Falkner-Skan IVP flat plate

integrate_fs.py

import scipy as sp
from scipy.integrate import ode 
from scipy.optimize import fsolve 
from matplotlib import rc
#rc('text', usetex=True)
tick_size = 'large'
label_size = 'xx-large'
rc('xtick', labelsize=8)
rc('ytick', labelsize=8)
rc('legend', fontsize=12)
rc('axes', labelsize=12)
fig_width_pt = 469.75502 
inches_per_pt = 1.0/72.27
golden_mean = (sp.sqrt(5.0)-1.0)/2.0
fig_width = fig_width_pt *inches_per_pt 
fig_height = fig_width * golden_mean
fig_size = [fig_width, fig_height] 
rc('figure', figsize=fig_size)

from matplotlib import pylab as plt

from fs import *

def int_fs(beta_0, beta, alpha, eta_inf):
    args = sp.array([beta_0, beta, eta_inf])    
    y0 = sp.array([0.0, 0.0, alpha])
    r = ode(fs.rates, fs.jac).set_integrator('zvode', rtol=1e-15, atol=1e-15, method='adams', with_jacobian=True, order=12)
    t0 = 0.0 
    t1 = 1.0 
    nt = 100
    dt = t1 / float(nt)
    r.set_initial_value(y0, t0).set_f_params(args).set_jac_params(args)
    for i in xrange(nt):
        r.integrate(r.t+dt)
    return(r)

def objective_func(x):
    eta_inf = x[0]
    alpha = x[1] 
    r = int_fs(1.0, 0.0, alpha, eta_inf)
    return(sp.array([(r.y[1]-1.0).real, (r.y[2]).real]))

eta_inf = 1.0 # ~6
alpha = 1.0  # ~0.46960 
x = sp.array([eta_inf, alpha]) 
foo = fsolve(objective_func, x, xtol=1e-15)

nt = 100 

args = sp.array([1.0, 0.0, foo[0]])    
y0 = sp.array([0.0, 0.0, foo[1]])
r = ode(fs.rates, fs.jac).set_integrator('zvode', rtol=1e-15, atol=1e-15, method='adams', with_jacobian=True, order=12)
t0 = 0.0 
t1 = 1.0 
nt = 100
dt = t1 / float(nt)
t = sp.linspace(0,1.0,nt+1)
y = sp.zeros((3,nt+1), dtype=float) 
y[:,0] = y0.copy() 
r.set_initial_value(y0, t0).set_f_params(args).set_jac_params(args)
for i in xrange(nt):
     r.integrate(r.t+dt)
     t[i+1] = r.t
     y[:,i+1] = r.y.copy().real 

plt.figure()
plt.plot(foo[0]*t,y[0,:])
plt.plot(foo[0]*t,y[1,:])
plt.plot(foo[0]*t,y[2,:])
plt.axis([0,foo[0],0,y.max()])
plt.xlabel('\eta')
plt.ylabel("f,f',f''")
plt.savefig("fs.png")
plt.show()