.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/stability/plot_03_lax_equivalence.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_03_lax_equivalence.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_03_lax_equivalence.py:


The Lax equivalence theorem: consistency + stability = convergence
==================================================================

Peter Lax and Robert Richtmyer (1956) proved that a *consistent* linear
difference scheme for a well-posed linear problem converges as the grid
is refined *if and only if* it is *stable*. Consistency alone is not
enough. This script refines the grid, at a fixed Courant number, for three
schemes for :math:`u_t + u_x = 0` that are all consistent. Stable
Lax-Friedrichs converges at first order, and Lax-Wendroff at second.
Unstable FTCS *looks* convergent on coarse grids, then explodes: finer
grids take more steps, and its growing modes, seeded by round-off,
eventually swamp the solution even though each step approximates the
PDE more accurately.

.. GENERATED FROM PYTHON SOURCE LINES 18-40

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import AdvectionEquation1D


    def profile(x):
        return np.sin(2 * np.pi * x)


    ns = np.array([25, 50, 100, 200, 400, 800])
    results = {}
    for scheme in ("lax_friedrichs", "lax_wendroff", "ftcs"):
        errors = []
        for n in ns:
            adv = AdvectionEquation1D(profile, n=n)
            sol = adv.solve(1.0, dt=0.5 * adv.dx, scheme=scheme)
            errors.append(np.max(np.abs(sol.final - adv.exact(1.0))))
        results[scheme] = np.array(errors)
        stable = AdvectionEquation1D(profile, n=50).solve(0.01, dt=0.01, scheme=scheme).extra["stability"].stable
        print(f"{scheme:>15}  stable at nu = 0.5: {stable!s:5}  errors: " + "  ".join(f"{e:.1e}" for e in errors))





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

 .. code-block:: none

     lax_friedrichs  stable at nu = 0.5: True   errors: 7.0e-01  4.5e-01  2.6e-01  1.4e-01  7.1e-02  3.6e-02
       lax_wendroff  stable at nu = 0.5: True   errors: 4.9e-02  1.2e-02  3.1e-03  7.8e-04  1.9e-04  4.8e-05
               ftcs  stable at nu = 0.5: False  errors: 4.8e-01  2.2e-01  1.0e-01  5.3e+03  6.6e+22  5.8e+61




.. GENERATED FROM PYTHON SOURCE LINES 41-43

Stable schemes converge, the unstable one diverges
--------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-56

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6.5, 4.5))
    dx = 1.0 / ns
    for scheme, marker in (("lax_friedrichs", "o"), ("lax_wendroff", "s"), ("ftcs", "x")):
        ax.loglog(dx, results[scheme], marker + "-", label=scheme)
    ax.invert_xaxis()
    ax.set_xlabel(r"$\Delta x$ (refining to the right)")
    ax.set_ylabel("max error at t = 1")
    ax.set_title("Consistent + stable converges; consistent + unstable does not")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/pde/stability/images/sphx_glr_plot_03_lax_equivalence_001.png
   :alt: Consistent + stable converges; consistent + unstable does not
   :srcset: /api/gallery/pde/stability/images/sphx_glr_plot_03_lax_equivalence_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.451 seconds)


.. _sphx_glr_download_api_gallery_pde_stability_plot_03_lax_equivalence.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_03_lax_equivalence.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_lax_equivalence.ipynb <plot_03_lax_equivalence.ipynb>`

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

      :download:`Download Python source code: plot_03_lax_equivalence.py <plot_03_lax_equivalence.py>`

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

      :download:`Download zipped: plot_03_lax_equivalence.zip <plot_03_lax_equivalence.zip>`


.. only:: html

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

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