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

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

.. _sphx_glr_examples_gallery_simple_pdes_multiplicative_noise.py:


Multiplicative noise
====================

This example solves an Allen-Cahn equation where the variance of the noise depends on
the field itself. Since the variance is given as an expression, its derivative is
determined automatically, so the equation can be interpreted in the anti-Itô sense
and integrated using the Milstein solver.

.. GENERATED FROM PYTHON SOURCE LINES 10-21



.. image-sg:: /examples_gallery/simple_pdes/images/sphx_glr_multiplicative_noise_001.png
   :alt: multiplicative noise
   :srcset: /examples_gallery/simple_pdes/images/sphx_glr_multiplicative_noise_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


      0%|          | 0/10.0 [00:00<?, ?it/s]
    Initializing:   0%|          | 0/10.0 [00:00<?, ?it/s]
      0%|          | 0/10.0 [00:06<?, ?it/s]              
      0%|          | 0.004/10.0 [00:06<4:16:25, 1539.18s/it]
      1%|          | 0.119/10.0 [00:06<08:33, 51.97s/it]    
      8%|▊         | 0.819/10.0 [00:06<01:11,  7.75s/it]
     25%|██▌       | 2.542/10.0 [00:06<00:19,  2.66s/it]
     53%|█████▎    | 5.257/10.0 [00:07<00:06,  1.41s/it]
     87%|████████▋ | 8.663/10.0 [00:08<00:01,  1.06it/s]
     87%|████████▋ | 8.663/10.0 [00:08<00:01,  1.02it/s]
    100%|██████████| 10.0/10.0 [00:08<00:00,  1.18it/s] 
    100%|██████████| 10.0/10.0 [00:08<00:00,  1.18it/s]






|

.. code-block:: Python


    from pde import PDE, ScalarField, UnitGrid

    eq = PDE(
        {"c": "laplace(c) + c - c**3"},
        noise={"c": "0.1 * c**2"},
        noise_interpretation="anti-ito",
    )
    state = ScalarField.random_uniform(UnitGrid([64, 64]), -1, 1)
    result = eq.solve(state, t_range=10, dt=1e-3, solver="milstein")
    result.plot()


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

   **Total running time of the script:** (0 minutes 8.601 seconds)


.. _sphx_glr_download_examples_gallery_simple_pdes_multiplicative_noise.py:

.. only:: html

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

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

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

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

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

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

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