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

# # Finite Volume Burgers
# 
# Copyright (C) 2010-2020 Luke Olson<br>
# Copyright (C) 2020 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
# 
# The above copyright notice and this permission notice shall be included in
# all copies or substantial portions of the Software.
# 
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
# THE SOFTWARE.
# </details>

# In[1]:


import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

from matplotlib import animation
from IPython.display import HTML


# In[2]:


def gaussian(x):
    u = np.exp(-100 * (x - 0.25)**2)
    return u


def step(x):
    u = np.zeros(x.shape)
    for j in range(len(x)):
        if (x[j] >= 0.6) and (x[j] <= 0.8):
            u[j] = 1.0

    return u

def g1(x):
    return 1+gaussian(x)

def g2(x):
    return 1+gaussian(x) + step(x)

g = g1


# In[3]:


nx = 164
x = np.linspace(0, 1, nx, endpoint=False)
dx = x[1] - x[0]
xx = np.linspace(0, 1, 1000, endpoint=False)

lmbda = 0.95
nt = 250
print('tsteps = %d' % nt)
print('    dx = %g' % dx)
print('lambda = %g' % lmbda)

J = np.arange(0, nx)  # all vertices
Jm1 = np.roll(J, 1)
Jp1 = np.roll(J, -1)

plt.plot(x, g(x))


# Plot the solution:

# In[4]:


if 1:
    # Burgers
    def f(u):
        return u**2/2
    def fprime(u):
        return u
else:
    # advection
    def f(u):
        return u
    def fprime(u):
        return 1+0*u

steps_per_frame = 2


# ## Part I: Lax-Friedrichs
# 
# Implement `rhs` for a Lax-Friedrichs flux:
# 
# Recall (local) Lax-Friedrichs:
# $$
# f^{\ast} (u^{_-}, u^+) = \frac{f (u^-) + f (u^+)}{2} - \frac{\alpha}{2} (u^+ - u^-) 
# \quad\text{with}\quad
# \alpha = \max \left( |f' (u^-)|, |f' (u^+)| \right).$$
# Recall FV:
# $$ \bar{u}_{j,\ell+1} = \bar{u}_{j,\ell} - \frac{h_t}{h_x} (f (u_{j + 1 / 2,\ell}) -
#    f (u_{j - 1 / 2,\ell})) . $$
# 

# In[5]:


def rhs(u):
    uplus = u[Jp1]
    uminus = u[J]
    alpha = np.maximum(np.abs(fprime(uplus)), np.abs(fprime(uminus)))
    # right-looking, between J and Jp1
    fluxes = (
        (f(uplus)+f(uminus))/2
        - alpha/2*(uplus-uminus)
        )
    return - 1/dx*(fluxes[J]-fluxes[Jm1])


# In[6]:


u = g(x)

def timestepper(n):
    for i in range(steps_per_frame):
        dt = dx*lmbda/np.max(fprime(np.abs(u)))
        u[:] = u + dt*rhs(u)

    line.set_data(x, u)
    return line

fig = plt.figure(figsize=(5,5))
line, = plt.plot(x, u)

ani = animation.FuncAnimation(
    fig, timestepper,
    frames=nt//steps_per_frame,
    interval=30)
html = HTML(ani.to_jshtml())
plt.clf()
html


# In[ ]: