source: trunk/anuga_core/source/anuga/shallow_water/shallow_water_ext.c @ 8219

// Python - C extension module for shallow_water.py
//
// To compile (Python2.3):
//  gcc -c domain_ext.c -I/usr/include/python2.3 -o domain_ext.o -Wall -O
//  gcc -shared domain_ext.o  -o domain_ext.so
//
// or use python compile.py
//
// See the module shallow_water_domain.py for more documentation on 
// how to use this module
//
//
// Ole Nielsen, GA 2004


#include "Python.h"
#include "numpy/arrayobject.h"
#include "math.h"
#include <stdio.h>
#include "numpy_shim.h"

// Shared code snippets
#include "util_ext.h"





const double pi = 3.14159265358979;

// Computational function for rotation
// Tried to inline, but no speedup was achieved 27th May 2009 (Ole)
// static inline int _rotate(double *q, double n1, double n2) {
int _rotate(double *q, double n1, double n2) {
  /*Rotate the momentum component q (q[1], q[2])
    from x,y coordinates to coordinates based on normal vector (n1, n2).

    Result is returned in array 3x1 r
    To rotate in opposite direction, call rotate with (q, n1, -n2)

    Contents of q are changed by this function */


  double q1, q2;

  // Shorthands
  q1 = q[1];  // uh momentum
  q2 = q[2];  // vh momentum

  // Rotate
  q[1] =  n1*q1 + n2*q2;
  q[2] = -n2*q1 + n1*q2;

  return 0;
}

int find_qmin_and_qmax(double dq0, double dq1, double dq2, 
               double *qmin, double *qmax){
  // Considering the centroid of an FV triangle and the vertices of its 
  // auxiliary triangle, find 
  // qmin=min(q)-qc and qmax=max(q)-qc, 
  // where min(q) and max(q) are respectively min and max over the
  // four values (at the centroid of the FV triangle and the auxiliary 
  // triangle vertices),
  // and qc is the centroid
  // dq0=q(vertex0)-q(centroid of FV triangle)
  // dq1=q(vertex1)-q(vertex0)
  // dq2=q(vertex2)-q(vertex0)
  
  if (dq0>=0.0){
    if (dq1>=dq2){
      if (dq1>=0.0)
    *qmax=dq0+dq1;
      else
    *qmax=dq0;
      
      *qmin=dq0+dq2;
      if (*qmin>=0.0) *qmin = 0.0;
    }
    else{// dq1<dq2
      if (dq2>0)
    *qmax=dq0+dq2;
      else
    *qmax=dq0;
      
      *qmin=dq0+dq1;    
      if (*qmin>=0.0) *qmin=0.0;
    }
  }
  else{//dq0<0
    if (dq1<=dq2){
      if (dq1<0.0)
    *qmin=dq0+dq1;
      else
    *qmin=dq0;
      
      *qmax=dq0+dq2;    
      if (*qmax<=0.0) *qmax=0.0;
    }
    else{// dq1>dq2
      if (dq2<0.0)
    *qmin=dq0+dq2;
      else
    *qmin=dq0;
      
      *qmax=dq0+dq1;
      if (*qmax<=0.0) *qmax=0.0;
    }
  }
  return 0;
}

int limit_gradient(double *dqv, double qmin, double qmax, double beta_w){
  // Given provisional jumps dqv from the FV triangle centroid to its 
  // vertices and jumps qmin (qmax) between the centroid of the FV 
  // triangle and the minimum (maximum) of the values at the centroid of 
  // the FV triangle and the auxiliary triangle vertices,
  // calculate a multiplicative factor phi by which the provisional 
  // vertex jumps are to be limited
  
  int i;
  double r=1000.0, r0=1.0, phi=1.0;
  static double TINY = 1.0e-100; // to avoid machine accuracy problems.
  // FIXME: Perhaps use the epsilon used elsewhere.
  
  // Any provisional jump with magnitude < TINY does not contribute to 
  // the limiting process.
  for (i=0;i<3;i++){
    if (dqv[i]<-TINY)
      r0=qmin/dqv[i];
      
    if (dqv[i]>TINY)
      r0=qmax/dqv[i];
      
    r=min(r0,r);
  }
  
  phi=min(r*beta_w,1.0);
  //for (i=0;i<3;i++)
  dqv[0]=dqv[0]*phi;
  dqv[1]=dqv[1]*phi;
  dqv[2]=dqv[2]*phi;
    
  return 0;
}


double compute_froude_number(double uh,
                 double h, 
                 double g,
                 double epsilon) {
              
  // Compute Froude number; v/sqrt(gh)
  // FIXME (Ole): Not currently in use
    
  double froude_number;
  
  //Compute Froude number (stability diagnostics)
  if (h > epsilon) { 
    froude_number = uh/sqrt(g*h)/h;
  } else {
    froude_number = 0.0;
    // FIXME (Ole): What should it be when dry??
  }
  
  return froude_number;
}



// Function to obtain speed from momentum and depth.
// This is used by flux functions
// Input parameters uh and h may be modified by this function.
// Tried to inline, but no speedup was achieved 27th May 2009 (Ole)
//static inline double _compute_speed(double *uh, 
double _compute_speed(double *uh, 
              double *h,
              double epsilon,
              double h0,
              double limiting_threshold) {
  
  double u;

  if (*h < limiting_threshold) {   
    // Apply limiting of speeds according to the ANUGA manual
    if (*h < epsilon) {
      *h = 0.0;  // Could have been negative
      u = 0.0;
    } else {
      u = *uh/(*h + h0/ *h);    
    }
  

    // Adjust momentum to be consistent with speed
    *uh = u * *h;
  } else {
    // We are in deep water - no need for limiting
    u = *uh/ *h;
  }
  
  return u;
}


// Optimised squareroot computation
//
//See 
//http://www.lomont.org/Math/Papers/2003/InvSqrt.pdf
//and http://mail.opensolaris.org/pipermail/tools-gcc/2005-August/000047.html
float fast_squareroot_approximation(float number) {
  float x; 
  const float f = 1.5;

  // Allow i and y to occupy the same memory
  union
  {
    long i;
    float y;
  } u;
  
  // Good initial guess  
  x = number * 0.5;  
  u.y  = number;
  u.i  = 0x5f3759df - ( u.i >> 1 );
  
  // Take a few iterations
  u.y  = u.y * ( f - ( x * u.y * u.y ) );
  u.y  = u.y * ( f - ( x * u.y * u.y ) );
    
  return number * u.y;
}



// Optimised squareroot computation (double version)
double Xfast_squareroot_approximation(double number) {
  double x;
  const double f = 1.5;
    
  // Allow i and y to occupy the same memory    
  union
  {
    long long i;
    double y;
  } u;

  // Good initial guess
  x = number * 0.5;
  u.y  = number;
  u.i  = 0x5fe6ec85e7de30daLL - ( u.i >> 1 );
  
  // Take a few iterations  
  u.y  = u.y * ( f - ( x * u.y * u.y ) );
  u.y  = u.y * ( f - ( x * u.y * u.y ) );

  return number * u.y;
}


// Innermost flux function (using stage w=z+h)
int _flux_function_central(double *q_left, double *q_right,
                           double z_left, double z_right,
                           double n1, double n2,
                           double epsilon, 
               double h0,
               double limiting_threshold,
               double g,
                           double *edgeflux, double *max_speed) 
{

  /*Compute fluxes between volumes for the shallow water wave equation
    cast in terms of the 'stage', w = h+z using
    the 'central scheme' as described in

    Kurganov, Noelle, Petrova. 'Semidiscrete Central-Upwind Schemes For
    Hyperbolic Conservation Laws and Hamilton-Jacobi Equations'.
    Siam J. Sci. Comput. Vol. 23, No. 3, pp. 707-740.

    The implemented formula is given in equation (3.15) on page 714
  */

  int i;

  double w_left, h_left, uh_left, vh_left, u_left;
  double w_right, h_right, uh_right, vh_right, u_right;
  double s_min, s_max, soundspeed_left, soundspeed_right;
  double denom, inverse_denominator, z;

  // Workspace (allocate once, use many)
  static double q_left_rotated[3], q_right_rotated[3], flux_right[3], flux_left[3];

  // Copy conserved quantities to protect from modification
  q_left_rotated[0] = q_left[0];
  q_right_rotated[0] = q_right[0];
  q_left_rotated[1] = q_left[1];
  q_right_rotated[1] = q_right[1];
  q_left_rotated[2] = q_left[2];
  q_right_rotated[2] = q_right[2];

  // Align x- and y-momentum with x-axis
  _rotate(q_left_rotated, n1, n2);
  _rotate(q_right_rotated, n1, n2);

  z = 0.5*(z_left + z_right); // Average elevation values. 
                              // Even though this will nominally allow 
                  // for discontinuities in the elevation data,
                  // there is currently no numerical support for
                  // this so results may be strange near
                  // jumps in the bed.

  // Compute speeds in x-direction
  w_left = q_left_rotated[0];          
  h_left = w_left - z;
  uh_left = q_left_rotated[1];
  u_left = _compute_speed(&uh_left, &h_left, 
              epsilon, h0, limiting_threshold);

  w_right = q_right_rotated[0];
  h_right = w_right - z;
  uh_right = q_right_rotated[1];
  u_right = _compute_speed(&uh_right, &h_right, 
               epsilon, h0, limiting_threshold);

  // Momentum in y-direction
  vh_left  = q_left_rotated[2];
  vh_right = q_right_rotated[2];

  // Limit y-momentum if necessary 
  // Leaving this out, improves speed significantly (Ole 27/5/2009)
  // All validation tests pass, so do we really need it anymore? 
  _compute_speed(&vh_left, &h_left, 
         epsilon, h0, limiting_threshold);
  _compute_speed(&vh_right, &h_right, 
         epsilon, h0, limiting_threshold);

  // Maximal and minimal wave speeds
  soundspeed_left  = sqrt(g*h_left);
  soundspeed_right = sqrt(g*h_right);  
  
  // Code to use fast square root optimisation if desired.
  // Timings on AMD 64 for the Okushiri profile gave the following timings
  //
  // SQRT           Total    Flux
  //=============================
  //
  // Ref            405s     152s
  // Fast (dbl)     453s     173s
  // Fast (sng)     437s     171s
  //
  // Consequently, there is currently (14/5/2009) no reason to use this 
  // approximation.
  
