#!/usr/bin/env python
# coding: utf-8

# # Linear Advection with Finite Differences
# ## CH EN 6355 - Computational Fluid Dynamics
# **Prof. Tony Saad (<a>www.tsaad.net</a>) <br/>Department of Chemical Engineering <br/>University of Utah**
# <hr/>

# We will be solving the linear advection equation
# \begin{equation}
# \frac{\partial u}{\partial t} = - c \frac{\partial u}{\partial x}
# \end{equation}
# subject to the initial condition $u(x,0)\equiv u_0(x)$ on the domain $[0,L]$. The exact solution for this equation is given by
# \begin{equation}
# u(x,t) = u_0(x-ct)
# \end{equation}
# 

# In[62]:


import numpy as np
get_ipython().run_line_magic('matplotlib', 'inline')
get_ipython().run_line_magic('config', "InlineBackend.figure_format = 'svg'")
import matplotlib.pyplot as plt
import matplotlib.animation as animation
plt.rcParams['animation.html'] = 'html5' # this is used to display animations in jupyter notebooks


# Let's start by defining an initial condition - let's take
# \begin{equation}
#     u_0(x) = \sin(\omega x)
# \end{equation}

# In[63]:


n = 41
L = 1.0
dx = L/(n-1)
x = np.linspace(0,L,n)
u0 = np.sin(2.0*np.pi*x)
plt.plot(x,u0,'o')
plt.grid()


# ## Forward in Time, Central in Space (FTCS)
# Using a forward Euler method for the time derivative and a central discretization for the spatial derivative, the FTCS scheme results in
# \begin{equation}
# u_i^{n + 1} = u_i^n + c\Delta t\frac{u_{i + 1}^n - u_{i - 1}^n}{2\Delta x}
# \end{equation}

# In[64]:


# the following implementation does not use ghost cells
dt = 0.001  # s
tend = 2.0 # s
c = 10.0 # m/s - wave speed
cfl = c*dt/2.0/dx

sol = []
sol.append(u0)

t = 0.0
while t < tend:
    un = sol[-1]
    unew = un.copy()
    unew[1:-1] = un[1:-1] - cfl * (un[2:] - un[:-2])
    unew[-1] = un[-1] - cfl*(un[1] - un[-2]) # compute last point on the right using periodicity
    unew[0] = unew[-1] # set periodic boundary on the left
    sol.append(unew)
    t += dt


# In[67]:


ims = []
fig = plt.figure(figsize=[4,3])
plt.grid()

i = 0
for solution in sol:
    if (i%10==0): # output frequency for frames  
        im = plt.plot(x,solution,'k-',animated=True)
        plt.ylim(-1,1)
        ims.append(im)
    i+=1
ani = animation.ArtistAnimation(fig, ims, interval=35, blit=True,
                                repeat_delay=1000)
# ani.save('ftcs.mp4')   
ani


# ## Forward in Time, Backward (Upwind) in Space (FTUS)

# We have shown in class that the previous FTCS scheme is unconditionally unstable. Here we will implement the upwind scheme in space

# In[69]:


tend = 1.0 # s
c = 1.0 # m/s - wave speed
dt = dx/c  # run just at the CFL condition
cfl = c*dt/dx

sol = []
sol.append(u0)

t = 0.0
while t < tend:
    un = sol[-1]
    unew = un.copy()
    unew[1:-1] = un[1:-1] - cfl * (un[1:-1] - un[:-2])
    unew[-1] = un[-1] - cfl*(un[-1] - un[-2]) # compute last point on the right using periodicity
    unew[0] = unew[-1] # set periodic boundary on the left
    sol.append(unew)
    t += dt


# In[72]:


ims = []
fig = plt.figure(figsize=[5,4],dpi = 150)
plt.grid()

i = 0
for solution in sol:
    if (i%2==0):        
        im = plt.plot(x,solution,'k-',animated=True)
        plt.ylim(-1,1)
        ims.append(im)
    i+=1
ani = animation.ArtistAnimation(fig, ims, interval=35, blit=True,
                                repeat_delay=1000)
# ani.save('ftus.mp4')   
ani


# In[73]:


import urllib
import requests
from IPython.core.display import HTML
def css_styling():
    styles = requests.get("https://raw.githubusercontent.com/saadtony/NumericalMethods/master/styles/custom.css")
    return HTML(styles.text)
css_styling()


# In[ ]: