{
  "id": "1609.08029",
  "version": "v1",
  "published": "2016-09-26T15:44:41.000Z",
  "updated": "2016-09-26T15:44:41.000Z",
  "title": "Shallow water equations: Split-form, entropy stable, well-balanced, and positivity preserving numerical methods",
  "authors": [
    "Hendrik Ranocha"
  ],
  "comment": "44 pages, 10 figures, submitted",
  "categories": [
    "math.NA"
  ],
  "abstract": "Entropy stable semidiscretisations of the shallow water equations are developed, based on summation-by-parts (SBP) operators and using split forms of the equations. The resulting two-parameter family of entropy conservative schemes for general SBP bases, especially using Gau{\\ss} nodes, is adapted to varying bottom topography in a well-balanced way, i.e. preserving the lake-at-rest steady state. Moreover, positivity preservation is ensured using the framework of Zhang and Shu (Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and recent developments, 2011. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, vol 467, pp. 2752--2766) and finite volume subcells, adapted to nodal SBP bases with diagonal mass matrix. Numerical tests of the proposed schemes are performed and some conclusions are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-26T15:44:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M70",
      "65M60",
      "65M06",
      "65M12"
    ],
    "keywords": [
      "shallow water equations",
      "positivity preserving numerical methods",
      "entropy stable",
      "royal society",
      "sbp bases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}