  //soundspeed_left  = fast_squareroot_approximation(g*h_left);
  //soundspeed_right = fast_squareroot_approximation(g*h_right);

  s_max = max(u_left + soundspeed_left, u_right + soundspeed_right);
  if (s_max < 0.0) 
  {
    s_max = 0.0;
  }

  s_min = min(u_left - soundspeed_left, u_right - soundspeed_right);
  if (s_min > 0.0)
  {
    s_min = 0.0;
  }
  
  // Flux formulas
  flux_left[0] = u_left*h_left;
  flux_left[1] = u_left*uh_left + 0.5*g*h_left*h_left;
  flux_left[2] = u_left*vh_left;

  flux_right[0] = u_right*h_right;
  flux_right[1] = u_right*uh_right + 0.5*g*h_right*h_right;
  flux_right[2] = u_right*vh_right;

  // Flux computation
  denom = s_max - s_min;
  if (denom < epsilon) 
  { // FIXME (Ole): Try using h0 here
    memset(edgeflux, 0, 3*sizeof(double));
    *max_speed = 0.0;
  } 
  else 
  {
    inverse_denominator = 1.0/denom;
    for (i = 0; i < 3; i++) 
    {
      edgeflux[i] = s_max*flux_left[i] - s_min*flux_right[i];
      edgeflux[i] += s_max*s_min*(q_right_rotated[i] - q_left_rotated[i]);
      edgeflux[i] *= inverse_denominator;
    }

    // Maximal wavespeed
    *max_speed = max(fabs(s_max), fabs(s_min));

    // Rotate back
    _rotate(edgeflux, n1, -n2);
  }
  
  return 0;
}

double erfcc(double x){
    double z,t,result;

    z=fabs(x);
    t=1.0/(1.0+0.5*z);
    result=t*exp(-z*z-1.26551223+t*(1.00002368+t*(.37409196+
         t*(.09678418+t*(-.18628806+t*(.27886807+t*(-1.13520398+
         t*(1.48851587+t*(-.82215223+t*.17087277)))))))));
    if (x < 0.0) result = 2.0-result;

    return result;
    }



// Computational function for flux computation (using stage w=z+h)
int flux_function_kinetic(double *q_left, double *q_right,
          double z_left, double z_right,
          double n1, double n2,
          double epsilon, double H0, double g,
          double *edgeflux, double *max_speed) {

  /*Compute fluxes between volumes for the shallow water wave equation
    cast in terms of the 'stage', w = h+z using
    the 'central scheme' as described in

    Zhang et. al., Advances in Water Resources, 26(6), 2003, 635-647.
  */

  int i;

  double w_left, h_left, uh_left, vh_left, u_left, F_left;
  double w_right, h_right, uh_right, vh_right, u_right, F_right;
  double s_min, s_max, soundspeed_left, soundspeed_right;
  double z;
  double q_left_rotated[3], q_right_rotated[3];

  double h0 = H0*H0; //This ensures a good balance when h approaches H0.
  double limiting_threshold = 10*H0; // Avoid applying limiter below this  
  
  //Copy conserved quantities to protect from modification
  for (i=0; i<3; i++) {
    q_left_rotated[i] = q_left[i];
    q_right_rotated[i] = q_right[i];
  }

  //Align x- and y-momentum with x-axis
  _rotate(q_left_rotated, n1, n2);
  _rotate(q_right_rotated, n1, n2);

  z = (z_left+z_right)/2; //Take average of field values

  //Compute speeds in x-direction
  w_left = q_left_rotated[0];          
  h_left = w_left-z;
  uh_left = q_left_rotated[1];
  u_left =_compute_speed(&uh_left, &h_left, 
             epsilon, h0, limiting_threshold);

  w_right = q_right_rotated[0];
  h_right = w_right-z;
  uh_right = q_right_rotated[1];
  u_right =_compute_speed(&uh_right, &h_right,
              epsilon, h0, limiting_threshold);


  //Momentum in y-direction
  vh_left  = q_left_rotated[2];
  vh_right = q_right_rotated[2];


  //Maximal and minimal wave speeds
  soundspeed_left  = sqrt(g*h_left);
  soundspeed_right = sqrt(g*h_right);

  s_max = max(u_left+soundspeed_left, u_right+soundspeed_right);
  if (s_max < 0.0) s_max = 0.0;

  s_min = min(u_left-soundspeed_left, u_right-soundspeed_right);
  if (s_min > 0.0) s_min = 0.0;


  F_left  = 0.0;
  F_right = 0.0;
  if (h_left > 0.0) F_left = u_left/sqrt(g*h_left);
  if (h_right > 0.0) F_right = u_right/sqrt(g*h_right);

  for (i=0; i<3; i++) edgeflux[i] = 0.0;
  *max_speed = 0.0;

  edgeflux[0] = h_left*u_left/2.0*erfcc(-F_left) +  \
          h_left*sqrt(g*h_left)/2.0/sqrt(pi)*exp(-(F_left*F_left)) + \
          h_right*u_right/2.0*erfcc(F_right) -  \
          h_right*sqrt(g*h_right)/2.0/sqrt(pi)*exp(-(F_right*F_right));

  edgeflux[1] = (h_left*u_left*u_left + g/2.0*h_left*h_left)/2.0*erfcc(-F_left) + \
          u_left*h_left*sqrt(g*h_left)/2.0/sqrt(pi)*exp(-(F_left*F_left)) + \
          (h_right*u_right*u_right + g/2.0*h_right*h_right)/2.0*erfcc(F_right) -  \
          u_right*h_right*sqrt(g*h_right)/2.0/sqrt(pi)*exp(-(F_right*F_right));

  edgeflux[2] = vh_left*u_left/2.0*erfcc(-F_left) + \
          vh_left*sqrt(g*h_left)/2.0/sqrt(pi)*exp(-(F_left*F_left)) + \
          vh_right*u_right/2.0*erfcc(F_right) - \
          vh_right*sqrt(g*h_right)/2.0/sqrt(pi)*exp(-(F_right*F_right));

  //Maximal wavespeed
  *max_speed = max(fabs(s_max), fabs(s_min));

  //Rotate back
  _rotate(edgeflux, n1, -n2);

  return 0;
}




void _manning_friction_flat(double g, double eps, int N,
               double* w, double* zv,
               double* uh, double* vh,
               double* eta, double* xmom, double* ymom) {

  int k, k3;
  double S, h, z, z0, z1, z2;

  for (k=0; k<N; k++) {
    if (eta[k] > eps) {
      k3 = 3*k;
      // Get bathymetry
      z0 = zv[k3 + 0];
      z1 = zv[k3 + 1];
      z2 = zv[k3 + 2];
      z = (z0+z1+z2)/3.0;
      h = w[k]-z;
      if (h >= eps) {
        S = -g * eta[k]*eta[k] * sqrt((uh[k]*uh[k] + vh[k]*vh[k]));
        S /= pow(h, 7.0/3);      //Expensive (on Ole's home computer)
        //S /= exp((7.0/3.0)*log(h));      //seems to save about 15% over manning_friction
        //S /= h*h*(1 + h/3.0 - h*h/9.0); //FIXME: Could use a Taylor expansion


        //Update momentum
        xmom[k] += S*uh[k];
        ymom[k] += S*vh[k];
      }
    }
  }
}


void _manning_friction_sloped(double g, double eps, int N,
               double* x, double* w, double* zv,
               double* uh, double* vh,
               double* eta, double* xmom_update, double* ymom_update) {

  int k, k3, k6;
  double S, h, z, z0, z1, z2, zs, zx, zy;
  double x0,y0,x1,y1,x2,y2;

  for (k=0; k<N; k++) {
    if (eta[k] > eps) {
      k3 = 3*k;
      // Get bathymetry
      z0 = zv[k3 + 0];
      z1 = zv[k3 + 1];
      z2 = zv[k3 + 2];

      // Compute bed slope
      k6 = 6*k;  // base index
  
      x0 = x[k6 + 0];
      y0 = x[k6 + 1];
      x1 = x[k6 + 2];
      y1 = x[k6 + 3];
      x2 = x[k6 + 4];
      y2 = x[k6 + 5];

      _gradient(x0, y0, x1, y1, x2, y2, z0, z1, z2, &zx, &zy);

      zs = sqrt(1.0 + zx*zx + zy*zy);
      z = (z0+z1+z2)/3.0;
      h = w[k]-z;
      if (h >= eps) {
        S = -g * eta[k]*eta[k] * zs * sqrt((uh[k]*uh[k] + vh[k]*vh[k]));
        S /= pow(h, 7.0/3);      //Expensive (on Ole's home computer)
        //S /= exp((7.0/3.0)*log(h));      //seems to save about 15% over manning_friction
        //S /= h*h*(1 + h/3.0 - h*h/9.0); //FIXME: Could use a Taylor expansion


        //Update momentum
        xmom_update[k] += S*uh[k];
        ymom_update[k] += S*vh[k];
      }
    }
  }
}



void _chezy_friction(double g, double eps, int N,
               double* x, double* w, double* zv,
               double* uh, double* vh,
               double* chezy, double* xmom_update, double* ymom_update) {

  int k, k3, k6;
  double S, h, z, z0, z1, z2, zs, zx, zy;
  double x0,y0,x1,y1,x2,y2;

  for (k=0; k<N; k++) {
    if (chezy[k] > eps) {
      k3 = 3*k;
      // Get bathymetry
      z0 = zv[k3 + 0];
      z1 = zv[k3 + 1];
      z2 = zv[k3 + 2];

      // Compute bed slope
      k6 = 6*k;  // base index

      x0 = x[k6 + 0];
      y0 = x[k6 + 1];
      x1 = x[k6 + 2];
      y1 = x[k6 + 3];
      x2 = x[k6 + 4];
      y2 = x[k6 + 5];

      _gradient(x0, y0, x1, y1, x2, y2, z0, z1, z2, &zx, &zy);

      zs = sqrt(1.0 + zx*zx + zy*zy);
      z = (z0+z1+z2)/3.0;
      h = w[k]-z;
      if (h >= eps) {
        S = -g * chezy[k] * zs * sqrt((uh[k]*uh[k] + vh[k]*vh[k]));
        S /= pow(h, 2.0);

        //Update momentum
        xmom_update[k] += S*uh[k];
        ymom_update[k] += S*vh[k];
      }
    }
  }
}

/*
void _manning_friction_explicit(double g, double eps, int N,
               double* w, double* z,
               double* uh, double* vh,
               double* eta, double* xmom, double* ymom) {

  int k;
  double S, h;

  for (k=0; k<N; k++) {
    if (eta[k] > eps) {
      h = w[k]-z[k];
      if (h >= eps) {
    S = -g * eta[k]*eta[k] * sqrt((uh[k]*uh[k] + vh[k]*vh[k]));
    S /= pow(h, 7.0/3);      //Expensive (on Ole's home computer)
    //S /= exp(7.0/3.0*log(h));      //seems to save about 15% over manning_friction
    //S /= h*h*(1 + h/3.0 - h*h/9.0); //FIXME: Could use a Taylor expansion


    //Update momentum
    xmom[k] += S*uh[k];
    ymom[k] += S*vh[k];
      }
    }
  }
}
*/




double velocity_balance(double uh_i, 
            double uh,
            double h_i, 
            double h, 
            double alpha,
            double epsilon) {
  // Find alpha such that speed at the vertex is within one
  // order of magnitude of the centroid speed   

  // FIXME(Ole): Work in progress
  
  double a, b, estimate;
  double m=10; // One order of magnitude - allow for velocity deviations at vertices

  
  printf("alpha = %f, uh_i=%f, uh=%f, h_i=%f, h=%f\n",
     alpha, uh_i, uh, h_i, h);
      
  
  
    
  // Shorthands and determine inequality
  if (fabs(uh) < epsilon) {
    a = 1.0e10; // Limit
  } else {
    a = fabs(uh_i - uh)/fabs(uh);
  }
      
  if (h < epsilon) {
    b = 1.0e10; // Limit
  } else {   
    b = m*fabs(h_i - h)/h;
  }

  printf("a %f, b %f\n", a, b); 

  if (a > b) {
    estimate = (m-1)/(a-b);     
    
    printf("Alpha %f, estimate %f\n", 
       alpha, estimate);    
       
    if (alpha < estimate) {
      printf("Adjusting alpha from %f to %f\n", 
         alpha, estimate);
      alpha = estimate;
    }
  } else {
  
    if (h < h_i) {
      estimate = (m-1)/(a-b);             
    
      printf("Alpha %f, estimate %f\n", 
         alpha, estimate);    
       
      if (alpha < estimate) {
    printf("Adjusting alpha from %f to %f\n", 
           alpha, estimate);
    alpha = estimate;
      }    
    }
    // Always fulfilled as alpha and m-1 are non negative
  }
  
  
  return alpha;
}


int _balance_deep_and_shallow(int N,
                  double* wc,
                  double* zc,
                  double* wv,
                  double* zv,
                  double* hvbar, // Retire this
                  double* xmomc,
                  double* ymomc,
                  double* xmomv,
                  double* ymomv,
                  double H0,
                  int tight_slope_limiters,
                  int use_centroid_velocities,
                  double alpha_balance) {

  int k, k3, i; 

  double dz, hmin, alpha, h_diff, hc_k;
  double epsilon = 1.0e-6; // FIXME: Temporary measure
  double hv[3]; // Depths at vertices
  double uc, vc; // Centroid speeds

  // Compute linear combination between w-limited stages and
  // h-limited stages close to the bed elevation.

  for (k=0; k<N; k++) {
    // Compute maximal variation in bed elevation
    // This quantitiy is
    //     dz = max_i abs(z_i - z_c)
    // and it is independent of dimension
    // In the 1d case zc = (z0+z1)/2
    // In the 2d case zc = (z0+z1+z2)/3

    k3 = 3*k;
    hc_k = wc[k] - zc[k]; // Centroid value at triangle k
    
    dz = 0.0;
    if (tight_slope_limiters == 0) {     
      // FIXME: Try with this one precomputed
      for (i=0; i<3; i++) {
    dz = max(dz, fabs(zv[k3+i]-zc[k]));
      }
    }

