.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples_gallery/simple_pdes/pde_brusselator_expression.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_brusselator_expression.py>`
        to download the full example code.

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

.. _sphx_glr_examples_gallery_simple_pdes_pde_brusselator_expression.py:


Brusselator - Using the `PDE` class
===================================

This example uses the :class:`~pde.pdes.pde.PDE` class to implement the
`Brusselator <https://en.wikipedia.org/wiki/Brusselator>`_ with spatial
coupling,

.. math::

    \partial_t u &= D_0 \nabla^2 u + a - (1 + b) u + v u^2 \\
    \partial_t v &= D_1 \nabla^2 v + b u - v u^2

Here, :math:`D_0` and :math:`D_1` are the respective diffusivity and the
parameters :math:`a` and :math:`b` are related to reaction rates.

Note that the same result can also be achieved with a
:doc:`full implementation of a custom class <../advanced_pdes/pde_brusselator_class>`,
which allows for more flexibility at the cost of code complexity.

.. GENERATED FROM PYTHON SOURCE LINES 21-43



.. image-sg:: /examples_gallery/simple_pdes/images/sphx_glr_pde_brusselator_expression_001.png
   :alt: Time: 20, Field $u$, Field $v$
   :srcset: /examples_gallery/simple_pdes/images/sphx_glr_pde_brusselator_expression_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    from pde import PDE, FieldCollection, PlotTracker, ScalarField, UnitGrid

    # define the PDE
    a, b = 1, 3
    d0, d1 = 1, 0.1
    eq = PDE(
        {
            "u": f"{d0} * laplace(u) + {a} - ({b} + 1) * u + u**2 * v",
            "v": f"{d1} * laplace(v) + {b} * u - u**2 * v",
        }
    )

    # initialize state
    grid = UnitGrid([64, 64])
    u = ScalarField(grid, a, label="Field $u$")
    v = b / a + 0.1 * ScalarField.random_normal(grid, label="Field $v$")
    state = FieldCollection([u, v])

    # simulate the pde
    tracker = PlotTracker(interrupts=1, plot_args={"vmin": 0, "vmax": 5})
    sol = eq.solve(state, t_range=20, dt=1e-3, tracker=tracker)


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

   **Total running time of the script:** (0 minutes 13.694 seconds)


.. _sphx_glr_download_examples_gallery_simple_pdes_pde_brusselator_expression.py:

.. only:: html

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

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

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

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

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

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

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