Convergence of Multistep Methods

If the multistep discretization defined by (J. 1.12) is consistent of order p and zero stable, then the method (5.2.6) converges with order p, i.e. the global error is 

Consequently, problems which are index-1 per se pose not more problems than explicit ODEs. Unfortunately mechanical problems can only be transformed into an index-1 formulation by differentiation, which results in instabilities in the problem, the drift-off effect. Thus, it is the stability of the problem and not of the method which causes problems. 

The index-2 case is the situation where the equations of motion are set up together with constraints on velocity level. We will see that the negative observation concerning the two step Adams-Moulton method holds in general for all higher order Adams-Moulton methods. The central convergence theorem requires oo-stability of the method. 

Definition 5.2.2 A method is called stable at infinity, if oo is in its stability region, i.e. if the roots of the a polynomial lie in the unit circle and those on its boundary are simple. If there are no roots on the boundary of the unit circle the method is called strictly stable at infinity. 

In light of the linear test equation this means that such a method is stable for all step sizes in the ultimately stiff case, where the stiffness parameter tends to infinity. Clearly, BDF methods are strictly stable at infinity as the generating polynomial cr( ) = has all its roots in the origin. 

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