    // Calculate depth at vertices (possibly negative here!)
    hv[0] = wv[k3] -   zv[k3];
    hv[1] = wv[k3+1] - zv[k3+1];
    hv[2] = wv[k3+2] - zv[k3+2];        
    
    // Calculate minimal depth across all three vertices
    hmin = min(hv[0], min(hv[1], hv[2]));

    //if (hmin < 0.0 ) {
    //  printf("hmin = %f\n",hmin);
    //}
    

    // Create alpha in [0,1], where alpha==0 means using the h-limited
    // stage and alpha==1 means using the w-limited stage as
    // computed by the gradient limiter (both 1st or 2nd order)
    if (tight_slope_limiters == 0) {     
      // If hmin > dz/alpha_balance then alpha = 1 and the bed will have no 
      // effect
      // If hmin < 0 then alpha = 0 reverting to constant height above bed.
      // The parameter alpha_balance==2 by default 

      
      if (dz > 0.0) {
    alpha = max( min( alpha_balance*hmin/dz, 1.0), 0.0 );
      } else {
    alpha = 1.0;  // Flat bed
      }
      //printf("Using old style limiter\n");
      
    } else {

      // Tight Slope Limiter (2007)
    
      // Make alpha as large as possible but still ensure that 
      // final depth is positive and that velocities at vertices
      // are controlled
    
      if (hmin < H0) {
    alpha = 1.0;
    for (i=0; i<3; i++) {

      h_diff = hc_k - hv[i];
      if (h_diff <= 0) {
        // Deep water triangle is further away from bed than
        // shallow water (hbar < h). Any alpha will do
      
      } else {
        // Denominator is positive which means that we need some of the
        // h-limited stage.

        alpha = min(alpha, (hc_k - H0)/h_diff);
      }
    }

    // Ensure alpha in [0,1]
    if (alpha>1.0) alpha=1.0;
    if (alpha<0.0) alpha=0.0;

      } else {
    // Use w-limited stage exclusively in deeper water.
    alpha = 1.0;
      }
    }
    
    
    //  Let
    //
    //    wvi be the w-limited stage (wvi = zvi + hvi)
    //    wvi- be the h-limited state (wvi- = zvi + hvi-)
    //
    //
    //  where i=0,1,2 denotes the vertex ids
    //
    //  Weighted balance between w-limited and h-limited stage is
    //
    //    wvi := (1-alpha)*(zvi+hvi-) + alpha*(zvi+hvi)
    //
    //  It follows that the updated wvi is
    //    wvi := zvi + (1-alpha)*hvi- + alpha*hvi
    //
    //   Momentum is balanced between constant and limited

    
    if (alpha < 1) {      
      for (i=0; i<3; i++) {

    wv[k3+i] = zv[k3+i] + (1-alpha)*hc_k + alpha*hv[i];

    // Update momentum at vertices
    if (use_centroid_velocities == 1) {
      // This is a simple, efficient and robust option
      // It uses first order approximation of velocities, but retains
      // the order used by stage.
    
      // Speeds at centroids
      if (hc_k > epsilon) {
        uc = xmomc[k]/hc_k;
        vc = ymomc[k]/hc_k;
      } else {
        uc = 0.0;
        vc = 0.0;
      }

      // Vertex momenta guaranteed to be consistent with depth guaranteeing
      // controlled speed
      hv[i] = wv[k3+i] - zv[k3+i]; // Recompute (balanced) vertex depth
      xmomv[k3+i] = uc*hv[i];
      ymomv[k3+i] = vc*hv[i];
      
    } else {
      // Update momentum as a linear combination of
      // xmomc and ymomc (shallow) and momentum
      // from extrapolator xmomv and ymomv (deep).
      // This assumes that values from xmomv and ymomv have
      // been established e.g. by the gradient limiter.

      // FIXME (Ole): I think this should be used with vertex momenta
      // computed above using centroid_velocities instead of xmomc
      // and ymomc as they'll be more representative first order
      // values.

      xmomv[k3+i] = (1-alpha)*xmomc[k] + alpha*xmomv[k3+i];
      ymomv[k3+i] = (1-alpha)*ymomc[k] + alpha*ymomv[k3+i];

    }
      }
    }
  }
  return 0;
}
  



int _protect(int N,
         double minimum_allowed_height,
         double maximum_allowed_speed,
         double epsilon,
         double* wc,
         double* zc,
         double* xmomc,
         double* ymomc) {

  int k;
  double hc;
  double u, v, reduced_speed;


  // Protect against initesimal and negative heights  
  if (maximum_allowed_speed < epsilon) {
    for (k=0; k<N; k++) {
      hc = wc[k] - zc[k];

      if (hc < minimum_allowed_height) {
        
    // Set momentum to zero and ensure h is non negative
    xmomc[k] = 0.0;
    ymomc[k] = 0.0;
    if (hc <= 0.0) wc[k] = zc[k];
      }
    }
  } else {
   
    // Protect against initesimal and negative heights
    for (k=0; k<N; k++) {
      hc = wc[k] - zc[k];
      
      if (hc < minimum_allowed_height) {

        //New code: Adjust momentum to guarantee speeds are physical
        //          ensure h is non negative
              
        if (hc <= 0.0) {
            wc[k] = zc[k];
        xmomc[k] = 0.0;
        ymomc[k] = 0.0;
        } else {
          //Reduce excessive speeds derived from division by small hc
      //FIXME (Ole): This may be unnecessary with new slope limiters
      //in effect.
          
          u = xmomc[k]/hc;
      if (fabs(u) > maximum_allowed_speed) {
        reduced_speed = maximum_allowed_speed * u/fabs(u);
        //printf("Speed (u) has been reduced from %.3f to %.3f\n",
        //   u, reduced_speed);
        xmomc[k] = reduced_speed * hc;
      }

          v = ymomc[k]/hc;
      if (fabs(v) > maximum_allowed_speed) {
        reduced_speed = maximum_allowed_speed * v/fabs(v);
        //printf("Speed (v) has been reduced from %.3f to %.3f\n",
        //   v, reduced_speed);
        ymomc[k] = reduced_speed * hc;
      }
        }
      }
    }
  }
  return 0;
}



int _assign_wind_field_values(int N,
                  double* xmom_update,
                  double* ymom_update,
                  double* s_vec,
                  double* phi_vec,
                  double cw) {

  // Assign windfield values to momentum updates

  int k;
  double S, s, phi, u, v;

  for (k=0; k<N; k++) {

    s = s_vec[k];
    phi = phi_vec[k];

    //Convert to radians
    phi = phi*pi/180;

    //Compute velocity vector (u, v)
    u = s*cos(phi);
    v = s*sin(phi);

    //Compute wind stress
    S = cw * sqrt(u*u + v*v);
    xmom_update[k] += S*u;
    ymom_update[k] += S*v;
  }
  return 0;
}



///////////////////////////////////////////////////////////////////
// Gateways to Python



PyObject *flux_function_central(PyObject *self, PyObject *args) {
  //
  // Gateway to innermost flux function.
  // This is only used by the unit tests as the c implementation is
  // normally called by compute_fluxes in this module.


  PyArrayObject *normal, *ql, *qr,  *edgeflux;
  double g, epsilon, max_speed, H0, zl, zr;
  double h0, limiting_threshold;

  if (!PyArg_ParseTuple(args, "OOOddOddd",
            &normal, &ql, &qr, &zl, &zr, &edgeflux,
            &epsilon, &g, &H0)) {
      report_python_error(AT, "could not parse input arguments");
      return NULL;
  }

  
  h0 = H0*H0; // This ensures a good balance when h approaches H0.
              // But evidence suggests that h0 can be as little as
              // epsilon!

  limiting_threshold = 10*H0; // Avoid applying limiter below this
                              // threshold for performance reasons.
                              // See ANUGA manual under flux limiting  
  
  _flux_function_central((double*) ql -> data, 
             (double*) qr -> data,
             zl,
             zr,
             ((double*) normal -> data)[0],
             ((double*) normal -> data)[1],
             epsilon, h0, limiting_threshold,
             g,
             (double*) edgeflux -> data,
             &max_speed);
  
  return Py_BuildValue("d", max_speed);  
}



PyObject *gravity(PyObject *self, PyObject *args) {
  //
  //  gravity(g, h, v, x, xmom, ymom)
  //


  PyArrayObject *h, *z, *x, *xmom, *ymom;
  int k, N, k3, k6;
  double g, avg_h, zx, zy;
  double x0, y0, x1, y1, x2, y2, z0, z1, z2;
  //double epsilon;

  if (!PyArg_ParseTuple(args, "dOOOOO",
            &g, &h, &z, &x,
            &xmom, &ymom)) {
    //&epsilon)) {
      report_python_error(AT, "could not parse input arguments");
      return NULL;
  }

  // check that numpy array objects arrays are C contiguous memory
  CHECK_C_CONTIG(h);
  CHECK_C_CONTIG(z);
  CHECK_C_CONTIG(x);
  CHECK_C_CONTIG(xmom);
  CHECK_C_CONTIG(ymom);

  N = h -> dimensions[0];
  for (k=0; k<N; k++) {
    k3 = 3*k;  // base index

    // Get bathymetry
    z0 = ((double*) z -> data)[k3 + 0];
    z1 = ((double*) z -> data)[k3 + 1];
    z2 = ((double*) z -> data)[k3 + 2];

    // Optimise for flat bed
    // Note (Ole): This didn't produce measurable speed up.
    // Revisit later
    //if (fabs(z0-z1)<epsilon && fabs(z1-z2)<epsilon) {
    //  continue;
    //} 

    // Get average depth from centroid values
    avg_h = ((double *) h -> data)[k];

    // Compute bed slope
    k6 = 6*k;  // base index
  
    x0 = ((double*) x -> data)[k6 + 0];
    y0 = ((double*) x -> data)[k6 + 1];
    x1 = ((double*) x -> data)[k6 + 2];
    y1 = ((double*) x -> data)[k6 + 3];
    x2 = ((double*) x -> data)[k6 + 4];
    y2 = ((double*) x -> data)[k6 + 5];


    _gradient(x0, y0, x1, y1, x2, y2, z0, z1, z2, &zx, &zy);

    // Update momentum
    ((double*) xmom -> data)[k] += -g*zx*avg_h;
    ((double*) ymom -> data)[k] += -g*zy*avg_h;
  }

  return Py_BuildValue("");
}


PyObject *manning_friction_sloped(PyObject *self, PyObject *args) {
  //
  // manning_friction_sloped(g, eps, x, h, uh, vh, z, eta, xmom_update, ymom_update)
  //


  PyArrayObject *x, *w, *z, *uh, *vh, *eta, *xmom, *ymom;
  int N;
  double g, eps;

  if (!PyArg_ParseTuple(args, "ddOOOOOOOO",
            &g, &eps, &x, &w, &uh, &vh, &z,  &eta, &xmom, &ymom)) {
      report_python_error(AT, "could not parse input arguments");
      return NULL;
  }

  // check that numpy array objects arrays are C contiguous memory
  CHECK_C_CONTIG(x);
  CHECK_C_CONTIG(w);
  CHECK_C_CONTIG(z);
  CHECK_C_CONTIG(uh);
  CHECK_C_CONTIG(vh);
  CHECK_C_CONTIG(eta);
  CHECK_C_CONTIG(xmom);
  CHECK_C_CONTIG(ymom);

  N = w -> dimensions[0];

  _manning_friction_sloped(g, eps, N,
            (double*) x -> data,
            (double*) w -> data,
            (double*) z -> data,
            (double*) uh -> data,
            (double*) vh -> data,
            (double*) eta -> data,
            (double*) xmom -> data,
            (double*) ymom -> data);

  return Py_BuildValue("");
}


PyObject *chezy_friction(PyObject *self, PyObject *args) {
  //
  // chezy_friction(g, eps, x, h, uh, vh, z, chezy, xmom_update, ymom_update)
  //


