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RungeKutta.cs
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RungeKutta.cs
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using BlackHoleRaytracer.Equation;
using System;
namespace BlackHoleRaytracer
{
public class RungeKutta
{
/// <summary>
/// Perform the integration.
/// </summary>
unsafe public static double Integrate(IODESystem equation, double* y, double* dydx, double hTry, double escal, double* yscal, out double hActual)
{
int i;
double errmax, h = hTry, htemp, hnext;
double* yerr = stackalloc double[equation.N];
double* ytemp = stackalloc double[equation.N];
while (true)
{
// Run a single step of integration using step h
IntegrateStep(equation, y, dydx, h, ytemp, yerr);
// Find the maximum calculation error of all equations after the calculation step
errmax = 0.0;
for (i = 0; i < equation.N; i++)
{
double temp = Math.Abs(yerr[i] / yscal[i]);
if (temp > errmax) errmax = temp;
}
errmax *= escal;
if (errmax <= 1.0) { break; }
htemp = 0.9 * h / Math.Sqrt(Math.Sqrt(errmax));
h *= 0.1;
if (h >= 0.0)
{
if (htemp > h) h = htemp;
}
else
{
if (htemp < h) h = htemp;
}
}
// Return next suggested step of integration.
// If error was small, increase step 5 times (to save computing time).
hnext = errmax > 1.89e-4 ? hnext = 0.9 * h * Math.Pow(errmax, -0.2) : hnext = 5.0 * h;
hActual = h;
Util.memcpy((IntPtr)y, (IntPtr)ytemp, equation.N * sizeof(double));
return hnext;
}
/// <summary>
/// Calculate single step of integration algorithm.
/// https://en.wikipedia.org/wiki/Cash%E2%80%93Karp_method
///
/// The Cash-Karp method uses the following Butcher tableau:
///
/// 0 |
/// 1/5 | 1/5
/// 3/10 | 3/40 9/40
/// 3/5 | 3/10 −9/10 6/5
/// 1 | −11/54 5/2 −70/27 35/27
/// 7/8 | 1631/55296 175/512 575/13824 44275/110592 253/4096
/// --------------------------------------------------------------------
/// | 37/378 0 250/621 125/594 0 512/1771
/// | 2825/27648 0 18575/48384 13525/55296 277/14336 1/4
///
/// </summary>
unsafe public static void IntegrateStep(IODESystem equation, double* y, double* dydx, double h, double* yout, double* yerr)
{
int i;
double* ak = stackalloc double[equation.N];
double* ytemp1 = stackalloc double[equation.N];
double* ytemp2 = stackalloc double[equation.N];
double* ytemp3 = stackalloc double[equation.N];
double* ytemp4 = stackalloc double[equation.N];
double* ytemp5 = stackalloc double[equation.N];
for (i = 0; i < equation.N; i++)
{
double hdydx = h * dydx[i];
double yi = y[i];
ytemp1[i] = yi + 0.2 * hdydx;
ytemp2[i] = yi + (3.0 / 40.0) * hdydx;
ytemp3[i] = yi + 0.3 * hdydx;
ytemp4[i] = yi - (11.0 / 54.0) * hdydx;
ytemp5[i] = yi + (1631.0 / 55296.0) * hdydx;
yout[i] = yi + (37.0 / 378.0) * hdydx;
yerr[i] = ((37.0 / 378.0) - (2825.0 / 27648.0)) * hdydx;
}
equation.Function(ytemp1, ak);
for (i = 0; i < equation.N; i++)
{
double yt = h * ak[i];
ytemp2[i] += (9.0 / 40.0) * yt;
ytemp3[i] -= 0.9 * yt;
ytemp4[i] += 2.5 * yt;
ytemp5[i] += (175.0 / 512.0) * yt;
}
equation.Function(ytemp2, ak);
for (i = 0; i < equation.N; i++)
{
double yt = h * ak[i];
ytemp3[i] += 1.2 * yt;
ytemp4[i] -= (70.0 / 27.0) * yt;
ytemp5[i] += (575.0 / 13824.0) * yt;
yout[i] += (250.0 / 621.0) * yt;
yerr[i] += ((250.0 / 621.0) - (18575.0 / 48384.0)) * yt;
}
equation.Function(ytemp3, ak);
for (i = 0; i < equation.N; i++)
{
double yt = h * ak[i];
ytemp4[i] += (35.0 / 27.0) * yt;
ytemp5[i] += (44275.0 / 110592.0) * yt;
yout[i] += (125.0 / 594.0) * yt;
yerr[i] += ((125.0 / 594.0) - (13525.0 / 55296.0)) * yt;
}
equation.Function(ytemp4, ak);
for (i = 0; i < equation.N; i++)
{
double yt = h * ak[i];
ytemp5[i] += (253.0 / 4096.0) * yt;
yerr[i] -= (277.0 / 14336.0) * yt;
}
equation.Function(ytemp5, ak);
for (i = 0; i < equation.N; i++)
{
double yt = h * ak[i];
yout[i] += (512.0 / 1771.0) * yt;
yerr[i] += ((512.0 / 1771.0) - 0.25) * yt;
}
}
}
}