{
  "id": "1802.00708",
  "version": "v1",
  "published": "2018-02-02T14:54:22.000Z",
  "updated": "2018-02-02T14:54:22.000Z",
  "title": "Numerical methods for conservation laws with rough flux",
  "authors": [
    "Håkon Hoel",
    "Kenneth Hvistendahl Karlsen",
    "Nils Henrik Risebro",
    "Erlend Briseid Storrøsten"
  ],
  "comment": "58 pages, 12 figures",
  "categories": [
    "math.NA"
  ],
  "abstract": "Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with a rough path dependent flux function. For a convex flux, it is demonstrated that rough path oscillations may lead to \"cancellations\" in the solution. Making use of this property, we show that for $\\alpha$-H{\\\"o}lder continuous rough paths the convergence rate of the numerical methods can improve from $\\mathcal{O}(\\text{COST}^{-\\gamma})$, for some $\\gamma \\in \\left[\\alpha/(12-8\\alpha), \\alpha/(10-6\\alpha)\\right]$, with $\\alpha\\in (0, 1)$, to $\\mathcal{O}(\\text{COST}^{-\\min(1/4,\\alpha/2)})$. Numerical examples support the theoretical results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-02T14:54:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L65",
      "65M06",
      "60H15",
      "65C30"
    ],
    "keywords": [
      "conservation laws",
      "numerical methods",
      "rough flux",
      "approximate pathwise entropy/kinetic solutions",
      "rough path dependent flux function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}