Linear Multistep Methods for DAEs

Example 5.1.13 We consider the mechanical system in its index-1 formulation (5.L15) with its invariants (5A.16). 

we demonstrate numerical problems of the various formulations when using Euler s method and the trapezoidal rule. 

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For simultaneous solution of (16), however, the equivalent set of DAEs (and the problem index) changes over the time domain as different constraints are active. Therefore, reformulation strategies cannot be applied since the active sets are unknown a priori. Instead, we need to determine a maximum index for (16) and apply a suitable discretization, if it exists. Moreover, BDF and other linear multistep methods are also not appropriate for (16), since they are not self-starting. Therefore, implicit Runge-Kutta (IRK) methods, including orthogonal collocation, need to be considered. 

AFS97] Arevalo C., Fiihrer C., and Soderlind G. (1997) -blocked multistep methods for Euler-Lagrange DAEs Linear analysis. Z. Angew. Math, Mech. 77 609-617. 

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