sandbox/bugs/two_layer_minimum.h

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/**
# A solver for the Two-layer equations

The [Saint-Venant equations](http://en.wikipedia.org/wiki/Shallow_water_equations)
can be written in integral form as the hyperbolic system of
conservation laws 
$$
\partial_t \int_{\Omega} \mathbf{q} d \Omega +
\int_{\partial \Omega} \mathbf{f} (
\mathbf{q}) \cdot \mathbf{n}d \partial
\Omega + \int_{\Omega} \mathbf{F} d \Omega= 0
$$
where $\Omega$ is a given subset of space, $\partial \Omega$ its boundary and
$\mathbf{n}$ the unit normal vector on this boundary. For
conservation of mass and momentum in the shallow-water context, $\Omega$ is a
subset of bidimensional space and $\mathbf{q}$ and
$\mathbf{f}$ are written
$$
\mathbf{q} = \left(\begin{array}{c}
h\						\
h u_x\					\
h u_y
h_s\						\
h_s u_s_x\					\
h_s u_s_y
\end{array}\right), 
\;\;\;\;\;\;
\mathbf{f} (\mathbf{q}) = \left(\begin{array}{cc}
h   u_x & h u_y\					\
h   u_x^2 + \frac{1}{2} g h^2 & h u_x u_y\	\
h   u_x u_y & h u_y^2 + \frac{1}{2} g h^2\	\
h_s u_s_x & h_s u_s_y\					\
h_s u_s_x^2 + \frac{1}{2} gh_s^2 & h_s u_s_x u_s_y\	\
h_s u_s_x u_sy & h_s u_s_y^2 + \frac{1}{2} g h_s^2
\end{array}\right),
\;\;\;\;\;\;
\mathbf{F} = \left(\begin{array}{c}
0\								\
g h \frac{\partial}{\partial x} \left( h_s + z_b \right)\	\
g h \frac{\partial}{\partial y} \left( h_s + z_b \right)\	\
0\									\
g h_s \frac{\partial}{\partial x} \left( z_b + \frac{\rho}{\rho_s} h \right)\ \
g h_s \frac{\partial}{\partial y} \left( z_b + \frac{\rho}{\rho_s} h \right)
\end{array}\right),
$$
where $h$ the water depth, $h_s$ is the landslide thickness, $\mathbf{u}$ is the velocity vector of the water, $\mathbf{u_s}$ is the velocity vector of the landslide, and
$z_b$ the height of the topography. See also [Popinet, 
2011](/src/references.bib#popinet2011) for a more detailed
introduction.

## User variables and parameters

The primary fields are the water depth $h$, the landslide thickness $h_s$, the bathymetry $z_b$ and
the flow speeds of water and landslide respectively $\mathbf{u}$ and $\mathbf{u_s}$. $\eta$ is the water level i.e. $z_b +
h + h_s$. Note that the order of the declarations is important as $z_b$
needs to be refined before $h_s$, before $h$, before $\eta$. */

scalar zb[], hs[], h[], eta[];
vector us[], u[];

/**
The only physical parameter is the acceleration of gravity `G` and the densities of the two fluids. 
Cells are considered "dry" when the water depth is less than the `dry` parameter (this 
should not require tweaking). */

double G = 1.;
double dry = 1e-10;
double RHOratio = 0.5;


/**
## Time-integration

### Setup

Time integration will be done with a generic
[predictor-corrector](predictor-corrector.h) scheme. */

#include "predictor-corrector.h"

/**
The generic time-integration scheme in predictor-corrector.h needs
to know which fields are updated. */

scalar * evolving = NULL;

/**
We need to overload the default *advance* function of the
predictor-corrector scheme, because the evolving variables ($h$, $\mathbf{u}, $h_s$ and
$\mathbf{u_s}$) are not the conserved variables $h$, $h\mathbf{u}, $h_s$ and
$h_s\mathbf{u_s}$. */

static void advance_two_layer (scalar * output, scalar * input, 
			       scalar * updates, double dt)
{
  //  fprintf(stderr,"advance_two_layer\n");
  // recover scalar and vector fields from lists
  scalar hi = input[0], ho = output[0], dh = updates[0];
  scalar hsi = input[1], hso = output[1], dhs = updates[1];
  vector uo = vector(output[2]), uso = vector(output[2+dimension]);

  /*
  Order within the scalar lists e.g. input, output, updates, evolving, [h, hs, u, us]
  Note that this differs from what I previously did...  But this should all occur within this so as long as it is self consistent it should be fine
  */

