RK4 and Euler

fluids2.2.py

+# -*- coding: utf-8 -*-
+"""
+Created on Tue May  5 18:45:25 2020
+
+@author: Marina
+"""
+from numpy import *
+
+def RK4(Xstart,Ustart,Xend,h,f):
+  imax = int((Xend-Xstart)/h)
+  X = Xstart + arange(imax+1)*h
+  U = zeros((size(Ustart), imax+1))
+  U[:, 0] = asarray(Ustart)
+  for i in range(1,imax+1):
+        K1 = asarray(f(U[:, i-1]))
+        K2 = asarray(f(U[:, i-1] + (h/2) * K1))
+        K3 = asarray(f(U[:, i-1] + (h/2)* K2))
+        K4 = asarray(f(U[:, i-1] + h * K3))
+        
+        U[:, i] = U[:, i-1] + (h/6) * (K1 + 2 * K2 + 2 * K3 + K4)
+        
+  U = transpose(U)
+  return X, U
+
+
+def Euler(Xstart,Ustart,Xend,h,f):
+    N = 5
+    X = arange(Xstart,Xend,h)
+    n = size(X)
+    U = zeros((N, n))
+    U[:, 0] = transpose(Ustart)
+    for j in range (1, n):
+      U[:,j] = U[:, j-1] + h*f(U[:,j-1])
+    
+    U = transpose(U)
+    return X,U
+
+
+def RK40(Xstart,Ustart,Xend,h,f):
+  imax = int((Xend-Xstart)/h)
+  X = Xstart + arange(imax+1)*h
+  U = zeros((imax+1,5)); U[0,:] = Ustart
+  for i in range(imax):  
+    K1 = f(U[i,:]       )
+    K2 = f(U[i,:]+K1*h/2)
+    K3 = f(U[i,:]+K2*h/2)
+    K4 = f(U[i,:]+K3*h  )
+    U[i+1,:] = U[i,:] + h*(K1+2*K2+2*K3+K4)/6     
+  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(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 = [0.5,1.5]
+delta2 = [-1,0]
+nmax = 20
+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)
+
+