{
  "id": "1602.06444",
  "version": "v1",
  "published": "2016-02-20T19:52:54.000Z",
  "updated": "2016-02-20T19:52:54.000Z",
  "title": "Superconvergence properties of an upwind-biased discontinuous Galerkin method",
  "authors": [
    "Daniel Frean",
    "Jennifer Ryan"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "In this paper we investigate the superconvergence properties of the discontinuous Galerkin method based on the upwind-biased flux for linear time-dependent hyperbolic equations. We prove that for even-degree polynomials, the method is locally $\\mathcal{O}(h^{k+2})$ superconvergent at roots of a linear combination of the left- and right-Radau polynomials. This linear combination depends on the value of $\\theta$ used in the flux. For odd-degree polynomials, the scheme is superconvergent provided that a proper global initial interpolation can be defined. We demonstrate numerically that, for decreasing $\\theta$, the discretization errors decrease for even polynomials and grow for odd polynomials. We prove that the use of Smoothness-Increasing Accuracy-Conserving (SIAC) filters is still able to draw out the superconvergence information and create a globally smooth and superconvergent solution of $\\mathcal{O}(h^{2k+1})$ for linear hyperbolic equations. Lastly, we briefly consider the spectrum of the upwind-biased DG operator and demonstrate that the price paid for the introduction of the parameter $\\theta$ is limited to a contribution to the constant attached to the post-processed error term.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-20T19:52:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upwind-biased discontinuous galerkin method",
      "superconvergence properties",
      "polynomials",
      "linear time-dependent hyperbolic equations",
      "linear combination"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}