Hyperbolic Systems Of Balance Laws Research Articles

Relative entropy in hyperbolic relaxation for balance laws

We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras [A. Tzavaras, Commun. Math. Sci., 3-2, 2005] for systems of hyperbolic conservation laws.

The generalized Riemann problems for compressible fluid flows: Towards high order

The generalized Riemann problems (GRP) for nonlinear hyperbolic systems of balance laws in one space dimension are now well-known and can be formulated as follows: Given initial data which are smooth on two sides of a discontinuity, determine the time evolution of the solution near the discontinuity. In particular, the GRP of (k+1)th order high-resolution is based on an analytical evaluation of time derivatives up to kth order, which turns out to be dependent only on the spatial derivatives up to kth order. While the classical Riemann problem serves as a primary “building block” in the construction of many numerical schemes (most notably the Godunov scheme), the analytic study of GRP will lead to an array of “GRP schemes”, which extend the Godunov scheme. Currently there are extensive studies on the second-order GRP scheme, which proves to be robust and is capable of resolving complex multidimensional fluid dynamic problems (Ben-Artzi and Falcovitz, 2003 [4]). In this paper, we provide a new approach for solving the GRP for the compressible flow system towards high order accuracy. The derivation of second-order GRP solver is more concise compared to those in previous works and the third-order GRP (or quadratic GRP) is resolved for the first time. The latter is shown to be necessary through numerical experiments with strong discontinuities. Our method relies heavily on the new treatment of rarefaction waves. Indeed, as a main technical step, the “propagation of singularities” argument for the rarefaction fan, is simplified by deriving the linear ODE systems for the “evolution” of “characteristic derivatives”, in the x–t coordinates, for generalized Riemann invariants. The case of sonic point is incorporated into a general treatment. The accuracy of the derived GRP solvers is justified and numerical examples are presented for the performance of the resulting schemes.

Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws

This work is concerned with ($N$-component) hyperbolic system of balance laws in arbitrary space dimensions. Under entropy dissipative assumption and the Shizuta-Kawashima algebraic condition, a general theory on the well-posedness of classical solutions in the framework of Chemin-Lerner's spaces with critical regularity is established. To do this, we first explore the functional space theory and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner's spaces. Then this fact allows to prove the local well-posedness for general data and global well-posedness for small data by using the Fourier frequency-localization argument. Finally, we apply the new existence theory to a specific fluid model-the compressible Euler equations with damping, and obtain the corresponding results in critical spaces.

BV solutions for hyperbolic systems of balance laws with relaxation

For hyperbolic systems of balance laws with source manifesting relaxation, it is shown that the Kawashima condition, which yields global classical solutions with smooth initial values near equilibrium, is also instrumental in inducing the existence of global admissible BV solutions, accommodating shocks.

Transient [formula omitted] error estimates for well-balanced schemes on non-resonant scalar balance laws

The ability of Well-Balanced (WB) schemes to capture very accurately steady-state regimes of non-resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by Greenberg and LeRoux (1996) [15] (see also the anterior WB Glimm scheme in E, 1992 [8]). This paper aims at showing, by means of rigorous Ct0(Lx1) estimates, that these schemes deliver an increased accuracy in transient regimes too. Namely, after explaining that for the vast majority of non-resonant scalar balance laws, the Ct0(Lx1) error of conventional fractional-step (Tang and Teng, 1995 [45]) numerical approximations grows exponentially in time like exp(max(g′)t)Δx (as a consequence of the use of Gronwallʼs lemma), it is shown that WB schemes involving an exact Riemann solver suffer from a much smaller error amplification: thanks to strict hyperbolicity, their error grows at most only linearly in time (see also Layton, 1984 [30]). Numerical results on several test-cases of increasing difficulty (including the classical LeVeque–Yeeʼs benchmark problem (LeVeque and Yee, 1990 [34]) in the non-stiff case) confirm the analysis.

HYPERBOLIC SYSTEMS OF BALANCE LAWS WITH WEAK DISSIPATION II

By extending the analysis in [C. M. Dafermos, Hyperbolic systems of balance laws with weak dissipation, J. Hyperbolic Differ. Equ.3 (2006) 505–527], this note constructs global BV solutions to the Cauchy problem for strictly hyperbolic systems of balance laws endowed with a convex entropy, under the assumption that the entropy production is positive definite.

