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PROFESSOR: Here is the
classical Runge-Kutta method.

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This was, by far and away,
the world's most popular

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numerical method for over 100
years for hand computation

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in the first half
of the 20th century,

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and then for computation
on digital computers

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in the latter half
of the 20th century.

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I suspect it's
still in use today.

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You evaluate the function
four times per step,

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first in the beginning
of the interval.

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And then use that to step into
the middle of the interval,

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to get s2.

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Then you use s2 to step into the
middle of the interval again.

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And evaluate the function
there again to get s3.

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And then use s3 to step
clear across the interval,

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and get s4.

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And then take a combination
of those four slopes,

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weighting the two in
the middle more heavily,

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to take your final step.

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That's the classical
Runge-Kutta method.

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Here's our MATLAB
implementation.

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And we will call
it ODE4, because it

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evaluates to function
four times per step.

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Same arguments, vector y out.

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Now we have four slopes-- s1
at the beginning, s2 halfway

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in the middle, s3
again in the middle,

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and then s4 at the right hand.

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1/6 of s1, 1/3 of s2,
1/3 of s3, and 1/6 of s4

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give you your final step.

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That's the classical
Runge-Kutta method.

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Carl Runge was a fairly
prominent German mathematician

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and physicist, who
published this method,

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along with several
others, in 1895.

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He produced a number of
other mathematical papers

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and was fairly well known.

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Martin Kutta discovered
this method independently

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and published it in 1901.

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He is not so nearly well
known for anything else.

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I'd like to pursue a
simple model of combustion.

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Because the model has some
important numerical properties.

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If you light a match,
the ball of flame

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grows rapidly until it
reaches a critical size.

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Then the remains at that size,
because the amount of oxygen

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being consumed by the combustion
in the interior of the ball

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balances the amount available
through the surface.

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Here's the dimensionless model.

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The match is a sphere,
and y is its radius.

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The y cubed term is the
volume of the sphere.

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And the y cubed accounts for
the combustion in the interior.

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The surface of the sphere
is proportional y squared.

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And the y squared term
accounts for the oxygen that's

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available through the surface.

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The critical parameter,
the important parameter,

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is the initial
radius, y0, y naught.

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The radius starts
at y0 and grows

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until the y cubed
and y squared terms

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balance each other, at which
point the rate of growth is 0.

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And the radius
doesn't grow anymore.

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We integrate over a long time.

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We integrate over a time
that's inversely proportional

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to the initial radius.

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That's the model.

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Here's an animation.

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We're starting with
a small flame here,

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a small spherical flame.

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You'll just see a
small radius there.

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The time and the radius are
shown at the top of the figure.

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It's beginning to grow.

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When time gets to 50,
we're halfway through.

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The flame sort of
explodes, and then gets up

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the radius 1, at which time the
two terms balance each other.

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And the flame stops growing.

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It's still growing
slightly here,

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although you can't see
it on this this scale.

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Let's set this up
for Runge-Kutta.

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The differential
equation is y prime is y

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squared minus y cubed.

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Starting at zero, with the
critical initial radius,

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I'm going to take to be 0.01.

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That means we're going to
integrate out to two over y0

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out to time 200.

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I'm going to choose the
step size to take 500 steps.

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I'm just going to pick
that somewhat arbitrarily.

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OK, now I'm ready to use ODE4.

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And I'll store the results in y.

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And it goes up to 1.

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I'm going to plot the results.

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So here's the
values of t I need.

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And here's the plot.

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Now, you can see the
flame starts to grow.

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It grows rather slowly.

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And then halfway through
the time interval,

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it's sort of explodes
and goes up quickly,

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until it reaches a radius
of 1, and then stays here.

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Now this transition
period is fairly narrow.

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And we're going to continue
to study this problem.

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And is this
transition area which

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is going to provide a challenge
for the numerical methods.

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Now here, we just
went through it.

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We had a step size h, that
we picked pretty arbitrarily.

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And we just generated
these values.

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We have really little idea how
accurate these numbers are.

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They look OK.

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But how accurate are they?

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This is the critical
question about the

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about the classical
Runge-Kutta method.

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How reliable are the values
we have here in our graph?

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I have four exercises
for your consideration.

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If the differential
equation does not involve y,

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then this solution
is just an integral.

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And the Runge-Kutta method
becomes a classic method

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of numerical integration.

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If you've studied
such methods, then you

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should be able to
recognize this method.

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Number. two-- find the exact
solution of y prime equals 1

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plus y squared, with
y of 0 equals zero.

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And then see what happens with
ODE4, when you try and solve it

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on the interval
from t from 0 to 2.

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Number three- what happens
if the length of the interval

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is not exactly divisible
by the step size?

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For example, if t final is
pi, and the step size is 0.1.

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Don't try and fix this.

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It's just one of the hazards
of a fixed step size.

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And finally, exercise four--
investigate the flame problem

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with an initial
radius of 1/1,000.

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For what value of
t does the radius

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reach 90% of its final value?