Problem Set # 8 Questions and answers

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Dear ...

> Dear Prof. Rosales
>
> I have a question about the pset problems.
> In the problem, it says to "are there any conditions
> that suggest when stability/ unstability applies." I
> just want to make sure I interpret this
> sentence correctly so I won't miss things.

 In this last problem set you are asked to calculate the Associated Equation
for a few numerical schemes for some p.d.e.'s.  Remember that, as I said
in the lectures, the AENS can point out to "problems" that a numerical
scheme can have if the parameters [e.g. lambda = (delta t)/(delta x)] are
not selected correctly --- you may end up with a negative diffusion
coefficient. HOWEVER, you cannot get sufficient conditions for stability
from them alone.

Hence, looking at the AENS you may be able to say stuff like:
 Unless I choose the scheme parameters to satisfy this or that condition,
 the scheme will be unstable [because, say, then the AENS has a negative
 diffusion coefficient]. If I choose the scheme parameters so that they
 satisfy the condition, my AENS behaves properly ... so that there is a
 chance that the scheme may be stable. But I cannot be certain, since the
 AENS does not tell me what happens with the "short" waves in the problem.

It is because of this that I say "suggest". You can find conditions that
will make the scheme unstable (because then the AENS is badly behaved),
and conditions were this does not happen. But this second set of conditions
may not be the whole story. It's a "maybe this is enough, if I am lucky".

> So in order to answer that sentence, we should do both: calculate |G| and

 What G? There is no G in the AENS stuff. The G is the growth rate that
 arises in the von Neumann stability analysis. But I am not asking you
 to do this. Just to get the AENS.

> observe the diffusion coefficient of Utt and Uxx the associated equation,

 In order to talk about "diffusion" coefficient, you have to get rid of all
 the time dependences. Remember how I eliminated the time derivatives in
 the correction terms, after Taylor expanding, to obtain the AENS? You have
 to do this here. There will not be any "u_tt" terms. In the end, your
 AENS will look like

     Equation that you are supposed to solve = "small error terms"
     \-------------------------------------/
         for example, u_t + u_x

 The  "small error terms" will only have space derivatives for the problems
 that I assigned, and you want to make sure that there is no term there
 that will cause unbounded growth:

   That is, if you plug u = A(t)*exp(i*k*x)

   into the AENS, and calculate A(t), you do not have a situation where
   A(t) grows arbitrarily fast.
   For example, negative difussion [i.e.: a term -u_xx on the right] will
   induce a exp(k^2*t) behavior in A. Namely, A grows, and arbitrarily
   fast, since k is arbitrary.

> and give the value for \lambda such that the scheme is stable; or prove that
> |G| cannot always be <= 1 so that the scheme is stable?

 No, this is only when doing von Neumann Stability!

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