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#include <stdio.h>
#include <string.h>
#include <math.h>
#include <assert.h>
#include "cctk.h"
#include "cctk_Arguments.h"
#include "cctk_Parameters.h"
/*
We will want to integrate functions F(th,ph) from th = 0 to pi, ph = 0
to 2 pi with a weighting function sin(th). Alternatively, we might
want to use u = cos(th) as the variable, in which case we will go from
u = -1 to 1 and ph = 0 to 2 pi. For simplicity, we implement an
integration routine with a weight function of 1, and require the user
to multiply the integrand by their own weight function. We divide the
interval [a,b] into nx subintervals of spacing h = (b-a)/nx. These
have coordinates [x_i-1, xi] where x_i = x_0 + i h. So i runs from 0
to nx. We require the function to integrate at the points F[x_i,
y_i]. We have x_0 = a and x_n = b. Check: x_n = x_0 + n (b-a)/n = a
+ b - a = b. Good.
If we are given these points in an array, we also need the width and
height of the array. To get an actual integral, we also need the grid
spacing hx and hy, but these are just multiplied by the result to give
the integral.
*/
#define idx(xx,yy) (assert((xx) <= nx), assert((xx) >= 0), assert((yy) <= ny), assert((yy) >= 0), ((xx) + (yy) * (nx+1)))
// Hard coded 2D integrals
static CCTK_REAL Trapezoidal2DIntegral(CCTK_REAL const *f, int nx, int ny, CCTK_REAL hx, CCTK_REAL hy)
{
CCTK_REAL integrand_sum = 0.0;
int ix = 0, iy = 0;
assert(nx > 0); assert(ny > 0); assert (f);
// Corners
integrand_sum += f[idx(0,0)] + f[idx(nx,0)] + f[idx(0,ny)] + f[idx(nx,ny)];
// Edges
for (ix = 1; ix <= nx-1; ix++)
integrand_sum += 2 * f[idx(ix,0)] + 2 * f[idx(ix,ny)];
for (iy = 1; iy <= ny-1; iy++)
integrand_sum += 2 * f[idx(0,iy)] + 2 * f[idx(nx,iy)];
// Interior
for (iy = 1; iy <= ny-1; iy++)
for (ix = 1; ix <= nx-1; ix++)
integrand_sum += 4 * f[idx(ix,iy)];
return (double) 1 / (double) 4 * hx * hy * integrand_sum;
}
CCTK_REAL Simpson2DIntegral(CCTK_REAL const *f, int nx, int ny, CCTK_REAL hx, CCTK_REAL hy)
{
CCTK_REAL integrand_sum = 0;
int ix = 0, iy = 0;
assert(nx > 0); assert(ny > 0); assert (f);
assert(nx % 2 == 0);
assert(ny % 2 == 0);
int px = nx / 2;
int py = ny / 2;
// Corners
integrand_sum += f[idx(0,0)] + f[idx(nx,0)] + f[idx(0,ny)] + f[idx(nx,ny)];
// Edges
for (iy = 1; iy <= py; iy++)
integrand_sum += 4 * f[idx(0,2*iy-1)] + 4 * f[idx(nx,2*iy-1)];
for (iy = 1; iy <= py-1; iy++)
integrand_sum += 2 * f[idx(0,2*iy)] + 2 * f[idx(nx,2*iy)];
for (ix = 1; ix <= px; ix++)
integrand_sum += 4 * f[idx(2*ix-1,0)] + 4 * f[idx(2*ix-1,ny)];
for (ix = 1; ix <= px-1; ix++)
integrand_sum += 2 * f[idx(2*ix,0)] + 2 * f[idx(2*ix,ny)];
// Interior
for (iy = 1; iy <= py; iy++)
for (ix = 1; ix <= px; ix++)
integrand_sum += 16 * f[idx(2*ix-1,2*iy-1)];
for (iy = 1; iy <= py-1; iy++)
for (ix = 1; ix <= px; ix++)
integrand_sum += 8 * f[idx(2*ix-1,2*iy)];
for (iy = 1; iy <= py; iy++)
for (ix = 1; ix <= px-1; ix++)
integrand_sum += 8 * f[idx(2*ix,2*iy-1)];
for (iy = 1; iy <= py-1; iy++)
for (ix = 1; ix <= px-1; ix++)
integrand_sum += 4 * f[idx(2*ix,2*iy)];
return ((double) 1 / (double) 9) * hx * hy * integrand_sum;
}
// 1D integrals
static CCTK_REAL Simpson1DIntegral(CCTK_REAL const *f, int n, CCTK_REAL h)
{
CCTK_REAL integrand_sum = 0;
int i = 0;
assert(n > 0); assert(f);
assert(n % 2 == 0);
int p = n / 2;
integrand_sum += f[0] + f[n];
for (i = 1; i <= p-1; i++)
integrand_sum += 4 * f[2*i-1] + 2 * f[2*i];
integrand_sum += 4 * f[2*p-1];
return 1.0/3.0 * h * integrand_sum;
}
// 2D integral built up from 1D
static CCTK_REAL Composite2DIntegral(CCTK_REAL const *f, int nx, int ny, CCTK_REAL hx, CCTK_REAL hy)
{
CCTK_REAL integrand_sum = 0;
assert(nx > 0); assert(ny > 0); assert (f);
assert(nx % 2 == 0);
assert(ny % 2 == 0);
CCTK_REAL *g = new CCTK_REAL[ny+1];
for (int i = 0; i <= ny; i++)
{
g[i] = Simpson1DIntegral(&f[idx(0,i)], nx, hx);
}
integrand_sum = Simpson1DIntegral(g, ny, hy);
delete [] g;
return integrand_sum;
}
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