/**
# Bouncing Saint-Venant bump

This test case is identical to [bump2D.c]() but using the generic
solver for systems of conservation laws rather than the Saint-Venant
solver. */

#include "conservation.h"

/**
The only conserved scalar is the water depth `h` and the only
conserved vector is the flow rate `q`. */

scalar h[];
vector q[];
scalar * scalars = {h};
vector * vectors = {q};

/**
Using these notations, the Saint-Venant system of conservation laws
(assuming a flat topography) can be written
$$
\partial_t\left(\begin{array}{c}
    h\\
    q_x\\
    q_y\\
 \end{array}\right) + \nabla\cdot\left(\begin{array}{c}
    q_x & q_y\\
    \frac{q_x^2}{h} + \frac{Gh^2}{2} & \frac{q_xq_y}{h}\\
    \frac{q_yq_x}{h} & \frac{q_y^2}{h} + \frac{Gh^2}{2}
 \end{array}\right) = 0
$$
with $G$ the acceleration of gravity.

This system is entirely defined by the `flux()` function called by the
generic solver for conservation laws. The parameter passed to the
function is the array `s` which contains the state variables for each
conserved field, in the order of their definition above (i.e. scalars
then vectors). In the function below, we first recover each value
(`h`, `qx` and `qy`) and then compute the corresponding fluxes
(`f[0]`, `f[1]` and `f[2]`). The minimum and maximum eigenvalues for
the Saint-Venant system are the characteristic speeds $u \pm
\sqrt(Gh)$. */

double G = 1.;

void flux (const double * s, double * f, double e[2])
{
  double h = s[0], qx = s[1], u = qx/h, qy = s[2];
  f[0] = qx;
  f[1] = qx*u + G*h*h/2.;
  f[2] = qy*u;
  // min/max eigenvalues
  double c = sqrt(G*h);
  e[0] = u - c; // min
  e[1] = u + c; // max
}

/**
The solver is now fully defined and we proceed with initial conditions
etc... as when using the standard Saint-Venant solver (see
[bump2D.c]() for details). */

#define LEVEL 7

int main()
{
  origin (-0.5, -0.5);
  init_grid (1 << LEVEL);
  run();
}

event init (i = 0)
{
  theta = 1.3; // tune limiting from the default minmod
  double b = 200.;
  foreach()
    h[] = 0.1 + exp(- b*(x*x + y*y));
}

event logfile (i++) {
  stats s = statsf (h);
  fprintf (stderr, "%g %d %g %g %.8f\n", t, i, s.min, s.max, s.sum);
}

event outputfile (t <= 2.5; t += 2.5/8) {
  static int nf = 0;
  printf ("file: eta-%d\n", nf);
  output_field ({h}, stdout, N, linear = true);

  scalar l[];
  foreach()
    l[] = level;
  printf ("file: level-%d\n", nf++);
  output_field ({l}, stdout, N);

  /* check symmetry */
  foreach() {
    double h0 = h[];
    point = locate (-x, -y);
    //    printf ("%g %g %g %g %g\n", x, y, h0, h[], h0 - h[]);
    assert (fabs(h0 - h[]) < 1e-12);
    point = locate (-x, y);
    assert (fabs(h0 - h[]) < 1e-12);
    point = locate (x, -y);
    assert (fabs(h0 - h[]) < 1e-12);
  }
}

#if TREE
event adapt (i++) {
  astats s = adapt_wavelet ({h}, (double[]){1e-3}, LEVEL);
  fprintf (stderr, "# refined %d cells, coarsened %d cells\n", s.nf, s.nc);
}
#endif

/**
## Results

The results are comparable to that of [bump2D.c](). They are not
identical mainly because the standard Saint-Venant solver applies
slope-limiting to velocity rather than flow rate in the present case.

![Evolution of water depth with time.](bump2D1/plot.png)

![Evolution of level of refinement with time.](bump2D1/level.png)

*/