.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/spectral/plot_03_hopf_cole_burgers.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_spectral_plot_03_hopf_cole_burgers.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_spectral_plot_03_hopf_cole_burgers.py:


Burgers' equation and the Hopf-Cole transformation
==================================================

J. M. Burgers (1948) proposed viscous Burgers' equation as the simplest
model of nonlinear steepening balanced by viscosity. Eberhard Hopf
(1950) and Julian Cole (1951) independently found that
the substitution :math:`u = -2\nu\,\varphi_x/\varphi` turns the
*nonlinear* viscous Burgers' equation :math:`u_t + u u_x = \nu u_{xx}`
into the *linear* heat equation :math:`\varphi_t = \nu\varphi_{xx}`. A
nonlinear PDE with shock-like fronts thus has an exact solution. This
script uses it to watch a sine wave form a viscous shock whose width
shrinks with :math:`\nu`, and to verify the pseudo-spectral Burgers
solver to near machine precision.

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import BurgersEquation1D, burgers_cole_hopf_solution








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Viscous shocks, exactly
-----------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-37

.. code-block:: Python


    fig1, ax1 = plt.subplots(figsize=(7, 4))
    for nu, color in ((0.3, "tab:blue"), (0.1, "tab:green"), (0.04, "tab:red")):
        x, u = burgers_cole_hopf_solution(np.sin, t=1.5, nu=nu, n=512)
        ax1.plot(x, u, color=color, label=rf"$\nu = {nu}$")
    ax1.plot(x, np.sin(x), "k--", lw=1, label="initial data")
    ax1.set_xlabel("$x$")
    ax1.set_title(r"Hopf-Cole: the front at $x = \pi$ sharpens as $\nu \to 0$")
    ax1.legend()
    fig1.tight_layout()




.. image-sg:: /api/gallery/pde/spectral/images/sphx_glr_plot_03_hopf_cole_burgers_001.png
   :alt: Hopf-Cole: the front at $x = \pi$ sharpens as $\nu \to 0$
   :srcset: /api/gallery/pde/spectral/images/sphx_glr_plot_03_hopf_cole_burgers_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 38-40

An exact benchmark for the numerical solver
-------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-48

.. code-block:: Python


    nu, t = 0.1, 1.0
    for n in (32, 64, 128):
        x, exact = burgers_cole_hopf_solution(np.sin, t=t, nu=nu, n=n)
        numerical = BurgersEquation1D(np.sin, n=n, nu=nu).solve(t, dt=1e-3).final
        print(f"n = {n:3d}: max |pseudo-spectral - Hopf-Cole| = {np.max(np.abs(numerical - exact)):.1e}")

    plt.show()




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

 .. code-block:: none

    n =  32: max |pseudo-spectral - Hopf-Cole| = 1.3e-04
    n =  64: max |pseudo-spectral - Hopf-Cole| = 4.5e-08
    n = 128: max |pseudo-spectral - Hopf-Cole| = 6.5e-13





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

   **Total running time of the script:** (0 minutes 0.159 seconds)


.. _sphx_glr_download_api_gallery_pde_spectral_plot_03_hopf_cole_burgers.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/spectral/plot_03_hopf_cole_burgers.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_hopf_cole_burgers.ipynb <plot_03_hopf_cole_burgers.ipynb>`

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

      :download:`Download Python source code: plot_03_hopf_cole_burgers.py <plot_03_hopf_cole_burgers.py>`

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

      :download:`Download zipped: plot_03_hopf_cole_burgers.zip <plot_03_hopf_cole_burgers.zip>`


.. only:: html

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

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