Stability at infinity

One can see here that all components are stabilized at infinity. The secular terms on this scale occur just in the second corrections... 

We defined consistency exclusively as a property of a discretization scheme which is independent of a given differential equations. So it is stability which causes convergence problems for higher index DAEs and which led to ask for stability at infinity. 

Starting with inconsistent initial values, i.e. Aq 0, leads to an increasing error and an unstable growth of A if jR(oo) > 1. The error is damped when R oo) < 1. For jR(oo) = 1 errors in the individual steps and the initial error are just summed up. Thus, we have at least to require stability at infinity for a method being stable in the DAE case. 

We have indeed the same requirements as in the multistep case where stability at infinity plays a central role (see Def. 5.2.2 and Th. 5.2.3). There, the method with optimal damping at infinity is the BDF. 

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See also in sourсe #XX -- [ ]

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