WEBVTT
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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?