  // new fields in ho[], uo[]
  foreach() {
    double hold = hi[];
    double hsold = hsi[];
    ho[] = hold + dt*dh[];
    hso[] = hsold + dt*dhs[];
    eta[] = ho[] + hso[] + zb[];
    if (hso[] > dry){
      //      fprintf(stderr,"HS wet");
	  vector usi = vector(input[2+dimension]);
	  vector dhus = vector(updates[2+dimension]);
      foreach_dimension()
	uso.x[] = (hsold*usi.x[] + dt*dhus.x[])/hso[];}
    else{
      //      fprintf(stderr,"HS dry");
      foreach_dimension()
	uso.x[] = 0.;}
    if (ho[] > dry){
      //      fprintf(stderr," H wet\n");
	  vector ui = vector(input[2]);
	  vector dhu = vector(updates[2]);
      foreach_dimension()
	uo.x[] = (hold*ui.x[] + dt*dhu.x[])/ho[];}
    else{
      //      fprintf(stderr," H dry\n");
      foreach_dimension()
	uo.x[] = 0.;}
  }
  //  fprintf(stderr,"boundary\n");
  //ho.prolongation = refine_linear;
  boundary ({hso, ho, eta, uso, uo});
  //  fprintf(stderr,"done\n");
}

/**
When using an adaptive discretisation (i.e. a quadtree)., we need
to make sure that $\eta$ is maintained as $z_b + h + h_s$ whenever cells are
refined or coarsened. */

/**
#if TREE
static void refine_eta (Point point, scalar eta)
{
  foreach_child()
    eta[] = zb[] + h[] + hs[];
}

static void coarsen_eta (Point point, scalar eta)
{
  eta[] = zb[] + h[] + hs[];
}
#endif
*/

/**
### Computing fluxes

Various approximate Riemann solvers are defined in [riemann.h](). */

#include "riemann.h"

double update_two_layer (scalar * evolving, scalar * updates, double dtmax)
{
  
/**
We first recover the currently evolving fields (as set by the
predictor-corrector scheme). 
NOTE change in order as mentioned above*/

  scalar h = evolving[0];
  scalar hs = evolving[1];
  vector u = vector(evolving[2]);
  vector us = vector(evolving[2+dimension]);
   
/**
`Fh`, `Fhs` `Fq` and `Fqs` will contain the fluxes for $h$, $h_s$, $h\mathbf{u}$ and $h_s\mathbf{u_s}$
respectively and `S1` and `S2` are necessary to store the asymmetric topographic
source terms. */

  face vector Fh[], Fhs[], S1[], S2[];
  tensor Fq[], Fqs[];
   
/**
The gradients are stored in locally-allocated fields. First-order
reconstruction is used for the gradient fields. */

  vector gh[], ghs[], geta[];
  tensor gu[], gus[];
  for (scalar s in {gh, ghs, geta, gu, gus}){
    s.gradient = zero;
#if TREE   //Is this where the problem is?  Do I need to refine some other way?
      s.prolongation = refine_linear;
    #endif
  }
  gradients ({h, hs, eta, u, us}, {gh, ghs, geta, gu, gus});
  //fprintf(stdout,"%% updates\n");
/**
The faces which are "wet" on at least one side are traversed. 
First we see whether "wet" in bottom fluid - if so look for lake at rest solution
$h_s+z_b=C_0$ $h=C$
If hs is dry look for lake at rest solution
$h_s=0$ $h+z_b=C_0$
*/
  int hswet;
  foreach_face (reduction (min:dtmax)) {
    // First the bottom layer
    double hi = hs[], hn = hs[-1,0];
    if (hi > dry || hn > dry) {
      //      fprintf(stderr,"HS wet");
	   hswet=1;
      
/**
#### Left/right state reconstruction

The gradients computed above are used to reconstruct the left and
right states of the primary fields $h$, $\mathbf{u}$, $z_b$. The
"interface" topography $z_{lr}$ is reconstructed using the hydrostatic
reconstruction of [Audusse et al, 2004](/src/references.bib#audusse2004) */
    
      double dx = Delta/2.;
      double zi = eta[] - hi - h[];
      double zl = zi - dx*(geta.x[] - ghs.x[]- gh.x[]);
      double zn = eta[-1,0] - hn - h[-1,0];
      double zr = zn + dx*(geta.x[-1,0] - ghs.x[-1,0] - gh.x[-1,0]);
      double zlr = max(zl, zr);
      
      double hl = hi - dx*ghs.x[];
      double up = us.x[] - dx*gus.x.x[];
      double hp = max(0., hl + zl - zlr);
      
      double hr = hn + dx*ghs.x[-1,0];
      double um = us.x[-1,0] + dx*gus.x.x[-1,0];
      double hm = max(0., hr + zr - zlr);
      
