Hyperbolic systems of conservation laws in one space dimension

Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak solutions and on the convergence of vanishing viscosity approximations

One Dimensional Hyperbolic Conservation Laws: Past and Future

Aim of these notes is provide a brief review of the current well-posedness theory for hyperbolic systems of conservation laws in one space dimension, also pointing out open problems and possible research directions. They supplement the slides of the short course given by the author in Erice, May 2023, available at: sites.google.com/view/erice23/speakers-and-slides.Comment: 38 pages, 25 figure

A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithm

Boundary layers in weak solutions to hyperbolic conservation laws

This paper is concerned with the initial-boundary value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing viscosity method and finite difference schemes (Lax-Friedrichs type schemes, Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform L\infty bounds on the approximate solutions and so dealing with L\infty solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. Form the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based a set of admissible boundary values, following Dubois and LeFloch's terminology in ``Boundary conditions for nonlinear hyperbolic systems of conservation laws'', J. Diff. Equa. 71 (1988), 93--122. The local structure of those sets and the well-posedness of the corresponding initial-boundary value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.Comment: 43 page

Hyperbolic Conservation Laws

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Euler hydrodynamics of one-dimensional attractive particle systems

We consider attractive irreducible conservative particle systems on $\mathbb{Z}$, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1--30], from which we relax the assumption that the process has explicit invariant measures.Comment: Published at http://dx.doi.org/10.1214/009117906000000115 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A locally quadratic Glimm functional and sharp convergence rate of the Glimm scheme for nonlinear hyperbolic systems

Consider the Cauchy problem for a strictly hyperbolic, $N\times N$ quasilinear system in one space dimension u_t+A(u) u_x=0,\qquad u(0,x)=\bar u(x), \eqno (1) where $u \mapsto A(u)$ is a smooth matrix-valued map, and the initial data $\overline u$ is assumed to have small total variation. We investigate the rate of convergence of approximate solutions of (1) constructed by the Glimm scheme, under the assumption that, letting $\lambda_k(u)$, $r_k(u)$ denote the $k$-th eigenvalue and a corresponding eigenvector of $A(u)$, respectively, for each $k$-th characteristic family the linearly degenerate manifold $$\mathcal{M}_k \doteq \big\{u\in\Omega : \nabla\lambda_k(u)\cdot r_k(u)=0\big\}$$ is either the whole space, or it is empty, or it consists of a finite number of smooth, $N-1$-dimensional, connected, manifolds that are transversal to the characteristic vector field $r_k$. We introduce a Glimm type functional which is the sum of the cubic interaction potential defined in \cite{sie}, and of a quadratic term that takes into account interactions of waves of the same family with strength smaller than some fixed threshold parameter. Relying on an adapted wave tracing method, and on the decrease amount of such a functional, we obtain the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems satisfying the classical Lax assumptions of genuine nonlinearity or linear degeneracy of the characteristic families.Comment: To appear on Archive for Rational Mechanics and Analysi

Wave front tracking in systems of conservation laws

summary:This paper contains several recent results about nonlinear systems of hyperbolic conservation laws obtained through the technique of Wave Front Tracking

Error control for statistical solutions

Statistical solutions have recently been introduced as a an alternative solution framework for hyperbolic systems of conservation laws. In this work we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of the residuals and verify their splitting.Comment: 25 pages, 2 figure