25.5 Examples

The following program solves the second-order nonlinear Van der Pol oscillator equation,

     x''(t) + \mu x'(t) (x(t)^2 - 1) + x(t) = 0

This can be converted into a first order system suitable for use with the routines described in this chapter by introducing a separate variable for the velocity, y = x'(t),

     x' = y
     y' = -x + \mu y (1-x^2)

The program begins by defining functions for these derivatives and their Jacobian,

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_odeiv.h>
     
     int
     func (double t, const double y[], double f[],
           void *params)
     {
       double mu = *(double *)params;
       f[0] = y[1];
       f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
       return GSL_SUCCESS;
     }
     
     int
     jac (double t, const double y[], double *dfdy, 
          double dfdt[], void *params)
     {
       double mu = *(double *)params;
       gsl_matrix_view dfdy_mat 
         = gsl_matrix_view_array (dfdy, 2, 2);
       gsl_matrix * m = &dfdy_mat.matrix; 
       gsl_matrix_set (m, 0, 0, 0.0);
       gsl_matrix_set (m, 0, 1, 1.0);
       gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
       gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
       dfdt[0] = 0.0;
       dfdt[1] = 0.0;
       return GSL_SUCCESS;
     }
     
     int
     main (void)
     {
       const gsl_odeiv_step_type * T 
         = gsl_odeiv_step_rk8pd;
     
       gsl_odeiv_step * s 
         = gsl_odeiv_step_alloc (T, 2);
       gsl_odeiv_control * c 
         = gsl_odeiv_control_y_new (1e-6, 0.0);
       gsl_odeiv_evolve * e 
         = gsl_odeiv_evolve_alloc (2);
     
       double mu = 10;
       gsl_odeiv_system sys = {func, jac, 2, &mu};
     
       double t = 0.0, t1 = 100.0;
       double h = 1e-6;
       double y[2] = { 1.0, 0.0 };
     
       while (t < t1)
         {
           int status = gsl_odeiv_evolve_apply (e, c, s,
                                                &sys, 
                                                &t, t1,
                                                &h, y);
     
           if (status != GSL_SUCCESS)
               break;
     
           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }
     
       gsl_odeiv_evolve_free (e);
       gsl_odeiv_control_free (c);
       gsl_odeiv_step_free (s);
       return 0;
     }

For functions with multiple parameters, the appropriate information can be passed in through the params argument using a pointer to a struct.

The main loop of the program evolves the solution from (y, y') = (1, 0) at t=0 to t=100. The step-size h is automatically adjusted by the controller to maintain an absolute accuracy of 10^{-6} in the function values y.

To obtain the values at user-specified positions, rather than those chosen automatically by the control function, the main loop can be modified to advance the solution from one chosen point to the next. For example, the following main loop prints the solution at the points t_i = 0, 1, 2, \dots, 100,

       for (i = 1; i <= 100; i++)
         {
           double ti = i * t1 / 100.0;
     
           while (t < ti)
             {
               gsl_odeiv_evolve_apply (e, c, s,
                                       &sys,
                                       &t, ti, &h,
                                       y);
             }
     
           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

Note that arbitrary values of t_i can be used for each stage of the integration. The equally spaced points in this example are just used as an illustration.

It is also possible to work with a non-adaptive integrator, using only the stepping function itself. The following program uses the rk4 fourth-order Runge-Kutta stepping function with a fixed stepsize of 0.01,

     int
     main (void)
     {
       const gsl_odeiv_step_type * T 
         = gsl_odeiv_step_rk4;
     
       gsl_odeiv_step * s 
         = gsl_odeiv_step_alloc (T, 2);
     
       double mu = 10;
       gsl_odeiv_system sys = {func, jac, 2, &mu};
     
       double t = 0.0, t1 = 100.0;
       double h = 1e-2;
       double y[2] = { 1.0, 0.0 }, y_err[2];
       double dydt_in[2], dydt_out[2];
     
       /* initialise dydt_in from system parameters */
       GSL_ODEIV_FN_EVAL(&sys, t, y, dydt_in);
     
       while (t < t1)
         {
           int status = gsl_odeiv_step_apply (s, t, h, 
                                              y, y_err, 
                                              dydt_in, 
                                              dydt_out, 
                                              &sys);
     
           if (status != GSL_SUCCESS)
               break;
     
           dydt_in[0] = dydt_out[0];
           dydt_in[1] = dydt_out[1];
     
           t += h;
     
           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }
     
       gsl_odeiv_step_free (s);
       return 0;
     }

The derivatives must be initialized for the starting point t=0 before the first step is taken. Subsequent steps use the output derivatives dydt_out as inputs to the next step by copying their values into dydt_in.