Analysis of the [formula omitted] model: Well-posedness and diffusion asymptotics

This paper is devoted to the analysis of the M1 model which arises in radiative transfer theory. The derivation of the model is based on the entropy minimization principle, which leads to a hyperbolic system of balance laws with relaxation. In the multi-dimensional case, we establish the existence–uniqueness of a globally defined smooth solution under a suitable smallness condition on the initial data. In the one-dimensional case we show that the smallness condition does not depend on the particles mean free path so that we can also rigorously justify the consistency of the model with the diffusion asymptotics. The result extends the analysis of Coulombel et al. [J.-F. Coulombel, F. Golse T. Goudon, Diffusion approximation and entropy-based moment closure for kinetic equations, Asymptotic Analysis, 45 (2005) 1–39] to the case where the entropy functional accounts for relaxation towards the Planckian state, which is physically more relevant.

The initial-boundary value problem of hyperbolic integro-differential systems of nonlinear balance laws

This research explores the initial-boundary value problem for the 2×2 hyperbolic systems of balance laws whose sources are the time-dependent and contain the integral of unknowns. Perturbed Riemann and boundary Riemann problems are provided to account for the time-dependence of sources. Their approximation solutions are constructed by modified Lax’s method. In addition, we introduce a new version of Glimm scheme (GGS) and study its stability which is proved by the wave interaction estimates in a dissipativity assumption. With the consistency of GGS, the existence of a global weak solution satisfying the entropy inequality is then achieved. Finally the Lipschitz continuous solution to the problem is established by the weak convergence of the residual.

ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws

A family of time-varying hyperbolic systems of balance laws is considered. The partial differential equations of this family can be stabilized by selecting suitable boundary conditions. For the stabilized systems, the classical technique of construction of Lyapunov functions provides a function which is a weak Lyapunov function in some cases, but is not in others. We transform this function through a strictification approach to obtain a time-varying strict Lyapunov function. It allows us to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of input-to-state stability (ISS) type. Two examples illustrate the results.

Large Time Behavior of Solutions to Nonlinear Viscoelastic Model with Fading Memory

We study the Cauchy problem of a one-dimensional nonlinear viscoelastic model with fading memory. By introducing appropriate new variables we convert the integro-partial differential equations into a hyperbolic system of balance laws. When it is a perturbation of a constant state, the solution is shown time asymptotically approaching to predetermined diffusion waves. Pointwise estimates on the convergence details are obtained.

Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws

Explicit boundary dissipative conditions are given for the exponential stability in L 2 -norm of one-dimensional linear hyperbolic systems of balance laws ∂ t ξ + Λ ∂ x ξ − M ξ = 0 over a finite interval, when the matrix M is marginally diagonally stable. The result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised Saint–Venant–Exner equations.

Relaxation Limit and Global Existence of Smooth Solutions of Compressible Euler–Maxwell Equations

We consider smooth periodic solutions for the Euler–Maxwell equations, which are a symmetrizable hyperbolic system of balance laws. We proved that as the relaxation time tends to zero, the Euler–Maxwell system converges to the drift-diffusion equations at least locally in time. The global existence of smooth solutions is established near a constant state with an asymptotic stability property.

Lyapunov exponential stability of linear hyperbolic systems of balance laws

AbstractExplicit boundary dissipative conditions are given for the exponential stability in L2-norm of one-dimensional linear hyperbolic sytems of balance laws ∂tξ+Λ∂xξ-Mξ=0 over a finite space interval, when the matrix M is marginally diagonally stable. The result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised Saint-Venant-Exner equations.

On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws

This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balance laws. We consider these systems as a particular case of hyperbolic systems in nonconservative form, for which we use the theory introduced by Dal Maso, LeFloch and Murat (J. Math. Pures Appl. 74:483, 1995) in order to define the concept of weak solutions. This theory is based on the prescription of a family of paths in the phases space. We also consider path-conservative schemes, that were introduced in Pares (SIAM J. Numer. Anal. 44:300, 2006). The first goal is to prove a Lax-Wendroff type convergence theorem. In Castro et al. (J. Comput. Phys. 227:8107, 2008) it was shown that, for general nonconservative systems a rather strong convergence assumption is needed to prove such a result. Here, we prove that the same hypotheses used in the classical Lax-Wendroff theorem are enough to ensure the convergence in the particular case of systems of balance laws, as the numerical results shown in Castro et al. (J. Comput. Phys. 227:8107, 2008) seemed to suggest. Next, we study the relationship between the well-balanced properties of path-conservative schemes applied to systems of balance laws and the family of paths.

