Why does ode15i work on my index-2 DAE?

Hello everyone,

I am simulating a differentiation-index 2 DAE with ode15i and everything works fine. My problem is: It shouldn't! ode15i is supposed to work only on index-1 DAEs and I wanna know why.

To give some context: The DAE in question stems from a discretization of a bulk-flow equation, which describes incrompressible fluid flow in narrow gaps (Navier-Stokes-like equation structure), which looks in its original form like this:

The first two equations are the momentum equations, the third one is the mass conservation law (index-2 algebraic equation) and the 3 last ones are boundary conditions (index-1 algebraic equations). V and U are fluid velocities and P is the fluid pressure. H is a known function. are some more complex dependencies. ε is a small parameter (). All other quantities are . θ and z are circumferential and axial coordinates and the respective partial derivatives are discretized using some finite differences scheme (I know... I know... FVM would be more appropriate). The resulting system after discretization is an index-2 DAE.

As mentioned above, the ode15i works like a charm, whilst ode15s and ode23t refuse to work with the (accurate) comment, that this is not an index-1 DAE.

The inevitable follow-up question is why I even bother, if it works. My problem is, that

a) I want to make sure, that I've got a reliable workflow, which I can trust and

b) that I am curious :-D

I already worked through the documentation and the originial paper from L. Shampine, which brought me no enlightment. Simulating a mathematical pendulum as DAE in index-3, index-2 and index-1 notation showed, that ode15i does not work on index-3, on index-2 the time steps get ridiculously/ impractiably small and index-1 works as excepted fine.

Has anyone the answer to that problem or could give me some hint on why this might work?

Thank you all in advance and greetings from Germany,

Simon