src/spherisym.h

    Spherically-symmetric coordinates

    This file defines the metric coefficients for a (one-dimensional) spherically-symmetric coordinate system.

    The radial coordinate r is x. Note that x (and so X0) cannot be negative.

    We first define a macro which will be used in some geometry-specific code (e.g. viscous stress tensor).

    #define SPHERISYM 1

    On trees we need refinement functions.

    #if TREE
static void refine_cm_spherisym (Point point, scalar cm)
{
  fine(cm,0) = sq (x - Delta/4.);
  fine(cm,1) = sq (x + Delta/4.);
}

static void refine_face_x_spherisym (Point point, scalar fm)
{
  if (!is_refined(neighbor(-1)))
    fine(fm,0) = sq (x - Delta/2.);
  if (!is_refined(neighbor(1)) && neighbor(1).neighbors)
    fine(fm,2) = sq (x + Delta/2.);
  fine(fm,1) = sq(x);
}
#endif // TREE

event metric (i = 0) {

    By default cm is a constant scalar field. To make it variable, we need to allocate a new field. We also move it at the begining of the list of variables: this is important to ensure the metric is defined before other fields.

      if (is_constant(cm)) {
    scalar * l = list_copy (all);
    cm = new scalar;
    free (all);
    all = list_concat ({cm}, l);
    free (l);
  }

    The volume/area of a cell is proportional to r^2 (i.e. x^2). We need to set boundary conditions at the top and bottom so that cm is interpolated properly when refining/coarsening the mesh.

      scalar cmv = cm;
  foreach()
    cmv[] = x*x;
  cm[left] = dirichlet(x*x);
  cm[right] = dirichlet(x*x);

    We do the same for the length scale factors. The “length” of faces on the center of spherical symmetry is zero (x=r=0 in the center). To avoid division by zero we set it to epsilon (note that mathematically the limit is well posed).

      if (is_constant(fm.x)) {
    scalar * l = list_copy (all);
    fm = new face vector;
    free (all);
    all = list_concat ((scalar *){fm}, l);
    free (l);
  }
  face vector fmv = fm;
  foreach_face()
    fmv.x[] = max(x*x, 1e-20);

    We set our refinement/prolongation functions on trees.

    #if TREE
  cm.refine = cm.prolongation = refine_cm_spherisym;
  fm.x.prolongation = refine_face_x_spherisym;
#endif
}

    Usage

    Tests