.. _sphx_glr_api_gallery_ode_dynamics_stiffness:

Stiffness and boundary-value problems
-------------------------------------

Why explicit integrators stall on stiff systems: Dahlquist's A-stability,
Curtiss and Hirschfelder's BDF, Robertson's stiff kinetics, Radau IIA
implicit Runge-Kutta, and boundary-value problems by collocation.


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    <div class="sphx-glr-thumbcontainer" tooltip="Applied to the test equation y&#x27; = \lambda y, a one-step method multiplies y by a stability function R(z), with z = \lambda h. Dahlquist (1963) called a method A-stable if |R(z)| \le 1 on the whole left half-plane, so that no decaying mode can ever be amplified, whatever the step size. Explicit methods never qualify, since their R is a polynomial and grows without bound. RK4&#x27;s region reaches only z \approx -2.785 on the real axis. Backward Euler&#x27;s R(z) = 1/(1 - z) is A-stable. Dahlquist also proved that no A-stable linear multistep method can exceed order 2 (his &quot;second barrier&quot;).">

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  .. image:: /api/gallery/ode_dynamics/stiffness/images/thumb/sphx_glr_plot_01_dahlquist_a_stability_thumb.png
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  :doc:`/api/gallery/ode_dynamics/stiffness/plot_01_dahlquist_a_stability`

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      <div class="sphx-glr-thumbnail-title">Dahlquist's A-stability: backward Euler vs RK4</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Curtiss and Hirschfelder (1952) named the difficulty &quot;stiffness&quot; while integrating chemical-kinetics equations, using the model problem">

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  .. image:: /api/gallery/ode_dynamics/stiffness/images/thumb/sphx_glr_plot_02_curtiss_hirschfelder_bdf_thumb.png
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  :doc:`/api/gallery/ode_dynamics/stiffness/plot_02_curtiss_hirschfelder_bdf`

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      <div class="sphx-glr-thumbnail-title">Curtiss and Hirschfelder: stiffness and BDF</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Robertson (1966) proposed three-species autocatalytic reactions">

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  .. image:: /api/gallery/ode_dynamics/stiffness/images/thumb/sphx_glr_plot_03_robertson_stiff_kinetics_thumb.png
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  :doc:`/api/gallery/ode_dynamics/stiffness/plot_03_robertson_stiff_kinetics`

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      <div class="sphx-glr-thumbnail-title">Robertson's stiff chemical kinetics</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Butcher (1964) built implicit Runge-Kutta methods on Gauss and Radau quadrature nodes. Ehle (1969) showed that their stability functions are Padé approximants of e^z, and that the Radau IIA family is A-stable and, moreover, L-stable: R(z) \to 0 as z \to -\infty, so infinitely stiff modes are damped in a single step. The three-stage, order-5 method used by stiff_integrate has">

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  .. image:: /api/gallery/ode_dynamics/stiffness/images/thumb/sphx_glr_plot_04_radau_iia_thumb.png
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  :doc:`/api/gallery/ode_dynamics/stiffness/plot_04_radau_iia`

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      <div class="sphx-glr-thumbnail-title">Butcher, Ehle and Radau IIA: L-stable implicit Runge-Kutta</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="A two-point boundary-value problem fixes conditions at both ends, so there is no initial state to march forward from. Collocation solves for the whole solution at once. It takes a piecewise polynomial on a mesh and requires it to satisfy the ODE exactly at a few collocation points per interval and to meet the boundary conditions, which gives one nonlinear system for Newton&#x27;s method. De Boor and Swartz (1973) proved that collocation with k Gauss points per interval converges with order 2k at the mesh nodes (&quot;superconvergence&quot;). This made it the basis of BVP codes such as COLSYS and scipy.integrate.solve_bvp, which uses the closely related 3-point Lobatto IIIA scheme.">

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  .. image:: /api/gallery/ode_dynamics/stiffness/images/thumb/sphx_glr_plot_05_de_boor_swartz_collocation_thumb.png
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  :doc:`/api/gallery/ode_dynamics/stiffness/plot_05_de_boor_swartz_collocation`

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      <div class="sphx-glr-thumbnail-title">De Boor and Swartz: collocation for boundary-value problems</div>
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.. toctree::
   :hidden:

   /api/gallery/ode_dynamics/stiffness/plot_01_dahlquist_a_stability
   /api/gallery/ode_dynamics/stiffness/plot_02_curtiss_hirschfelder_bdf
   /api/gallery/ode_dynamics/stiffness/plot_03_robertson_stiff_kinetics
   /api/gallery/ode_dynamics/stiffness/plot_04_radau_iia
   /api/gallery/ode_dynamics/stiffness/plot_05_de_boor_swartz_collocation