: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 id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

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

.. thumbnail-parent-div-open

.. thumbnail-parent-div-close

.. raw:: html

    </div>

Advanced PDEs
-------------

These examples demonstrate more advanced usage of the package.

.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

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

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Demonstrate the simple hook function in PDE, which is called after each time step and may modify the state and abort the simulation.">

.. only:: html

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

  :doc:`/examples_gallery/advanced_pdes/post_step_hook`

.. 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 diffusion equation with a boundary condition specified by a function, which can in principle depend on time.">

.. only:: html

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

  :doc:`/examples_gallery/advanced_pdes/heterogeneous_bcs`

.. raw:: html

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


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example loads an example image and uses it as the source field for a simple reaction-diffusion equation.">

.. only:: html

  .. image:: /examples_gallery/advanced_pdes/images/thumb/sphx_glr_heterogeneous_pde_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/heterogeneous_pde`

.. raw:: html

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


.. raw:: html

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

.. only:: html

  .. image:: /examples_gallery/advanced_pdes/images/thumb/sphx_glr_pde_brusselator_rd_pde_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/pde_brusselator_rd_pde`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Brusselator - Using the ReactionDiffusionPDE class</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/advanced_pdes/images/thumb/sphx_glr_custom_noise_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/custom_noise`

.. 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. We here consider the Kuramoto–Sivashinsky equation, which for instance describes the dynamics of flame fronts:">

.. only:: html

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

  :doc:`/examples_gallery/advanced_pdes/pde_custom_class`

.. 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/advanced_pdes/images/thumb/sphx_glr_pde_coupled_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/pde_coupled`

.. 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 solves a stochastic diffusion equation with Stratonovich interpretation">

.. only:: html

  .. image:: /examples_gallery/advanced_pdes/images/thumb/sphx_glr_stratonovich_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/stratonovich`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">SDE with Stratonovich interpretation</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/advanced_pdes/images/thumb/sphx_glr_pde_1d_class_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/pde_1d_class`

.. 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 shows how to set up solvers explicitly and how to extract diagnostic information.">

.. only:: html

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

  :doc:`/examples_gallery/advanced_pdes/solver_comparison`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Solver comparison</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/advanced_pdes/images/thumb/sphx_glr_pde_custom_numba_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/pde_custom_numba`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kuramoto-Sivashinsky - Compiled methods</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/advanced_pdes/images/thumb/sphx_glr_post_step_hook_class_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/post_step_hook_class`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Post-step hook function in a custom 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/advanced_pdes/images/thumb/sphx_glr_pde_sir_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/pde_sir`

.. 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/advanced_pdes/images/thumb/sphx_glr_pde_brusselator_class_thumb.png
    :alt:

  :doc:`/examples_gallery/advanced_pdes/pde_brusselator_class`

.. raw:: html

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


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Grids and fields
----------------

These examples show how to define and manipulate discretized fields.

.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. 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 a polar grid with a hole inside and angular symmetry, so that fields only depend on the radial coordinate.">

.. only:: html

  .. image:: /examples_gallery/fields/images/thumb/sphx_glr_plot_polar_grid_thumb.png
    :alt:

  :doc:`/examples_gallery/fields/plot_polar_grid`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Plot a polar grid</div>
    </div>


.. 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/fields/images/thumb/sphx_glr_plot_vector_field_thumb.png
    :alt:

  :doc:`/examples_gallery/fields/plot_vector_field`

.. 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/fields/images/thumb/sphx_glr_plot_cylindrical_field_thumb.png
    :alt:

  :doc:`/examples_gallery/fields/plot_cylindrical_field`

.. 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 showcases several random fields">

.. only:: html

  .. image:: /examples_gallery/fields/images/thumb/sphx_glr_random_fields_thumb.png
    :alt:

  :doc:`/examples_gallery/fields/random_fields`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Random scalar fields</div>
    </div>


.. raw:: html

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

.. only:: html

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

  :doc:`/examples_gallery/fields/analyze_scalar_field`

.. raw:: html

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


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example displays various finite difference (FD) approximations of derivatives of simple harmonic function.">

.. only:: html

  .. image:: /examples_gallery/fields/images/thumb/sphx_glr_finite_differences_thumb.png
    :alt:

  :doc:`/examples_gallery/fields/finite_differences`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Finite differences approximation</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Output and analysis
-------------------

These examples demonstrate how to store and analyze output.

.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

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

.. thumbnail-parent-div-open

.. raw:: html

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

.. only:: html

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

  :doc:`/examples_gallery/output/storages`

.. raw:: html

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


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example demonstrates a space-time plot with a logarithmic time axis (using utilitiez.densityplot), which is useful to analyze coarsening processes.">

.. only:: html

  .. image:: /examples_gallery/output/images/thumb/sphx_glr_logarithmic_kymograph_thumb.png
    :alt:

  :doc:`/examples_gallery/output/logarithmic_kymograph`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Logarithmic kymograph</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/output/images/thumb/sphx_glr_trajectory_io_thumb.png
    :alt:

  :doc:`/examples_gallery/output/trajectory_io`

.. 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/output/images/thumb/sphx_glr_trackers_thumb.png
    :alt:

  :doc:`/examples_gallery/output/trackers`

.. raw:: html

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


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Simple PDEs
-----------

These examples demonstrate basic usage of the package to solve PDEs.

.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

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

.. thumbnail-parent-div-open

.. 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/simple_pdes/images/thumb/sphx_glr_laplace_eq_2d_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/laplace_eq_2d`

.. 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/simple_pdes/images/thumb/sphx_glr_poisson_eq_1d_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/poisson_eq_1d`

.. 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/simple_pdes/images/thumb/sphx_glr_simple_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/simple`

.. 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/simple_pdes/images/thumb/sphx_glr_pde_custom_expression_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/pde_custom_expression`

.. 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/simple_pdes/images/thumb/sphx_glr_spherical_grid_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/spherical_grid`

.. 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/simple_pdes/images/thumb/sphx_glr_cartesian_grid_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/cartesian_grid`

.. 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 solves the Klein-Gordon equation in one dimension, showing the effect of the mass term on wave propagation.  With mass=0 the equation reduces to the standard wave equation; increasing the mass introduces dispersion.">

.. only:: html

  .. image:: /examples_gallery/simple_pdes/images/thumb/sphx_glr_klein_gordon_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/klein_gordon`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Klein-Gordon equation</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/simple_pdes/images/thumb/sphx_glr_stochastic_simulation_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/stochastic_simulation`

.. 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/simple_pdes/images/thumb/sphx_glr_boundary_conditions_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/boundary_conditions`

.. 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/simple_pdes/images/thumb/sphx_glr_time_dependent_bcs_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/time_dependent_bcs`

.. 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/simple_pdes/images/thumb/sphx_glr_pde_1d_expression_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/pde_1d_expression`

.. 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 uses the PDE class to implement the Brusselator with spatial coupling,">

.. only:: html

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

  :doc:`/examples_gallery/simple_pdes/pde_brusselator_expression`

.. 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/simple_pdes/images/thumb/sphx_glr_pde_heterogeneous_diffusion_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/pde_heterogeneous_diffusion`

.. 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 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/simple_pdes/images/thumb/sphx_glr_pde_schroedinger_thumb.png
    :alt:

  :doc:`/examples_gallery/simple_pdes/pde_schroedinger`

.. raw:: html

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


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:
   :includehidden:


   /examples_gallery/advanced_pdes/index.rst
   /examples_gallery/fields/index.rst
   /examples_gallery/output/index.rst
   /examples_gallery/simple_pdes/index.rst