Scalar Balance Laws Research Articles

SBV-Like Regularity of Entropy Solutions for a Scalar Balance Law

SBV-Like Regularity of Entropy Solutions for a Scalar Balance Law

Initial boundary value problem for 1D scalar balance laws with strictly convex flux

A Lax-Oleĭnik type explicit formula for 1D scalar balance laws has been recently obtained for the pure initial value problem by Adimurthi et al. in [2]. In this article, by introducing a suitable boundary functional, we establish a Lax-Oleĭnik type formula for the initial-boundary value problem. For the pure initial value problem, the solution for the corresponding Hamilton-Jacobi equation turns out to be the minimizer of a functional on the set of curves known as h-curves. In the present situation, part of the h-curve joining any two points in the quarter plane may cross the boundary x=0. This phenomenon breaks the simplicity of the minimization process through the boundary functional compared to the case of conservation laws. Moreover, this complicates the verification of the boundary condition in the sense of Bardos, le Roux, and Nédélec [4]. To verify the boundary condition, the boundary points are classified into three types depending on the structure of the minimizers at those points. Finally, by introducing characteristic triangles, we construct generalized characteristics and show that the explicit solution is entropy admissible.

Uniform asymptotic stability for convection–reaction–diffusion equations in the inviscid limit towards Riemann shocks

The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter \varepsilon . The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale \varepsilon -dependent topology reduces to the classical W^{1, \infty} -topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine–Hugoniot conditions and the nonuniform shift arising merely from phasing out the nondecaying 0-mode, as in the classical stability analysis for fronts of reaction–diffusion equations.

Sticky particle Cucker–Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker–Smale dynamics. Our approach is inspired by the work of Brenier and Grenier on the pressureless Euler equations.

A well‐balanced and entropy stable scheme for a reduced blood flow model

AbstractA well‐known reduced model of the flow of blood in arteries can be formulated as a strictly hyperbolic system of two scalar balance laws in one space dimension where the unknowns are the cross‐sectional area of the artery and the average blood flow velocity as functions of the axial coordinate and time. This system is endowed with an entropy pair such that solutions of the balance equations satisfy an entropy inequality in the distributional sense. It is demonstrated that this property can be utilized to construct an entropy stable finite difference scheme for the blood flow model based on the general framework by Tadmor (E. Tadmor, Math. Comput., 49 (1987), 91–103). Furthermore, a fourth‐order extension of the resulting entropy conservative flux and a fourth‐order sign‐preserving reconstruction of the scaled entropy variables are employed as well as a second‐order strong stability preserving Runge–Kutta method for time discretization. The result is a computationally inexpensive and easy‐to‐implement explicit entropy stable scheme for the blood flow model. It is proven that the scheme is well‐balanced (i.e., preserves certain steady solutions of the model) and numerical examples are presented.

Stability and instability in scalar balance laws: fronts and periodic waves

We complete a full classification of non-degenerate traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under (piecewise) smooth perturbations. A striking feature of our analysis is the elucidation of the prominent role of characteristic points in the determination of both the spectra of the linearized operators and the phase dynamics involved in the nonlinear large-time evolution. For a generic class of equations an upshot of our analysis is a dramatic reduction from a tremendously wide variety of entropic traveling waves to a relatively small range of stable entropic traveling waves.

Well-balanced high-order finite difference methods for systems of balance laws

In this paper, high order well-balanced finite difference weighted essentially non-oscillatory methods to solve general systems of balance laws are presented. Two different families are introduced: while the methods in the first one preserve every stationary solution, those in the second family only preserve a given set of stationary solutions that depend on some parameters. The accuracy, well-balancedness, and conservation properties of the methods are discussed, as well as their application to systems with singular source terms. The strategy is applied to derive third and fifth order well-balanced methods for a linear scalar balance law, Burgers' equation with a nonlinear source term, and for the shallow water model. In particular, numerical methods that preserve every stationary solution or only water at rest equilibria are derived for the latter.

Large-Time Asymptotic Stability of Riemann Shocks of Scalar Balance Laws

We prove the large-time asymptotic orbital stability of strictly entropic Riemann shock solutions of first-order scalar hyperbolic balance laws under piecewise regular perturbations provided that the source term is dissipative about endstates of the shock. Moreover, the convergence toward a shifted reference state is exponential with a rate predicted by the linearized equations about constant endstates.

