Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1402.0909

Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws

Ulrik S. Fjordholm·Roger Käppeli·Siddhartha Mishra·Eitan Tadmor
View original source ↗Read source document ↗Export metadata ↓

Abstract

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes \emph{may not} necessarily converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. Furthermore, we propose a more general notion, that of \emph{entropy measure valued solutions}, as an appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, Eitan Tadmor. 2015-06-29. Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws. https://doi.org/10.1007/s10208-015-9299-z

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