{
  "id": "1307.8167",
  "version": "v2",
  "published": "2013-07-30T22:55:55.000Z",
  "updated": "2020-07-22T12:07:59.000Z",
  "title": "A computational methodology for two-dimensional fluid flows",
  "authors": [
    "Jahrul Alam",
    "Raymond Walsh",
    "Alamgir Hossain",
    "Andrew Rose"
  ],
  "comment": "29 pages, 7 figures",
  "journal": "International Journal of Numerical Methods in Fluids, 30 August 2014",
  "doi": "10.1002/fld.3917",
  "categories": [
    "physics.flu-dyn",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "A weighted residual collocation methodology for simulating two-dimensional shear-driven and natural convection flows has been presented. Using a dyadic mesh refinement, the methodology generates a basis and a multiresolution scheme to approximate a fluid flow. To extend the benefits of the dyadic mesh refinement approach to the field of computational fluid dynamics, this article has studied an iterative interpolation scheme for the construction and differentiation of a basis function in a two-dimensional mesh that is a finite collection of rectangular elements. We have verified that, on a given mesh, the discretization error is controlled by the order of the basis function. The potential of this novel technique has been demonstrated with some representative examples of the Poisson equation. We have also verified the technique with a dynamical core of two-dimensional flow in primitive variables. An excellent result has been observed-on resolving a shear layer and on the conservation of the potential and the kinetic energies with respect to previously reported benchmark simulations. In particular, the shear-driven simulation at CFL = 2.5 (Courant-Friedrichs-Lewy) and $\\mathcal Re = 1\\,000$ (Reynolds number) exhibits a linear speedup of CPU time with an increase of the time step, $Delta t$. For the natural convection flow, the conversion of the potential energy to the kinetic energy and the conservation of total energy is resolved by the proposed method. The computed streamlines and the velocity fields have been demonstrated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-30T22:55:55.000Z",
      "title": "Discretization of the Poisson equation using the interpolating scaling function with applications",
      "abstract": "Dyadic translations of the interpolating scaling function generate a basis that can be used to approximate functions and develop a multiresolution methodology for constructing smooth surfaces or curves. Many wavelet methods for solving par- tial differential equations are also derived from the interpolating scaling function. However, little is done for developing a higher order numerical discretization methodology using the scaling function. In this article, we have employed an iterative interpolation scheme for the construction of scaling functions in a two- dimensional mesh that is a finite collection of rectangles. We have studied the development of a weighted residual collocation method for approximating partial derivatives. We show that the discretization error is controlled by the order of the scaling function. The potential of this novel technique has been verified with some representative examples of the Poisson equation. We have extended the technique for solving nonlinear advection-diffusion equations, and simulated a shear driven flow in a square cavity at CFL = 2.5 (Courant Friedrichs Lewy) and Re = 1 000 (Reynolds number). Agreement with the reference solution at a large CFL = 2.5 explores the potential of this development for advection dominated problems.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2020-07-22T12:07:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "interpolating scaling function",
      "poisson equation",
      "applications",
      "higher order numerical discretization methodology",
      "courant friedrichs lewy"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.8167A"
    }
  }
}