:orphan:

Examples
========

These are example scripts using the `py-pde` package, which illustrates some of the most
important features of the package.


.. raw:: html

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

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how to initialize and visualize the vector field \boldsymbol u = \bigl(\sin(x), \cos(x)\bigr).">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_plot_vector_field_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_plot_vector_field.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Plotting a vector field</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how to initialize and visualize the scalar field u = \sqrt{z} \, \exp(-r^2) in cylindrical coordinates.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_plot_cylindrical_field_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_plot_cylindrical_field.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Plotting a scalar field in cylindrical coordinates</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how to solve a 2d Laplace equation with spatially varying boundary conditions.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_laplace_eq_2d_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_laplace_eq_2d.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Solving Laplace's equation in 2d</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how to solve a 1d Poisson equation with boundary conditions.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_poisson_eq_1d_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_poisson_eq_1d.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Solving Poisson's equation in 1d</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example solves a simple diffusion equation in two dimensions.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_simple_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_simple.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Simple diffusion equation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements a scalar PDE using the PDE. We here consider the Kuramoto–Sivashinsky equation, which for instance describes the dynamics of flame fronts:">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_custom_expression_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_custom_expression.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kuramoto-Sivashinsky - Using PDE class</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates how to solve a PDE in a spherically symmetric geometry.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_spherical_grid_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_spherical_grid.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Spherically symmetric PDE</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how to solve the diffusion equation on a Cartesian grid.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_cartesian_grid_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_cartesian_grid.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Diffusion on a Cartesian grid</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates how a stochastic simulation can be done.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_stochastic_simulation_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_stochastic_simulation.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Stochastic simulation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how different boundary conditions can be specified.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_boundary_conditions_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_boundary_conditions.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Setting boundary conditions</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example solves a simple diffusion equation in one dimensions with time-dependent boundary conditions.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_time_dependent_bcs_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_time_dependent_bcs.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Time-dependent boundary conditions</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements a PDE that is only defined in one dimension. Here, we chose the Korteweg-de Vries equation, given by">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_1d_expression_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_1d_expression.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">1D problem - Using PDE class</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements a spatially coupled SIR model  with  the following dynamics for the density of susceptible, infected, and recovered individuals:">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_heterogeneous_bcs_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_heterogeneous_bcs.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Heterogeneous boundary conditions</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how to use storage to store data persistently.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_storages_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_storages.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Storage examples</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example uses the PDE class to implement the Brusselator with spatial coupling,">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_brusselator_expression_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_brusselator_expression.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Brusselator - Using the PDE class</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example solve the Diffusion equation with a heterogeneous diffusivity:">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_heterogeneous_diffusion_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_heterogeneous_diffusion.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Diffusion equation with spatial dependence</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates how to store intermediate data to a file for later post-processing. The storage frequency is an argument to the tracker.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_trajectory_io_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_trajectory_io.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Writing and reading trajectory data</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates how trackers can be used to analyze simulations.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_trackers_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_trackers.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Using simulation trackers</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements a complex PDE using the PDE. We here chose the Schrödinger equation without a spatial potential in non-dimensional form:">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_schroedinger_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_schroedinger.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Schrödinger's Equation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements a scalar PDE using a custom class. We here consider the Kuramoto–Sivashinsky equation, which for instance describes the dynamics of flame fronts:">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_custom_class_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_custom_class.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kuramoto-Sivashinsky - Using custom class</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how to solve a set of coupled PDEs, the spatially coupled FitzHugh–Nagumo model , which is a simple model for the excitable dynamics of coupled Neurons:">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_coupled_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_coupled.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Custom Class for coupled PDEs</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements a PDE that is only defined in one dimension. Here, we chose the Korteweg-de Vries equation, given by">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_1d_class_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_1d_class.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">1D problem - Using custom class</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example displays methods for visualizing scalar fields.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_analyze_scalar_field_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_analyze_scalar_field.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Visualizing a scalar field</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example solves a diffusion equation with a custom noise.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_custom_noise_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_custom_noise.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Custom noise</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements a scalar PDE using a custom class with a numba-compiled method for accelerated calculations. We here consider the Kuramoto–Sivashinsky equation, which for instance describes the dynamics of flame fronts:">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_custom_numba_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_custom_numba.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kuramoto-Sivashinsky - Compiled methods</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example shows how to set up solvers explicitly and how to extract diagnostic information.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_solver_comparison_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_solver_comparison.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Solver comparison</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The hook function created by make_post_step_hook is called after each time step. The function can modify the state, keep track of additional information and abort the simulation.">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_post_step_hook_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_post_step_hook.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Post-step hook function</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements a spatially coupled SIR model  with  the following dynamics for the density of susceptible, infected, and recovered individuals:">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_sir_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_sir.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Custom PDE class: SIR model</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example implements the Brusselator  with spatial coupling,">

.. only:: html

  .. image:: /examples_gallery/images/thumb/sphx_glr_pde_brusselator_class_thumb.png
    :alt:

  :ref:`sphx_glr_examples_gallery_pde_brusselator_class.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Brusselator - Using custom class</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /examples_gallery/plot_vector_field
   /examples_gallery/plot_cylindrical_field
   /examples_gallery/laplace_eq_2d
   /examples_gallery/poisson_eq_1d
   /examples_gallery/simple
   /examples_gallery/pde_custom_expression
   /examples_gallery/spherical_grid
   /examples_gallery/cartesian_grid
   /examples_gallery/stochastic_simulation
   /examples_gallery/boundary_conditions
   /examples_gallery/time_dependent_bcs
   /examples_gallery/pde_1d_expression
   /examples_gallery/heterogeneous_bcs
   /examples_gallery/storages
   /examples_gallery/pde_brusselator_expression
   /examples_gallery/pde_heterogeneous_diffusion
   /examples_gallery/trajectory_io
   /examples_gallery/trackers
   /examples_gallery/pde_schroedinger
   /examples_gallery/pde_custom_class
   /examples_gallery/pde_coupled
   /examples_gallery/pde_1d_class
   /examples_gallery/analyze_scalar_field
   /examples_gallery/custom_noise
   /examples_gallery/pde_custom_numba
   /examples_gallery/solver_comparison
   /examples_gallery/post_step_hook
   /examples_gallery/pde_sir
   /examples_gallery/pde_brusselator_class


.. only:: html

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

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

      :download:`Download all examples in Python source code: examples_gallery_python.zip </examples_gallery/examples_gallery_python.zip>`

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

      :download:`Download all examples in Jupyter notebooks: examples_gallery_jupyter.zip </examples_gallery/examples_gallery_jupyter.zip>`