.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/stability/plot_02_von_neumann_analysis.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_stability_plot_02_von_neumann_analysis.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_stability_plot_02_von_neumann_analysis.py:


Von Neumann stability analysis: one Fourier mode at a time
==========================================================

John von Neumann's method substitutes a single Fourier mode
:math:`u_j^n = G^n e^{ij\xi}` into a linear scheme. Each step multiplies
the mode by the *amplification factor* :math:`G(\xi)`, so the scheme is
stable exactly when :math:`|G(\xi)| \le 1` for every wavenumber. This
script plots :math:`|G|` for heat and advection schemes, recovers the
FTCS limit :math:`r \le 1/2` numerically, and shows why FTCS advection
is unstable for *every* time step.

.. GENERATED FROM PYTHON SOURCE LINES 15-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import max_amplification
    from mathematicskit.pde.visualizers import plot_amplification_factors








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Heat-equation schemes
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-29

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    plot_amplification_factors(["ftcs_heat", "crank_nicolson_heat", "backward_euler_heat"], 0.6, ax=axes[0])
    axes[0].set_title("Heat equation, diffusion number r = 0.6")




.. image-sg:: /api/gallery/pde/stability/images/sphx_glr_plot_02_von_neumann_analysis_001.png
   :alt: Heat equation, diffusion number r = 0.6
   :srcset: /api/gallery/pde/stability/images/sphx_glr_plot_02_von_neumann_analysis_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Heat equation, diffusion number r = 0.6')



.. GENERATED FROM PYTHON SOURCE LINES 30-32

Advection schemes
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 32-37

.. code-block:: Python


    plot_amplification_factors(["upwind", "lax_friedrichs", "lax_wendroff", "ftcs_advection"], 0.8, ax=axes[1])
    axes[1].set_title("Advection, Courant number 0.8")
    fig.tight_layout()








.. GENERATED FROM PYTHON SOURCE LINES 38-40

Finding the stability limits numerically
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-50

.. code-block:: Python


    rs = np.linspace(0.3, 0.7, 401)
    limit = rs[np.argmax([max_amplification("ftcs_heat", r) > 1 + 1e-12 for r in rs]) - 1]
    print(f"FTCS heat is stable up to r = {limit:.3f} (theory: 1/2)")
    for nu in (0.01, 0.5, 1.0):
        print(f"FTCS advection, nu = {nu}: max |G| = {max_amplification('ftcs_advection', nu):.6f}  (> 1 for every nu > 0)")
    for r in (1.0, 100.0):
        print(f"Crank-Nicolson, r = {r}: max |G| = {max_amplification('crank_nicolson_heat', r):.3f}")

    plt.show()




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

 .. code-block:: none

    FTCS heat is stable up to r = 0.500 (theory: 1/2)
    FTCS advection, nu = 0.01: max |G| = 1.000050  (> 1 for every nu > 0)
    FTCS advection, nu = 0.5: max |G| = 1.118034  (> 1 for every nu > 0)
    FTCS advection, nu = 1.0: max |G| = 1.414214  (> 1 for every nu > 0)
    Crank-Nicolson, r = 1.0: max |G| = 1.000
    Crank-Nicolson, r = 100.0: max |G| = 1.000





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

   **Total running time of the script:** (0 minutes 0.191 seconds)


.. _sphx_glr_download_api_gallery_pde_stability_plot_02_von_neumann_analysis.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/stability/plot_02_von_neumann_analysis.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_von_neumann_analysis.ipynb <plot_02_von_neumann_analysis.ipynb>`

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

      :download:`Download Python source code: plot_02_von_neumann_analysis.py <plot_02_von_neumann_analysis.py>`

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

      :download:`Download zipped: plot_02_von_neumann_analysis.zip <plot_02_von_neumann_analysis.zip>`


.. only:: html

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

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