.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/advection/plot_03_lax_wendroff.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_03_lax_wendroff.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_03_lax_wendroff.py:


The Lax-Wendroff scheme: second order, and dispersive wiggles
=============================================================

Lax and Wendroff (1960) used the PDE itself to replace time derivatives
in a Taylor expansion, :math:`u_{tt} = c^2 u_{xx}`, giving
:math:`u_j^{n+1} = u_j^n - \tfrac{\nu}{2}(u_{j+1}^n - u_{j-1}^n) + \tfrac{\nu^2}{2}(u_{j+1}^n - 2u_j^n + u_{j-1}^n)`.
It is second-order accurate and far less diffusive than upwind on smooth
waves, but, as Godunov later proved every linear second-order scheme
must, it oscillates near discontinuities.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import AdvectionEquation1D








.. GENERATED FROM PYTHON SOURCE LINES 20-22

A smooth wave after ten periods
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-34

.. code-block:: Python


    adv = AdvectionEquation1D(lambda x: np.sin(2 * np.pi * x), n=50)
    fig, ax = plt.subplots()
    ax.plot(adv.x, adv.exact(10.0), "k--", lw=2, label="exact")
    for scheme in ("upwind", "lax_wendroff"):
        sol = adv.solve(10.0, dt=0.5 * adv.dx, scheme=scheme)
        print(f"smooth, {scheme:>12}: max error {np.max(np.abs(sol.final - adv.exact(10.0))):.3f}")
        ax.plot(sol.x, sol.final, "o-", ms=3, label=scheme)
    ax.set_xlabel("$x$")
    ax.set_title("Ten periods on 50 points: second order keeps the amplitude")
    ax.legend()




.. image-sg:: /api/gallery/pde/advection/images/sphx_glr_plot_03_lax_wendroff_001.png
   :alt: Ten periods on 50 points: second order keeps the amplitude
   :srcset: /api/gallery/pde/advection/images/sphx_glr_plot_03_lax_wendroff_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    smooth,       upwind: max error 0.860
    smooth, lax_wendroff: max error 0.123

    <matplotlib.legend.Legend object at 0x7fd7bf850c20>



.. GENERATED FROM PYTHON SOURCE LINES 35-37

A square wave: Lax-Wendroff overshoots, upwind smears
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-50

.. code-block:: Python


    square = AdvectionEquation1D(lambda x: np.where((x > 0.3) & (x < 0.6), 1.0, 0.0), n=200)
    fig, ax = plt.subplots()
    ax.plot(square.x, square.exact(1.0), "k--", lw=2, label="exact")
    for scheme in ("upwind", "lax_wendroff"):
        sol = square.solve(1.0, dt=0.5 * square.dx, scheme=scheme)
        print(f"square, {scheme:>12}: min {sol.final.min():+.3f}, max {sol.final.max():.3f}")
        ax.plot(sol.x, sol.final, label=scheme)
    ax.set_xlabel("$x$")
    ax.set_title("Dispersive oscillations at discontinuities")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/pde/advection/images/sphx_glr_plot_03_lax_wendroff_002.png
   :alt: Dispersive oscillations at discontinuities
   :srcset: /api/gallery/pde/advection/images/sphx_glr_plot_03_lax_wendroff_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    square,       upwind: min +0.000, max 0.997
    square, lax_wendroff: min -0.232, max 1.232





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

   **Total running time of the script:** (0 minutes 0.222 seconds)


.. _sphx_glr_download_api_gallery_pde_advection_plot_03_lax_wendroff.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_03_lax_wendroff.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_lax_wendroff.ipynb <plot_03_lax_wendroff.ipynb>`

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

      :download:`Download Python source code: plot_03_lax_wendroff.py <plot_03_lax_wendroff.py>`

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

      :download:`Download zipped: plot_03_lax_wendroff.zip <plot_03_lax_wendroff.zip>`


.. only:: html

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

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