#!/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 MIT license.  (c) Lorena A. Barba, Gilbert F. Forsyth 2015. Thanks to NSF for support via CAREER award #1149784.
# [@LorenaABarba](https://twitter.com/LorenaABarba)

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

# You see where this is going ... we'll do 2D diffusion now and next we will combine steps 6 and 7 to solve Burgers' equation. So make sure your previous steps work well before continuing.

# Step 7: 2D Diffusion
# ----
# ***

# And here is the 2D-diffusion equation:
# 
# $$\frac{\partial u}{\partial t} = \nu \frac{\partial ^2 u}{\partial x^2} + \nu \frac{\partial ^2 u}{\partial y^2}$$
# 
# You will recall that we came up with a method for discretizing second order derivatives in Step 3, when investigating 1-D diffusion.  We are going to use the same scheme here, with our forward difference in time and two second-order derivatives. 

# $$\frac{u_{i,j}^{n+1} - u_{i,j}^n}{\Delta t} = \nu \frac{u_{i+1,j}^n - 2 u_{i,j}^n + u_{i-1,j}^n}{\Delta x^2} + \nu \frac{u_{i,j+1}^n-2 u_{i,j}^n + u_{i,j-1}^n}{\Delta y^2}$$
# 
# Once again, we reorganize the discretized equation and solve for $u_{i,j}^{n+1}$

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

# In[1]:


import numpy
from matplotlib import pyplot
from mpl_toolkits.mplot3d import Axes3D ##library for 3d projection plots
from matplotlib import cm ##cm = "colormap" for changing the 3d plot color palette
get_ipython().run_line_magic('matplotlib', 'inline')

###variable declarations
nx = 31
ny = 31
nt = 17
nu=.05
dx = 2/(nx-1)
dy = 2/(ny-1)
sigma = .25
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
un = numpy.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

fig = pyplot.figure()
ax = fig.gca(projection='3d')
X,Y = numpy.meshgrid(x,y)
surf = ax.plot_surface(X,Y,u, rstride=1, cstride=1, cmap=cm.coolwarm,
        linewidth=0, antialiased=False)
ax.set_xlim(0,2)
ax.set_ylim(0,2)
ax.set_zlim(1,2.5);


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

# In[2]:


###Run through nt timesteps
def diffuse(nt):
    u[.5/dy:1/dy+1,.5/dx:1/dx+1]=2
    
    for n in range(nt+1): 
        un = u.copy()
        u[1:-1,1:-1]=un[1:-1,1:-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])
        u[0,:]=1
        u[-1,:]=1
        u[:,0]=1
        u[:,-1]=1

    
    fig = pyplot.figure()
    ax = fig.gca(projection='3d')
    surf = ax.plot_surface(X,Y,u[:], rstride=1, cstride=1, cmap=cm.coolwarm,
        linewidth=0, antialiased=True)
    ax.set_zlim(1,2.5)
    


# In[3]:


diffuse(10)


# In[4]:


diffuse(14)


# In[5]:


diffuse(50)


# ## 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.)