pencil.math.poisson_solver
==========================

.. py:module:: pencil.math.poisson_solver

.. autoapi-nested-parse::

   This code contains various functions to solve the vector form of the Poisson
   equation in various coordinate systems (Cartesian, cylindrical and spherical)
   using finite differences.



Functions
---------

.. autoapisummary::

   pencil.math.poisson_solver.poisson_vector_cartesian
   pencil.math.poisson_solver.poisson_scalar_cylindrical
   pencil.math.poisson_solver.poisson_vector_cylindrical
   pencil.math.poisson_solver.poisson_vector_spherical


Module Contents
---------------

.. py:function:: poisson_vector_cartesian(bx, by, bz, x, y, z, hx, hy, hz, niter=1000)

       poisson_vector_cartesian(bx, by, bz, x, y, z, hx, hy, hz, niter=1000)

       Solve the vector form of the Poisson equation in 3D Cartesian coordinates,
       $
   abla^2 u = h$, using finite differences.

       Parameters
       ----------
       bx, by, bz : ndarray of shape [nz, ny, nx]
           Boundary conditions on exterior points for each component.
           Keep the inner points 0.

       x, y, z : ndarrays of shape [nx], [ny] and [nz]
           Coordinate arrays to calculate grid spacing.

       hx, hy, hz : ndarray of shape [nz, ny, nx]
           Representing the components of the known function h.

       niter : int
           Number of iterations.

       Returns
       ----------
       ndarray with the shape [3, nz, ny, nx], representing solution to the Poisson equation.



.. py:function:: poisson_scalar_cylindrical(bc, r, theta, z, h, niter=200)

   poisson_scalar_cylindrical(bc, r, theta, z, h, niter=200)

   Solve the scalar form of the Poisson equation in cylindical coordinates using finite differences.

   :param bc: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type bc: ndarray of shape [nz, ny, nx]
   :param r: Coordinate arrays to calculate grid spacing.
   :type r: ndarrays of shape [nx], [ny] and [nz]
   :param theta: Coordinate arrays to calculate grid spacing.
   :type theta: ndarrays of shape [nx], [ny] and [nz]
   :param z: Coordinate arrays to calculate grid spacing.
   :type z: ndarrays of shape [nx], [ny] and [nz]
   :param h: Representing the known function h.
   :type h: ndarray of shape [nz, ny, nx]
   :param niter: Number of iterations.
   :type niter: int

   :rtype: ndarray with the shape [nz, ny, nx], representing solution to the Poisson equation.


.. py:function:: poisson_vector_cylindrical(br, btheta, bz, r, theta, z, hr, htheta, hz, niter=1000)

       poisson_vector_cylindrical(br, btheta, bz, r, theta, z, hr, htheta, hz, niter=1000)

       Solve the vector form of the Poisson equation, $
   abla^2 u = h$, in cylindrical
       coordinates using finite diffferences.

       Parameters:
       ----------
       br, btheta, bz : ndarray of shape [nz, ny, nx]
           Boundary conditions on exterior points for each component.
           Keep the inner points 0.

       r, theta, z : ndarrays of shape [nx], [ny] and [nz]
           Coordinate arrays to calculate grid spacing.

       hr, htheta, hz : ndarray of shape [nz, ny, nx]
           Representing the components of the known function h.

       niter : int
           Number of iterations.

       Returns
       ----------
       ndarray with the shape [3, nz, ny, nx], representing solution to the Poisson equation.



.. py:function:: poisson_vector_spherical(br, btheta, bphi, r, theta, phi, hr, htheta, hphi, niter=200)

       poisson_vector_spherical(br, btheta, bphi, r, theta, phi, hr, htheta, hphi, niter=200)

       Solve the vector form of the Poisson equation, $
   abla^2 u = h$,
       in spherical coordinates using finite differences.

       Parameters:
       ----------
       br, btheta, bphi :  ndarray of shape [nz, ny, nx]
           Boundary conditions on exterior points for each component.
           Keep the inner points 0.

       r, theta, phi :  ndarrays of shape [nx], [ny] and [nz]
           Coordinate arrays, where the convention is taken that theta ranges from 0 to pi,
           phi ranges from 0 to 2pi.
           Note that singularities arise in the equation when theta = 0.
           This may produce unintended results.

       hr, htheta, hphi : ndarray of shape [nz, ny, nx]
           Representing the components of the known function h.

       niter : int
           Number of iterations.

       Returns
       ----------
       ndarray with the shape [3, nz, ny, nx], representing solution to the Poisson equation.