src/elevation.h

    Conservation of water surface elevation

    When using the default adaptive reconstruction of variables, the Saint-Venant solver will conserve the water depth when cells are refined or coarsened. However, this will not necessarily ensure that the “lake-at-rest” condition (i.e. a constant water surface elevation) is also preserved. In what follows, we redefine the prolongation() and restriction() methods of the water depth h so that the water surface elevation \eta is conserved.

    We start with the reconstruction of fine “wet” cells:

    #if TREE
    static const double default_sea_level = 0.;
    
    static void refine_elevation (Point point, scalar h)
    {
      // reconstruction of fine cells using elevation (rather than water depth)
      // (default refinement conserves mass but not lake-at-rest)
      if (h[] >= dry) {
        double eta = zb[] + h[];   // water surface elevation  
        coord g; // gradient of eta
        if (gradient)
          foreach_dimension()
    	g.x = gradient (zb[-1] + h[-1], eta, zb[1] + h[1])/4.;
        else
          foreach_dimension()
    	g.x = (zb[1] - zb[-1])/(2.*Delta);
        // reconstruct water depth h from eta and zb
        foreach_child() {
          double etac = eta;
          foreach_dimension()
    	etac += g.x*child.x;
          h[] = max(0., etac - zb[]);
        }
      }
      else {

    The “dry” case is a bit more complicated. We look in a 3x3 neighborhood of the coarse parent cell and compute a depth-weighted average of the “wet” surface elevation \eta. We need to do this because we cannot assume a priori that the surrounding wet cells are necessarily close to e.g. \eta = 0.

        double v = 0., eta = 0.; // water surface elevation
        // 3x3 neighbourhood
        foreach_neighbor(1)
          if (h[] >= dry) {
    	eta += h[]*(zb[] + h[]);
    	v += h[];
          }
        if (v > 0.)
          eta /= v; // volume-averaged eta of neighbouring wet cells
        else

    If none of the surrounding cells is wet, we set a default sealevel.

          eta = default_sea_level;

    We then reconstruct the water depth in each child using \eta (of the parent cell i.e. a first-order interpolation in contrast to the wet case above) and z_b of the child cells.

        // reconstruct water depth h from eta and zb
        foreach_child()
          h[] = max(0., eta - zb[]);
      }
    }

    Cell restriction is simpler. We first compute the depth-weighted average of \eta over all the children…

    static void restriction_elevation (Point point, scalar h)
    {
      double eta = 0., v = 0.;
      foreach_child()
        if (h[] > dry) {
          eta += h[]*(zb[] + h[]);
          v += h[];
        }

    … and use this in combination with z_b (of the coarse cell) to compute the water depth h.

      if (v > 0.)
        h[] = max(0., eta/v - zb[]);
      else // dry cell
        h[] = 0.;
    }

    We also need to define a consistent prolongation function. For cells which are entirely surrounded by wet cells, we can use the standard linear refinement function, otherwise we use straight injection from the parent cell.

    static void prolongation_elevation (Point point, scalar h)
    {
      bool wet = true;
      foreach_neighbor(1)
        if (h[] <= dry)
          wet = false, break;
      if (wet)
        refine_linear (point, h);
      else {
        double hc = h[], zc = zb[];
        foreach_child() {
          h[] = hc;
          zb[] = zc;
        }
      }
    }

    Finally we define a function which will be called by the user to apply these reconstructions.

    void conserve_elevation (void)
    {
      h.refine  = refine_elevation;
      h.prolongation = prolongation_elevation;
      h.restriction = restriction_elevation;
    }
    #else // Cartesian
    void conserve_elevation (void) {}
    #endif

    Usage