TS grand plan

[Jed Brown]

Barry Smith wrote:
> 
>    Jed,
> 
>     I suggest you "start from scratch" with "f(t,x,x')" and design what
> makes sense for you and ignore the current Arhs etc.
> Then we can try to transition the simple ones we have now as you
> suggest.  Please look at the explicit RK in TS and see how
> that may fit into your new grand plan.

Thanks Barry.  Thinking out loud:

So explicit integrators really can't use the f(t,x,x') form, at least
without a hint.  For example, if you knew that df/dx' was the identity
then -f(t,x,0) is the RHS and you can try an explicit method.  I've
never heard of an explicit integrator done this way, but I don't see why
it wouldn't work.  If df/dx' really is the identity, then the cost of
evaluating f(t,x,0) should be negligibly more than a RHS since the
latter involves stencil operations/fluxes/integrals.  If this is true
then it would be possible to remove the implicit/explicit and
linear/nonlinear dichotomies and have only Sundials and GL methods
(explicit and diagonally implicit, of which RK and multistep are special
cases that can be defined in a tableau) written exclusively in terms of
f(t,x,x') with a little sugar so that the user doesn't have to modify
their linear/RHS code.

I'll stick to variable order diagonally implicit GL for now.  It would
be easy to make it handle explicit as well, if that turned out to be
desirable down the road.

Jed

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