Aslam Extrapolations

    We try to use the constant and linear Aslam extrapolations (Aslam 2003), defined in aslam.h. The problem is characterized by a squared domain, with dimensions (-\pi,\pi)\times(-\pi,\pi), and by a level set function \phi = \sqrt{x^2 + y^2} - 2. The field u exist just in a region of the domain, and must be extrapolated:

    \displaystyle u = \begin{cases} 0 & \text{if } \phi > 0,\\ \cos(x)\sin(x) & \text{if } \phi \leq 0. \end{cases}

    #include "utils.h"
    #include "aslam.h"
    #include "view.h"

    We define a function that writes a picture with the map and isolines of the extended scalar fields.

    void write_picture (char* name, scalar u) {
      vertex scalar phi[];
        phi[] = (u[] + u[-1] + u[0,-1] + u[-1,-1])/4.;
      isoline ("levelset", val = 0., lw = 2.);
      isoline ("phi", n = 20, min = -1.5, max = 1.5);
      squares ("phi", spread = -1);
      save (name);

    We declare the level set field levelset, and the field to extrapolate u.

    scalar levelset[], u[];
    int main (void) {

    We set the domain geometry and we initialize the grid.

      size (2.*pi);
      origin (-pi,-pi);
      init_grid (1 << 8);
      double R0 = 2.;

    We initialize the level set function.

        levelset[] = sqrt (sq (x) + sq (y)) - R0;

    We initialize the function u to be extrapolated and we call the constant_extrapolation() function. The extrapolations are perfomed using a \Delta t of 0.01 (it must be small enough to guarantee the stability of the explicit in time discretization), and using a total number of time steps equal to 300, in order to obtain a steady-state solution.

        u[] = (levelset[] <= 0.) ? cos(x)*sin(y) : 0.;
      write_picture ("initial.png", u);
      constant_extrapolation (u, levelset, 0.5, 300);
      write_picture ("constant.png", u);
      fprintf (stderr, "constant = %g\n", statsf(u).sum);

    We re-initialize the function u and we apply the linear_extrapolation().

        u[] = (levelset[] <= 0.) ? cos(x)*sin(y) : 0.;
      linear_extrapolation (u, levelset, 0.5, 300);
      write_picture ("linear.png", u);
      fprintf (stderr, "linear = %g\n", statsf(u).sum);


    Initial field

    Initial field

    Constant extrapolation

    Constant extrapolation

    Linear extrapolation

    Linear extrapolation



    Tariq D Aslam. A partial differential equation approach to multidimensional extrapolation. Journal of Computational Physics, 193(1):349–355, 2004.