src/bcg.h

Bell-Collela-Glaz advection scheme

The function below implements the 2nd-order, unsplit, upwind scheme of Bell-Collela-Glaz, 1989. Given a centered scalar field f, a face vector field uf (possibly weighted by a face metric), a timestep dt and a source term field src, it fills the face vector field flux with the components of the advection fluxes of f.

void tracer_fluxes (scalar f,
		    face vector uf,
		    face vector flux,
		    double dt,
		    (const) scalar src)
{

We first compute the cell-centered gradient of f in a locally-allocated vector field.

  vector g[];
  gradients ({f}, {g});

For each face, the flux is composed of two parts…

  foreach_face() {

A normal component… (Note that we cheat a bit here, un should strictly be dt*(uf.x[i] + uf.x[i+1])/((fm.x[] + fm.x[i+1])*Delta) but this causes trouble with boundary conditions (when using narrow ‘1 ghost cell’ stencils)).

    double un = dt*uf.x[]/(fm.x[]*Δ), s = sign(un);
    int i = -(s + 1.)/2.;
    double f2 = f[i] + (src[] + src[-1])*dt/4. + s*(1. - s*un)*g.x[i]*Δ/2.;

and tangential components…

    #if dimension > 1
      double vn = uf.y[i,0]/fm.y[i,0] + uf.y[i,1]/fm.y[i,1];
      double fyy = vn < 0. ? f[i,1] - f[i,0] : f[i,0] - f[i,-1];
      f2 -= dt*vn*fyy/(4.*Δ);
    #endif
    #if dimension > 2
      double wn = uf.z[i,0,0]/fm.z[i,0,0] + uf.z[i,0,1]/fm.z[i,0,1];
      double fzz = wn < 0. ? f[i,0,1] - f[i,0,0] : f[i,0,0] - f[i,0,-1];
      f2 -= dt*wn*fzz/(4.*Δ);
    #endif

    flux.x[] = f2*uf.x[];
  }

Boundary conditions ensure the consistency of fluxes across variable-resolution boundaries (on adaptive meshes).

  boundary_flux ({flux});
}

The function below uses the tracer_fluxes function to integrate the advection equation, using an explicit scheme with timestep dt, for each tracer in the list.

struct Advection {
  scalar * tracers;
  face vector u;
  double dt;
  scalar * src; // optional
};

void advection (struct Advection p)
{

If src is not provided we set all the source terms to zero.

  scalar * lsrc = p.src;
  if (!lsrc) {
    const scalar zero[] = 0.;
    for (scalar s in p.tracers)
      lsrc = list_append (lsrc, zero);
  }

  assert (list_len(p.tracers) == list_len(lsrc));
  scalar f, src;
  for (f,src in p.tracers,lsrc) {
    face vector flux[];
    tracer_fluxes (f, p.u, flux, p.dt, src);
    foreach()
      foreach_dimension()
        f[] += p.dt*(flux.x[] - flux.x[1])/(Δ*cm[]);
  }
  boundary (p.tracers);

  if (!p.src)
    free (lsrc);
}

Usage

Tests