Well-posedness of general 1D initial boundary value problems for scalar balance laws

We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax–Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kružkov's doubling of variables technique, together with a careful treatment of the boundary terms.

Definitions of solutions to the IBVP for multi-dimensional scalar balance laws

We consider four definitions of solution to the initial-boundary value problem (IBVP) for a scalar balance laws in several space dimensions. These definitions are extended to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.

Relative entropy and L2 stability to a shock for scalar balance laws

We consider a scalar balance law with a strict convex flux. In this paper, we study L2 stability to a shock for a scalar balance law up to a shift function, which is based on the relative entropy. This result generalizes Leger?s works [18] and provides more a simple proof than Leger?s proof.

Discontinuous traveling waves for scalar hyperbolic-parabolic balance law

This paper is concerned with the existence of traveling waves for the scalar hyperbolic-parabolic balance law. Using a phase-plane analysis method, we first prove the existence of an increasing traveling wave solution in $C^{1}(\mathbb{R})$ . Then we construct a family of discontinuous periodic traveling wave entropy solutions.

On the attainable set for scalar balance laws with distributed control

The paper deals with the set of attainable profiles of a solution u to a scalar balance law in one space dimension with strictly convex flux function ∂ t u + ∂ x f (u ) = z (t, x ). Here the function z is regarded as a bounded measurable control. We are interested in studying the set of attainable profiles at a fixed time T > 0, both in case z (t, ·) is supported in the all real line, and in case z (t, ·) is supported in a compact interval [a, b ] independent on the time variable t .

Multi-dimensional scalar balance laws with discontinuous flux

We consider scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect to both the space variable x and the unknown quantity u. We formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem. The problem is formulated in the framework of multi-valued mappings. The notion of entropy measure-valued solutions is used to prove the so-called contraction principle and comparison principle.

Asymptotic Limit to Shocks for Scalar Balance Laws Using Relative Entropy

We consider a scalar balance law with a strict convex flux. In this paper, we study inviscid limit to shocks for scalar balance laws up to a shift function, which is based on the relative entropy.

A Two-Dimensional Version of the Godunov Scheme for Scalar Balance Laws

A Godunov scheme is derived for two-dimensional scalar conservation laws without or with source terms following ideas originally proposed by Boukadida and LeRoux [Math. Comput., 63 (1994), pp. 541--553] in the context of a staggered Lax--Friedrichs scheme. In both situations, the numerical fluxes are obtained at each interface of a uniform Cartesian computational grid just by means of the “external waves” involved in the entropy solution of the elementary two-dimensional (2D) Riemann problems; in particular, all the wave-interaction phenomena are overlooked. This restriction of the wave pattern suffices for deriving the exact numerical fluxes of the staggered Lax--Friedrichs scheme, but it furnishes only an approximation for the Godunov scheme: we show that under convenient assumptions, these flux functions are smooth and the resulting discretization process is stable under nearly optimal CFL restriction. A well-balanced extension is presented, relying on the Curl-free component of the Helmholtz decomposi...

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.

Vanishing viscosity for mixed systems with moving boundaries

We consider a system of scalar balance laws in one space dimension coupled with a system of ordinary differential equations. The coupling acts through the (moving) boundary condition of the balance laws and the vector fields of the ordinary differential equations. We prove the existence of solutions for such systems passing to the limit in a vanishing viscosity approximation.

Vehicular Traffic Flow Dynamics on a Bus Route

We propose a new model in the analysis of car traffic flow phenomena which in particular refers to a bus route, namely, a closed path embedded in the urban network of roads. The model consists in a scalar balance law for the (macroscopic) density of cars, with source terms representing incoming and outgoing roads (with respect to the bus route), and a coupled system of ODEs describing the dynamics of (microscopic) buses along the route. The coupling takes into account both the buses as moving bottlenecks in the car flow and the effect of the car flow on the buses' ODEs as microscopic vehicles. The resulting fully coupled micro--macro model promises to give an appropriate description for the car flow as well as for the buses. Finally, we study the model numerically.

N-WAVES IN HYPERBOLIC BALANCE LAWS

It is shown that, as time tends to infinity, solutions to the Cauchy problem for a class of genuinely nonlinear scalar balance laws attain N-wave profiles, when the initial data have compact support, or saw-toothed profiles, when the initial data are periodic. The amplitude and length of these waves results from the synergy between flux and source.