  PyArrayObject *x, *w, *z, *uh, *vh, *chezy, *xmom, *ymom;
  int N;
  double g, eps;

  if (!PyArg_ParseTuple(args, "ddOOOOOOOO",
            &g, &eps, &x, &w, &uh, &vh, &z,  &chezy, &xmom, &ymom)) {
      report_python_error(AT, "could not parse input arguments");
      return NULL;
  }

  // check that numpy array objects arrays are C contiguous memory
  CHECK_C_CONTIG(x);
  CHECK_C_CONTIG(w);
  CHECK_C_CONTIG(z);
  CHECK_C_CONTIG(uh);
  CHECK_C_CONTIG(vh);
  CHECK_C_CONTIG(chezy);
  CHECK_C_CONTIG(xmom);
  CHECK_C_CONTIG(ymom);

  N = w -> dimensions[0];

  _chezy_friction(g, eps, N,
            (double*) x -> data,
            (double*) w -> data,
            (double*) z -> data,
            (double*) uh -> data,
            (double*) vh -> data,
            (double*) chezy -> data,
            (double*) xmom -> data,
            (double*) ymom -> data);

  return Py_BuildValue("");
}


PyObject *manning_friction_flat(PyObject *self, PyObject *args) {
  //
  // manning_friction_flat(g, eps, h, uh, vh, z, eta, xmom_update, ymom_update)
  //


  PyArrayObject *w, *z, *uh, *vh, *eta, *xmom, *ymom;
  int N;
  double g, eps;

  if (!PyArg_ParseTuple(args, "ddOOOOOOO",
            &g, &eps, &w, &uh, &vh, &z,  &eta, &xmom, &ymom)) {
      report_python_error(AT, "could not parse input arguments");
      return NULL;
  }

  // check that numpy array objects arrays are C contiguous memory
  CHECK_C_CONTIG(w);
  CHECK_C_CONTIG(z);
  CHECK_C_CONTIG(uh);
  CHECK_C_CONTIG(vh);
  CHECK_C_CONTIG(eta);
  CHECK_C_CONTIG(xmom);
  CHECK_C_CONTIG(ymom);

  N = w -> dimensions[0];

  _manning_friction_flat(g, eps, N,
            (double*) w -> data,
            (double*) z -> data,
            (double*) uh -> data,
            (double*) vh -> data,
            (double*) eta -> data,
            (double*) xmom -> data,
            (double*) ymom -> data);

  return Py_BuildValue("");
}


/*
PyObject *manning_friction_explicit(PyObject *self, PyObject *args) {
  //
  // manning_friction_explicit(g, eps, h, uh, vh, eta, xmom_update, ymom_update)
  //


  PyArrayObject *w, *z, *uh, *vh, *eta, *xmom, *ymom;
  int N;
  double g, eps;

  if (!PyArg_ParseTuple(args, "ddOOOOOOO",
            &g, &eps, &w, &z, &uh, &vh, &eta,
            &xmom, &ymom))
    return NULL;

  // check that numpy array objects arrays are C contiguous memory
  CHECK_C_CONTIG(w);
  CHECK_C_CONTIG(z);
  CHECK_C_CONTIG(uh);
  CHECK_C_CONTIG(vh);
  CHECK_C_CONTIG(eta);
  CHECK_C_CONTIG(xmom);
  CHECK_C_CONTIG(ymom);

  N = w -> dimensions[0];
  _manning_friction_explicit(g, eps, N,
            (double*) w -> data,
            (double*) z -> data,
            (double*) uh -> data,
            (double*) vh -> data,
            (double*) eta -> data,
            (double*) xmom -> data,
            (double*) ymom -> data);

  return Py_BuildValue("");
}
*/



// Computational routine
int _extrapolate_second_order_sw(int number_of_elements,
                                 double epsilon,
                                 double minimum_allowed_height,
                                 double beta_w,
                                 double beta_w_dry,
                                 double beta_uh,
                                 double beta_uh_dry,
                                 double beta_vh,
                                 double beta_vh_dry,
                                 long* surrogate_neighbours,
                                 long* number_of_boundaries,
                                 double* centroid_coordinates,
                                 double* stage_centroid_values,
                                 double* xmom_centroid_values,
                                 double* ymom_centroid_values,
                                 double* elevation_centroid_values,
                                 double* vertex_coordinates,
                                 double* stage_vertex_values,
                                 double* xmom_vertex_values,
                                 double* ymom_vertex_values,
                                 double* elevation_vertex_values,
                                 int optimise_dry_cells) {
                  
                  

  // Local variables
  double a, b; // Gradient vector used to calculate vertex values from centroids
  int k, k0, k1, k2, k3, k6, coord_index, i;
  double x, y, x0, y0, x1, y1, x2, y2, xv0, yv0, xv1, yv1, xv2, yv2; // Vertices of the auxiliary triangle
  double dx1, dx2, dy1, dy2, dxv0, dxv1, dxv2, dyv0, dyv1, dyv2, dq0, dq1, dq2, area2, inv_area2;
  double dqv[3], qmin, qmax, hmin, hmax;
  double hc, h0, h1, h2, beta_tmp, hfactor;

    
  for (k = 0; k < number_of_elements; k++) 
  {
    k3=k*3;
    k6=k*6;

    if (number_of_boundaries[k]==3)
    {
      // No neighbours, set gradient on the triangle to zero
      
      stage_vertex_values[k3]   = stage_centroid_values[k];
      stage_vertex_values[k3+1] = stage_centroid_values[k];
      stage_vertex_values[k3+2] = stage_centroid_values[k];
      xmom_vertex_values[k3]    = xmom_centroid_values[k];
      xmom_vertex_values[k3+1]  = xmom_centroid_values[k];
      xmom_vertex_values[k3+2]  = xmom_centroid_values[k];
      ymom_vertex_values[k3]    = ymom_centroid_values[k];
      ymom_vertex_values[k3+1]  = ymom_centroid_values[k];
      ymom_vertex_values[k3+2]  = ymom_centroid_values[k];
      
      continue;
    }
    else 
    {
      // Triangle k has one or more neighbours. 
      // Get centroid and vertex coordinates of the triangle
      
      // Get the vertex coordinates
      xv0 = vertex_coordinates[k6];   
      yv0 = vertex_coordinates[k6+1];
      xv1 = vertex_coordinates[k6+2]; 
      yv1 = vertex_coordinates[k6+3];
      xv2 = vertex_coordinates[k6+4]; 
      yv2 = vertex_coordinates[k6+5];
      
      // Get the centroid coordinates
      coord_index = 2*k;
      x = centroid_coordinates[coord_index];
      y = centroid_coordinates[coord_index+1];
      
      // Store x- and y- differentials for the vertices of 
      // triangle k relative to the centroid
      dxv0 = xv0 - x; 
      dxv1 = xv1 - x; 
      dxv2 = xv2 - x;
      dyv0 = yv0 - y; 
      dyv1 = yv1 - y; 
      dyv2 = yv2 - y;
    }


            
    if (number_of_boundaries[k]<=1)
    {
      //==============================================
      // Number of boundaries <= 1
      //==============================================    
    
    
      // If no boundaries, auxiliary triangle is formed 
      // from the centroids of the three neighbours
      // If one boundary, auxiliary triangle is formed 
      // from this centroid and its two neighbours
      
      k0 = surrogate_neighbours[k3];
      k1 = surrogate_neighbours[k3 + 1];
      k2 = surrogate_neighbours[k3 + 2];
      
      // Get the auxiliary triangle's vertex coordinates 
      // (really the centroids of neighbouring triangles)
      coord_index = 2*k0;
      x0 = centroid_coordinates[coord_index];
      y0 = centroid_coordinates[coord_index+1];
      
      coord_index = 2*k1;
      x1 = centroid_coordinates[coord_index];
      y1 = centroid_coordinates[coord_index+1];
      
      coord_index = 2*k2;
      x2 = centroid_coordinates[coord_index];
      y2 = centroid_coordinates[coord_index+1];
      
      // Store x- and y- differentials for the vertices 
      // of the auxiliary triangle
      dx1 = x1 - x0; 
      dx2 = x2 - x0;
      dy1 = y1 - y0; 
      dy2 = y2 - y0;
      
      // Calculate 2*area of the auxiliary triangle
      // The triangle is guaranteed to be counter-clockwise      
      area2 = dy2*dx1 - dy1*dx2; 
      
      // If the mesh is 'weird' near the boundary, 
      // the triangle might be flat or clockwise
      // Default to zero gradient
      if (area2 <= 0) 
      {
          //printf("Error negative triangle area \n");
          //report_python_error(AT, "Negative triangle area");
          //return -1;

          stage_vertex_values[k3]   = stage_centroid_values[k];
          stage_vertex_values[k3+1] = stage_centroid_values[k];
          stage_vertex_values[k3+2] = stage_centroid_values[k];
          xmom_vertex_values[k3]    = xmom_centroid_values[k];
          xmom_vertex_values[k3+1]  = xmom_centroid_values[k];
          xmom_vertex_values[k3+2]  = xmom_centroid_values[k];
          ymom_vertex_values[k3]    = ymom_centroid_values[k];
          ymom_vertex_values[k3+1]  = ymom_centroid_values[k];
          ymom_vertex_values[k3+2]  = ymom_centroid_values[k];

          continue;
      }  
      
      // Calculate heights of neighbouring cells
      hc = stage_centroid_values[k]  - elevation_centroid_values[k];
      h0 = stage_centroid_values[k0] - elevation_centroid_values[k0];
      h1 = stage_centroid_values[k1] - elevation_centroid_values[k1];
      h2 = stage_centroid_values[k2] - elevation_centroid_values[k2];
      hmin = min(min(h0, min(h1, h2)), hc);
      //hfactor = hc/(hc + 1.0);

      hfactor = 0.0;
      if (hmin > 0.001) 
      {
        hfactor = (hmin - 0.001)/(hmin + 0.004);
      }
      
      if (optimise_dry_cells) 
      {      
        // Check if linear reconstruction is necessary for triangle k
        // This check will exclude dry cells.

        hmax = max(h0, max(h1, h2));      
        if (hmax < epsilon) 
        {
            continue;
        }
      }
    
      //-----------------------------------
      // stage
      //-----------------------------------      
      
      // Calculate the difference between vertex 0 of the auxiliary 
      // triangle and the centroid of triangle k
      dq0 = stage_centroid_values[k0] - stage_centroid_values[k];
      
      // Calculate differentials between the vertices 
      // of the auxiliary triangle (centroids of neighbouring triangles)
      dq1 = stage_centroid_values[k1] - stage_centroid_values[k0];
      dq2 = stage_centroid_values[k2] - stage_centroid_values[k0];
      
      inv_area2 = 1.0/area2;
      // Calculate the gradient of stage on the auxiliary triangle
      a = dy2*dq1 - dy1*dq2;
      a *= inv_area2;
      b = dx1*dq2 - dx2*dq1;
      b *= inv_area2;
      
      // Calculate provisional jumps in stage from the centroid 
      // of triangle k to its vertices, to be limited
      dqv[0] = a*dxv0 + b*dyv0;
      dqv[1] = a*dxv1 + b*dyv1;
      dqv[2] = a*dxv2 + b*dyv2;
      
      // Now we want to find min and max of the centroid and the 
      // vertices of the auxiliary triangle and compute jumps 
      // from the centroid to the min and max
      find_qmin_and_qmax(dq0, dq1, dq2, &qmin, &qmax);
      
      // Playing with dry wet interface
      //hmin = qmin;
      //beta_tmp = beta_w_dry;
      //if (hmin>minimum_allowed_height)
      beta_tmp = beta_w_dry + (beta_w - beta_w_dry) * hfactor;
    
      //printf("min_alled_height = %f\n",minimum_allowed_height);
      //printf("hmin = %f\n",hmin);
      //printf("beta_w = %f\n",beta_w);
      //printf("beta_tmp = %f\n",beta_tmp);
      // Limit the gradient
      limit_gradient(dqv, qmin, qmax, beta_tmp); 
      
