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

# # Dispersion and Dissipation
# 
# 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


# Consider
# $$u_t+au_x=0$$
# with periodic boundary conditions.
# 
# Set up parameters:
# 
# - `a` for the advection speed
# - `lmbda` for the CFL number
# - `dx` for the grid spacing in $x$
# - `dt` for the time step
# - `ks` for the range of wave numbers to consider

# In[2]:


a = 1
lmbda = 0.6/a
dx = .1
dt = dx*lmbda
ks = np.arange(1,16)


# Find $\omega(\kappa)$. Recall $\lambda = ah_t / h_x$.
# 
# ETBS:
# $$ u_{k, \ell + 1} = \lambda u_{k - 1 , \ell} + (1 - \lambda) u_{k, \ell} $$
# 
# Recall:
# * $r_k=\delta_{k,j}\Leftrightarrow\hat{\boldsymbol{r}} (\varphi) =  e^{- i \theta j}$.
# * Index sign flip between matrix and Toeplitz vector.
# * $e^{- i \omega (\kappa) h_t} = s (\kappa)$.

# In[9]:


#clear
kappa = ks*dx
p_ETBS = 1
q_ETBS = lmbda*np.exp(-1j*kappa) + (1-lmbda)
s_ETBS = q_ETBS/p_ETBS

omega_ETBS = 1j*np.log(s_ETBS)/dt


# Again recall $\lambda = ah_t / h_x$.
# 
# Lax-Wendroff:
# $$
#   u_{k, \ell + 1} - u_{k, \ell}
#   = -\frac{\lambda}2 (u_{k + 1, \ell} - u_{k - 1, \ell}) +
#   \frac{\lambda^2}{2} ( u_{k + 1, \ell} - 2 u_{k, \ell} + u_{k - 1, \ell})
# $$

# In[10]:


#clear
p_LW = 1
q_LW = (
    # u_{k,l}
    1 - 2*lmbda**2/2
    
    # u_{k+1,l}
    + np.exp(1j*kappa) * (-lmbda/2 + lmbda**2/2)
    
    # u_{k-1,l}
    + np.exp(-1j*kappa) * (lmbda/2 + lmbda**2/2)
    )
s_LW = q_LW/p_LW

omega_LW = 1j*np.log(s_LW)/dt


# In[11]:


plt.plot(ks, omega_ETBS.real, label="ETBS")
plt.plot(ks, omega_LW.real, label="Lax-Wendroff")
plt.plot(ks, a*ks, color='black', label='exact')
plt.legend(loc="best")


# In[12]:


plt.plot( ks, omega_ETBS.imag, label="ETBS")
plt.plot( ks, omega_LW.imag, label="Lax-Wendroff")
plt.legend(loc="best")


# In[ ]: