On symmetric schemes and differential-algebraic equations

An example is given that demonstrates a potential risk in using symmetric difference schemes for initial value differential-algebraic equations (DAEs) or for very stiff ordinary differential equations (ODEs). The basic difficulty is that the stability of the scheme is controlled by the stability of an auxiliary (ghost) ODE problem that is not necessarily stable even when the given problem is.The stability of symmetric schemes is better understood in the context of boundary value problems. In this context, such schemes are more naturally applied as well. For initial value problems, better alternatives may exist. A computational algorithm is proposed for boundary value index-1 DAEs.