      //for (i=0;i<3;i++)
      stage_vertex_values[k3+0] = stage_centroid_values[k] + dqv[0];
      stage_vertex_values[k3+1] = stage_centroid_values[k] + dqv[1];
      stage_vertex_values[k3+2] = stage_centroid_values[k] + dqv[2];
      
      
      //-----------------------------------
      // xmomentum
      //-----------------------------------            

      // Calculate the difference between vertex 0 of the auxiliary 
      // triangle and the centroid of triangle k      
      dq0 = xmom_centroid_values[k0] - xmom_centroid_values[k];
      
      // Calculate differentials between the vertices 
      // of the auxiliary triangle
      dq1 = xmom_centroid_values[k1] - xmom_centroid_values[k0];
      dq2 = xmom_centroid_values[k2] - xmom_centroid_values[k0];
      
      // Calculate the gradient of xmom on the auxiliary triangle
      a = dy2*dq1 - dy1*dq2;
      a *= inv_area2;
      b = dx1*dq2 - dx2*dq1;
      b *= inv_area2;
      
      // Calculate provisional jumps in stage from the centroid 
      // of triangle k to its vertices, to be limited      
      dqv[0] = a*dxv0+b*dyv0;
      dqv[1] = a*dxv1+b*dyv1;
      dqv[2] = a*dxv2+b*dyv2;
      
      // Now we want to find min and max of the centroid and the 
      // vertices of the auxiliary triangle and compute jumps 
      // from the centroid to the min and max
      find_qmin_and_qmax(dq0, dq1, dq2, &qmin, &qmax);
      //beta_tmp = beta_uh;
      //if (hmin<minimum_allowed_height)
      //beta_tmp = beta_uh_dry;
      beta_tmp = beta_uh_dry + (beta_uh - beta_uh_dry) * hfactor;

      // Limit the gradient
      limit_gradient(dqv, qmin, qmax, beta_tmp);

      for (i=0; i < 3; i++)
      {
        xmom_vertex_values[k3+i] = xmom_centroid_values[k] + dqv[i];
      }
      
      //-----------------------------------
      // ymomentum
      //-----------------------------------                  

      // Calculate the difference between vertex 0 of the auxiliary 
      // triangle and the centroid of triangle k
      dq0 = ymom_centroid_values[k0] - ymom_centroid_values[k];
      
      // Calculate differentials between the vertices 
      // of the auxiliary triangle
      dq1 = ymom_centroid_values[k1] - ymom_centroid_values[k0];
      dq2 = ymom_centroid_values[k2] - ymom_centroid_values[k0];
      
      // Calculate the gradient of xmom on the auxiliary triangle
      a = dy2*dq1 - dy1*dq2;
      a *= inv_area2;
      b = dx1*dq2 - dx2*dq1;
      b *= inv_area2;
      
      // Calculate provisional jumps in stage from the centroid 
      // of triangle k to its vertices, to be limited
      dqv[0] = a*dxv0 + b*dyv0;
      dqv[1] = a*dxv1 + b*dyv1;
      dqv[2] = a*dxv2 + b*dyv2;
      
      // Now we want to find min and max of the centroid and the 
      // vertices of the auxiliary triangle and compute jumps 
      // from the centroid to the min and max
      find_qmin_and_qmax(dq0, dq1, dq2, &qmin, &qmax);
      
      //beta_tmp = beta_vh;
      //
      //if (hmin<minimum_allowed_height)
      //beta_tmp = beta_vh_dry;
      beta_tmp = beta_vh_dry + (beta_vh - beta_vh_dry) * hfactor;   

      // Limit the gradient
      limit_gradient(dqv, qmin, qmax, beta_tmp);
      
      for (i=0;i<3;i++)
      {
        ymom_vertex_values[k3 + i] = ymom_centroid_values[k] + dqv[i];
      }
    } // End number_of_boundaries <=1 
    else
    {

      //==============================================
      // Number of boundaries == 2
      //==============================================        
        
      // One internal neighbour and gradient is in direction of the neighbour's centroid
      
      // Find the only internal neighbour (k1?)
      for (k2 = k3; k2 < k3 + 3; k2++)
      {
      // Find internal neighbour of triangle k      
      // k2 indexes the edges of triangle k   
      
          if (surrogate_neighbours[k2] != k)
          {
             break;
          }
      }
      
      if ((k2 == k3 + 3)) 
      {
        // If we didn't find an internal neighbour
        report_python_error(AT, "Internal neighbour not found");
        return -1;
      }
      
      k1 = surrogate_neighbours[k2];
      
      // The coordinates of the triangle are already (x,y). 
      // Get centroid of the neighbour (x1,y1)
      coord_index = 2*k1;
      x1 = centroid_coordinates[coord_index];
      y1 = centroid_coordinates[coord_index + 1];
      
      // Compute x- and y- distances between the centroid of 
      // triangle k and that of its neighbour
      dx1 = x1 - x; 
      dy1 = y1 - y;
      
      // Set area2 as the square of the distance
      area2 = dx1*dx1 + dy1*dy1;
      
      // Set dx2=(x1-x0)/((x1-x0)^2+(y1-y0)^2) 
      // and dy2=(y1-y0)/((x1-x0)^2+(y1-y0)^2) which
      // respectively correspond to the x- and y- gradients 
      // of the conserved quantities
      dx2 = 1.0/area2;
      dy2 = dx2*dy1;
      dx2 *= dx1;
      
      
      //-----------------------------------
      // stage
      //-----------------------------------            

      // Compute differentials
      dq1 = stage_centroid_values[k1] - stage_centroid_values[k];
      
      // Calculate the gradient between the centroid of triangle k 
      // and that of its neighbour
      a = dq1*dx2;
      b = dq1*dy2;
      
      // Calculate provisional vertex jumps, to be limited
      dqv[0] = a*dxv0 + b*dyv0;
      dqv[1] = a*dxv1 + b*dyv1;
      dqv[2] = a*dxv2 + b*dyv2;
      
      // Now limit the jumps
      if (dq1>=0.0)
      {
        qmin=0.0;
        qmax=dq1;
      }
      else
      {
        qmin = dq1;
        qmax = 0.0;
      }
      
      // Limit the gradient
      limit_gradient(dqv, qmin, qmax, beta_w);
      
      //for (i=0; i < 3; i++)
      //{
      stage_vertex_values[k3] = stage_centroid_values[k] + dqv[0];
      stage_vertex_values[k3 + 1] = stage_centroid_values[k] + dqv[1];
      stage_vertex_values[k3 + 2] = stage_centroid_values[k] + dqv[2];
      //}
      
      //-----------------------------------
      // xmomentum
      //-----------------------------------                        
      
      // Compute differentials
      dq1 = xmom_centroid_values[k1] - xmom_centroid_values[k];
      
      // Calculate the gradient between the centroid of triangle k 
      // and that of its neighbour
      a = dq1*dx2;
      b = dq1*dy2;
      
      // Calculate provisional vertex jumps, to be limited
      dqv[0] = a*dxv0+b*dyv0;
      dqv[1] = a*dxv1+b*dyv1;
      dqv[2] = a*dxv2+b*dyv2;
      
      // Now limit the jumps
      if (dq1 >= 0.0)
      {
        qmin = 0.0;
        qmax = dq1;
      }
      else
      {
        qmin = dq1;
        qmax = 0.0;
      }
      
      // Limit the gradient      
      limit_gradient(dqv, qmin, qmax, beta_w);
      
      //for (i=0;i<3;i++)
      //xmom_vertex_values[k3] = xmom_centroid_values[k] + dqv[0];
      //xmom_vertex_values[k3 + 1] = xmom_centroid_values[k] + dqv[1];
      //xmom_vertex_values[k3 + 2] = xmom_centroid_values[k] + dqv[2];
      
      for (i = 0; i < 3;i++)
      {
          xmom_vertex_values[k3 + i] = xmom_centroid_values[k] + dqv[i];
      }
      
      //-----------------------------------
      // ymomentum
      //-----------------------------------                        

      // Compute differentials
      dq1 = ymom_centroid_values[k1] - ymom_centroid_values[k];
      
      // Calculate the gradient between the centroid of triangle k 
      // and that of its neighbour
      a = dq1*dx2;
      b = dq1*dy2;
      
      // Calculate provisional vertex jumps, to be limited
      dqv[0] = a*dxv0 + b*dyv0;
      dqv[1] = a*dxv1 + b*dyv1;
      dqv[2] = a*dxv2 + b*dyv2;
      
      // Now limit the jumps
      if (dq1>=0.0)
      {
        qmin = 0.0;
        qmax = dq1;
      } 
      else
      {
        qmin = dq1;
        qmax = 0.0;
      }
      
      // Limit the gradient
      limit_gradient(dqv, qmin, qmax, beta_w);
      
      //for (i=0;i<3;i++)
      //ymom_vertex_values[k3] = ymom_centroid_values[k] + dqv[0];
      //ymom_vertex_values[k3 + 1] = ymom_centroid_values[k] + dqv[1];
      //ymom_vertex_values[k3 + 2] = ymom_centroid_values[k] + dqv[2];

for (i=0;i<3;i++)
        {
ymom_vertex_values[k3 + i] = ymom_centroid_values[k] + dqv[i];
        }
//ymom_vertex_values[k3] = ymom_centroid_values[k] + dqv[0];
//ymom_vertex_values[k3 + 1] = ymom_centroid_values[k] + dqv[1];
//ymom_vertex_values[k3 + 2] = ymom_centroid_values[k] + dqv[2];
    } // else [number_of_boundaries==2]
  } // for k=0 to number_of_elements-1
  
  return 0;
}           


PyObject *extrapolate_second_order_sw(PyObject *self, PyObject *args) {
  /*Compute the vertex values based on a linear reconstruction 
    on each triangle
    
    These values are calculated as follows:
    1) For each triangle not adjacent to a boundary, we consider the 
       auxiliary triangle formed by the centroids of its three 
       neighbours.
    2) For each conserved quantity, we integrate around the auxiliary 
       triangle's boundary the product of the quantity and the outward 
       normal vector. Dividing by the triangle area gives (a,b), the 
       average of the vector (q_x,q_y) on the auxiliary triangle. 
       We suppose that the linear reconstruction on the original 
       triangle has gradient (a,b).
    3) Provisional vertex jumps dqv[0,1,2] are computed and these are 
       then limited by calling the functions find_qmin_and_qmax and 
       limit_gradient

    Python call:
    extrapolate_second_order_sw(domain.surrogate_neighbours,
                                domain.number_of_boundaries
                                domain.centroid_coordinates,
                                Stage.centroid_values
                                Xmom.centroid_values
                                Ymom.centroid_values
                                domain.vertex_coordinates,
                                Stage.vertex_values,
                                Xmom.vertex_values,
                                Ymom.vertex_values)

    Post conditions:
            The vertices of each triangle have values from a 
        limited linear reconstruction
        based on centroid values

  */
  PyArrayObject *surrogate_neighbours,
    *number_of_boundaries,
    *centroid_coordinates,
    *stage_centroid_values,
    *xmom_centroid_values,
    *ymom_centroid_values,
    *elevation_centroid_values,
    *vertex_coordinates,
    *stage_vertex_values,
    *xmom_vertex_values,
    *ymom_vertex_values,
    *elevation_vertex_values;
  
  PyObject *domain;

  
  double beta_w, beta_w_dry, beta_uh, beta_uh_dry, beta_vh, beta_vh_dry;    
  double minimum_allowed_height, epsilon;
  int optimise_dry_cells, number_of_elements, e;
  
