yv = yy[1:]
yv = yv[idx]
A = vstack([ones(size(ya)), log(abs(yv))])
[knot, kk2] = dot(inv(dot(A, A.T)), dot(A, log(ya + 9.8)))
kk1 = exp(knot)
def f4(t, y):
return array([y[1], -9.8 + kk1 * (abs(y[1])) ** kk2])
drag_string = "Drag model:$F_d(v) = k_1 |v|^{{k_2}},\quad k_1 = {:.4f}, \quad k_2 = {:.4f}$".format(kk1, kk2)
titles = ["Linear Drag", "Quadratic Drag", "Two Term Quadratic Drag", drag_string]
titles.reverse()
for f in (f1, f2, f3, f4):
integrator = RungeKutta(f, [y[0], yy[0]], data[0][0], data[0][-1], 0.001)
(mt, my) = integrator.integrate()
idx = find(my[0] > 0)
mt = mt[idx]
my = my[:, idx]
f = figure()
subplot(121)
plot(t, y, ".")
plot(mt, my[0])
xlabel("Time (sec)")
ylabel("Displacement (meters)")
subplot(122)
plot(tt, yy, ".")
plot(mt, my[1])
xlabel("Time (sec)")
ylabel("Velocity (meters)")
a1 = -GM/r1**3*x1 + m2*GM/r21**3*(x2-x1)
a2 = -GM/r2**3*x2 + m1*GM/r21**3*(x1-x2)
derivatives = array([v1,a1,v2,a2]).T.flatten(1)
return derivatives
scipy_result = odeint(two_body,xinit,times)
scipy_result = scipy_result.T
atoll=10**-13
rtoll=10**-14
high_prec_result = odeint(two_body,xinit,times,atol=atoll,rtol=rtoll)
high_prec_result = high_prec_result.T
######## Do RK4 HERE ############
f = lambda t,x: two_body(x,t)
rk4 = RungeKutta(f,xinit,0.,years,years/num_samples)
(t,rk4_result) = rk4.integrate()
rk4_result = rk4_result[:,:-2]
#################################
iterates = ((scipy_result,"Scipy odeint default tolerances"),
(rk4_result,"Runge Kutta 4th Order"),
(high_prec_result,"odeint rtol={} atol={}".format(rtoll,atoll)))
for x,t in iterates:
figure()
r1 = sqrt(x[0]**2 + x[1]**2)
r2 = sqrt(x[4]**2 + x[5]**2)
r21 = sqrt( (x[4]-x[0])**2 + (x[5]-x[1])**2)
E = .5*(m1*(x[2]**2 + x[3]**2) + m2*(x[6]**2+x[7]**2)) - GM*( m1/r1 + m2/r2 + m1*m2/r21 )
deltaE = (E-E[0])/E[0]
