lorenz_ode.cpp

# include <cstdlib>
# include <iostream>
# include <fstream>
# include <iomanip>
# include <cmath>
# include <ctime>

using namespace std;

int main ( );
double *lorenz_rhs ( double t, int m, double x[] );
double *r8vec_linspace_new ( int n, double a, double b );
double *rk4vec ( double t0, int n, double u0[], double dt, 
  double *f ( double t, int n, double u[] ) );
void timestamp ( );

//****************************************************************************80

int main ( )

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for LORENZ_ODE.
//
//  Discussion:
//
//    Thanks to Ben Whitney for pointing out an error in specifying a loop,
//    24 May 2016.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    24 May 2016
//
//  Author:
//
//    John Burkardt
//
{
  string command_filename = "lorenz_ode_commands.txt";
  ofstream command_unit;
  string data_filename = "lorenz_ode_data.txt";
  ofstream data_unit;
  double dt;
  int i;
  int j;
  int m = 3;
  int n = 200000;
  double *t;
  double t_final;
  double *x;
  double *xnew;

  timestamp ( );
  cout << "\n";
  cout << "LORENZ_ODE\n";
  cout << "  C++ version\n";
  cout << "  Compute solutions of the Lorenz system.\n";
  cout << "  Write data to a file for use by gnuplot.\n";
//
//  Data
//
  t_final = 40.0;
  dt = t_final / ( double ) ( n );
//
//  Store the initial conditions in entry 0.
//
  t = r8vec_linspace_new ( n + 1, 0.0, t_final );
  x = new double[m*(n+1)];
  x[0+0*m] = 8.0;
  x[0+1*m] = 1.0;
  x[0+2*m] = 1.0;
//
//  Compute the approximate solution at equally spaced times.
//
  for ( j = 0; j < n; j++ )
  {
    xnew = rk4vec ( t[j], m, x+j*m, dt, lorenz_rhs );
    for ( i = 0; i < m; i++ )
    {
      x[i+(j+1)*m] = xnew[i];
    }
    delete [] xnew; 
  }
//
//  Create the plot data file.
//
  data_unit.open ( data_filename.c_str ( ) );
  for ( j = 0; j <= n; j = j + 50 )
  {
    data_unit << "  " << setw(14) << t[j]
              << "  " << setw(14) << x[0+j*m]
              << "  " << setw(14) << x[1+j*m]
              << "  " << setw(14) << x[2+j*m] << "\n";
  }
  data_unit.close ( );
  cout << "  Created data file \"" << data_filename << "\".\n";
/*
  Create the plot command file.
*/
  command_unit.open ( command_filename.c_str ( ) );
  command_unit << "# " << command_filename << "\n";
  command_unit << "#\n";
  command_unit << "# Usage:\n";
  command_unit << "#  gnuplot < " << command_filename << "\n";
  command_unit << "#\n";
  command_unit << "set term png\n";
  command_unit << "set output 'xyz_time.png'\n";
  command_unit << "set xlabel '<--- T --->'\n";
  command_unit << "set ylabel '<--- X(T), Y(T), Z(T) --->'\n";
  command_unit << "set title 'X, Y and Z versus Time'\n";
  command_unit << "set grid\n";
  command_unit << "set style data lines\n";
  command_unit << "plot '" << data_filename 
               << "' using 1:2 lw 3 linecolor rgb 'blue',";
  command_unit << "'' using 1:3 lw 3 linecolor rgb 'red',";
  command_unit << "'' using 1:4 lw 3 linecolor rgb 'green'\n";
  command_unit << "set output 'xyz_3d.png'\n";
  command_unit << "set xlabel '<--- X(T) --->'\n";
  command_unit << "set ylabel '<--- Y(T) --->'\n";
  command_unit << "set zlabel '<--- Z(T) --->'\n";
  command_unit << "set title '(X(T),Y(T),Z(T)) trajectory'\n";
  command_unit << "set grid\n";
  command_unit << "set style data lines\n";
  command_unit << "splot '" << data_filename 
               << "' using 2:3:4 lw 1 linecolor rgb 'blue'\n";
  command_unit << "quit\n";
  command_unit.close ( );
  cout << "  Created command file '" << command_filename << "'\n";
//
//  Terminate.
//
  cout << "\n";
  cout << "LORENZ_ODE:\n";
  cout << "  Normal end of execution.\n";
  cout << "\n";
  timestamp ( );

