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INSTRUCTOR: I want to
illustrate the important notion

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of stiffness by running ode45,
the primary MATLAB ODE solver,

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on our flame example.

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The differential
equation is y prime is y

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squared minus y
cubed, and I'm going

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to choose a fairly--
an extremely small

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initial condition, 10
to the minus sixth.

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The final value of t
is 2 over y naught,

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and I'm going to impose a
modest accuracy requirement, 10

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to the minus fifth.

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Now let's run ode45
with its default output.

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Now, see it's taking-- it's
moving very slowly here.

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It's taking lots of steps.

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So I'm take- pressing
the stop button here.

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It's working very hard.

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Let's zoom in and see why it's
taking so many steps, very

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densely packed steps here.

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This is stiffness.

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It's satisfying the accuracy
requirements we imposed.

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All these steps are within
10 to the minus sixth of one,

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but it's taken very
small steps to do it.

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These steps are so small
that the graphics can't even

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discern the step size.

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This is stiffness.

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It's an efficiency issue.

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It's doing what we asked for.

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It's meeting the
accuracy requirements,

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but it's having to take
very small steps to do it.

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Let's try another
ODE solver-- ode23.

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Just change this to 23
and see what it does.

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It's also taking very small
steps for the same reason.

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If we zoom in on here, we'll
see the same kind of behavior.

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But it's taking very
small steps in order

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to achieve the desired accuracy.

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Now let me introduce
a new solver, ode23s.

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The s for stiffness.

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This was designed to
solve stiff problems.

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And boom, it goes
up, turns the corner,

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and it takes just a few steps
to get to the final result.

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There it turns the
corner very quickly.

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We'll see how ode23s
works in a minute,

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but first let's try
to define stiffness.

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It's a qualitative
notion that doesn't

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have a precise
mathematical definition.

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It depends upon the
problem, but also

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on the solver and the
accuracy requirements.

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But it's an important notion.

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We say that a problem is stiff
if the solution being sought

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very slowly, but there are
nearby solutions that very

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rapidly.

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So the numerical method
must take small steps

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to obtain satisfactory results.

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Stiff methods for ordinary
differential equations

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must be implicit.

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They must involve formulas
that involve looking backward

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from the forward timestep.

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The prototype of these methods
is the backward Euler method,

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or the implicit Euler method.

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This formula, it involves--
defines y n plus 1,

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but doesn't tell us
how to compute it.

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We have to solve this
equation for y n plus 1.

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And I'm not going to go into
detail about how we actually

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do it.

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It involves something
like a Newton method

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that would-- requires
knowing the derivative,

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or an approximation to
the derivative of f.

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But this gives you
an idea of what you

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can expect in stiff methods.

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I like to make an
analogy with taking

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a hike in one of
the slot canyons

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we have here in the Southwest.

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Explicit methods
like ode23 and 45

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take steps on the
walls of the canyon

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and go back and forth across
the sides of the canyon,

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make very slow progress
down the canyon.

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Whereas implicit
methods, like ode15s,

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look ahead down the
canyon and look ahead

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to where you want to go and make
rapid progress of the canyon.

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The stiff solver, ode23s,
uses an implicit second-order

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formula and an associated
third-order error estimator.

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It evaluates the partial
derivatives of f with respect

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to both t and f at each
step, so that's expensive.

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It's efficient at
crude error tolerances,

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like graphic accuracy.

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And it has relatively
low overhead.

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By way of comparison,
the stiff solver ode15s,

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can be configured to use
either the variable order

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numerical
differentiation formula,

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NDF, or the related to backward
differentiation formula BDF.

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Neither case it
saves several values

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of the function
over previous steps.

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The order varies automatically
between one and five,

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it evaluates the partial
derivatives less frequently,

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and did see efficient at
higher tolerances then 23s.