{
  "id": "2206.06495",
  "version": "v1",
  "published": "2022-06-13T21:57:47.000Z",
  "updated": "2022-06-13T21:57:47.000Z",
  "title": "RBF-FD discretization of the Navier-Stokes equations using staggered nodes",
  "authors": [
    "Tianyi Chu",
    "Oliver T. Schmidt"
  ],
  "categories": [
    "physics.flu-dyn",
    "physics.comp-ph"
  ],
  "abstract": "A semi-implicit fractional-step method that uses a staggered node layout and radial basis function-finite differences (RBF-FD) to solve the incompressible Navier-Stokes equations is developed. Polyharmonic splines (PHS) with polynomial augmentation (PHS+poly) are used to construct the global differentiation matrices. A systematic parameter study identifies a combination of stencil size, PHS exponent, and polynomial degree that minimizes the truncation error for a wave-like test function on scattered nodes. Classical modified wavenumber analysis is extended to RBF-FDs on heterogeneous node distributions and used to confirm that the accuracy of the selected 28-point stencil is comparable to that of spectral-like, 6th-order Pad\\'e-type finite differences. The Navier-Stokes solver is demonstrated on two benchmark problems, internal flow in a lid-driven cavity in the Reynolds number regime $10^2 \\leq \\mathrm{Re}\\leq 10^4$, and open flow around a cylinder at $\\mathrm{Re}= 100$ and $200$. The combination of grid staggering and careful parameter selection facilitates accurate and stable simulations at significantly lower resolutions than previously reported, using more compact RBF-FD stencils, without special treatment near solid walls, and without the need for hyperviscosity or other means of regularization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-13T21:57:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "navier-stokes equations",
      "staggered node",
      "rbf-fd discretization",
      "radial basis function-finite differences",
      "parameter selection facilitates accurate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}