{
  "id": "2304.07809",
  "version": "v1",
  "published": "2023-04-16T15:24:47.000Z",
  "updated": "2023-04-16T15:24:47.000Z",
  "title": "A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations: A Combination of Conservative and Primitive Formulations",
  "authors": [
    "Remi Abgrall",
    "Yongle Liu"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this paper, we introduce a new approach for constructing robust well-balanced numerical methods for the one-dimensional Saint-Venant system with and without the Manning friction term. Following the idea presented in [R. Abgrall, Commun. Appl. Math. Comput. 5(2023), pp. 370-402], we first combine the conservative and non-conservative (primitive) formulations of the studied conservative hyperbolic system in a natural way. The solution is globally continuous and described by a combination of point values and average values. The point values and average values will then be evolved by two different forms of PDEs: a conservative version of the cell averages and a possibly non-conservative one for the points. We show how to deal with both the conservative and non-conservative forms of PDEs in a well-balanced manner. The developed schemes are capable of exactly preserving both the still-water and moving-water equilibria. Compared with existing well-balanced methods, this new class of scheme is nonlinear-equations-solver-free. This makes the developed schemes less computationally costly and easier to extend to other models. We demonstrate the behavior of the proposed new scheme on several challenging examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-16T15:24:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "shallow water equations",
      "designing well-balanced schemes",
      "primitive formulations",
      "robust well-balanced numerical methods",
      "conservative"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}