.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/stability/plot_01_cfl_condition.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_01_cfl_condition.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_01_cfl_condition.py:


The Courant-Friedrichs-Lewy condition
=====================================

Courant, Friedrichs and Lewy (1928) observed that an explicit scheme
for :math:`u_t + c\,u_x = 0` computes each new value from a few
neighbors, so information can travel at most one cell per step. If the
true wave moves farther than that -- Courant number
:math:`\nu = c\,\Delta t/\Delta x > 1` -- the scheme cannot possibly see
where the solution came from, and it diverges. This script advects a
pulse with the upwind scheme below and above :math:`\nu = 1`,
and with Lax-Wendroff, whose limit is the same.

.. GENERATED FROM PYTHON SOURCE LINES 16-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import AdvectionEquation1D, check_cfl

    adv = AdvectionEquation1D(lambda x: np.exp(-200 * (x - 0.5) ** 2), n=200, c=1.0)








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Below and above nu = 1
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-42

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4), sharey=True)
    for ax, scheme in zip(axes, ("upwind", "lax_wendroff"), strict=False):
        ax.plot(adv.x, adv.exact(1.0), "k--", lw=2, label="exact (one period)")
        for nu, color in ((0.5, "tab:blue"), (0.95, "tab:green"), (1.2, "tab:red")):
            sol = adv.solve(1.0, dt=nu * adv.dx, scheme=scheme)
            check = sol.extra["stability"]
            print(f"{scheme:>12}  nu = {nu:4.2f}  stable = {check.stable!s:5}  max |u| = {np.max(np.abs(sol.final)):.3g}")
            ax.plot(sol.x, np.clip(sol.final, -0.5, 1.5), color=color, label=f"nu = {nu}")
        ax.set_title(scheme)
        ax.set_xlabel("$x$")
        ax.set_ylim(-0.5, 1.5)
    axes[0].legend(fontsize=8)
    fig.suptitle("CFL: the Courant number must not exceed 1")
    fig.tight_layout()




.. image-sg:: /api/gallery/pde/stability/images/sphx_glr_plot_01_cfl_condition_001.png
   :alt: CFL: the Courant number must not exceed 1, upwind, lax_wendroff
   :srcset: /api/gallery/pde/stability/images/sphx_glr_plot_01_cfl_condition_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

          upwind  nu = 0.50  stable = True   max |u| = 0.707
          upwind  nu = 0.95  stable = True   max |u| = 0.952
          upwind  nu = 1.20  stable = False  max |u| = 7.95e+07
    lax_wendroff  nu = 0.50  stable = True   max |u| = 0.995
    lax_wendroff  nu = 0.95  stable = True   max |u| = 0.999
    lax_wendroff  nu = 1.20  stable = False  max |u| = 5.95e+28




.. GENERATED FROM PYTHON SOURCE LINES 43-45

The domain-of-dependence argument, in numbers
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-51

.. code-block:: Python


    for dt in (0.004, 0.005, 0.006):
        r = check_cfl(adv.c, dt, adv.dx, "upwind")
        print(f"dt = {dt}: Courant number {r.number:.2f} (limit {r.limit:.0f}) -> {'stable' if r.stable else 'UNSTABLE'}")

    plt.show()




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

 .. code-block:: none

    dt = 0.004: Courant number 0.80 (limit 1) -> stable
    dt = 0.005: Courant number 1.00 (limit 1) -> stable
    dt = 0.006: Courant number 1.20 (limit 1) -> UNSTABLE





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

   **Total running time of the script:** (0 minutes 0.250 seconds)


.. _sphx_glr_download_api_gallery_pde_stability_plot_01_cfl_condition.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_01_cfl_condition.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_cfl_condition.ipynb <plot_01_cfl_condition.ipynb>`

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

      :download:`Download Python source code: plot_01_cfl_condition.py <plot_01_cfl_condition.py>`

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

      :download:`Download zipped: plot_01_cfl_condition.zip <plot_01_cfl_condition.zip>`


.. only:: html

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

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