.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluids/compressible_flow/plot_sod_shock_tube.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_fluids_compressible_flow_plot_sod_shock_tube.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_fluids_compressible_flow_plot_sod_shock_tube.py:


The Sod shock tube
=====================

Sod's (1978) shock-tube problem is the standard test Riemann problem for a
compressible-flow solver: a diaphragm at :math:`x_0` initially separates
high-pressure, high-density gas at rest from low-pressure, low-density gas
at rest,

.. math::

    (\rho, u, p) = \begin{cases}
        (1,\ 0,\ 1) & x < x_0 \\
        (0.125,\ 0,\ 0.1) & x \geq x_0,
    \end{cases}

on the domain :math:`x\in[0,1]`. Removing the diaphragm at :math:`t=0`
means integrating the 1D Euler equations for an inviscid, compressible
ideal gas (conservation of mass, momentum, and energy) forward from this
discontinuous initial state, which produces, for :math:`t>0`, exactly three
simple waves -- a left-running rarefaction fan, a right-running contact
discontinuity, and a right-running shock satisfying
:func:`~physicskit.fluids.systems.compressible_flow.rankine_hugoniot_jump_conditions`.
:func:`~physicskit.fluids.systems.compressible_flow.sod_shock_tube` resolves
all three at once with a Lax-Friedrichs finite-volume scheme.

.. GENERATED FROM PYTHON SOURCE LINES 27-35

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.fluids.systems.compressible_flow import sod_shock_tube
    from physicskit.fluids.visualizers import theme
    from physicskit.fluids.visualizers.compressible import plot_shock_tube_profiles








.. GENERATED FROM PYTHON SOURCE LINES 36-38

Solve the Riemann problem
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-41

.. code-block:: Python


    result = sod_shock_tube(nx=400, t_final=0.2)








.. GENERATED FROM PYTHON SOURCE LINES 42-49

Plot the three waves
-----------------------
From left to right: the rarefaction fan (density and pressure falling off
smoothly), the contact discontinuity (a density jump with continuous
pressure and velocity -- look for the density "step" where pressure and
velocity stay flat), and the shock (a sharp jump in all three fields,
smeared over a few cells by the scheme's numerical diffusion).

.. GENERATED FROM PYTHON SOURCE LINES 49-52

.. code-block:: Python


    fig, axes = plot_shock_tube_profiles(result["x"], result["rho"], result["u"], result["p"])




.. image-sg:: /api/gallery/fluids/compressible_flow/images/sphx_glr_plot_sod_shock_tube_001.png
   :alt: Sod shock tube
   :srcset: /api/gallery/fluids/compressible_flow/images/sphx_glr_plot_sod_shock_tube_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 53-62

How the three waves emerge: a density space-time diagram
---------------------------------------------------------------
The single snapshot above is one horizontal slice through this picture.
Re-solving :func:`~physicskit.fluids.systems.compressible_flow.sod_shock_tube`
at a sequence of increasing `t_final` (each an independent solve from the
same initial diaphragm, at a coarser resolution to keep the sweep cheap)
traces out how the three waves fan out from `x0` over time: the rarefaction
fan spreading left as a broadening pale wedge, and the contact discontinuity
and shock both running right, at their own distinct, constant speeds.

.. GENERATED FROM PYTHON SOURCE LINES 62-77

.. code-block:: Python


    times = np.linspace(0.01, 0.2, 60)
    history = [sod_shock_tube(nx=200, t_final=float(t)) for t in times]
    x_coarse = history[0]["x"]
    rho_history = np.array([h["rho"] for h in history])

    fig, ax = plt.subplots(figsize=(7, 5))
    im = ax.pcolormesh(x_coarse, times, rho_history, cmap=theme.SEQUENTIAL_CMAP, shading="auto")
    fig.colorbar(im, ax=ax, label=r"$\rho$")
    ax.set_xlabel("x")
    ax.set_ylabel("t")
    ax.set_title("Sod shock tube: density space-time diagram")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/fluids/compressible_flow/images/sphx_glr_plot_sod_shock_tube_002.png
   :alt: Sod shock tube: density space-time diagram
   :srcset: /api/gallery/fluids/compressible_flow/images/sphx_glr_plot_sod_shock_tube_002.png
   :class: sphx-glr-single-img






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

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


.. _sphx_glr_download_api_gallery_fluids_compressible_flow_plot_sod_shock_tube.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

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

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

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

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

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


.. only:: html

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

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