.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/advection/plot_02_lax_friedrichs.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_advection_plot_02_lax_friedrichs.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_advection_plot_02_lax_friedrichs.py:


The Lax-Friedrichs scheme: stabilizing centered differences
===========================================================

The centered FTCS scheme :math:`u_j^{n+1} = u_j^n - \tfrac{\nu}{2}(u_{j+1}^n - u_{j-1}^n)`
is unconditionally unstable for advection. Peter Lax (1954), following
Friedrichs, replaced :math:`u_j^n` by the neighbour average
:math:`\tfrac12(u_{j+1}^n + u_{j-1}^n)`. That one change makes the scheme
stable for :math:`\nu \le 1`, and its conservation form carries over to
nonlinear conservation laws and shocks -- at the price of heavy
numerical diffusion.

.. GENERATED FROM PYTHON SOURCE LINES 15-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import AdvectionEquation1D, max_amplification

    G_NAME = {"ftcs": "ftcs_advection"}  # amplification-factor name of each advection scheme

    adv = AdvectionEquation1D(lambda x: np.exp(-200 * (x - 0.5) ** 2), n=200)








.. GENERATED FROM PYTHON SOURCE LINES 25-27

FTCS blows up; averaging tames it
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-39

.. code-block:: Python


    fig, ax = plt.subplots()
    ax.plot(adv.x, adv.exact(0.5), "k--", lw=2, label="exact")
    for scheme in ("ftcs", "lax_friedrichs"):
        sol = adv.solve(0.5, dt=0.8 * adv.dx, scheme=scheme)
        print(f"{scheme:>15}: max |u| = {np.max(np.abs(sol.final)):.3g}, max |G| = {max_amplification(G_NAME.get(scheme, scheme), 0.8):.4f}")
        ax.plot(sol.x, np.clip(sol.final, -1.0, 2.0), label=scheme)
    ax.set_ylim(-1.0, 2.0)
    ax.set_xlabel("$x$")
    ax.set_title("Same centered difference, with and without averaging (nu = 0.8)")
    ax.legend()




.. image-sg:: /api/gallery/pde/advection/images/sphx_glr_plot_02_lax_friedrichs_001.png
   :alt: Same centered difference, with and without averaging (nu = 0.8)
   :srcset: /api/gallery/pde/advection/images/sphx_glr_plot_02_lax_friedrichs_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

               ftcs: max |u| = 1.88, max |G| = 1.2806
     lax_friedrichs: max |u| = 0.831, max |G| = 1.0000

    <matplotlib.legend.Legend object at 0x7fd7bfdf3320>



.. GENERATED FROM PYTHON SOURCE LINES 40-43

Lax-Friedrichs' numerical diffusion grows as nu shrinks
-------------------------------------------------------
Its modified equation has diffusion coefficient dx^2 (1 - nu^2) / (2 dt).

.. GENERATED FROM PYTHON SOURCE LINES 43-54

.. code-block:: Python


    fig, ax = plt.subplots()
    ax.plot(adv.x, adv.exact(1.0), "k--", lw=2, label="exact")
    for nu in (0.3, 0.6, 0.95):
        sol = adv.solve(1.0, dt=nu * adv.dx, scheme="lax_friedrichs")
        ax.plot(sol.x, sol.final, label=f"nu = {nu}")
    ax.set_xlabel("$x$")
    ax.set_title("Lax-Friedrichs after one period")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/pde/advection/images/sphx_glr_plot_02_lax_friedrichs_002.png
   :alt: Lax-Friedrichs after one period
   :srcset: /api/gallery/pde/advection/images/sphx_glr_plot_02_lax_friedrichs_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.182 seconds)


.. _sphx_glr_download_api_gallery_pde_advection_plot_02_lax_friedrichs.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/advection/plot_02_lax_friedrichs.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_lax_friedrichs.ipynb <plot_02_lax_friedrichs.ipynb>`

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

      :download:`Download Python source code: plot_02_lax_friedrichs.py <plot_02_lax_friedrichs.py>`

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

      :download:`Download zipped: plot_02_lax_friedrichs.zip <plot_02_lax_friedrichs.zip>`


.. only:: html

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

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