pencil.math.laplace_solver
==========================

.. py:module:: pencil.math.laplace_solver

.. autoapi-nested-parse::

   This code contains various functions to solve the vector form of the Laplace
   equation in various coordinate systems (Cartesian, cylindrical and spherical)
   using finite differences.
   Additionally, one can find functions to solve the scalar Laplace equation
   in Cartesian, cylidrindical and spherical coordinate systems, since these are used
   (at least in the case of Cartesian and cylindrical) in the vector solvers.



Functions
---------

.. autoapisummary::

   pencil.math.laplace_solver.laplace_scalar_cartesian
   pencil.math.laplace_solver.laplace_vector_cartesian
   pencil.math.laplace_solver.laplace_scalar_cylindrical
   pencil.math.laplace_solver.laplace_vector_cylindrical
   pencil.math.laplace_solver.laplace_scalar_spherical
   pencil.math.laplace_solver.laplace_vector_spherical


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

.. py:function:: laplace_scalar_cartesian(bc, dx, dy, dz, niter=200)

   laplace_scalar_cartesian(bc, dx, dy, dz, niter=100)

   Solve the scalar Laplace equation in Cartesian coordinates in 3 dimensions
   using finite differences.

   :param bc: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type bc: ndarray of shape [nz, ny, nx]

   dx, dy, dz : floats
       Grid spacing in eaach direction.

   niter : int
       Number of iterations.

   :rtype: ndarray with the same shape as bc, representing solution to the Laplace equation.


.. py:function:: laplace_vector_cartesian(bx, by, bz, dx, dy, dz, niter=200)

   laplace_vector_cartesian(bx, by, bz, dx, dy, dz, niter=1000)

   Solve the vector Laplace equation in Cartesian coordinates in 3 dimensions
   using finite differences. This function simply applies the scalar function
   to the three components.

   :param bx: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type bx: ndarrays of shape [nz, ny, nx]
   :param by: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type by: ndarrays of shape [nz, ny, nx]
   :param bz: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type bz: ndarrays of shape [nz, ny, nx]

   dx, dy, dz : float
       Grid spacing in eaach direction.

   niter : int
       Number of iterations.

   :rtype: ndarray with the shape [3, nz, ny, nx], representing solution to the Laplace equation.


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

   laplace_scalar_cylindrical(bc, r, theta, z, niter=1000)

   Solve the scalar Laplace equation in cylindical coordinates in 3 dimensions
   using finite differences.

   :param bc: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type bc: ndarray of shape [nz, ny, nx]

   r, theta, z : ndarrays of shape [nx], [ny] and [nz]
       1D coordinate arrays of r, theta and z.

   niter : int
       Number of iterations.

   :rtype: ndarray with the same shape as bc, representing solution to the Laplace equation.


.. py:function:: laplace_vector_cylindrical(br, btheta, bz, r, theta, z, niter=200)

   laplace_vector_cylindrical(br, btheta, bz, r, theta, z, niter=200)

   Solve the vector Laplace equation in cylindrical coordinates in 3 dimensions
   using finite differences.

   :param br: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type br: ndarrays of shape [nz, ny, nx]
   :param btheta: Boundary conditions on exterior points.
                  Keep the inner points 0.
   :type btheta: ndarrays of shape [nz, ny, nx]
   :param bz: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type bz: ndarrays of shape [nz, ny, nx]

   r, theta, z : ndarrays of shape [nx], [ny] and [nz]
       1D coordinate arrays of r, theta and z.

   niter : int
       Number of iterations.

   :rtype: ndarray with the shape [3, nz, ny, nx], representing solution to the Laplace equation.


.. py:function:: laplace_scalar_spherical(bc, r, theta, phi, niter=200)

   laplace_scalar_spherical(bc, r, theta, phi, niter=200)

   Solve the scalar Laplace equation in spherical coordinates in 3 dimensions
   using finite differences.

   :param bc: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type bc: ndarray of shape [nz, ny, nx]

   r, theta, phi : ndarrays of shape [nx], [ny] and [nz]
       1D coordinate arrays of r, theta and phi.
       The convention is taken that theta ranges from 0 to pi and
       phi ranges from 0 to 2pi.
       Note that singularities arise in the equation when theta = 0 and r = 0.
       This may produce unintended results.

   niter : int
       Number of iterations.

   :rtype: ndarray with the same shape as bc, representing solution to the Laplace equation.


.. py:function:: laplace_vector_spherical(br, btheta, bphi, r, theta, phi, niter=200)

   laplace_vector_spherical(br, btheta, bphi, r, theta, phi, niter=200)

   Solve the scalar Laplace equation in spherical coordinates in 3 dimensions
   using finite differences.

   :param bc: Boundary conditions on exterior points.
              Keep the inner points 0.
   :type bc: ndarray of shape [nz, ny, nx]
   :param btheta: Boundary conditions on exterior points.
                  Keep the inner points 0.
   :type btheta: ndarray of shape [nz, ny, nx]
   :param bphi: Boundary conditions on exterior points.
                Keep the inner points 0.
   :type bphi: ndarray of shape [nz, ny, nx]

   r, theta, phi : ndarrays of shape [nx], [ny] and [nz]
       1D coordinate arrays of r, theta and phi.
       The convention is taken that theta ranges from 0 to pi and
       phi ranges from 0 to 2pi.
       Note that singularities arise in the equation when theta = 0 and r = 0.
       This may produce unintended results.

   niter : int
       Number of iterations.

   :rtype: ndarray with the shape [3, nz, ny, nx], representing solution to the Laplace equation.