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#include <blitz/vector.h>
#include <blitz/tinyvec.h>
BZ_USING_NAMESPACE(blitz)
/*
* Test a 12th order symmetric multistep method for solving the equations
* of motion of a single planet circling the Sun. The Sun is fixed in
* space.
*
* Original F77 version written by John K. Prentice, Quetzal Computational
* Associates, 21 Decmber 1992
*
* Blitz++ version by Todd Veldhuizen, 17 August 1997
* The C++ version is a faithful translation of the Fortran 90 version,
* so apologies for the "C++Tran" style.
*/
inline double relativeError(double a, double b)
{
if (b != 0.0)
return (a - b) / b;
else
return a;
}
int main()
{
Vector<double> x_position_numerical(13), y_position_numerical(13),
alpha(13), beta(13), gamma(13), x_acceleration(13), y_acceleration(13);
/*
* 12th order symmetric method coefficients
*
* Reference: "Symmetric Multistep Methods for the Numerical
* Integration of Planetary Orbits", G. D. Quinlan and
* S. Tremaine, The Astronomical Journal, 100 (1990), page 1695.
*
* Note!! The beta below are actually 53,222,400 times the
* real beta. This common factor is divided out in the
* symmetric multistep calculation itself, in order to minimize
* round-off
*/
const double beta_factor = 53222400.0;
alpha = 1.0, -2.0, 2.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 2.0, -2.0, 1.0;
beta = 0.0, 90987349.0, -229596838.0, 812627169.0, -1628539944.0,
2714971338.0, -3041896548.0, 2714971338.0, -1628539944.0,
812627169.0, -229596838.0, 90987349.0, 0.0;
/*
* 12th order Cowell predictor coefficients
*
* Reference: "Astronomical Papers Prepared for the Use of the
* American Ephemeris and Nautical Almanac", C. J. Cohen, E. C.
* Hubbard, and C. Oesterwinter, 22 (1973), page 20-21.
*
* Note!! The gamma below are actually 1,743,565,824,000 times
* the real gamma. This common factor is divided out in the
* Cowell predictor calculation itself, in order to minimize
* round-off
*/
const double gamma_factor = 1743565824000.0;
gamma = 9072652009253.0, -39726106418680.0, 140544566352762.0,
-344579280210129.0, 613137294629235.0, -811345852376496.0,
807012356281740.0, -602852367932304.0, 333888089374395.0,
-133228219027160.0, 36262456774618.0, -6033724094760.0,
463483373517.0;
// Initialize variables
const double time_step = 0.25,
stop_time = 365000.0,
radius = 1.0;
double time = - time_step;
cout << " Position solution via 12th order symmetric multistep method\n"
<< " Velocity solution via 12th order Cowell predictor method\n"
<< " radius = " << radius << ", time step = " << time_step
<< endl;
// Define a constant which is needed later by the exact solution
const double gaussian_constant = 0.01720209895;
const double gravitational_constant = pow(gaussian_constant,2);
const double constant = sqrt(gravitational_constant/pow(radius,3));
// Initialize the first 12 numerical values using the exact values
double x_position_exact, y_position_exact;
for (int j=-1; j <= 11; ++j)
{
if (j >= 0)
time += time_step;
x_position_exact = radius * cos(constant * time);
y_position_exact = radius * sin(constant * time);
if (j >= 0)
{
x_position_numerical(j) = x_position_exact;
y_position_numerical(j) = y_position_exact;
}
x_acceleration(j+1) = -gravitational_constant/pow(radius,3)
* x_position_exact;
y_acceleration(j+1) = -gravitational_constant/pow(radius,3)
* y_position_exact;
}
/*
* Compute exact kinetic and potential energies, and the
* angular momentum. These values are all divided by the mass
* of the object. Since they are conserved, they will never change
* and hence do not have to be recalculated later.
