.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_ch7\ch7_heat_eq.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_heat_eq.py>`
        to download the full example code.

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

.. _sphx_glr_auto_ch7_ch7_heat_eq.py:


====================================
7.03 Heat Equation
====================================
We reproduce here the figure 7.4 & 7.5 of the book.
Utilitary functions can be found next to this file. Here, we only define codpy-related functions.

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

.. GENERATED FROM PYTHON SOURCE LINES 12-27

.. 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 PDE3, PDE4








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Problem statement
------------------------
let's consider the heat equation in a fixed geometry $\Omega$ with null Dirichlet conditions:
$$\partial_t u(t,x) = \Delta u(t,x), \quad u(0,x)=u_0(x),\quad x \in \Omega, \quad u_{\partial \Omega}=0$$
To approximate this equation, we follow the following steps:
Select a mesh $X \in \mathbb{R}^{N_x,D}$ for the domain $\Omega$.
Pick up a kernel $k$ generating a space of vanishing trace functions.
We represent this equation as $\frac{d}{dt} u(t) = \Delta_k u(t)$, with evolution operator $u^{n+1} = B(\Delta_k,u^{n},dt,\theta)$ and $\theta = 1$. This corresponds to the fully implicit case

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Regular mesh
------------------------
This image provides a 3-D representation of the initial condition and time evolution of the heat equation on a fixed square.

.. GENERATED FROM PYTHON SOURCE LINES 42-45

.. code-block:: Python

    PDE3() 
    plt.show()




.. image-sg:: /auto_ch7/images/sphx_glr_ch7_heat_eq_001.png
   :alt: initial condition, time evolution
   :srcset: /auto_ch7/images/sphx_glr_ch7_heat_eq_001.png
   :class: sphx-glr-single-img





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Irregular mesh
------------------------
This image shows the heat equation on an irregular mesh generated by a bimodal Gaussian process.

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.. code-block:: Python

    PDE4() 
    plt.show()
    pass


.. image-sg:: /auto_ch7/images/sphx_glr_ch7_heat_eq_002.png
   :alt: initial condition, time evolution
   :srcset: /auto_ch7/images/sphx_glr_ch7_heat_eq_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.404 seconds)


.. _sphx_glr_download_auto_ch7_ch7_heat_eq.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: ch7_heat_eq.ipynb <ch7_heat_eq.ipynb>`

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

      :download:`Download Python source code: ch7_heat_eq.py <ch7_heat_eq.py>`

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

      :download:`Download zipped: ch7_heat_eq.zip <ch7_heat_eq.zip>`


.. only:: html

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

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