Nonhomogeneous Hyperbolic Systems Research Articles

The dichotomy property in stabilizability of 2 × 2 linear hyperbolic systems

This paper is devoted to discuss the stabilizability of a class of 2 × 2 non-homogeneous hyperbolic systems. Motivated by the example in the Section 5.6 of Bastin and Coron’s book in 2016, we analyze the influence of the interval length L on stabilizability of the system. By spectral analysis, we prove that either the system is stabilizable for all L > 0 or it possesses the dichotomy property: there exists a critical length Lc > 0 such that the system is stabilizable for L ∈ (0, Lc) but unstabilizable for L ∈ [Lc,+∞). In addition, for L ∈ [Lc, +∞), we obtain that the system can reach equilibrium state in finite time by backstepping control combined with observer. Finally, we also provide some numerical simulations to confirm our developed analytical criteria.

Numerical schemes for quasi-1D steady nozzle flows

In this work we construct high-resolution numerical schemes for the calculation of the quasi-1D unsteady flow in pipes with variable cross-sectional area. This is an example of non-homogeneous hyperbolic systems of conservation laws that admit stationary solutions. We use the strategy developed by the authors in [1] which is to transform the non-homogeneous system into homogeneous writing the source term in divergence form, so that it can be incorporated into the flux vector of the homogeneous system and discretized in the same way. As a result, the source terms are automatically discretized to achieve perfect balance with flux terms, obtaining well-balanced schemes that produce very robust and accurate solutions. Concretely, the mentioned strategy will be used to extend the flux limiter technique [2] and the Harten, Lax and van Leer (HLL) Riemann solver [3] to the quasi-1D flow in ducts of variable cross-section. The numerical results confirm the capacity of these methods to construct well-balanced schemes.

Iterative approximate factorization for difference operators of high-order bicompact schemes for multidimensional nonhomogeneous hyperbolic systems

An iterative method for solving equations of multidimensional bicompact schemes based on an approximate factorization of their difference operators is proposed for the first time. Its algorithm is described as applied to a system of two-dimensional nonhomogeneous quasilinear hyperbolic equations. The convergence of the iterative method is proved in the case of the two-dimensional homogeneous linear advection equation. The performance of the method is demonstrated on two numerical examples. It is shown that the method preserves a high (greater than the second) order of accuracy in time and performs 3–4 times faster than Newton’s method. Moreover, the method can be efficiently parallelized.

Central finite volume schemes on nonuniform grids and applications

We propose a new one-dimensional unstaggered central scheme on nonuniform grids for the numerical solution of homogeneous hyperbolic systems of conservation laws with applications in two-phase flows and in hydrodynamics with and without gravitational effect. The numerical base scheme is a generalization of the original Lax–Friedrichs scheme and an extension of the Nessyahu and Tadmor central scheme to the case of nonuniform irregular grids. The main feature that characterizes the proposed scheme is its simplicity and versatility. In fact, the developed scheme evolves a piecewise linear numerical solution defined at the cell centers of a nonuniform grid, and avoids the resolution of the Riemann problems arising at the cell interfaces, thanks to a layer of staggered cells used intermediately. Spurious oscillations are avoided using a slopes limiting procedure. The developed scheme is then validated and used to solve classical problems arising in gas–solid two phase flow problems. The proposed scheme is then extended to the case of non-homogenous hyperbolic systems with a source term, in particular to the case of Euler equations with a gravitational source term. The obtained numerical results are in perfect agreement with corresponding ones appearing in the recent literature, thus confirming the efficiency and potential of the proposed method to handle both homogeneous and non-homogeneous hyperbolic systems.

Differential Constraints and Exact Solution to Riemann Problems for a Traffic Flow Model

In view of solving in a closed form initial and/or boundary value problems of interest in nonlinear hyperbolic and dissipative wave processes it is considered a reduction approach based upon appending differential constraints to quasilinear nonhomogeneous hyperbolic systems of first order PDEs. In this context a governing model of traffic flow is analyzed thoroughly and a classification of possible constraints along with sets of consistent response functions involved therein is worked out whereupon the classes of corresponding exact solutions are determined. To some extent these solutions generalize the classical simple wave solutions and may also incorporate dissipative effects. Furthermore, in order to solve a Riemann Problem, an exact rarefaction wave-like solution is built. Finally an application of the results to the so-called “green traffic light problem” is also illustrated.

