In [61]:

from __future__ import division

import numpy as np
import matplotlib.pyplot as pt

In [62]:

def rk4_step(y, t, h, f):
    k1 = f(t, y)
    k2 = f(t+h/2, y + h/2*k1)
    k3 = f(t+h/2, y + h/2*k2)
    k4 = f(t+h, y + h*k3)
    return y + h/6*(k1 + 2*k2 + 2*k3 + k4)

Want to solve:

\[w''(t)=\frac 32w^2\]

with \(w(0)=4\) and \(w(1)=1\). (Example due to Stoer and Bulirsch)

In [63]:

def f(t, y):
    w, w_prime = y
    return np.array([w_prime, 3/2*w**2])

In [64]:

def shoot(w_prime):
    times = [0]
    y_values = [np.array([4, w_prime])]
    
    h = 1/2**7
    t_end = 1
    
    while times[-1] < t_end:
        y_values.append(rk4_step(y_values[-1], times[-1], h, f))
        times.append(times[-1]+h)
    
    y_values = np.array(y_values)
    
    # actually floating-point-equal due to power-of-2 h
    assert times[-1] == t_end
    
    print "w'(0) = %g  ->  w(1)= %.5g" % (w_prime, y_values[-1,0])

    pt.plot(times, y_values[:, 0], label="$w'(0)=%.2g$" % w_prime)

In [56]:

shoot(0)
shoot(-5)
shoot(-7)
shoot(-7.5)

pt.grid()
pt.legend(loc="best")

w'(0) = 0  ->  w(1)= 87.08
w'(0) = -5  ->  w(1)= 12.058
w'(0) = -7  ->  w(1)= 3.6442
w'(0) = -7.5  ->  w(1)= 2.2233

Out[56]:

<matplotlib.legend.Legend at 0x575e110>

In [60]:

shoot(-30)

pt.grid()
pt.legend(loc="best")

w'(0) = -30  ->  w(1)= -1.4668

Out[60]:

<matplotlib.legend.Legend at 0x628da10>

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