:orphan:

Inner Products
==============

Numerical solutions to differential equations frequently make use
of the **weak formulation**. That is, we take the inner product
of each PDE with some test function. There are many different ways to
evaluate inner products numerically; i.e. trapezoidal rule, midpoint
rule, or higher-order approximations. A simple method for evaluating
inner products on a numerical grid is to apply the midpoint rule; 
which is used by the *discretize* package.

Here, we demonstrate how to approximate various classes of inner
products numerically. If this is known, the user will be capable
of properly discretizing any term in a problem specific PDE.


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    <div class="sphx-glr-thumbnails">


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    <div class="sphx-glr-thumbcontainer" tooltip="Inner products between two scalar or vector quantities represents the most basic class of inner...">

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  .. image:: /tutorials/inner_products/images/thumb/sphx_glr_1_basic_thumb.png
    :alt:

  :ref:`sphx_glr_tutorials_inner_products_1_basic.py`

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      <div class="sphx-glr-thumbnail-title">Basic Inner Products</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="When solving PDEs using the finite volume approach, inner products may contain constitutive rel...">

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  .. image:: /tutorials/inner_products/images/thumb/sphx_glr_2_physical_properties_thumb.png
    :alt:

  :ref:`sphx_glr_tutorials_inner_products_2_physical_properties.py`

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      <div class="sphx-glr-thumbnail-title">Constitutive Relations</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="When solving PDEs using the finite volume approach, inner products may contain differential ope...">

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  .. image:: /tutorials/inner_products/images/thumb/sphx_glr_3_calculus_thumb.png
    :alt:

  :ref:`sphx_glr_tutorials_inner_products_3_calculus.py`

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      <div class="sphx-glr-thumbnail-title">Differential Operators</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In this section, we demonstrate how to go from the inner product to the discrete approximation ...">

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  .. image:: /tutorials/inner_products/images/thumb/sphx_glr_4_advanced_thumb.png
    :alt:

  :ref:`sphx_glr_tutorials_inner_products_4_advanced.py`

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      <div class="sphx-glr-thumbnail-title">Advanced Examples</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /tutorials/inner_products/1_basic
   /tutorials/inner_products/2_physical_properties
   /tutorials/inner_products/3_calculus
   /tutorials/inner_products/4_advanced


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  .. container:: sphx-glr-footer sphx-glr-footer-gallery

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

      :download:`Download all examples in Python source code: inner_products_python.zip </tutorials/inner_products/inner_products_python.zip>`

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

      :download:`Download all examples in Jupyter notebooks: inner_products_jupyter.zip </tutorials/inner_products/inner_products_jupyter.zip>`


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

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