#!/usr/bin/env python
# coding: utf-8
Text provided under a Creative Commons Attribution license, CC-BY.  All code is made available under the FSF-approved BSD-3 license.  (c) Lorena A. Barba, Gilbert F. Forsyth 2017. Thanks to NSF for support via CAREER award #1149784.
# [@LorenaABarba](https://twitter.com/LorenaABarba)

# 12 steps to Navier–Stokes
# =====
# ***

# This will be a milestone! We now get to Step 8: Burgers' equation. We can learn so much more from this equation. It plays a very important role in fluid mechanics, because it contains the full convective nonlinearity of the flow equations, and at the same time there are many known analytical solutions.
# 

# Step 8: Burgers' Equation in 2D
# ----
# ***

# Remember, Burgers' equation can generate discontinuous solutions from an initial condition that is smooth, i.e., can develop "shocks." We want to see this in two dimensions now!
# 
# Here is our coupled set of PDEs:
# 
# $$
# \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \; \left(\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2}\right)$$
# 
# $$
# \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = \nu \; \left(\frac{\partial ^2 v}{\partial x^2} + \frac{\partial ^2 v}{\partial y^2}\right)$$

# We know how to discretize each term: we've already done it before!
# 
# $$
# \begin{split}
# & \frac{u_{i,j}^{n+1} - u_{i,j}^n}{\Delta t} + u_{i,j}^n \frac{u_{i,j}^n-u_{i-1,j}^n}{\Delta x} + v_{i,j}^n \frac{u_{i,j}^n - u_{i,j-1}^n}{\Delta y} = \\
# & \qquad \nu \left( \frac{u_{i+1,j}^n - 2u_{i,j}^n+u_{i-1,j}^n}{\Delta x^2} + \frac{u_{i,j+1}^n - 2u_{i,j}^n + u_{i,j-1}^n}{\Delta y^2} \right)
# \end{split}
# $$

# $$
# \begin{split}
# & \frac{v_{i,j}^{n+1} - v_{i,j}^n}{\Delta t} + u_{i,j}^n \frac{v_{i,j}^n-v_{i-1,j}^n}{\Delta x} + v_{i,j}^n \frac{v_{i,j}^n - v_{i,j-1}^n}{\Delta y} = \\
# & \qquad \nu \left( \frac{v_{i+1,j}^n - 2v_{i,j}^n+v_{i-1,j}^n}{\Delta x^2} + \frac{v_{i,j+1}^n - 2v_{i,j}^n + v_{i,j-1}^n}{\Delta y^2} \right)
# \end{split}
# $$

# And now, we will rearrange each of these equations for the only unknown: the two components $u,v$ of the solution at the next time step:

# $$
# \begin{split}
# u_{i,j}^{n+1} = & u_{i,j}^n - \frac{\Delta t}{\Delta x} u_{i,j}^n (u_{i,j}^n - u_{i-1,j}^n)  - \frac{\Delta t}{\Delta y} v_{i,j}^n (u_{i,j}^n - u_{i,j-1}^n) \\
# &+ \frac{\nu \Delta t}{\Delta x^2}(u_{i+1,j}^n-2u_{i,j}^n+u_{i-1,j}^n) + \frac{\nu \Delta t}{\Delta y^2} (u_{i,j+1}^n - 2u_{i,j}^n + u_{i,j-1}^n)
# \end{split}
# $$

# $$
# \begin{split}
# v_{i,j}^{n+1} = & v_{i,j}^n - \frac{\Delta t}{\Delta x} u_{i,j}^n (v_{i,j}^n - v_{i-1,j}^n) - \frac{\Delta t}{\Delta y} v_{i,j}^n (v_{i,j}^n - v_{i,j-1}^n) \\
# &+ \frac{\nu \Delta t}{\Delta x^2}(v_{i+1,j}^n-2v_{i,j}^n+v_{i-1,j}^n) + \frac{\nu \Delta t}{\Delta y^2} (v_{i,j+1}^n - 2v_{i,j}^n + v_{i,j-1}^n)
# \end{split}
# $$

# In[1]:


import numpy
from matplotlib import pyplot, cm
from mpl_toolkits.mplot3d import Axes3D
get_ipython().run_line_magic('matplotlib', 'inline')


# In[2]:


###variable declarations
nx = 41
ny = 41
nt = 120
c = 1
dx = 2 / (nx - 1)
dy = 2 / (ny - 1)
sigma = .0009
nu = 0.01
dt = sigma * dx * dy / nu


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

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

###Assign initial conditions

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


# In[3]:


###(plot ICs)
fig = pyplot.figure(figsize=(11, 7), dpi=100)
ax = fig.gca(projection='3d')
X, Y = numpy.meshgrid(x, y)
ax.plot_surface(X, Y, u[:], cmap=cm.viridis, rstride=1, cstride=1)
ax.plot_surface(X, Y, v[:], cmap=cm.viridis, rstride=1, cstride=1)
ax.set_xlabel('$x$')
ax.set_ylabel('$y$');


# In[4]:


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

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


# In[5]:


fig = pyplot.figure(figsize=(11, 7), dpi=100)
ax = fig.gca(projection='3d')
X, Y = numpy.meshgrid(x, y)
ax.plot_surface(X, Y, u, cmap=cm.viridis, rstride=1, cstride=1)
ax.plot_surface(X, Y, v, cmap=cm.viridis, rstride=1, cstride=1)
ax.set_xlabel('$x$')
ax.set_ylabel('$y$');


# ## Learn More

# The video lesson that walks you through the details for Steps 5 to 8 is **Video Lesson 6** on You Tube:

# In[6]:


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


# In[7]:


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


# > (The cell above executes the style for this notebook.)