Welcome to a short video
tutorial of the Eigenvalue
Stability applet.
You can get to the applet by
browsing to math.mit.edu/dai
mp/EigenvalueStability.html.
In eigenvalue stability,
there are two complex planes
of interest.
There is the g plane, shown
here in the upper right.
And the lambda delta t
plane, shown on the left.
The function that
relates these two plots
is the amplification factor.
You can see here when you
browse to the page already
the eigenvalue stability region
for the forward Euler method.
If you want to know the formula
for the amplification factor,
you can click this
radio button down here.
And you find that g
is equal to 1 plus z.
z here is lambda delta t.
And this is exactly what
our analysis showed before.
Now, if you want to plot the
eigenvalue stability region,
remember we parameterised
g in the unit circle.
So g equaled e to the I theta.
Here is the theta scrollbar.
So if we now scroll for
theta from 0 to 2 pi,
you can see the
yellow dot traces
out the unit circle
in the g plane.
And then also traces out
the stability boundary
in the lambda delta t
plane on the left here.
Now once you've decided that you
want to use the forward Euler
method to solve a
particular problem,
you first need to find the
eigenvalue of the problem.
Let's suppose, for example, that
the eigenvalue is a real number
sitting on the negative axis.
So each eigenvalue will have
a magnitude and a phase.
You can set the phase of
the eigenvalue over here.
That's phi.
So for our negative real
number, phi will be equal to pi.
You can see now the green
ray corresponding to e
to the I phi.
And now just click
anywhere down here
in the bottom right and
an orange dot appears.
This plot here in
the bottom right
shows the magnitude of the
eigenvalue against the timestep
delta t.
So now we can set over here the
magnitude of the eigenvalue.
Let's suppose the magnitude
is equal to, say 100.
Now we can see, for
different values of delta t,
where the corresponding
eigenvalue
will be for the
difference equation using
the forward Euler method.
So as I drag delta
t around here,
the orange dot moves
in all three plots.
So you can see here if I
go for a delta t that's
very small, I will have
eigenvalue stability
because the difference equation
will have an eigenvalue here,
at this orange dot.
The stability boundary is
indicated in all three plots,
using this blue shaded region.
So now you can see that
for a small delta t,
the simulation will
be eigenvalue stable.
But as I increase delta t,
eventually that orange dot
crosses over the
line and outside
of the stability boundary.
If you'd like to know what
is the maximum timestep delta
t for which you still
have eigenvalue stability
for this problem
using forward Euler,
you just place the orange
dot on the blue line,
and read off what
the delta t is.
In this case, delta t equals
0.02 is the maximum timestep
you can take to maintain
eigenvalue stability
for the problem with
eigenvalue minus 100
using the forward Euler method.
