{
  "id": "1208.0309",
  "version": "v1",
  "published": "2012-07-10T12:39:46.000Z",
  "updated": "2012-07-10T12:39:46.000Z",
  "title": "A finite volume scheme for a Keller-Segel model with additional cross-diffusion",
  "authors": [
    "Marianne Bessemoulin-Chatard",
    "Ansgar Jüngel"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "A finite volume scheme for the (Patlak-) Keller-Segel model in two space dimensions with an additional cross-diffusion term in the elliptic equation for the chemical signal is analyzed. The main feature of the model is that there exists a new entropy functional yielding gradient estimates for the cell density and chemical concentration. The main features of the numerical scheme are positivity preservation, mass conservation, entropy stability, and - under additional assumptions - entropy dissipation. The existence of a discrete solution and its numerical convergence to the continuous solution is proved. Furthermore, temporal decay rates for convergence of the discrete solution to the homogeneous steady state is shown using a new discrete logarithmic Sobolev inequality. Numerical examples point out that the solutions exhibit intermediate states and that there exist nonhomogeneous stationary solutions with a finite cell density peak at the domain boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-10T12:39:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M08",
      "65M12",
      "92C17"
    ],
    "keywords": [
      "finite volume scheme",
      "keller-segel model",
      "additional cross-diffusion",
      "entropy functional yielding gradient estimates",
      "discrete logarithmic sobolev inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.0309B"
    }
  }
}