A hybrid Godunov method for radiation hydrodynamics

From a mathematical perspective, radiation hydrodynamics can be thought of as a system of hyperbolic balance laws with dual multiscale behavior (multiscale behavior associated with the hyperbolic wave speeds as well as multiscale behavior associated with source term relaxation). With this outlook in mind, this paper presents a hybrid Godunov method for one-dimensional radiation hydrodynamics that is uniformly well behaved from the photon free streaming (hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and to the strong equilibrium diffusion (hyperbolic) limit. Moreover, one finds that the technique preserves certain asymptotic limits. The method incorporates a backward Euler upwinding scheme for the radiation energy density E r and flux F r as well as a modified Godunov scheme for the material density ρ, momentum density m, and energy density E. The backward Euler upwinding scheme is first-order accurate and uses an implicit HLLE flux function to temporally advance the radiation components according to the material flow scale. The modified Godunov scheme is second-order accurate and directly couples stiff source term effects to the hyperbolic structure of the system of balance laws. This Godunov technique is composed of a predictor step that is based on Duhamel’s principle and a corrector step that is based on Picard iteration. The Godunov scheme is explicit on the material flow scale but is unsplit and fully couples matter and radiation without invoking a diffusion-type approximation for radiation hydrodynamics. This technique derives from earlier work by Miniati and Colella (2007) [41]. Numerical tests demonstrate that the method is stable, robust, and accurate across various parameter regimes.

Large Time Asymptotics for Partially Dissipative Hyperbolic Systems

This work is concerned with (n-component) hyperbolic systems of balance laws in m space dimensions. First, we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t → ∞. Using the Fourier transform, we examine the role that control theoretical tools, such as the classical Kalman rank condition, play. We build Lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable. In this way we extend the previous analysis by Shizuta and Kawashima under the so-called algebraic condition (SK). In particular, we show the existence of systems exhibiting more complex behavior than the one that the (SK) condition allows. We also discuss links between this analysis and previous literature in the context of damped wave equations, hypoellipticity and hypocoercivity. To conclude, we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws. Our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover.

Wave Propagation in Multicomponent Flow Models

We consider systems of hyperbolic balance laws governing flows of an arbitrary number of components equipped with general equations of state. The components are assumed to be immiscible. We compare two such models: one in which thermal equilibrium is attained through a relaxation procedure, and a fully relaxed model in which equal temperatures are instantaneously imposed. We describe how the relaxation procedure may be made consistent with the second law of thermodynamics. Exact wave velocities for both models are obtained and compared. In particular, our formulation directly proves a general subcharacteristic condition: For an arbitrary number of components and thermodynamically stable equations of state, the mixture sonic velocity of the relaxed system can never exceed the sonic velocity of the relaxation system.

Car-following and the macroscopic Aw-Rascle traffic flow model

We consider a semi-discrete car-following model and the macroscopic Aw-Rascle model for traffic flow given in Lagrangian form. The solution of the car-following model converges to a weak entropy solution of the system of hyperbolic balance laws with Cauchy initial data. For the homogeneous system, we allow vacuum in the initial data. By using properties of the semi-discrete model, we show that this solution of the hyperbolic system is stable in the $L^1$-norm.

Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow

We consider a strictly hyperbolic system of balance laws in one space variable, that represents a simple model for a fluid flow in the presence of phase transitions. The state variables are specific volume, velocity and mass–density fraction λ of the vapor in the fluid. A reactive source term drives the dynamics of the phase mixtures; such a term depends on a relaxation parameter and involves an equilibrium pressure, allowing for metastable states. First we prove the global existence of weak solutions to the Cauchy problem, where the initial datum for λ is close either to 0 or 1 (the pure phases) and has small total variation, while the initial variations of pressure and velocity are not necessarily small. Then we consider the relaxation limit and prove that the weak solutions of the full system converge to those of the reduced system.

Global Existence of Large BV Solutions in a Model of Granular Flow

In this paper we analyze a set of equations proposed by Hadeler and Kuttler [20], describing the flow of granular matter in terms of the heights of a standing layer and of a moving layer. By a suitable change of variables, the system can be written as a 2 × 2 hyperbolic system of balance laws, which we study in the one-dimensional case. The system is linearly degenerate along two straight lines in the phase plane, and therefore is weakly linearly degenerate at the point of the intersection. The source term is quadratic, consisting of product of two quantities, which are transported with strictly different speeds. Assuming that the initial height of the moving layer is sufficiently small, we prove the global existence of entropy-weak solutions to the Cauchy problem, for a class of initial data with bounded but possibly large total variation.