On Tue, Mar 2, 2010 at 11:44 AM, Jed Brown <jed@59a2.org> wrote:
* At each stage of the implicit system, integrate the explicit system up
 to the current stage using interpolated values of the implicit system.
 If the final abscissa is 1 and the implicit method has full stage
 order, then I think this is nominally of full accuracy
 (i.e. min(implicit_order,explicit order)).  This is burying the
 explicit integrator in function evaluation for the implicit system.
 Dana told me that he has actually done this in the past, albeit for
 simpler implicit integrators.

This is what we do now.... and we match the implicit and explicit integration orders.  This is really what we'd need to be able to do before I could use TSGL.

TSGL is not currently set up to offer the high-order interpolation at
arbitrary points within the interval, especially before all the stages
have been solved, but the method can supply these values (I or
(preferably) the inventors of the method just need to work out the
details for this operation).  I would like to have it anyway for
integration of continuous adjoints (discrete adjoints don't need

Bummer... if you have a trial implementation at some point that you want me to try out... I'm all game!