.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/conservation_laws/plot_01_godunov_shocks.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_conservation_laws_plot_01_godunov_shocks.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_conservation_laws_plot_01_godunov_shocks.py:


Godunov's method: capturing shocks without oscillations
=======================================================

In inviscid Burgers' equation :math:`u_t + (u^2/2)_x = 0`, faster fluid
overtakes slower fluid and smooth data steepens into a *shock*. Sergei
Godunov (1959) computed the flux between neighboring cells from the
exact solution of the Riemann problem they pose. The result is a
conservative, monotone scheme that moves shocks at the right speed
without spurious wiggles. He also proved that no linear scheme can be
both monotone and better than first order, which is why second-order
Lax-Wendroff oscillates. This script shows a sine wave breaking into a
shock, then compares three schemes on a Riemann problem.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import BurgersConservationLaw1D, burgers_riemann_solution, total_variation
    from mathematicskit.pde.visualizers import plot_snapshots








.. GENERATED FROM PYTHON SOURCE LINES 24-26

A smooth wave breaks into a shock
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-34

.. code-block:: Python


    wave = BurgersConservationLaw1D(lambda x: 0.5 + np.sin(2 * np.pi * x), n=400, bc="periodic")
    sol = wave.solve(0.6, dt=0.5 * wave.dx / 1.5, save_every=40)
    fig1, ax1 = plt.subplots(figsize=(7, 4))
    plot_snapshots(sol, n_snapshots=5, ax=ax1)
    ax1.set_title("Godunov: the wave steepens and a shock forms (breaking time 1/(2 pi))")
    fig1.tight_layout()




.. image-sg:: /api/gallery/pde/conservation_laws/images/sphx_glr_plot_01_godunov_shocks_001.png
   :alt: Godunov: the wave steepens and a shock forms (breaking time 1/(2 pi))
   :srcset: /api/gallery/pde/conservation_laws/images/sphx_glr_plot_01_godunov_shocks_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 35-37

The Riemann problem: shock and rarefaction
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-55

.. code-block:: Python


    u0 = lambda x: np.where(x < -0.3, -0.5, np.where(x < 0.3, 1.0, 0.0))  # a rarefaction, then a shock
    law = BurgersConservationLaw1D(u0, x_range=(-1.0, 1.0), n=200)
    fig2, ax2 = plt.subplots(figsize=(7, 4))
    xs = np.linspace(-1, 1, 1000)
    t = 0.6
    exact = np.where(xs < 0.0, burgers_riemann_solution(-0.5, 1.0, xs, t, x0=-0.3), burgers_riemann_solution(1.0, 0.0, xs, t, x0=0.3))
    ax2.plot(xs, exact, color="black", lw=2, label="exact entropy solution")
    for scheme, style in (("godunov", "o"), ("lax_friedrichs", "s"), ("lax_wendroff", "^")):
        s = law.solve(t, dt=0.8 * law.dx, scheme=scheme)
        ax2.plot(s.x, s.final, style, ms=3, label=scheme)
        print(f"{scheme:>15}: max u = {s.final.max():.3f}, total variation {total_variation(law.u0):.2f} -> {total_variation(s.final):.2f}")
    ax2.set_xlabel("$x$")
    ax2.set_title("Only Godunov is both sharp and free of overshoots")
    ax2.legend(fontsize=8)
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/pde/conservation_laws/images/sphx_glr_plot_01_godunov_shocks_002.png
   :alt: Only Godunov is both sharp and free of overshoots
   :srcset: /api/gallery/pde/conservation_laws/images/sphx_glr_plot_01_godunov_shocks_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

            godunov: max u = 1.000, total variation 2.50 -> 2.50
     lax_friedrichs: max u = 1.000, total variation 2.50 -> 2.50
       lax_wendroff: max u = 1.129, total variation 2.50 -> 3.53





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

   **Total running time of the script:** (0 minutes 0.269 seconds)


.. _sphx_glr_download_api_gallery_pde_conservation_laws_plot_01_godunov_shocks.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/conservation_laws/plot_01_godunov_shocks.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_godunov_shocks.ipynb <plot_01_godunov_shocks.ipynb>`

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

      :download:`Download Python source code: plot_01_godunov_shocks.py <plot_01_godunov_shocks.py>`

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

      :download:`Download zipped: plot_01_godunov_shocks.zip <plot_01_godunov_shocks.zip>`


.. only:: html

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

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