On Thu, 2 Oct 2008, Nasser Mohieddin Abukhdeir wrote:
> What now confuses me is whether or not EulerSolver with theta=0.5 is
> actually Crank-Nicolson. It makes sense to me that Euler2Solver with
> theta=0.5 is...but not EulerSolver.
That actually depends on your definition of Crank-Nicholson. For
linear problems, F((u^n+u^{n+1})/2) = (F(u^n)+F(u^{n+1}))/2 and
they're both the same. For nonlinear problems, I think the former
method (what EulerSolver does) is usually what's called
Crank-Nicholson and the latter method (what Euler2Solver does) is just
called "trapezoidal integration".
> Moving back to John's point about the previously mentioned
> Predictor/Corrector algorithm (assuming we compute everything
> correctly), I'm assuming that idea is trashed.
The idea that we can use the first Newton step as an explicit solution
is wrong, but using a forward Euler solve for the predictor and
following it up by a backward Euler Newton solve for the corrector
should still work just fine.
> I did some digging
> around, looked up the methods that John suggested, and found a variable
> coefficient multistep method (allows for unequally spaced data):
>
> Chapter 12
> http://books.google.com/books?id=aqfMgehyGnEC
That looks like an excellent adaptive method for us to use in the
future. I'd still suggest doing a single-step predictor/corrector
first before biting off something more complicated.
---
Roy
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