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

# In[1]:


get_ipython().run_line_magic('pylab', 'inline')
from scipy.fftpack import dst, idst, dct, idct, fft, ifft, fftfreq


# # Fourier Spectral Methods

# ### Periodic Domains (fft)

# In[2]:


help(ifft)


# In[3]:


n = 99
N = (99 -1) /2
x = arange(n)/n * 2*pi
y = sin(3 * x)
k = fftfreq((n), 1/(n))
plot(x,y,label="sin 3x")

dy = real(ifft(fft(y) * k * 1j ))
plot(x,dy, label="fourier deriv")
legend()


# ### Neumann/Dirichclet Boundary Conditions (dct, dst)
# 
# Fourier derivative doesn't work near boundaries, so we have to use Sin/Cos basis, which is more difficult to use

# In[4]:


n = 100
x = arange(n+1)/n * pi
y = sin(3*x)


dy = real(ifft(fft(y) * fftfreq(n+1, 1/(n+1)) * 1j ))

plot(x,y, label='sin 3x')
plot(x,dy, label="fourier deriv")
legend()


# In[5]:


help(dst)


# Which of the three DST types looks like it can be written as $f(x) = \sum_k sin(x k)$? 
# 
# By the way, there are also three DCTs:

# In[6]:


help(dct)


# In[7]:


# remove end points
yy = y[1:-1]

# idst type-1
y_dst = idst(yy, type=1)

# take deriv
dy_dct = arange(1,len(y_dst)+1) * y_dst

# dct type-1 needs boundaries
dy_dct = hstack((0, dy_dct, 0))

# need to normalize properly
# normalization is not handled automatically
# in other words:
#    dst(idst(x)) != x
dy = dct(dy_dct, type=1)/dy_dct.shape[0]/2

plot(x,y, label='sin 3x')
plot(x,dy, label='Sin/Cos deriv')
legend()


# # Finite differencing for Barotropic Equations
# 
# $$ \zeta_t + J(\psi, \zeta) = 0 $$
# $$ \Delta \psi = \zeta$$
# 
# The variable are defined by
# $$\nabla \psi = (-u, v)$$
# $$ J(a,b) = b_x a_y - a_x b_y$$ 
# 
# This jacobian is hard to approximate, and Arakawa (1966) is a famous finite difference scheme.

# In[8]:


get_ipython().run_cell_magic('bash', '', "\nopen /Users/noah/Dropbox/Papers/Journal\\ Article/Arakawa_1966_Computational\\ design\\ for\\ long-term\\ numerical\\ integration\\ of\\ the\\ equations\\ of.pdf  /Users/noah/Dropbox/Papers/Journal\\ Article/Fix_1975_Finite\\ Element\\ Models\\ for\\ Ocean\\ Circulation\\ Problems.pdf /Users/noah/Dropbox/Papers/Journal\\ Article/Jespersen_1974_Arakawa\\'s\\ method\\ is\\ a\\ finite-element\\ method.pdf /Users/noah/Dropbox/Papers/Journal\\ Article/Naulin_Nielsen_2003_Accuracy\\ of\\ Spectral\\ and\\ Finite\\ Difference\\ Schemes\\ in\\ 2D\\ Advection\\ Problems.pdf\n")


# # Demonstration
# 
# Spectral code
# ```
# python -m gnl.pdes.barotropic_spec
# ```
# 
# FD code
# ```
# python -m gnl.pdes.barotropic_fd_par
# ```

# In[9]:


get_ipython().system('open -a ITerm.app')