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

.. only:: html

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

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

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

.. _sphx_glr_examples_gallery_simple_pdes_pde_heterogeneous_diffusion.py:


Diffusion equation with spatial dependence
==========================================

This example solve the
`Diffusion equation <https://en.wikipedia.org/wiki/Diffusion_equation>`_ with a
heterogeneous diffusivity:

.. math::
    \partial_t c = \nabla\bigr( D(\boldsymbol r) \nabla c \bigr)

using the :class:`~pde.pdes.pde.PDE` class. In particular, we consider
:math:`D(x) = 1.01 + \tanh(x)`, which gives a low diffusivity on the left side of the
domain.

Note that the naive implementation,
:code:`PDE({"c": "divergence((1.01 + tanh(x)) * gradient(c))"})`, has numerical
instabilities. This is because two finite difference approximations are nested. To
arrive at a more stable numerical scheme, it is advisable to expand the divergence,

.. math::
    \partial_t c = D \nabla^2 c + \nabla D . \nabla c

.. GENERATED FROM PYTHON SOURCE LINES 24-41



.. image-sg:: /examples_gallery/simple_pdes/images/sphx_glr_pde_heterogeneous_diffusion_001.png
   :alt: pde heterogeneous diffusion
   :srcset: /examples_gallery/simple_pdes/images/sphx_glr_pde_heterogeneous_diffusion_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph

    # Expanded definition of the PDE
    diffusivity = "1.01 + tanh(x)"
    term_1 = f"({diffusivity}) * laplace(c)"
    term_2 = f"dot(gradient({diffusivity}), gradient(c))"
    eq = PDE({"c": f"{term_1} + {term_2}"}, bc={"value": 0})


    grid = CartesianGrid([[-5, 5]], 64)  # generate grid
    field = ScalarField(grid, 1)  # generate initial condition

    storage = MemoryStorage()  # store intermediate information of the simulation
    res = eq.solve(field, 100, dt=1e-3, tracker=storage.tracker(1))  # solve the PDE

    plot_kymograph(storage)  # visualize the result in a space-time plot


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

   **Total running time of the script:** (0 minutes 9.407 seconds)


.. _sphx_glr_download_examples_gallery_simple_pdes_pde_heterogeneous_diffusion.py:

.. only:: html

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

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

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

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

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

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

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