.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/stiffness/plot_03_robertson_stiff_kinetics.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_ode_dynamics_stiffness_plot_03_robertson_stiff_kinetics.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_ode_dynamics_stiffness_plot_03_robertson_stiff_kinetics.py:


Robertson's stiff chemical kinetics
===================================

Robertson (1966) proposed three-species autocatalytic reactions

.. math::

    y_1' = -0.04\,y_1 + 10^4\,y_2 y_3,\quad
    y_2' = 0.04\,y_1 - 10^4\,y_2 y_3 - 3\times10^7\,y_2^2,\quad
    y_3' = 3\times10^7\,y_2^2

whose rate constants span nine orders of magnitude. It became the
standard stiff test problem. The intermediate :math:`y_2` jumps to a
quasi-steady value of about :math:`3.6\times10^{-5}` within about
:math:`10^{-4}` time units, then evolves slowly for over :math:`10^{5}`
units. Mass is conserved, :math:`y_1 + y_2 + y_3 = 1`.

.. GENERATED FROM PYTHON SOURCE LINES 21-41

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.integrators import dopri5_integrate, njit, stiff_integrate


    @njit
    def robertson(state, t, params):
        k1, k2, k3 = params
        y1, y2, y3 = state
        out = np.empty(3)
        out[0] = -k1 * y1 + k3 * y2 * y3
        out[1] = k1 * y1 - k3 * y2 * y3 - k2 * y2**2
        out[2] = k2 * y2**2
        return out


    params = np.array([0.04, 3e7, 1e4])
    state0 = np.array([1.0, 0.0, 0.0])








.. GENERATED FROM PYTHON SOURCE LINES 42-44

Eleven decades of time with an implicit solver
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-55

.. code-block:: Python


    ts, ys = stiff_integrate(robertson, state0, 0.0, 1e5, params, rtol=1e-6, atol=1e-10)
    print(f"implicit solver: {len(ts) - 1} steps to t = 1e5, max |y1+y2+y3-1| = {np.max(np.abs(ys.sum(axis=1) - 1)):.1e}")

    fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(11, 4.2))
    for i, label in enumerate(["$y_1$", r"$10^4\,y_2$", "$y_3$"]):
        ax0.semilogx(ts[1:], ys[1:, i] * (1e4 if i == 1 else 1.0), label=label)
    ax0.set_xlabel("t")
    ax0.set_title("Robertson kinetics, t up to 1e5")
    ax0.legend()




.. image-sg:: /api/gallery/ode_dynamics/stiffness/images/sphx_glr_plot_03_robertson_stiff_kinetics_001.png
   :alt: Robertson kinetics, t up to 1e5
   :srcset: /api/gallery/ode_dynamics/stiffness/images/sphx_glr_plot_03_robertson_stiff_kinetics_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    implicit solver: 188 steps to t = 1e5, max |y1+y2+y3-1| = 2.2e-16

    <matplotlib.legend.Legend object at 0x7fd7c0b99220>



.. GENERATED FROM PYTHON SOURCE LINES 56-58

An explicit solver's step is pinned by the fast reaction
--------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 58-71

.. code-block:: Python


    ts_dp, _ = dopri5_integrate(robertson, state0, 0.0, 40.0, 1e-6, params, rtol=1e-6, atol=1e-10)
    ts_im, _ = stiff_integrate(robertson, state0, 0.0, 40.0, params, rtol=1e-6, atol=1e-10)
    print(f"to t = 40: explicit Dormand-Prince {len(ts_dp) - 1} steps, implicit {len(ts_im) - 1} steps")
    ax1.loglog(ts_dp[1:], np.diff(ts_dp), label=f"Dormand-Prince ({len(ts_dp) - 1} steps)")
    ax1.loglog(ts_im[1:], np.diff(ts_im), label=f"implicit ({len(ts_im) - 1} steps)")
    ax1.set_xlabel("t")
    ax1.set_ylabel(r"accepted $\Delta t$")
    ax1.set_title("Step size up to t = 40")
    ax1.legend()
    fig.tight_layout()

    plt.show()




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

 .. code-block:: none

    to t = 40: explicit Dormand-Prince 34562 steps, implicit 78 steps





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

   **Total running time of the script:** (0 minutes 2.474 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_stiffness_plot_03_robertson_stiff_kinetics.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/ode_dynamics/stiffness/plot_03_robertson_stiff_kinetics.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_robertson_stiff_kinetics.ipynb <plot_03_robertson_stiff_kinetics.ipynb>`

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

      :download:`Download Python source code: plot_03_robertson_stiff_kinetics.py <plot_03_robertson_stiff_kinetics.py>`

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

      :download:`Download zipped: plot_03_robertson_stiff_kinetics.zip <plot_03_robertson_stiff_kinetics.zip>`


.. only:: html

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

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