  // Provisional jumps from centroids to v'tices and safety factor re limiting
  // by which these jumps are limited
  // Convert Python arguments to C
  if (!PyArg_ParseTuple(args, "OOOOOOOOOOOOOi",
            &domain,
            &surrogate_neighbours,
            &number_of_boundaries,
            &centroid_coordinates,
            &stage_centroid_values,
            &xmom_centroid_values,
            &ymom_centroid_values,
            &elevation_centroid_values,
            &vertex_coordinates,
            &stage_vertex_values,
            &xmom_vertex_values,
            &ymom_vertex_values,
            &elevation_vertex_values,
            &optimise_dry_cells)) {         

    report_python_error(AT, "could not parse input arguments");
    return NULL;
  }

  // check that numpy array objects arrays are C contiguous memory
  CHECK_C_CONTIG(surrogate_neighbours);
  CHECK_C_CONTIG(number_of_boundaries);
  CHECK_C_CONTIG(centroid_coordinates);
  CHECK_C_CONTIG(stage_centroid_values);
  CHECK_C_CONTIG(xmom_centroid_values);
  CHECK_C_CONTIG(ymom_centroid_values);
  CHECK_C_CONTIG(elevation_centroid_values);
  CHECK_C_CONTIG(vertex_coordinates);
  CHECK_C_CONTIG(stage_vertex_values);
  CHECK_C_CONTIG(xmom_vertex_values);
  CHECK_C_CONTIG(ymom_vertex_values);
  CHECK_C_CONTIG(elevation_vertex_values);
  
  // Get the safety factor beta_w, set in the config.py file. 
  // This is used in the limiting process
  

  beta_w                 = get_python_double(domain,"beta_w");
  beta_w_dry             = get_python_double(domain,"beta_w_dry");
  beta_uh                = get_python_double(domain,"beta_uh");
  beta_uh_dry            = get_python_double(domain,"beta_uh_dry");
  beta_vh                = get_python_double(domain,"beta_vh");
  beta_vh_dry            = get_python_double(domain,"beta_vh_dry");  

  minimum_allowed_height = get_python_double(domain,"minimum_allowed_height");
  epsilon                = get_python_double(domain,"epsilon");

  number_of_elements = stage_centroid_values -> dimensions[0];  

  // Call underlying computational routine
  e = _extrapolate_second_order_sw(number_of_elements,
                   epsilon,
                   minimum_allowed_height,
                   beta_w,
                   beta_w_dry,
                   beta_uh,
                   beta_uh_dry,
                   beta_vh,
                   beta_vh_dry,
                   (long*) surrogate_neighbours -> data,
                   (long*) number_of_boundaries -> data,
                   (double*) centroid_coordinates -> data,
                   (double*) stage_centroid_values -> data,
                   (double*) xmom_centroid_values -> data,
                   (double*) ymom_centroid_values -> data,
                   (double*) elevation_centroid_values -> data,
                   (double*) vertex_coordinates -> data,
                   (double*) stage_vertex_values -> data,
                   (double*) xmom_vertex_values -> data,
                   (double*) ymom_vertex_values -> data,
                   (double*) elevation_vertex_values -> data,
                   optimise_dry_cells);
  if (e == -1) {
    // Use error string set inside computational routine
    return NULL;
  }                   
  
  
  return Py_BuildValue("");
  
}// extrapolate_second-order_sw




PyObject *rotate(PyObject *self, PyObject *args, PyObject *kwargs) {
  //
  // r = rotate(q, normal, direction=1)
  //
  // Where q is assumed to be a Float numeric array of length 3 and
  // normal a Float numeric array of length 2.

  // FIXME(Ole): I don't think this is used anymore

  PyObject *Q, *Normal;
  PyArrayObject *q, *r, *normal;

  static char *argnames[] = {"q", "normal", "direction", NULL};
  int dimensions[1], i, direction=1;
  double n1, n2;

  // Convert Python arguments to C
  if (!PyArg_ParseTupleAndKeywords(args, kwargs, "OO|i", argnames,
                   &Q, &Normal, &direction)) {
    report_python_error(AT, "could not parse input arguments");
    return NULL;
  }  

  // Input checks (convert sequences into numeric arrays)
  q = (PyArrayObject *)
    PyArray_ContiguousFromObject(Q, PyArray_DOUBLE, 0, 0);
  normal = (PyArrayObject *)
    PyArray_ContiguousFromObject(Normal, PyArray_DOUBLE, 0, 0);


  if (normal -> dimensions[0] != 2) {
    report_python_error(AT, "Normal vector must have 2 components");
    return NULL;
  }

  // Allocate space for return vector r (don't DECREF)
  dimensions[0] = 3;
  r = (PyArrayObject *) anuga_FromDims(1, dimensions, PyArray_DOUBLE);

  // Copy
  for (i=0; i<3; i++) {
    ((double *) (r -> data))[i] = ((double *) (q -> data))[i];
  }

  // Get normal and direction
  n1 = ((double *) normal -> data)[0];
  n2 = ((double *) normal -> data)[1];
  if (direction == -1) n2 = -n2;

  // Rotate
  _rotate((double *) r -> data, n1, n2);

  // Release numeric arrays
  Py_DECREF(q);
  Py_DECREF(normal);

  // Return result using PyArray to avoid memory leak
  return PyArray_Return(r);
}


// Computational function for flux computation

double _compute_fluxes_central(int number_of_elements,
        double timestep,
        double epsilon,
        double H0,
        double g,
        long* neighbours,
        long* neighbour_edges,
        double* normals,
        double* edgelengths,
        double* radii,
        double* areas,
        long* tri_full_flag,
        double* stage_edge_values,
        double* xmom_edge_values,
        double* ymom_edge_values,
        double* bed_edge_values,
        double* stage_boundary_values,
        double* xmom_boundary_values,
        double* ymom_boundary_values,
        double* stage_explicit_update,
        double* xmom_explicit_update,
        double* ymom_explicit_update,
        long* already_computed_flux,
        double* max_speed_array,
        int optimise_dry_cells) {
    // Local variables
    double max_speed, length, inv_area, zl, zr;
    double h0 = H0*H0; // This ensures a good balance when h approaches H0.

    double limiting_threshold = 10 * H0; // Avoid applying limiter below this
    // threshold for performance reasons.
    // See ANUGA manual under flux limiting
    int k, i, m, n;
    int ki, nm = 0, ki2; // Index shorthands


    // Workspace (making them static actually made function slightly slower (Ole))
    double ql[3], qr[3], edgeflux[3]; // Work array for summing up fluxes

    static long call = 1; // Static local variable flagging already computed flux

    // Start computation
    call++; // Flag 'id' of flux calculation for this timestep

    // Set explicit_update to zero for all conserved_quantities.
    // This assumes compute_fluxes called before forcing terms
    memset((char*) stage_explicit_update, 0, number_of_elements * sizeof (double));
    memset((char*) xmom_explicit_update, 0, number_of_elements * sizeof (double));
    memset((char*) ymom_explicit_update, 0, number_of_elements * sizeof (double));

    // For all triangles
    for (k = 0; k < number_of_elements; k++) {
        // Loop through neighbours and compute edge flux for each
        for (i = 0; i < 3; i++) {
            ki = k * 3 + i; // Linear index to edge i of triangle k

            if (already_computed_flux[ki] == call) {
                // We've already computed the flux across this edge
                continue;
            }

            // Get left hand side values from triangle k, edge i
            ql[0] = stage_edge_values[ki];
            ql[1] = xmom_edge_values[ki];
            ql[2] = ymom_edge_values[ki];
            zl = bed_edge_values[ki];

            // Get right hand side values either from neighbouring triangle
            // or from boundary array (Quantities at neighbour on nearest face).
            n = neighbours[ki];
            if (n < 0) {
                // Neighbour is a boundary condition
                m = -n - 1; // Convert negative flag to boundary index

                qr[0] = stage_boundary_values[m];
                qr[1] = xmom_boundary_values[m];
                qr[2] = ymom_boundary_values[m];
                zr = zl; // Extend bed elevation to boundary
            }
            else {
                // Neighbour is a real triangle
                m = neighbour_edges[ki];
                nm = n * 3 + m; // Linear index (triangle n, edge m)

                qr[0] = stage_edge_values[nm];
                qr[1] = xmom_edge_values[nm];
                qr[2] = ymom_edge_values[nm];
                zr = bed_edge_values[nm];
            }

            // Now we have values for this edge - both from left and right side.

            if (optimise_dry_cells) {
                // Check if flux calculation is necessary across this edge
                // This check will exclude dry cells.
                // This will also optimise cases where zl != zr as
                // long as both are dry

                if (fabs(ql[0] - zl) < epsilon &&
                        fabs(qr[0] - zr) < epsilon) {
                    // Cell boundary is dry

                    already_computed_flux[ki] = call; // #k Done
                    if (n >= 0) {
                        already_computed_flux[nm] = call; // #n Done
                    }

                    max_speed = 0.0;
                    continue;
                }
            }


            if (fabs(zl-zr)>1.0e-10) {
                report_python_error(AT,"Discontinuous Elevation");
                return 0.0;
            }
            
            // Outward pointing normal vector (domain.normals[k, 2*i:2*i+2])
            ki2 = 2 * ki; //k*6 + i*2

            // Edge flux computation (triangle k, edge i)
            _flux_function_central(ql, qr, zl, zr,
                    normals[ki2], normals[ki2 + 1],
                    epsilon, h0, limiting_threshold, g,
                    edgeflux, &max_speed);


            // Multiply edgeflux by edgelength
            length = edgelengths[ki];
            edgeflux[0] *= length;
            edgeflux[1] *= length;
            edgeflux[2] *= length;


            // Update triangle k with flux from edge i
            stage_explicit_update[k] -= edgeflux[0];
            xmom_explicit_update[k] -= edgeflux[1];
            ymom_explicit_update[k] -= edgeflux[2];

            already_computed_flux[ki] = call; // #k Done


            // Update neighbour n with same flux but reversed sign
            if (n >= 0) {
                stage_explicit_update[n] += edgeflux[0];
                xmom_explicit_update[n] += edgeflux[1];
                ymom_explicit_update[n] += edgeflux[2];

                already_computed_flux[nm] = call; // #n Done
            }

            // Update timestep based on edge i and possibly neighbour n
            if (tri_full_flag[k] == 1) {
                if (max_speed > epsilon) {
                    // Apply CFL condition for triangles joining this edge (triangle k and triangle n)

                    // CFL for triangle k
                    timestep = min(timestep, radii[k] / max_speed);

                    if (n >= 0) {
                        // Apply CFL condition for neigbour n (which is on the ith edge of triangle k)
                        timestep = min(timestep, radii[n] / max_speed);
                    }

                    // Ted Rigby's suggested less conservative version
                    //if (n>=0) {
                    //  timestep = min(timestep, (radii[k]+radii[n])/max_speed);
                    //} else {
                    //  timestep = min(timestep, radii[k]/max_speed);
                    // }
                }
            }

        } // End edge i (and neighbour n)


        // Normalise triangle k by area and store for when all conserved
        // quantities get updated
        inv_area = 1.0 / areas[k];
        stage_explicit_update[k] *= inv_area;
        xmom_explicit_update[k] *= inv_area;
        ymom_explicit_update[k] *= inv_area;


        // Keep track of maximal speeds
        max_speed_array[k] = max_speed;

    } // End triangle k

    return timestep;
}

//=========================================================================
// Python Glue
//=========================================================================

PyObject *compute_fluxes_ext_central_new(PyObject *self, PyObject *args) {
  /*Compute all fluxes and the timestep suitable for all volumes
    in domain.

    Compute total flux for each conserved quantity using "flux_function_central"

    Fluxes across each edge are scaled by edgelengths and summed up
    Resulting flux is then scaled by area and stored in
    explicit_update for each of the three conserved quantities
    stage, xmomentum and ymomentum

    The maximal allowable speed computed by the flux_function for each volume
    is converted to a timestep that must not be exceeded. The minimum of
    those is computed as the next overall timestep.

    Python call:
    timestep = compute_fluxes(timestep, domain, stage, xmom, ymom, bed)


    Post conditions:
      domain.explicit_update is reset to computed flux values
      returns timestep which is the largest step satisfying all volumes.


