.. _EB-C12:

C12. Smooth periodic problem
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Case description
################

This is the standard smooth advection problem with :math:`u_0=1`, :math:`p_0=1`, and

.. math::
   \rho(x) = 1 + 0.2 \sin (\pi x)

This is an exact solution to the Euler equations and reduces them to
simple linear advection with a constant velocity. Periodic boundary
conditions are imposed. The gas constant is :math:`\gamma=1.4`. Usage
of such a test case can be found in references such as Guan-Shan Jiang
and Chi-Wang Shu. "Efficient implementation of weighted eno
schemes". J.  Comp. Phys., 126:202–228 (1996) and Liska and Wendroff
"Comparison of several difference schemes on 1D and 2D test problems
for the Euler equations". SIAM J. Sci. Comput., 25(3), 995–1017
(2006).

Simulations are performed in an :math:`x`-direction aligned circular
channel. The cylinder radius, :math:`R`, is 0.1cm and the cylinder
length, :math:`L`, is 2cm. The simulations are performed until
:math:`t=2`, at which point the wave is back at its starting location.

Density profiles in centerline
##############################

.. image:: /ebverification/C12/rho.png
   :height: 300pt


:math:`L_2` error norm of density
#################################

.. image:: /ebverification/C12/error.png
   :height: 300pt

.. note::
   The second order convergence observed here is expected for this
   test case as all relevant physics happen in the direction
   perpendicular to the EB surface.

Running study
#############

.. code-block:: bash

   paren=`pwd`
   pelec="${paren}/PeleC3d.gnu.MPI.ex"
   mpi_ranks=36

   res=( 4 8 16 32 )
   for i in "${res[@]}"
   do
       rm -rf "${i}"
       mkdir "${i}"
       cd "${i}" || exit
       cp "${paren}/inputs_3d" .
       ny="$((i*2))"
       nx="$((i*10))"
       srun -n ${mpi_ranks} "${pelec}" inputs_3d amr.n_cell="${nx} ${ny} ${ny}" pelec.fixed_dt=1e-5 > out
       ls -1v ./*plt*/Header | tee movie.visit
       cd "${paren}" || exit
   done