/**
#### Riemann solver

We can now call one of the approximate Riemann solvers to get the fluxes. */

      double fh, fu, fv;
      kurganov (hm, hp, um, up, Delta*cm[]/fm.x[], &fh, &fu, &dtmax);
      fv = (fh > 0. ? us.y[-1,0] + dx*gus.y.x[-1,0] : us.y[] - dx*gus.y.x[])*fh;
      
/**
#### Topographic source term

In the case of adaptive refinement, care must be taken to ensure
well-balancing at coarse/fine faces (see [notes/balanced.tm]()). */

#if TREE
      if (is_prolongation(cell)) {
	hi = coarse(hs,0,0,0);
	zi = coarse(zb,0,0,0);
      }
      if (is_prolongation(neighbor(-1,0))) {
	hn = coarse(hs,-1,0,0);
	zn = coarse(zb,-1,0,0);
      }
#endif
      double sl = G/2.*(sq(hp) - sq(hl) + (hl + hi)*(zi - zl+dx*RHOratio*gh.x[]));
      double sr = G/2.*(sq(hm) - sq(hr) + (hr + hn)*(zn - zr-dx*RHOratio*gh.x[-1,0]));
      
/**
#### Flux update */

      Fhs.x[]   = fm.x[]*fh;
      Fqs.x.x[] = fm.x[]*(fu - sl);
      S2.x[]    = fm.x[]*(fu - sr);
      Fqs.y.x[] = fm.x[]*fv;
      //fprintf(stdout,"2 %g %g %g %g %g\n",x,y,fh,fu,fv);

    }   
    else {//h_s is dry - Note that h_s is not necessarily dry in the neighbouring cell...
      Fhs.x[] = Fqs.x.x[] = S2.x[] = Fqs.y.x[] = 0.;
	   hswet=0;
      //      fprintf(stderr,"HS dry");
	  }
/**
Now we must calculate fluxes for $h$.
If $h_s>dry$ then we must satisfy $h_s+z_b=C_0$ $h=C$;
If $h_s=0$ then we must satisfy $h+z_b=C_0$
*/
      hi = h[], hn = h[-1,0];
      if (hi > dry || hn > dry) {
	//	fprintf(stderr," H wet\n");
/**
#### Left/right state reconstruction

The gradients computed above are used to reconstruct the left and
right states of the primary fields $h$, $\mathbf{u}$, $z_b$. The
"interface" topography $z_{lr}$ is reconstructed using the hydrostatic
reconstruction of [Audusse et al, 2004](/src/references.bib#audusse2004) */
    
	double dx = Delta/2.;
	double zi = eta[] - hi;
	double zl = zi - dx*(geta.x[] - gh.x[]);
	double zn = eta[-1,0] - hn;
	double zr = zn + dx*(geta.x[-1,0] - gh.x[-1,0]);
	double zlr = max(zl, zr);
	
	double hl = hi - dx*gh.x[];
	double up = u.x[] - dx*gu.x.x[];
	double hp = max(0., hl + zl - zlr);
	
	double hr = hn + dx*gh.x[-1,0];
	double um = u.x[-1,0] + dx*gu.x.x[-1,0];
	double hm = max(0., hr + zr - zlr);
	
/**
#### Riemann solver

We can now call one of the approximate Riemann solvers to get the fluxes. */

	double fh, fu, fv;
	kurganov (hm, hp, um, up, Delta*cm[]/fm.x[], &fh, &fu, &dtmax);
	fv = (fh > 0. ? u.y[-1,0] + dx*gu.y.x[-1,0] : u.y[] - dx*gu.y.x[])*fh;
	
/**
#### Topographic source term

In the case of adaptive refinement, care must be taken to ensure
well-balancing at coarse/fine faces (see [notes/balanced.tm]()). */

#if TREE
	if (is_prolongation(cell)) {
	  hi = coarse(h,0,0,0);
	  zi = coarse(zb,0,0,0) + hswet?coarse(hs,0,0,0):0.;
	}
	if (is_prolongation(neighbor(-1,0))) {
	  hn = coarse(h,-1,0,0);
	  zn = coarse(zb,-1,0,0) + hswet?coarse(hs,-1,0,0):0.;
	}
#endif
	double sl = G/2.*(sq(hp) - sq(hl) + (hl + hi)*(zi - zl));
	double sr = G/2.*(sq(hm) - sq(hr) + (hr + hn)*(zn - zr));
	
