.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/elliptic/plot_02_richardson_relaxation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_pde_elliptic_plot_02_richardson_relaxation.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_pde_elliptic_plot_02_richardson_relaxation.py:


Richardson's relaxation: iterating toward Laplace's solution
============================================================

Lewis Fry Richardson (1911) replaced Laplace's equation by finite
differences and solved the resulting equations *by hand*, sweeping over
the grid and replacing each value by the average of its neighbors until
nothing changed. That idea became the Jacobi iteration; updating values
in place gives Gauss-Seidel, and over-relaxing each update gives SOR
(Young, 1950). This script heats one side of a square plate and compares
how fast the three relaxation schemes from :mod:`mathematicskit.linalg`,
and conjugate gradient, converge.

.. GENERATED FROM PYTHON SOURCE LINES 16-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.linalg.visualizers import plot_residual_history
    from mathematicskit.pde import solve_laplace_2d
    from mathematicskit.pde.visualizers import plot_field_2d

    n = 17
    boundary = np.zeros((n, n))
    boundary[:, -1] = np.sin(np.pi * np.linspace(0.0, 1.0, n))  # warm top edge








.. GENERATED FROM PYTHON SOURCE LINES 28-30

The steady temperature
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-37

.. code-block:: Python


    direct = solve_laplace_2d(boundary, n=(n, n))
    fig1, ax1 = plt.subplots(figsize=(5.5, 4.5))
    plot_field_2d(direct, ax=ax1, levels=20, cmap="inferno")
    ax1.set_title("Laplace's equation: a plate warmed along its top edge")
    fig1.tight_layout()




.. image-sg:: /api/gallery/pde/elliptic/images/sphx_glr_plot_02_richardson_relaxation_001.png
   :alt: Laplace's equation: a plate warmed along its top edge
   :srcset: /api/gallery/pde/elliptic/images/sphx_glr_plot_02_richardson_relaxation_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 38-40

Relaxation sweeps
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 40-52

.. code-block:: Python


    fig2, ax2 = plt.subplots(figsize=(7, 4.5))
    for method in ("jacobi", "gauss_seidel", "sor", "cg"):
        sol = solve_laplace_2d(boundary, n=(n, n), method=method, tol=1e-8)
        result = sol.solver_result
        print(f"{method:>12}: {result.iterations:5d} iterations, max |u - direct| = {np.max(np.abs(sol.u - direct.u)):.1e}")
        plot_residual_history(result, ax=ax2, label=method)
    ax2.set_xscale("log")
    ax2.set_title("Jacobi (Richardson) < Gauss-Seidel < SOR < CG")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/pde/elliptic/images/sphx_glr_plot_02_richardson_relaxation_002.png
   :alt: Jacobi (Richardson) < Gauss-Seidel < SOR < CG
   :srcset: /api/gallery/pde/elliptic/images/sphx_glr_plot_02_richardson_relaxation_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

          jacobi:   812 iterations, max |u - direct| = 4.6e-08
    gauss_seidel:   411 iterations, max |u - direct| = 4.4e-08
             sor:    58 iterations, max |u - direct| = 8.7e-09
              cg:    16 iterations, max |u - direct| = 2.9e-11





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.762 seconds)


.. _sphx_glr_download_api_gallery_pde_elliptic_plot_02_richardson_relaxation.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/pde/elliptic/plot_02_richardson_relaxation.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_richardson_relaxation.ipynb <plot_02_richardson_relaxation.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_richardson_relaxation.py <plot_02_richardson_relaxation.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_richardson_relaxation.zip <plot_02_richardson_relaxation.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_