  return 0;
}
//****************************************************************************80

double *lorenz_rhs ( double t, int m, double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    LORENZ_RHS evaluates the right hand side of the Lorenz ODE.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    10 October 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double T, the value of the independent variable.
//
//    Input, int M, the spatial dimension.
//
//    Input, double X[M], the values of the dependent variables
//    at time T.
//
//    Output, double DXDT[M], the values of the derivatives
//    of the dependent variables at time T.
//
{
  double beta = 8.0 / 3.0;
  double *dxdt;
  double rho = 28.0;
  double sigma = 10.0;

  dxdt = new double[m];

  dxdt[0] = sigma * ( x[1] - x[0] );
  dxdt[1] = x[0] * ( rho - x[2] ) - x[1];
  dxdt[2] = x[0] * x[1] - beta * x[2];

  return dxdt;
}
//****************************************************************************80

double *r8vec_linspace_new ( int n, double a_first, double a_last )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//    4 points evenly spaced between 0 and 12 will yield 0, 4, 8, 12.
//
//    In other words, the interval is divided into N-1 even subintervals,
//    and the endpoints of intervals are used as the points.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    29 March 2011
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, double A_FIRST, A_LAST, the first and last entries.
//
//    Output, double R8VEC_LINSPACE_NEW[N], a vector of linearly spaced data.
//
{
  double *a;
  int i;

  a = new double[n];

  if ( n == 1 )
  {
    a[0] = ( a_first + a_last ) / 2.0;
  }
  else
  {
    for ( i = 0; i < n; i++ )
    {
      a[i] = ( ( double ) ( n - 1 - i ) * a_first 
             + ( double ) (         i ) * a_last ) 
             / ( double ) ( n - 1     );
    }
  }
  return a;
}
//****************************************************************************80

double *rk4vec ( double t0, int m, double u0[], double dt, 
  double *f ( double t, int m, double u[] ) )

//****************************************************************************80
//
//  Purpose:
//
//    RK4VEC takes one Runge-Kutta step for a vector ODE.
//
//  Discussion:
//
//    It is assumed that an initial value problem, of the form
//
//      du/dt = f ( t, u )
//      u(t0) = u0
//
//    is being solved.
//
//    If the user can supply current values of t, u, a stepsize dt, and a
//    function to evaluate the derivative, this function can compute the
//    fourth-order Runge Kutta estimate to the solution at time t+dt.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    09 October 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double T0, the current time.
//
//    Input, int M, the spatial dimension.
//
//    Input, double U0[M], the solution estimate at the current time.
//
//    Input, double DT, the time step.
//
//    Input, double *F ( double T, int M, double U[] ), a function which evaluates
//    the derivative, or right hand side of the problem.
//
//    Output, double RK4VEC[M], the fourth-order Runge-Kutta solution estimate
//    at time T0+DT.
//
{
  double *f0;
  double *f1;
  double *f2;
  double *f3;
  int i;
  double t1;
  double t2;
  double t3;
  double *u;
  double *u1;
  double *u2;
  double *u3;
//
//  Get four sample values of the derivative.
//
  f0 = f ( t0, m, u0 );

  t1 = t0 + dt / 2.0;
  u1 = new double[m];
  for ( i = 0; i < m; i++ )
  {
    u1[i] = u0[i] + dt * f0[i] / 2.0;
  }
  f1 = f ( t1, m, u1 );

  t2 = t0 + dt / 2.0;
  u2 = new double[m];
  for ( i = 0; i < m; i++ )
  {
    u2[i] = u0[i] + dt * f1[i] / 2.0;
  }
  f2 = f ( t2, m, u2 );

  t3 = t0 + dt;
  u3 = new double[m];
  for ( i = 0; i < m; i++ )
  {
     u3[i] = u0[i] + dt * f2[i];
  }
  f3 = f ( t3, m, u3 );
//
//  Combine them to estimate the solution.
//
  u = new double[m];
  for ( i = 0; i < m; i++ )
  {
     u[i] = u0[i] + dt * ( f0[i] + 2.0 * f1[i] + 2.0 * f2[i] + f3[i] ) / 6.0;
  }
//
//  Free memory.
//
  delete [] f0;
  delete [] f1;
  delete [] f2;
  delete [] f3;
  delete [] u1;
  delete [] u2;
  delete [] u3;

  return u;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  size_t len;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  len = std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}