Quasilinear Hyperbolic Systems Research Articles

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.

A new model for fluid transients in piping systems taking into account the fluid–structure interaction

The present work extends a recently developed quasi-2D flow model for fluid transients in elastic pipes to accommodate fluid–structure interaction mechanisms. In this context, the primary goal of the proposed approach is to analyze energy transfer and dissipative effects in the fluid–pipe system. The fluid–structure interaction couples the flow dynamics with the axial movement of the tube, giving rise to friction, Poisson, and junction coupling mechanisms to take place. The mechanical model mathematical structure forms a quasi-linear hyperbolic system of partial differential equations for which approximated solutions are sought by employing the method of characteristics. The proposed model reproduces classical benchmark solutions and presents a good agreement with experimental data. In addition, the whole thermomechanical consistent framework in which the model is established allows an accurate description of the flow’s internal structure. This feature allows a better comprehension of the phenomenon when compared to the classic frictionless and quasi-steady-friction-based four-equation models. The present approach allows a better description of the friction coupling mechanism and unveils uneven shear stress distributions in the unsteady flow. As a result, the energy dissipation in the fluid is captured with accuracy. Due to the absence or limited ability to describe these dispersive and dissipative effects, the traditional approaches are proved to lose their accuracy as the fluid transient goes. The coupled and uncoupled models are also analyzed and are proven to respond differently due to localized pipe–fluid interface effects in addition to the bulk mechanisms of transfer of energy that occur differently in both approaches.

Global classical solutions of a kind of boundary value problem for quasilinear hyperbolic systems

In this paper, we consider a stability problem of a kind of boundary value problem for quasilinear hyperbolic systems. For small boundary data, we prove that the solution exists globally in time when the system is weakly linearly degenerate. In the special case of linear degeneracy, the smallness assumption on the boundary data is weakened. The error estimate in space between two different solutions with different boundary data is also obtained. In our proof, an important estimate which describes the interaction of different waves is established by constructing a continuous Glimm function. This estimate together with uniform a prior estimates with respect to the time leads to the global‐in‐time existence of smooth solutions. Finally, we apply the stability results to the isentropic Euler equations and the system of the motion of an elastic string.

Two-dimensional non-self-similar Riemann solutions for a thin film model of a perfectly soluble anti-surfactant solution

In this article, we construct non-self-similar Riemann solutions for a two-dimensional quasilinear hyperbolic system of conservation laws which describes the fluid flow in a thin film for a perfectly soluble anti-surfactant solution. The initial Riemann data consists of two different constant states separated by a smooth curve inx−yx-yplane, so without using self-similarity transformation and dimension reduction, we establish solutions for five different cases. Further, we consider interaction of all possible nonlinear waves by taking initial discontinuity curve as a parabola to develop the structure of global entropy solutions explicitly.

Low regularity ill-posedness for non-strictly hyperbolic systems in three dimensions

In this paper, we survey a new approach combining algebraic and geometric ideas, with which we prove low regularity ill-posedness for quasilinear hyperbolic systems with non-strict hyperbolicity in three dimensions. These systems are also associated with multiple wave-speeds.

Lyapunov functions and finite-time stabilization in optimal time for homogeneous linear and quasilinear hyperbolic systems

Hyperbolic systems in one-dimensional space are frequently used in the modeling of many physical systems. In our recent works we introduced time-independent feedbacks leading to finite stabilization in optimal time of homogeneous linear and quasilinear hyperbolic systems. In this work we present Lyapunov’s functions for these feedbacks and use estimates for Lyapunov’s functions to rediscover the finite stabilization results.

Asymptotic behavior of global solutions to some multidimensional quasilinear hyperbolic systems

Asymptotic behavior of global solutions to some multidimensional quasilinear hyperbolic systems

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Symmetry breaking in ideal magnetohydrodynamics: the role of the velocity

The ideal magnetohydrodynamic equations are, roughly speaking, a quasi-linear symmetric hyperbolic system of PDEs, but not all the unknowns play the same role in this system. Indeed, in the regime of small magnetic fields, the equations are close to the incompressible Euler equations. In the present paper, we adopt this point of view to study questions linked with the lifespan of strong solutions to the ideal magnetohydrodynamic equations. First of all, we prove a continuation criterion in terms of the velocity field only. Secondly, we refine the explicit lower bound for the lifespan of 2-D flows found in Cobb and Fanelli (arXiv, 2020), by relaxing the regularity assumptions on the initial magnetic field.

