{
  "id": "1402.0909",
  "version": "v2",
  "published": "2014-02-04T23:17:54.000Z",
  "updated": "2015-06-29T15:19:36.000Z",
  "title": "Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws",
  "authors": [
    "Ulrik S. Fjordholm",
    "Roger Käppeli",
    "Siddhartha Mishra",
    "Eitan Tadmor"
  ],
  "categories": [
    "math.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-04T23:17:54.000Z",
      "abstract": "Numerical evidence is presented to demonstrate that state of the art numerical schemes need \\emph{not} converge to entropy solutions of systems of hyperbolic conservation laws in several space dimensions. Combined with recent results on the lack of stability of these solutions, we advocate the more general notion of \\emph{entropy measure valued solutions} as the appropriate paradigm for solutions of such multi-dimensional systems. We propose a detailed numerical procedure which constructs approximate entropy measure valued solutions, and we prove sufficient criteria that ensure their (narrow) convergence, thus providing a viable numerical framework for the approximation of entropy measure valued solutions. Examples of schemes satisfying these criteria are presented. A number of numerical experiments, illustrating our proposed procedure and examining interesting properties of the entropy measure valued solutions, are also provided.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-29T15:19:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M06",
      "35L65",
      "35R06"
    ],
    "keywords": [
      "approximate entropy measure valued solutions",
      "hyperbolic systems",
      "constructs approximate entropy measure",
      "construction",
      "hyperbolic conservation laws"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.0909F"
    }
  }
}