston in a Van der Waals magnetogasdynamics with dust particles

In this study, by using the perturbation series approach, a global solution to the implosion problem is obtained for a quasi-linear hyperbolic system of partial differential equations (PDEs) describing a problem of strong converging cylindrical shock waves collapsing at the axis of symmetry in a Van der Waals dusty gas under the effect of axial magnetic field. This global solution provides the results of Guderley’s local self-similar solution which is valid only in the neighborhood of the axis of implosion. The similarity exponents and corresponding amplitudes are obtained in the neighborhood of the shock collapse. In addition, the values of leading similarity exponents are compared with the results obtained by Whitham’s method. The shock trajectory and flow variables (i.e., density, velocity, pressure and magnetic pressure) have been drawn for different values of the relative specific heat, the mass concentration of dust particles, the ratio of the density of solid particles to the initial density of the gas, Van der Waals excluded volume and shock Cowling number.

Global piecewise classical solutions to quasilinear hyperbolic systems on a tree-like network

In this paper, we discuss the existence, uniqueness and asymptotic stability of global piecewise [Formula: see text] solution to the mixed initial-boundary value problem for 1-D quasilinear hyperbolic systems on a tree-like network. Under the assumption of boundary dissipation, when the given boundary and interface functions possess suitably small [Formula: see text] norm, we obtain the existence and uniqueness of global piecewise [Formula: see text] solution. Moreover, when they further possess a polynomial or exponential decaying property with respect to [Formula: see text], then the corresponding global piecewise [Formula: see text] solution possesses the same or similar decaying property. These results will be used to show the asymptotic stability of the exact boundary controllability of nodal profile on a tree-like network.

On the development of shocks in fluids

In my 2007 work, I studied the formation of shocks in the context of Eulerian equations of the mechanics of compressible fluids. That work studied the maximal classical development of smooth initial data. The present review article is a presentation of my 2019 work, which addresses the problem of the physical continuation of the solution past the point of shock formation. The problem requires the construction of a hypersurface, the shock hypersurface, originating from a given singular spacelike surface, which is acoustically spacelike as viewed from its past, and the construction of a new solution in the future of the union of a given regular null hypersurface with the shock hypersurface, a solution which extends continuously the prior maximal classical solution across the given regular null hypersurface, but which displays discontinuities in physical variables across the shock hypersurface in accordance with the mass, momentum, and energy conservation laws. Moreover, the shock hypersurface is to be acoustically timelike as viewed from its future. Mathematically, this is a free boundary problem, with nonlinear conditions at the free boundary, for a first order quasilinear hyperbolic system of PDE, with characteristic initial data, which are singular at the past boundary of the initial null hypersurface, that boundary being the singular surface of origin. My 2019 work solved a restricted form of the problem through the introduction of new geometric and analytic methods. This Review is focused on the presentation of these methods.