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    ##New Library required for projected 3d plots

import numpy as np
import matplotlib.pyplot as plt

###variable declarations
nx = 81
ny = 81
nt = 100
c = 1
dx = 2.0/(nx-1)
dy = 2.0/(ny-1)
sigma = .2
dt = sigma*dx

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

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

###Plot Initial Condition
fig = plt.figure(figsize=(11,7), dpi=100)          ##the figsize parameter can be used to produce different sized images
ax = fig.gca(projection='3d')                      
X, Y = np.meshgrid(x,y)                            
surf = ax.plot_surface(X,Y,u[:])



u = np.ones((ny,nx))
u[.5/dy:1/dy+1,.5/dx:1/dx+1]=2

for n in range(nt+1): ##loop across number of time steps
    un[:] = u[:]
    for i in range(1, len(u)):
        for j in range(1, len(u)):
            u[i,j] = un[i, j] - (c*dt/dx*(un[i,j] - un[i-1,j]))-(c*dt/dy*(un[i,j]-un[i,j-1]))
            u[0,:] = 1
            u[-1,:] = 1
            u[:,0] = 1
            u[:,-1] = 1

fig = plt.figure(figsize=(11,7), dpi=100)
ax = fig.gca(projection='3d')
surf2 = ax.plot_surface(X,Y,u[:])

    

u = np.ones((ny,nx))
u[.5/dy:1/dy+1,.5/dx:1/dx+1]=2

for n in range(nt+1): ##loop across number of time steps
    un[:] = u[:]
    u[1:,1:]=un[1:,1:]-(c*dt/dx*(un[1:,1:]-un[0:-1,1:]))-(c*dt/dy*(un[1:,1:]-un[1:,0:-1]))
    u[0,:] = 1
    u[-1,:] = 1
    u[:,0] = 1
    u[:,-1] = 1

fig = plt.figure(figsize=(11,7), dpi=100)
ax = fig.gca(projection='3d')
surf2 = ax.plot_surface(X,Y,u[:])

    

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()