# Exact solution used in MES runs

We would like to MES the operation

\begin{eqnarray} \frac{\int_0^{2\pi} f \rho d\theta}{\int_0^{2\pi} \rho d\theta} = \frac{\int_0^{2\pi} f d\theta}{\int_0^{2\pi} d\theta} = \frac{\int_0^{2\pi} f d\theta}{2\pi} \end{eqnarray}

Using cylindrical geometry.



In [1]:

%matplotlib notebook

from sympy import init_printing
from sympy import S
from sympy import sin, cos, tanh, exp, pi, sqrt
from sympy import integrate
import numpy as np

from boutdata.mms import x, y, z, t

import os, sys
# If we add to sys.path, then it must be an absolute path
common_dir = os.path.abspath('./../../../common')
# Sys path is a list of system paths
sys.path.append(common_dir)
from CELMAPy.MES import get_metric, make_plot, BOUT_print

init_printing()



## Initialize



In [2]:

folder = '../zHat/'
metric = get_metric()



## Define the variables



In [3]:

# Initialization
the_vars = {}



### Define the function to take the derivative of

NOTE:

These do not need to be fulfilled in order to get convergence

1. z must be periodic
2. The field $f(\rho, \theta)$ must be of class infinity in $z=0$ and $z=2\pi$
3. The field $f(\rho, \theta)$ must be continuous in the $\rho$ direction with $f(\rho, \theta + \pi)$

But this needs to be fulfilled:

1. The field $f(\rho, \theta)$ must be single valued when $\rho\to0$
2. Eventual BC in $\rho$ must be satisfied


In [4]:

# We need Lx
from boututils.options import BOUTOptions
myOpts = BOUTOptions(folder)
Lx = eval(myOpts.geom['Lx'])




In [5]:

# Z hat function

# NOTE: The function is not continuous over origo

s = 2
c = pi
w = pi/2
the_vars['f'] = ((1/2)*(tanh(s*(z-(c-w/2)))-tanh(s*(z-(c+w/2)))))*sin(3*2*pi*x/Lx)



Calculating the solution



In [6]:

the_vars['S'] = (integrate(the_vars['f'], (z, 0, 2*np.pi))/(2*np.pi)).evalf()



## Plot



In [7]:

make_plot(folder=folder, the_vars=the_vars, plot2d=True, include_aux=False)




In [8]:

BOUT_print(the_vars, rational=False)