Text and code provided under a Creative Commons Attribution license, CC-BY. (c) Lorena A. Barba, 2013. Thanks: Gilbert Forsyth for help writing the notebooks. NSF for support via CAREER award #1149784.
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
import matplotlib.pyplot as plt
import numpy as np

###variable declarations
nx = 41
ny = 41
nt = 120
c = 1
dx = 2.0/(nx-1)
dy = 2.0/(ny-1)
sigma = .0009
nu = 0.01
dt = sigma*dx*dy/nu


x = np.linspace(0,2,nx)
y = np.linspace(0,2,ny)

u = np.ones((ny,nx)) ##create a 1xn vector of 1's
v = np.ones((ny,nx))
un = np.ones((ny,nx)) ##
vn = np.ones((ny,nx))
comb = np.ones((ny,nx))

###Assign initial conditions

u[.5/dy:1/dy+1,.5/dx:1/dx+1]=2 ##set hat function I.C. : u(.5<=x<=1 && .5<=y<=1 ) is 2
v[.5/dy:1/dy+1,.5/dx:1/dx+1]=2 ##set hat function I.C. : u(.5<=x<=1 && .5<=y<=1 ) is 2

###(plot ICs)
fig = plt.figure(figsize=(11,7), dpi=100)
ax = fig.gca(projection='3d')
X,Y = np.meshgrid(x,y)
wire1 = ax.plot_wireframe(X,Y,u[:], cmap=cm.coolwarm)
wire2 = ax.plot_wireframe(X,Y,v[:], cmap=cm.coolwarm)
#ax.set_xlim(1,2)
#ax.set_ylim(1,2)
#ax.set_zlim(1,5)
plt.show()

for n in range(nt+1): ##loop across number of time steps
    un[:] = u[:]
    vn[:] = v[:]

    u[1:-1,1:-1] = un[1:-1,1:-1] - dt/dx*un[1:-1,1:-1]*(un[1:-1,1:-1]-un[0:-2,1:-1])-dt/dy*vn[1:-1,1:-1]* \
                   (un[1:-1,1:-1]-un[1:-1,0:-2])+nu*dt/dx**2*(un[2:,1:-1]-2*un[1:-1,1:-1]+un[0:-2,1:-1])+ \
                   nu*dt/dy**2*(un[1:-1,2:]-2*un[1:-1,1:-1]+un[1:-1,0:-2])
    
    v[1:-1,1:-1] = vn[1:-1,1:-1] - dt/dx*un[1:-1,1:-1]*(vn[1:-1,1:-1]-vn[0:-2,1:-1])-dt/dy*vn[1:-1,1:-1]* \
                   (vn[1:-1,1:-1]-vn[1:-1,0:-2])+nu*dt/dx**2*(vn[2:,1:-1]-2*vn[1:-1,1:-1]+vn[0:-2,1:-1])+ \
                   nu*dt/dy**2*(vn[1:-1,2:]-2*vn[1:-1,1:-1]+vn[1:-1,0:-2])
     
    u[0,:] = 1
    u[-1,:] = 1
    u[:,0] = 1
    u[:,-1] = 1
    
    v[0,:] = 1
    v[-1,:] = 1
    v[:,0] = 1
    v[:,-1] = 1

fig = plt.figure(figsize=(11,7), dpi=100)
ax = fig.gca(projection='3d')
X,Y = np.meshgrid(x,y)
wire1 = ax.plot_wireframe(X,Y,u[:])
wire2 = ax.plot_wireframe(X,Y,v[:])
#ax.set_xlim(1,2)
#ax.set_ylim(1,2)
#ax.set_zlim(1,5)
plt.show()

from IPython.display import YouTubeVideo
YouTubeVideo('tUg_dE3NXoY')

from IPython.core.display import HTML
def css_styling():
    styles = open("../styles/custom.css", "r").read()
    return HTML(styles)
css_styling()