Boundary Feedback Control of 2×2 Quasilinear Hyperbolic Systems: Predictive Synthesis and Robustness Analysis

We present a predictive feedback control method for a class of quasilinear hyperbolic systems with one boundary control input. Assuming exact model knowledge, convergence to the origin, or tracking at the uncontrolled boundary, are achieved in finite time. A robustness certificate is provided, showing that at least under more restrictive assumptions on the system coefficients, the control method has inherent robustness properties with respect to small errors in the model, measurements and control input. Rigorous, although conservative conditions on the time derivative of the initial condition and on the design parameter controlling the convergence speed are given to ensure global existence of the solution for initial conditions with arbitrary infinity-norm.

Bounded and almost periodic solvability of nonautonomous quasilinear hyperbolic systems

The paper concerns boundary value problems for general nonautonomous first-order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right-hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the “loss of smoothness” property of hyperbolic PDEs.

Propagation of one-dimensional planar and nonplanar shock waves in nonideal radiating gas

The present study seeks to investigate a quasilinear hyperbolic system of partial differential equations which describes the unsteady one-dimensional motion of a shock wave of arbitrary strength propagating through a nonideal radiating gas. We have derived an infinite hierarchy of the transport equation which is based on the kinematics of one-dimensional motion of shock front. By using the truncation approximation method, an infinite hierarchy of transport equations, which governs the shock strength and the induced discontinuities behind it, is derived to study the kinematics of the shock front. The first three transport equations (i.e., first, second, and third-orders) are used to study the growth and decay behavior of shocks in van der Waals radiating gas. The decay laws for weak shock waves in nonradiating gas are entirely recovered in the second-order truncation approximation. The results obtained by the first three approximations for shock waves of arbitrary strength are compared with the results predicted by the characteristic rule. Also, the effect of nonideal parameters and radiation on the evolutionary behavior of shock waves are discussed and depicted pictorially.

Input-to-State Stability in sup norms for hyperbolic systems with boundary disturbances

We give sufficient conditions for local Input-to-State Stability in C1 norm of general quasilinear hyperbolic systems with boundary input disturbances. In particular the derivation of explicit Input-to-State Stability conditions is discussed for the special case of 2 × 2 systems.

High-Gain Observer Design for a Class of Quasi-Linear Integro-Differential Hyperbolic Systems—Application to an Epidemic Model

This article addresses the problem of high-gain observer design for a class of quasi-linear hyperbolic systems (with one characteristic velocity), possibly including nonlocal terms, making them systems of partial integro-differential equations. The design relies on distributed measurement of a part of the state vector. The observer is presented and discussed and the exponential stability in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C^1$</tex-math></inline-formula> spatial norm of the origin for the error system is fully established via Lyapunov-based analysis. Its use is illustrated via an application to an age-dependent susceptible, infected, recovered (SIR) epidemic model.

Propagation and interaction of weakly nonlinear plane waves in transversely isotropic elastic materials

The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media. It begins with a definition of stored energy functions of considered hyperelastic models. The equation of elastodynamics as well as the first-order quasilinear hyperbolic system for plane waves are provided. The eigensystem for this system is determined to study three-wave interaction coefficients. The main part of the paper concerns a discussion of these coefficients. Applying the weakly nonlinear asymptotics method, it is shown that in the case of transverse isotropy the inviscid Burgers’ equation describes an evolution of a single quasi-shear wave. The result contradicts the case of isotropy, where the equation with quadratic nonlinearity cannot describe any shear wave propagation. The paper ends with an example of numerical solutions for the obtained evolution equation.

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Interaction of waves in one-dimensional dusty gas flow

Abstract The present study uses the theory of weakly nonlinear geometrical acoustics to derive the high-frequency small amplitude asymptotic solution of the one-dimensional quasilinear hyperbolic system of partial differential equations characterizing compressible, unsteady flow with generalized geometry in ideal gas flow with dust particles. The method of multiple time scales is applied to derive the transport equations for the amplitude of resonantly interacting high-frequency waves in a dusty gas. These transport equations are used for the qualitative analysis of nonlinear wave interaction process and self-interaction of nonlinear waves which exist in the system under study. Further, the evolutionary behavior of weak shock waves propagating in ideal gas flow with dust particles is examined here. The progressive wave nature of nonresonant waves terminating into the shock wave and its location is also studied. Further, we analyze the effect of the small solid particles on the propagation of shock wave.