TS grand plan

[Jed Brown]

I'm putting in support for DAEs written in fully implicit form,
i.e. F(t,x,x')=0.  (I brought this up ages ago, but just got to work
implementing it.)  TS uses a frustrating mixed design in an attempt to
reuse code between the different implementations.  Is there clear intent
behind A, B, Alhs, Arhs?  My expectation was that we would be calling
KSPSetOperators(ksp,A,B,...) and SNESSetJacobian(snes,A,B,...) where the
other matrices are for the user, but the usage is much less clear.

TSSetUp:
  KSPSetOperators(ksp,Arhs,B,...)
  SNESSetJacobian(snes,Arhs,B,...)

TS_BEULER:
  A = Arhs; (aliasing)
  KSPSetOperators(ksp,A,A,...)
  SNESSetJacobian(snes,Arhs,B,...)  [shift applied to A=Arhs]
  
  * -snes_mf_operator doesn't do anything matrix-free, see
    tutorials/ex2 -nox -ts_max_steps 1 -ts_type beuler -snes_mf_operator -snes_view
    There was a hack to detect MFFD which was introduced in 2004 and
    removed a day later (although the associated comment remains)
  * No support for shifting a preconditioning matrix
    (also made an appearance in 2004)
  * No way to get a different preconditioning matrix to the KSP

TS_CN:
  KSPSetOperators(ksp,A,A,...)
  SNESSetJacobian(snes,A,B,...)

  * linear variable matrix calls ts->ops->rhsmatrix with NULL preconditioning
  matrix
  * For linear problems with variable matrix, A is destroyed and
  recreated every step
  * For nonlinear problems, only B is shifted


Is there some consistent grand plan?


More trivially, I ripped isExplicit and Iindex out of _p_TS because they
are never used in PETSc.


As a very basic DAE integrator, I wrote an implementation of the theta
method as a 1-step Runge-Kutta which only uses the implicit form
F(t,x,xdot)=0.  With the implicit formulation, the matrices Alhs, Arhs
never appear and instead the user assembles exactly the matrix that is
needed by SNES.  Of course, we can serve up the various explicit
representations (xdot = f(t,x) and the linear cases) with light
wrappers, but I feel like I'm fighting the use of A,B,Alhs,Arhs in
interface/ts.c.

It seems to me that with a little caching, we should be able to make all
the implicit integrators use only the F(t,x,xdot)=0 form and still get
comparable efficiency to the specialized versions that we have now.
Since TS_THETA includes the functionality of TS_BEULER and TS_CN as
special cases, we could remove the latter if we can demonstrate that the
fully implicit formulation doesn't lose efficiency for the RHS and
linear cases (I expect it will be more efficient when the user provides
the implicit form because we get to skip scaling and shifting).

Jed

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