  */

    PyObject 
    *domain,
    *stage, 
    *xmom, 
    *ymom, 
    *bed;

    PyArrayObject 
    *neighbours, 
    *neighbour_edges,
    *normals, 
    *edgelengths, 
    *radii, 
    *areas,
    *tri_full_flag,
    *stage_edge_values,
    *xmom_edge_values,
    *ymom_edge_values,
    *bed_edge_values,
    *stage_boundary_values,
    *xmom_boundary_values,
    *ymom_boundary_values,
    *stage_explicit_update,
    *xmom_explicit_update,
    *ymom_explicit_update,
    *already_computed_flux, //Tracks whether the flux across an edge has already been computed
    *max_speed_array; //Keeps track of max speeds for each triangle

    
    double timestep, epsilon, H0, g;
    int optimise_dry_cells;
    
    // Convert Python arguments to C
    if (!PyArg_ParseTuple(args, "dOOOO", &timestep, &domain, &stage, &xmom, &ymom, &bed )) {
      report_python_error(AT, "could not parse input arguments");
      return NULL;
    }

    epsilon           = get_python_double(domain,"epsilon");
    H0                = get_python_double(domain,"H0");
    g                 = get_python_double(domain,"g");
    optimise_dry_cells = get_python_integer(domain,"optimse_dry_cells");
    
    neighbours        = get_consecutive_array(domain, "neighbours");
    neighbour_edges   = get_consecutive_array(domain, "neighbour_edges"); 
    normals           = get_consecutive_array(domain, "normals");
    edgelengths       = get_consecutive_array(domain, "edge_lengths");    
    radii             = get_consecutive_array(domain, "radii");    
    areas             = get_consecutive_array(domain, "areas");    
    tri_full_flag     = get_consecutive_array(domain, "tri_full_flag");
    already_computed_flux  = get_consecutive_array(domain, "already_computed_flux");
    max_speed_array   = get_consecutive_array(domain, "max_speed");
    
    stage_edge_values = get_consecutive_array(stage, "edge_values");    
    xmom_edge_values  = get_consecutive_array(xmom, "edge_values");    
    ymom_edge_values  = get_consecutive_array(ymom, "edge_values"); 
    bed_edge_values   = get_consecutive_array(bed, "edge_values");    

    stage_boundary_values = get_consecutive_array(stage, "boundary_values");    
    xmom_boundary_values  = get_consecutive_array(xmom, "boundary_values");    
    ymom_boundary_values  = get_consecutive_array(ymom, "boundary_values"); 

    stage_explicit_update = get_consecutive_array(stage, "explicit_update");    
    xmom_explicit_update  = get_consecutive_array(xmom, "explicit_update");    
    ymom_explicit_update  = get_consecutive_array(ymom, "explicit_update"); 


  int number_of_elements = stage_edge_values -> dimensions[0];

  // Call underlying flux computation routine and update 
  // the explicit update arrays 
  timestep = _compute_fluxes_central(number_of_elements,
                     timestep,
                     epsilon,
                     H0,
                     g,
                     (long*) neighbours -> data,
                     (long*) neighbour_edges -> data,
                     (double*) normals -> data,
                     (double*) edgelengths -> data, 
                     (double*) radii -> data, 
                     (double*) areas -> data,
                     (long*) tri_full_flag -> data,
                     (double*) stage_edge_values -> data,
                     (double*) xmom_edge_values -> data,
                     (double*) ymom_edge_values -> data,
                     (double*) bed_edge_values -> data,
                     (double*) stage_boundary_values -> data,
                     (double*) xmom_boundary_values -> data,
                     (double*) ymom_boundary_values -> data,
                     (double*) stage_explicit_update -> data,
                     (double*) xmom_explicit_update -> data,
                     (double*) ymom_explicit_update -> data,
                     (long*) already_computed_flux -> data,
                     (double*) max_speed_array -> data,
                     optimise_dry_cells);

  Py_DECREF(neighbours);
  Py_DECREF(neighbour_edges);
  Py_DECREF(normals);
  Py_DECREF(edgelengths);
  Py_DECREF(radii);
  Py_DECREF(areas);
  Py_DECREF(tri_full_flag);
  Py_DECREF(already_computed_flux);
  Py_DECREF(max_speed_array);
  Py_DECREF(stage_edge_values);
  Py_DECREF(xmom_edge_values);
  Py_DECREF(ymom_edge_values);
  Py_DECREF(bed_edge_values);
  Py_DECREF(stage_boundary_values);
  Py_DECREF(xmom_boundary_values);
  Py_DECREF(ymom_boundary_values);
  Py_DECREF(stage_explicit_update);
  Py_DECREF(xmom_explicit_update);
  Py_DECREF(ymom_explicit_update);

  
  // Return updated flux timestep
  return Py_BuildValue("d", timestep);
}






PyObject *compute_fluxes_ext_central(PyObject *self, PyObject *args) {
  /*Compute all fluxes and the timestep suitable for all volumes
    in domain.

    Compute total flux for each conserved quantity using "flux_function_central"

    Fluxes across each edge are scaled by edgelengths and summed up
    Resulting flux is then scaled by area and stored in
    explicit_update for each of the three conserved quantities
    stage, xmomentum and ymomentum

    The maximal allowable speed computed by the flux_function for each volume
    is converted to a timestep that must not be exceeded. The minimum of
    those is computed as the next overall timestep.

    Python call:
    domain.timestep = compute_fluxes(timestep,
                                     domain.epsilon,
                     domain.H0,
                                     domain.g,
                                     domain.neighbours,
                                     domain.neighbour_edges,
                                     domain.normals,
                                     domain.edgelengths,
                                     domain.radii,
                                     domain.areas,
                                     tri_full_flag,
                                     Stage.edge_values,
                                     Xmom.edge_values,
                                     Ymom.edge_values,
                                     Bed.edge_values,
                                     Stage.boundary_values,
                                     Xmom.boundary_values,
                                     Ymom.boundary_values,
                                     Stage.explicit_update,
                                     Xmom.explicit_update,
                                     Ymom.explicit_update,
                                     already_computed_flux,
                     optimise_dry_cells)                     


    Post conditions:
      domain.explicit_update is reset to computed flux values
      domain.timestep is set to the largest step satisfying all volumes.


  */


  PyArrayObject *neighbours, *neighbour_edges,
    *normals, *edgelengths, *radii, *areas,
    *tri_full_flag,
    *stage_edge_values,
    *xmom_edge_values,
    *ymom_edge_values,
    *bed_edge_values,
    *stage_boundary_values,
    *xmom_boundary_values,
    *ymom_boundary_values,
    *stage_explicit_update,
    *xmom_explicit_update,
    *ymom_explicit_update,
    *already_computed_flux, //Tracks whether the flux across an edge has already been computed
    *max_speed_array; //Keeps track of max speeds for each triangle

    
  double timestep, epsilon, H0, g;
  int optimise_dry_cells;
    
  // Convert Python arguments to C
  if (!PyArg_ParseTuple(args, "ddddOOOOOOOOOOOOOOOOOOOi",
            &timestep,
            &epsilon,
            &H0,
            &g,
            &neighbours,
            &neighbour_edges,
            &normals,
            &edgelengths, &radii, &areas,
            &tri_full_flag,
            &stage_edge_values,
            &xmom_edge_values,
            &ymom_edge_values,
            &bed_edge_values,
            &stage_boundary_values,
            &xmom_boundary_values,
            &ymom_boundary_values,
            &stage_explicit_update,
            &xmom_explicit_update,
            &ymom_explicit_update,
            &already_computed_flux,
            &max_speed_array,
            &optimise_dry_cells)) {
    report_python_error(AT, "could not parse input arguments");
    return NULL;
  }

  // check that numpy array objects arrays are C contiguous memory
  CHECK_C_CONTIG(neighbours);
  CHECK_C_CONTIG(neighbour_edges);
  CHECK_C_CONTIG(normals);
  CHECK_C_CONTIG(edgelengths);
  CHECK_C_CONTIG(radii);
  CHECK_C_CONTIG(areas);
  CHECK_C_CONTIG(tri_full_flag);
  CHECK_C_CONTIG(stage_edge_values);
  CHECK_C_CONTIG(xmom_edge_values);
  CHECK_C_CONTIG(ymom_edge_values);
  CHECK_C_CONTIG(bed_edge_values);
  CHECK_C_CONTIG(stage_boundary_values);
  CHECK_C_CONTIG(xmom_boundary_values);
  CHECK_C_CONTIG(ymom_boundary_values);
  CHECK_C_CONTIG(stage_explicit_update);
  CHECK_C_CONTIG(xmom_explicit_update);
  CHECK_C_CONTIG(ymom_explicit_update);
  CHECK_C_CONTIG(already_computed_flux);
  CHECK_C_CONTIG(max_speed_array);

  int number_of_elements = stage_edge_values -> dimensions[0];

  // Call underlying flux computation routine and update 
  // the explicit update arrays 
  timestep = _compute_fluxes_central(number_of_elements,
                     timestep,
                     epsilon,
                     H0,
                     g,
                     (long*) neighbours -> data,
                     (long*) neighbour_edges -> data,
                     (double*) normals -> data,
                     (double*) edgelengths -> data, 
                     (double*) radii -> data, 
                     (double*) areas -> data,
                     (long*) tri_full_flag -> data,
                     (double*) stage_edge_values -> data,
                     (double*) xmom_edge_values -> data,
                     (double*) ymom_edge_values -> data,
                     (double*) bed_edge_values -> data,
                     (double*) stage_boundary_values -> data,
                     (double*) xmom_boundary_values -> data,
                     (double*) ymom_boundary_values -> data,
                     (double*) stage_explicit_update -> data,
                     (double*) xmom_explicit_update -> data,
                     (double*) ymom_explicit_update -> data,
                     (long*) already_computed_flux -> data,
                     (double*) max_speed_array -> data,
                     optimise_dry_cells);
  
  // Return updated flux timestep
  return Py_BuildValue("d", timestep);
}





PyObject *compute_fluxes_ext_kinetic(PyObject *self, PyObject *args) {
  /*Compute all fluxes and the timestep suitable for all volumes
    in domain.

    THIS IS AN EXPERIMENTAL FUNCTION - NORMALLY flux_function_central IS USED.
    
    Compute total flux for each conserved quantity using "flux_function_kinetic"

    Fluxes across each edge are scaled by edgelengths and summed up
    Resulting flux is then scaled by area and stored in
    explicit_update for each of the three conserved quantities
    stage, xmomentum and ymomentum

    The maximal allowable speed computed by the flux_function for each volume
    is converted to a timestep that must not be exceeded. The minimum of
    those is computed as the next overall timestep.

    Python call:
    domain.timestep = compute_fluxes(timestep,
                                     domain.epsilon,
                                     domain.H0,
                                     domain.g,
                                     domain.neighbours,
                                     domain.neighbour_edges,
                                     domain.normals,
                                     domain.edgelengths,
                                     domain.radii,
                                     domain.areas,
                                     Stage.edge_values,
                                     Xmom.edge_values,
                                     Ymom.edge_values,
                                     Bed.edge_values,
                                     Stage.boundary_values,
                                     Xmom.boundary_values,
                                     Ymom.boundary_values,
                                     Stage.explicit_update,
                                     Xmom.explicit_update,
                                     Ymom.explicit_update,
                                     already_computed_flux)


    Post conditions:
      domain.explicit_update is reset to computed flux values
      domain.timestep is set to the largest step satisfying all volumes.


