.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/heat/plot_03_method_of_lines_2d_heat.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_pde_heat_plot_03_method_of_lines_2d_heat.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_pde_heat_plot_03_method_of_lines_2d_heat.py:


The method of lines: a hot plate as a system of ODEs
====================================================

The method of lines discretizes only space. On a grid, the 2D heat
equation :math:`u_t = u_{xx} + u_{yy}` becomes one ODE per interior
point, :math:`dU/dt = L U`, which any ODE integrator can then advance.
Here that integrator is :func:`mathematicskit.integrators.rk4_integrate`
(the same one :mod:`mathematicskit.ode_dynamics` uses) and, for
comparison, the adaptive :func:`~mathematicskit.integrators.dopri5_integrate`.
The ODE system is *stiff*, so RK4's stable step shrinks like
:math:`\Delta x^2`.

.. GENERATED FROM PYTHON SOURCE LINES 16-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import HeatEquation2D
    from mathematicskit.pde.visualizers import plot_field_2d


    def hot_square(X, Y):
        return np.where((np.abs(X - 0.5) < 0.2) & (np.abs(Y - 0.5) < 0.2), 1.0, 0.0)


    heat = HeatEquation2D(hot_square, n=(41, 41))
    print(f"ODE system size: {heat.state0.size} unknowns (one per interior grid point)")
    print(f"RK4 stable step on this grid: dt <= {heat.max_stable_dt('rk4'):.2e}")





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

 .. code-block:: none

    ODE system size: 1521 unknowns (one per interior grid point)
    RK4 stable step on this grid: dt <= 2.18e-04




.. GENERATED FROM PYTHON SOURCE LINES 32-34

Integrate the semi-discrete system with RK4
-------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-44

.. code-block:: Python


    sol = heat.solve(0.04, dt=0.9 * heat.max_stable_dt("rk4"), save_every=40)
    fig, axes = plt.subplots(1, 3, figsize=(12, 3.8))
    for ax, t in zip(axes, (0.0, 0.01, 0.04), strict=False):
        k = int(np.argmin(np.abs(sol.t - t)))
        plot_field_2d(sol, time_index=k, ax=ax)
        ax.set_title(f"t = {sol.t[k]:.3f}")
    fig.suptitle("Method of lines + RK4: the hot square spreads and cools")
    fig.tight_layout()




.. image-sg:: /api/gallery/pde/heat/images/sphx_glr_plot_03_method_of_lines_2d_heat_001.png
   :alt: Method of lines + RK4: the hot square spreads and cools, t = 0.000, t = 0.008, t = 0.040
   :srcset: /api/gallery/pde/heat/images/sphx_glr_plot_03_method_of_lines_2d_heat_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 45-47

An adaptive integrator agrees
-----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 47-53

.. code-block:: Python


    adaptive = heat.solve(0.04, method="dopri5", rtol=1e-6, atol=1e-9)
    print(f"dopri5 accepted steps: {len(adaptive.t) - 1}")
    print(f"max |RK4 - dopri5| at t = 0.04: {np.max(np.abs(sol.final - adaptive.final)):.2e}")

    plt.show()




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

 .. code-block:: none

    dopri5 accepted steps: 170
    max |RK4 - dopri5| at t = 0.04: 3.68e-08





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

   **Total running time of the script:** (0 minutes 3.596 seconds)


.. _sphx_glr_download_api_gallery_pde_heat_plot_03_method_of_lines_2d_heat.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/pde/heat/plot_03_method_of_lines_2d_heat.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_method_of_lines_2d_heat.ipynb <plot_03_method_of_lines_2d_heat.ipynb>`

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

      :download:`Download Python source code: plot_03_method_of_lines_2d_heat.py <plot_03_method_of_lines_2d_heat.py>`

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

      :download:`Download zipped: plot_03_method_of_lines_2d_heat.zip <plot_03_method_of_lines_2d_heat.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_