Changing the coordinate system
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By default, the coordinate system is just the normal Cartesian one. This holds true for :math:`1`\ d, :math:`2`\ d and :math:`3`\ d geometries. However, a lot of physical problems are in fact :math:`3`\ d, but have a rotational symmetry, i.e. can be considered to be axisymmetric so that axisymmetric cylindrical coordinates can effectively reduce the problem to a computationally favorable :math:`2`\ d problem. With a single line of code, i.e. by calling :py:meth:`~pyoomph.generic.problem.Problem.set_coordinate_system`, one can change the coordinate system from :math:`2`\ d Cartesian to axisymmetric cylindrical coordinates:

.. code:: python

   # Load the previous code
   from poisson_2d_adaptive import *


   if __name__=="__main__":
       with AdaptivePoissonProblem2d() as problem:
           # Change the coordinate system to axisymmetric
           problem.set_coordinate_system(axisymmetric)
           # The rest is the same
           problem.max_refinement_level = 5
           problem.max_permitted_error = 0.0005
           problem.min_permitted_error = 0.00005
           problem.solve(spatial_adapt=problem.max_refinement_level)
           problem.output()

Note that the radial coordinate :math:`r` is given by the :math:`x`-direction, which can be accessed in both cases by ``var("coordinate_x")``, whereas the :math:`y`-axis becomes the coordinate along the axis of symmetry (often called :math:`z` in cylindrical coordinates). In pyoomph, this coordinate is hence accessible via ``var("coordinate_y")``. The variable ``var("coordinate")`` will expand to the vector consisting of these coordinates.

When changing the coordinate system, pyoomph will intrinsically modify the weighting in the spatial integral terms in the weak form and will evaluate the spatial derivatives accordingly.

Besides the coordinate system ``axisymmetric``, which works in :math:`1`\ d and :math:`2`\ d geometries, there is also the ``radialsymmetric`` coordinate system, which works in :math:`1`\ d geometries only. It can be used to simulate problems that have the full rotational symmetry of a sphere by just solving for the radial direction. In both cases, it is important to make sure that the geometry (i.e. the mesh) has only non-negative :math:`x`-coordinates (i.e. radial coordinates). Furthermore, one cannot impose Neumann conditions at :math:`x=0` (i.e. :math:`r=0`), since the spatial integrals will :math:`\int \ldots \mathrm{d}S` will expand to :math:`2\pi\int \ldots r\mathrm{d}l`. Since :math:`r=0`, there cannot be any contribution from Neumann conditions here. However, the symmetry assumption usually requires to have a vanishing Neumann flux at :math:`r=0` anyhow.

..  figure:: axisymm_poisson.*
	:name: figspatialaxisymmpoisson
	:align: center
	:alt: Axisymmetric coordinate system
	:class: with-shadow
	:width: 50%
	
	Poisson equation with an axisymmetric coordinate system.

.. only:: html

	.. container:: downloadbutton

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