/**
#### Flux update */

	Fh.x[]   = fm.x[]*fh;
	Fq.x.x[] = fm.x[]*(fu - sl);
	S1.x[]    = fm.x[]*(fu - sr);
	Fq.y.x[] = fm.x[]*fv;
	//fprintf(stdout,"1 %g %g %g %g %g\n",x,y,fh,fu,fv);
      }
      else {// dry
	Fh.x[] = Fq.x.x[] = S1.x[] = Fq.y.x[] = 0.;
	//	fprintf(stderr," H dry\n");
      }
    }
  boundary_flux ({Fh, Fhs, S1, S2, Fq, Fqs});

/**
#### Updates for evolving quantities

We store the divergence of the fluxes in the update fields. Note that
these are updates for $h$ and $h\mathbf{u}$ (not $\mathbf{u}$). */

  scalar dh = updates[0], dhs = updates[1];
  vector dhu = vector( updates[2]), dhus = vector(updates[2+dimension]);
  
  foreach() {
    dh[] = (Fh.x[] + Fh.y[] - Fh.x[1,0] - Fh.y[0,1])/(cm[]*Delta);
    dhs[] = (Fhs.x[] + Fhs.y[] - Fhs.x[1,0] - Fhs.y[0,1])/(cm[]*Delta);
    foreach_dimension(){
      dhu.x[] = (Fq.x.x[] + Fq.x.y[] - S1.x[1,0] - Fq.x.y[0,1])/(cm[]*Delta);
      dhus.x[] = (Fqs.x.x[] + Fqs.x.y[] - S2.x[1,0] - Fqs.x.y[0,1])/(cm[]*Delta);
    }

    /**
    We also need to add the metric terms. They can be written (see
    eq. (8) of [Popinet, 2011](references.bib#popinet2011)) 
    $$
    S_g = h \left(\begin{array}{c}
    0\								\
    \frac{g}{2} h \partial_{\lambda} m_{\theta} + f_G u_y\	\
    \frac{g}{2} h \partial_{\theta} m_{\lambda} - f_G u_x
    \end{array}\right)
    $$
    with
    $$
    f_G = u_y \partial_{\lambda} m_{\theta} - u_x \partial_{\theta} m_{\lambda}
    $$

    */

    double dmdl = (fm.x[1,0] - fm.x[])/(cm[]*Delta);
    double dmdt = (fm.y[0,1] - fm.y[])/(cm[]*Delta);
    double fG1 = u.y[]*dmdl - u.x[]*dmdt;
    double fG2 = us.y[]*dmdl - us.x[]*dmdt;
    dhu.x[] += h[]*(G*h[]/2.*dmdl + fG1*u.y[]);
    dhu.y[] += h[]*(G*h[]/2.*dmdt - fG1*u.x[]);
    dhus.x[] += hs[]*(G*hs[]/2.*dmdl + fG2*us.y[]);
    dhus.y[] += hs[]*(G*hs[]/2.*dmdt - fG2*us.x[]);

  }

  return dtmax;
}



/**
We use the main time loop (in the predictor-corrector scheme) to setup
the initial defaults. */

event defaults (i = 0){
  evolving =(scalar *) {h, hs, u, us};
  foreach()
    for (scalar s in evolving)
      s[]=0.;
  boundary(evolving);
  for (int ii=0; ii<6; ii++) {
    printf (" evolving[ %d ]: %s ", ii, _attribute[evolving[ii].i].name); 
  }
  printf ("\n");
/**
We overload the default 'advance' and 'update' function of the predictor-corrector
scheme and setup the refinement and coarsening methods on quadtrees. */

  advance = advance_two_layer; 
  update = update_two_layer;  
#if QUADTREE
  for (scalar s in {h,zb,us,u,eta}) {
    s.refine = s.prolongation = refine_linear;
  }
#endif
}

/**
The event below will happen after all the other initial events to take
into account user-defined field initialisations. */

event init0 (i = 0)
{
  printf("In init0: ");
  for (int ii=0; ii<6; ii++) {
    printf (" evolving[ %d ]: %s ", ii, _attribute[evolving[ii].i].name); 
  }
  foreach()
    eta[] = zb[] + h[] + hs[];
  boundary (all);
}