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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;
}
// See: J.R. Driscoll and D.M. Healy Jr., Computing Fourier transforms
// and convolutions on the 2-sphere. Advances in Applied Mathematics,
// 15(2):202–250, 1994.
CCTK_REAL DriscollHealy2DIntegral(CCTK_REAL const *const f,
int const nx, int const ny,
CCTK_REAL const hx, CCTK_REAL const hy)
{
assert(f);
assert(nx >= 0);
assert(ny >= 0);
assert(nx % 2 == 0);
CCTK_REAL integrand_sum = 0.0;
// Skip the poles (ix=0 and ix=nx), since the weight there is zero
// anyway
#pragma omp parallel for reduction(+: integrand_sum)
for (int ix = 1; ix < nx; ++ ix)
{
// These weights lead to an almost spectral convergence
CCTK_REAL const theta = M_PI * ix / nx;
CCTK_REAL weight = 0.0;
for (int l = 0; l < nx/2; ++ l)
{
weight += sin((2*l+1)*theta)/(2*l+1);
}
weight *= 4.0 / M_PI;
CCTK_REAL local_sum = 0.0;
// Skip the last point (iy=ny), since we assume periodicity and
// therefor it has the same value as the first point. We don't use
// weights in this direction, which leads to spectral convergence.
// (Yay periodicity!)
for (int iy = 0; iy < ny; ++ iy)
{
local_sum += f[idx(ix,iy)];
}
integrand_sum += weight * local_sum;
}
return 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(f);
assert(n > 0);
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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