.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_ch7\ch7_heatLagrange.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_ch7_ch7_heatLagrange.py>`
        to download the full example code.

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

.. _sphx_glr_auto_ch7_ch7_heatLagrange.py:


====================================
7.04 Lagrange Heat Equation
====================================
We reproduce here the figure 7.6 of the book.
Utilitary functions can be found next to this file. Here, we only define codpy-related functions.

.. GENERATED FROM PYTHON SOURCE LINES 10-12

Necessary Imports
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 12-28

.. code-block:: Python

    import os 
    import sys 

    import matplotlib.pyplot as plt 

    try:
        CURRENT_DIR = os.path.dirname(os.path.abspath(__file__))
    except NameError:
        CURRENT_DIR = os.getcwd()
    data_path = os.path.join(CURRENT_DIR, "data")
    PARENT_DIR = os.path.abspath(os.path.join(CURRENT_DIR, ".."))
    sys.path.insert(0, PARENT_DIR)

    from utils.ch7.ch7_utils import Lagrangian









.. GENERATED FROM PYTHON SOURCE LINES 29-37

Problem statement
------------------------
We consider the following semi-discrete scheme for $t \mapsto Y(t) \in \mathbb{R}^{N,D}$
$$\frac{d}{dt} Y = \nabla_k \cdot (\nabla_k Y)^{-1} = \nabla_k \cdot \Big( \Delta_k\Big)^{-1} \nabla_k Y, \quad Y(0,x) = X, $$
This figure shows our results with this numerical scheme. 
In the left=hand picture the initial condition, taken as a two-dimensional variate of a standard normal law. 
The figure in the middle displays the evolution at the time $t=1$.
The right-hand picture is a standard scaling of this last to unit variance.

.. GENERATED FROM PYTHON SOURCE LINES 37-39

.. code-block:: Python


    Lagrangian()
    plt.show()


.. image-sg:: /auto_ch7/images/sphx_glr_ch7_heatLagrange_001.png
   :alt: initial condition, time evolution, sharp sequences
   :srcset: /auto_ch7/images/sphx_glr_ch7_heatLagrange_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 1.061 seconds)


.. _sphx_glr_download_auto_ch7_ch7_heatLagrange.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: ch7_heatLagrange.ipynb <ch7_heatLagrange.ipynb>`

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

      :download:`Download Python source code: ch7_heatLagrange.py <ch7_heatLagrange.py>`

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

      :download:`Download zipped: ch7_heatLagrange.zip <ch7_heatLagrange.zip>`


.. only:: html

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

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