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# -*- coding: utf-8 -*-
"""
Created on Sun May 10 00:27:43 2020
@author: Marina
"""
from numpy import *
from scipy.integrate import solve_ivp
def RK4(Xstart,Ustart,Xend,h,f):
sol = solve_ivp(f,(Xstart,Xend),Ustart, t_eval=arange(Xstart,Xend,h))
X = transpose(sol.t)
U = transpose(sol.y)
return X,U
# We define:
# f = u1; f' = v1; f''= w1
# \u03B8 = u2; \u03B8'= v2
# u = [u1, v1, w1, u2, v2]
# In that case the system to solve is:
# u1' = v1
# v1' = w1
# w1' = -3*u1*w1 + 2*(v1)**2 - u2
# u2' = v2
# v2' = -3*Pr*u1*v2
def f(eta,u):
du1dt = u[1]
dv1dt = u[2]
dw1dt = - 3*u[0]*u[2] + 2*(u[1]**2) - u[3]
du2dt = u[4]
dv2dt = - 3 * 0.01 * u[0]*u[4]
return array([du1dt,dv1dt,dw1dt,du2dt,dv2dt])
# Shoot method
def shoot(a,b,h,integrator):
X,U = integrator(0,[0,0,a,1,b],30,h,f)
print(-U)
return array([-U[-1,2], -U[-1,4]])
# We solve the problem in the same way we solved the Blasius equation,
# aplaying the bisection method.
# We are supposing that we want to get the an error equal to the tolerance in both cases,
# to calculate alfa1 and alfa2, and that we can choose different intervals.
def blasius(delta1,delta2,nmax,tol,h,integrator):
delta0 = array([delta1, delta2])
a = zeros(2)
b = zeros(2)
a[0] = delta0[0,0]
a[1] = delta0[1,0]
b[0] = delta0[0,1]
b[1] = delta0[1,1]
x = zeros(2)
delta = zeros(2)
print('a1,a2',a[0],a[1])
fa = shoot(a[0],a[1],h,integrator)
print('fa', fa)
print('b1,b2',b[0],b[1])
fb = shoot(b[0],b[1],h,integrator)
print('fb', fb)
n = 1
delta[0] = (b[0]-a[0])/2
delta[1] = (b[1]-a[1])/2
for i in range (0,2):
if (fa[i]*fb[i] > 0) :
return 0, 0,'Bad initial interval :-( for i = %d' % (i)
while (abs(delta[0]) >= tol and abs(delta[1]) >= tol and n < nmax) :
n = n + 1
for i in range (0,2):
delta[i] = (b[i]-a[i])/2
x[i] = a[i] + delta[i]
print('x%d = %13.7e'%(i,x[i]))
fx = shoot(x[0],x[1],h,integrator)
print('fx', fx)
print(" x1 = %14.7e (Estimated error %13.7e at iteration %d)" % (x[0],abs(delta[0]),n))
print(" x2 = %14.7e (Estimated error %13.7e at iteration %d)" % (x[1],abs(delta[1]),n))
for i in range (0,2):
if (fx[i]*fa[i] > 0) :
a[i] = x[i]
fa[i] = fx[i]
else:
b[i] = x[i]
fb[i] = fx[i]
if (n == nmax) :
return x[0], x[1],'Increase nmax : more iterations are needed :-('
return x[0], x[1],'Convergence observed :-)'
# These are the initial values we have choosen
Xend = 30
h = 0.5
Pr = 0.01
delta1 = [1,1.1]
delta2 = [-0.07,-0.1]
nmax = 1
tol = 1e-4
a, b, message = blasius(delta1,delta2,nmax,tol,h,RK4)
X,U = RK4(0,[0,0,a,1,b],Xend,h,f)
print(U,shape(U))
print('For RK4:')
print(" === Requested f''(0) = %.4f === %s" % (a,message))
print(" === Requested \u03B8'(0) = %.4f === %s" % (b,message))
print(" === Obtained final value for f'(0) = %13.7e " % U[-1,2])
print(" === Obtained final value for \u03B8(0) = %13.7e " % U[-1,4])
from matplotlib import pyplot as plt
import matplotlib
matplotlib.rcParams['toolbar'] = 'None'
plt.rcParams['figure.facecolor'] = 'lavender'
plt.rcParams['axes.facecolor'] = 'lavender'
fig = plt.figure("Fluids")
plt.plot(X,U[:,2],'-b',X,U[:,4],'-r')
plt.text(4,2e48,"f'(x)",color='blue',fontsize=12)
plt.text(0.5,1e48,"\u03B8 (x)",color='red',fontsize=12)