The Modified Nodal Analysis leads to differential algebraic equations
with properly stated leading terms. In this article a special structure of the DAEs
modelling electrical circuits is exploited in order to derive a new decoupling for
nonlinear index-2 DAEs. This decoupling procedure leads to a solvability result and
is also used to study general linear methods, a class of numerical schemes that covers
both Runge-Kutta and linear multistep methods. Convergence for index-2 DAEs is
proved.
Classical results about the local existence and uniqueness of
DAE solutions are based on the derivative array [2] or on a geometrical
approach [13]. Thus these results can't be applied to equations with nonsmooth
coefficients. Also, sufficient conditions that guarantee solvability
are hard to check in general [6, 13]. In this paper a new approach to proving
local existence and uniqueness of DAE solutions is presented. The
main tool is a decoupling procedure that makes it possible to split DAE
solutions into their characteristic parts. Thus it is possible to weaken
the smoothness requirements considerably. In order for the decoupling
procedure to work we require a certain structural condition to hold. In
contrast to results already known, this condition can be easily verified.