*/
double x_dot_exact = -radius * constant * sin(constant*time),
y_dot_exact = radius * constant * cos(constant*time),
exact_velocity_squared = pow(x_dot_exact,2) + pow(y_dot_exact,2),
exact_kinetic_energy = 0.5 * exact_velocity_squared,
exact_potential_energy = -gravitational_constant / radius,
exact_total_energy = exact_potential_energy + exact_kinetic_energy,
exact_angular_momentum = x_position_exact * y_dot_exact
- y_position_exact * x_dot_exact;
double x_dot_numerical, y_dot_numerical;
// Perform loop over time
while (time <= stop_time)
{
// Advance time step (eek!)
time += time_step;
// Calculate new acceleration of body at time=time-time_step
double numerical_radius_squared = pow(x_position_numerical(11),2)
+ pow(y_position_numerical(11),2);
x_acceleration(12) = -gravitational_constant
/ pow(numerical_radius_squared, 1.5) * x_position_numerical(11);
y_acceleration(12) = -gravitational_constant
/ pow(numerical_radius_squared, 1.5) * y_position_numerical(11);
// Numerically solve for the new positions using a 12th order
// symmetric multistep method.
// First sum the first and second terms
double x_alpha_sum = dot(alpha(Range(0,11)),
x_position_numerical(Range(0,11)));
double y_alpha_sum = dot(alpha(Range(0,11)),
y_position_numerical(Range(0,11)));
double x_beta_sum = dot(beta(Range(0,11)), x_acceleration(Range(1,12)));
double y_beta_sum = dot(beta(Range(0,11)), y_acceleration(Range(1,12)));
x_position_numerical(12) = (-x_alpha_sum) + pow(time_step,2)
* (x_beta_sum / beta_factor);
y_position_numerical(12) = (-y_alpha_sum) + pow(time_step,2)
* (y_beta_sum / beta_factor);
// Numerically solve for the new velocities using a 12th order
// Cowell predictor method.
// First sum the gamma terms
double x_gamma_sum = dot(gamma, x_acceleration.reverse()),
y_gamma_sum = dot(gamma, y_acceleration.reverse());
x_dot_numerical = (x_position_numerical(11)
- x_position_numerical(10)) / time_step + time_step
* (x_gamma_sum / gamma_factor);
y_dot_numerical = (y_position_numerical(11)
- y_position_numerical(10)) / time_step + time_step
* (y_gamma_sum / gamma_factor);
// Push the stack down one
for (int j=0; j <= 11; ++j)
{
x_position_numerical(j) = x_position_numerical(j+1);
y_position_numerical(j) = y_position_numerical(j+1);
x_acceleration(j) = x_acceleration(j+1);
y_acceleration(j) = y_acceleration(j+1);
}
}
// Print results
// First compute energies and angular momenta (add divided by the mass
// of the object)
double numerical_velocity_squared = pow(x_dot_numerical,2) +
pow(y_dot_numerical,2),
numerical_radius = sqrt(pow(x_position_numerical(12),2)
+ pow(y_position_numerical(12),2)),
numerical_kinetic_energy = 0.5 * numerical_velocity_squared,
numerical_potential_energy = -gravitational_constant
/ numerical_radius,
numerical_total_energy = numerical_potential_energy
+ numerical_kinetic_energy,
numerical_angular_momentum = x_position_numerical(12)
* y_dot_numerical - y_position_numerical(12) * x_dot_numerical;
// Compute exact results for comparison to the numerical results
x_position_exact = radius * cos(constant * time);
y_position_exact = radius * sin(constant * time);
x_dot_exact = -radius * constant * sin(constant * time);
y_dot_exact = radius * constant * cos(constant * time);
// Next compute relative errors
double radius_error = relativeError(numerical_radius, radius),
x_error = relativeError(x_position_numerical(12), x_position_exact),
y_error = relativeError(y_position_numerical(12), y_position_exact),
x_dot_error = relativeError(x_dot_numerical, x_dot_exact),
y_dot_error = relativeError(y_dot_numerical, y_dot_exact);
double kinetic_energy_error = relativeError(numerical_kinetic_energy,
exact_kinetic_energy),
potential_energy_error = relativeError(numerical_potential_energy,
exact_potential_energy),
total_energy_error = relativeError(numerical_total_energy,
exact_total_energy),
angular_momentum_error = relativeError(numerical_angular_momentum,
exact_angular_momentum);
cout << " Time = " << time << endl
<< " x rel error = " << x_error << " y rel error = " << y_error
<< endl
<< " vx rel error = " << x_dot_error << " vy rel error = "
<< y_dot_error << endl
<< " KE rel error = " << kinetic_energy_error
<< " PE rel error = " << potential_energy_error << endl
<< " TE rel error = " << total_energy_error << " AM rel error = "
<< angular_momentum_error << endl
<< " numerical radius = " << numerical_radius
<< " radius rel error = " << radius_error << endl;
return 0;
}
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