  */


  PyArrayObject *neighbours, *neighbour_edges,
    *normals, *edgelengths, *radii, *areas,
    *tri_full_flag,
    *stage_edge_values,
    *xmom_edge_values,
    *ymom_edge_values,
    *bed_edge_values,
    *stage_boundary_values,
    *xmom_boundary_values,
    *ymom_boundary_values,
    *stage_explicit_update,
    *xmom_explicit_update,
    *ymom_explicit_update,
    *already_computed_flux; // Tracks whether the flux across an edge has already been computed


  // Local variables
  double timestep, max_speed, epsilon, g, H0;
  double normal[2], ql[3], qr[3], zl, zr;
  double edgeflux[3]; //Work arrays for summing up fluxes

  int number_of_elements, k, i, m, n;
  int ki, nm=0, ki2; //Index shorthands
  static long call=1;


  // Convert Python arguments to C
  if (!PyArg_ParseTuple(args, "ddddOOOOOOOOOOOOOOOOOO",
            &timestep,
            &epsilon,
            &H0,
            &g,
            &neighbours,
            &neighbour_edges,
            &normals,
            &edgelengths, &radii, &areas,
            &tri_full_flag,
            &stage_edge_values,
            &xmom_edge_values,
            &ymom_edge_values,
            &bed_edge_values,
            &stage_boundary_values,
            &xmom_boundary_values,
            &ymom_boundary_values,
            &stage_explicit_update,
            &xmom_explicit_update,
            &ymom_explicit_update,
            &already_computed_flux)) {
    report_python_error(AT, "could not parse input arguments");
    return NULL;
  }
  number_of_elements = stage_edge_values -> dimensions[0];
  call++;//a static local variable to which already_computed_flux is compared
  //set explicit_update to zero for all conserved_quantities.
  //This assumes compute_fluxes called before forcing terms
  for (k=0; k<number_of_elements; k++) {
    ((double *) stage_explicit_update -> data)[k]=0.0;
    ((double *) xmom_explicit_update -> data)[k]=0.0;
    ((double *) ymom_explicit_update -> data)[k]=0.0;
  }
  //Loop through neighbours and compute edge flux for each
  for (k=0; k<number_of_elements; k++) {
    for (i=0; i<3; i++) {
      ki = k*3+i;
      if (((long *) already_computed_flux->data)[ki]==call)//we've already computed the flux across this edge
    continue;
      ql[0] = ((double *) stage_edge_values -> data)[ki];
      ql[1] = ((double *) xmom_edge_values -> data)[ki];
      ql[2] = ((double *) ymom_edge_values -> data)[ki];
      zl =    ((double *) bed_edge_values -> data)[ki];

      //Quantities at neighbour on nearest face
      n = ((long *) neighbours -> data)[ki];
      if (n < 0) {
    m = -n-1; //Convert negative flag to index
    qr[0] = ((double *) stage_boundary_values -> data)[m];
    qr[1] = ((double *) xmom_boundary_values -> data)[m];
    qr[2] = ((double *) ymom_boundary_values -> data)[m];
    zr = zl; //Extend bed elevation to boundary
      } else {
    m = ((long *) neighbour_edges -> data)[ki];
    nm = n*3+m;
    qr[0] = ((double *) stage_edge_values -> data)[nm];
    qr[1] = ((double *) xmom_edge_values -> data)[nm];
    qr[2] = ((double *) ymom_edge_values -> data)[nm];
    zr =    ((double *) bed_edge_values -> data)[nm];
      }
      // Outward pointing normal vector
      // normal = domain.normals[k, 2*i:2*i+2]
      ki2 = 2*ki; //k*6 + i*2
      normal[0] = ((double *) normals -> data)[ki2];
      normal[1] = ((double *) normals -> data)[ki2+1];
      //Edge flux computation
      flux_function_kinetic(ql, qr, zl, zr,
            normal[0], normal[1],
            epsilon, H0, g,
            edgeflux, &max_speed);
      //update triangle k
      ((long *) already_computed_flux->data)[ki]=call;
      ((double *) stage_explicit_update -> data)[k] -= edgeflux[0]*((double *) edgelengths -> data)[ki];
      ((double *) xmom_explicit_update -> data)[k] -= edgeflux[1]*((double *) edgelengths -> data)[ki];
      ((double *) ymom_explicit_update -> data)[k] -= edgeflux[2]*((double *) edgelengths -> data)[ki];
      //update the neighbour n
      if (n>=0){
    ((long *) already_computed_flux->data)[nm]=call;
    ((double *) stage_explicit_update -> data)[n] += edgeflux[0]*((double *) edgelengths -> data)[nm];
    ((double *) xmom_explicit_update -> data)[n] += edgeflux[1]*((double *) edgelengths -> data)[nm];
    ((double *) ymom_explicit_update -> data)[n] += edgeflux[2]*((double *) edgelengths -> data)[nm];
      }
      ///for (j=0; j<3; j++) {
    ///flux[j] -= edgeflux[j]*((double *) edgelengths -> data)[ki];
    ///}
    //Update timestep
    //timestep = min(timestep, domain.radii[k]/max_speed)
    //FIXME: SR Add parameter for CFL condition
    if ( ((long *) tri_full_flag -> data)[k] == 1) {
        if (max_speed > epsilon) {
            timestep = min(timestep, ((double *) radii -> data)[k]/max_speed);
            //maxspeed in flux_function is calculated as max(|u+a|,|u-a|)
            if (n>=0)
                timestep = min(timestep, ((double *) radii -> data)[n]/max_speed);
        }
    }
    } // end for i
    //Normalise by area and store for when all conserved
    //quantities get updated
    ((double *) stage_explicit_update -> data)[k] /= ((double *) areas -> data)[k];
    ((double *) xmom_explicit_update -> data)[k] /= ((double *) areas -> data)[k];
    ((double *) ymom_explicit_update -> data)[k] /= ((double *) areas -> data)[k];
  } //end for k
  return Py_BuildValue("d", timestep);
}

PyObject *protect(PyObject *self, PyObject *args) {
  //
  //    protect(minimum_allowed_height, maximum_allowed_speed, wc, zc, xmomc, ymomc)


  PyArrayObject
  *wc,            //Stage at centroids
  *zc,            //Elevation at centroids
  *xmomc,         //Momentums at centroids
  *ymomc;


  int N;
  double minimum_allowed_height, maximum_allowed_speed, epsilon;

  // Convert Python arguments to C
  if (!PyArg_ParseTuple(args, "dddOOOO",
            &minimum_allowed_height,
            &maximum_allowed_speed,
            &epsilon,
            &wc, &zc, &xmomc, &ymomc)) {
    report_python_error(AT, "could not parse input arguments");
    return NULL;
  }  

  N = wc -> dimensions[0];

  _protect(N,
       minimum_allowed_height,
       maximum_allowed_speed,
       epsilon,
       (double*) wc -> data,
       (double*) zc -> data,
       (double*) xmomc -> data,
       (double*) ymomc -> data);

  return Py_BuildValue("");
}



PyObject *balance_deep_and_shallow(PyObject *self, PyObject *args) {
  // Compute linear combination between stage as computed by
  // gradient-limiters limiting using w, and stage computed by
  // gradient-limiters limiting using h (h-limiter).
  // The former takes precedence when heights are large compared to the
  // bed slope while the latter takes precedence when heights are
  // relatively small.  Anything in between is computed as a balanced
  // linear combination in order to avoid numerical disturbances which
  // would otherwise appear as a result of hard switching between
  // modes.
  //
  //    balance_deep_and_shallow(wc, zc, wv, zv,
  //                             xmomc, ymomc, xmomv, ymomv)


  PyArrayObject
    *wc,            //Stage at centroids
    *zc,            //Elevation at centroids
    *wv,            //Stage at vertices
    *zv,            //Elevation at vertices
    *hvbar,         //h-Limited depths at vertices
    *xmomc,         //Momentums at centroids and vertices
    *ymomc,
    *xmomv,
    *ymomv;
  
  PyObject *domain, *Tmp;
    
  double alpha_balance = 2.0;
  double H0;

  int N, tight_slope_limiters, use_centroid_velocities; //, err;

  // Convert Python arguments to C
  if (!PyArg_ParseTuple(args, "OOOOOOOOOO",
            &domain,
            &wc, &zc, 
            &wv, &zv, &hvbar,
            &xmomc, &ymomc, &xmomv, &ymomv)) {
    report_python_error(AT, "could not parse input arguments");
    return NULL;
  }  
      
      
  // FIXME (Ole): I tested this without GetAttrString and got time down
  // marginally from 4.0s to 3.8s. Consider passing everything in 
  // through ParseTuple and profile.
  
  // Pull out parameters
  Tmp = PyObject_GetAttrString(domain, "alpha_balance");
  if (!Tmp) {
    report_python_error(AT, "could not obtain object alpha_balance from domain");
    return NULL;
  }  
  alpha_balance = PyFloat_AsDouble(Tmp);
  Py_DECREF(Tmp);

  
  Tmp = PyObject_GetAttrString(domain, "H0");
  if (!Tmp) {
    report_python_error(AT, "could not obtain object H0 from domain");
    return NULL;
  }  
  H0 = PyFloat_AsDouble(Tmp);
  Py_DECREF(Tmp);

  
  Tmp = PyObject_GetAttrString(domain, "tight_slope_limiters");
  if (!Tmp) {
    report_python_error(AT, "could not obtain object tight_slope_limiters from domain");
    return NULL;
  }  
  tight_slope_limiters = PyInt_AsLong(Tmp);
  Py_DECREF(Tmp);
  
  
  Tmp = PyObject_GetAttrString(domain, "use_centroid_velocities");
  if (!Tmp) {
    report_python_error(AT, "could not obtain object use_centroid_velocities from domain");
    return NULL;
  }  
  use_centroid_velocities = PyInt_AsLong(Tmp);
  Py_DECREF(Tmp);
  
  
  
  N = wc -> dimensions[0];
  _balance_deep_and_shallow(N,
                (double*) wc -> data,
                (double*) zc -> data,
                (double*) wv -> data,
                (double*) zv -> data,
                (double*) hvbar -> data,
                (double*) xmomc -> data,
                (double*) ymomc -> data,
                (double*) xmomv -> data,
                (double*) ymomv -> data,
                H0,
                (int) tight_slope_limiters,
                (int) use_centroid_velocities,              
                alpha_balance);


  return Py_BuildValue("");
}




PyObject *assign_windfield_values(PyObject *self, PyObject *args) {
  //
  //      assign_windfield_values(xmom_update, ymom_update,
  //                              s_vec, phi_vec, self.const)



  PyArrayObject   //(one element per triangle)
  *s_vec,         //Speeds
  *phi_vec,       //Bearings
  *xmom_update,   //Momentum updates
  *ymom_update;


  int N;
  double cw;

  // Convert Python arguments to C
  if (!PyArg_ParseTuple(args, "OOOOd",
            &xmom_update,
            &ymom_update,
            &s_vec, &phi_vec,
            &cw)) {
    report_python_error(AT, "could not parse input arguments");
    return NULL;
  }  
            

  N = xmom_update -> dimensions[0];

  _assign_wind_field_values(N,
       (double*) xmom_update -> data,
       (double*) ymom_update -> data,
       (double*) s_vec -> data,
       (double*) phi_vec -> data,
       cw);

  return Py_BuildValue("");
}




//-------------------------------
// Method table for python module
//-------------------------------
static struct PyMethodDef MethodTable[] = {
  /* The cast of the function is necessary since PyCFunction values
   * only take two PyObject* parameters, and rotate() takes
   * three.
   */

  {"rotate", (PyCFunction)rotate, METH_VARARGS | METH_KEYWORDS, "Print out"},
  {"extrapolate_second_order_sw", extrapolate_second_order_sw, METH_VARARGS, "Print out"},
  {"compute_fluxes_ext_central", compute_fluxes_ext_central, METH_VARARGS, "Print out"},
  {"compute_fluxes_ext_central_new", compute_fluxes_ext_central_new, METH_VARARGS, "Print out"},
  {"compute_fluxes_ext_kinetic", compute_fluxes_ext_kinetic, METH_VARARGS, "Print out"},
  {"gravity", gravity, METH_VARARGS, "Print out"},
  {"manning_friction_flat", manning_friction_flat, METH_VARARGS, "Print out"},
  {"manning_friction_sloped", manning_friction_sloped, METH_VARARGS, "Print out"},
  {"chezy_friction", chezy_friction, METH_VARARGS, "Print out"},
  {"flux_function_central", flux_function_central, METH_VARARGS, "Print out"},  
  {"balance_deep_and_shallow", balance_deep_and_shallow,
   METH_VARARGS, "Print out"},
  {"protect", protect, METH_VARARGS | METH_KEYWORDS, "Print out"},
  {"assign_windfield_values", assign_windfield_values,
   METH_VARARGS | METH_KEYWORDS, "Print out"},
  {NULL, NULL}
};

// Module initialisation
void initshallow_water_ext(void){
  Py_InitModule("shallow_water_ext", MethodTable);

  import_array(); // Necessary for handling of NumPY structures
}