Well-balanced finite volume schemes for 2D non-homogeneous hyperbolic systems. Application to the dam break of Aznalcóllar

In this paper, we introduce a class of well-balanced finite volume schemes for 2D non-homogeneous hyperbolic systems. We extend the derivation of standard finite volume solvers for homogeneous systems to non-homogeneous ones using the method of lines. We study conservation and some well-balanced properties of the numerical scheme. We apply our solvers to shallow water equations: we prove that these exactly compute the water at rest solutions. We also perform some numerical tests, by comparing with 1D solutions, simulating the formation of a hydraulic drop and a hydraulic jump, and studying a real dam break: Aznalcóllar, an ecological disaster happened in the province of Seville, Spain in 1998.

Couplage interfacial instationnaire de modèles diphasiques

The primary coolant circuit in a nuclear power plant contains several distinct components (vessel, core, pipes,...). To these components correspond one or several codes which have been developed to solve specific systems of partial differential equations. In order to perform two-phase flow simulations over the whole primary circuit, the coupling of these codes has to be considered. The approach that has been examined within this work consists in coupling the codes by providing information through the coupling interface, taking unsteady effects into account. For that purpose, numerical fluxes calculated at the interface are used as boundary conditions for each code. The numerical technique relies on the use of an interface model, which is combined with the basic strategy that was introduced by Greenberg and LeRoux in order to compute approximations of steady solutions of non-homogeneous hyperbolic systems. Three different coupling cases have been examined : the coupling of a one-dimensional Euler system with a two-dimensional Euler system ; the coupling of two distinct homogeneous two-phase flow models ; the coupling of a four-equation homogeneous model with the standard six-equation two-fluid model.

On Well‐Balanced Finite Volume Methods for Nonconservative Nonhomogeneous Hyperbolic Systems

In this work we introduce a general family of finite volume methods for nonhomogeneous hyperbolic systems with nonconservative terms. We prove that all of them are “asymptotically well‐balanced”: they preserve all smooth stationary solutions in all the domain except for a set whose measure tends to zero as $\Delta x$ tends to zero. This theory is applied to solve the bilayer shallow‐water equations with arbitrary cross‐section. Finally, some numerical tests are presented for simplified but meaningful geometries, comparing the computed solution with approximated asymptotic analytical solutions.

MULTIVARIABLE BOUNDARY CONTROL APPROACH BY INTERNAL MODEL, APPLIED TO IRRIGATION CANALS REGULATION

This paper deals with the regulation problem of a class of irrigation canals using the Saint-Venant partial differential equations (pde). The Internal Model Boundary Control (IMBC) approach is used and the multireaches case is considered (several reaches in cascade). Perturbation theory of exponential semigroup used for control synthesis is extended here to nonhomogeneous hyperbolic systems, as the multireaches regulation model is described by hyperbolic pde's. Experimental results (on the Valence experimental canal) are encouraging for an extension to the real case. Additionally a multi-model approach was introduced to allow wider water level variations.

Asymptotically balanced schemes for non-homogeneous hyperbolic systems – application to the Shallow Water equations

In this work we introduce a class of balanced numerical schemes, up to second order, for the solution of general non-homogeneous hyperbolic systems of conservation laws. We give a general technique to build such schemes. We also prove that they balance up to second order a large class of steady solutions in the whole domain but some subset whose measure tends to zero as the grid size decreases to zero. We finally present an application to Shallow Water equations that exhibit the good performances of some of the schemes introduced. To cite this article: T. Chacón Rebollo et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems

<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

Hyperbolic to Parabolic Relaxation Theory for Quasilinear First Order Systems

In this paper we study the limiting behavior of nonhomogeneous hyperbolic systems of balance laws when the relaxed equilibria are described by means of systems of parabolic type. In particular we obtain a complete theory for the 2×2 systems of genuinely nonlinear hyperbolic balance laws in 1≳D with a strong dissipative term. A different method, which combines the div–curl lemma with accretive operators, is then applied to study the limiting profiles in the case of nonhomogeneous isentropic gas dynamics. We also investigate relaxation results for some 2≳D cases, which include the Cattaneo model for nonlinear heat conduction and the compressible Euler flow. Moreover, convergence result is also obtained for general semilinear systems in 1≳D.

Solutions in the large for nonhomogeneous quasilinear hyperbolic systems of equations

Solutions in the large for nonhomogeneous quasilinear hyperbolic systems of equations