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Many different methods have been suggested for the numerical solution of index 2 and 3 Euler‐Lagrange equations. We focus on 0‐stability of multistep methods (φ, σ) and investigate the relations between some well‐known computational techniques. By various modifications, referred to as β‐blocking of the σ polynomial, some basic shortcomings of multistep methods may be overcome. This approach is related to projection techniques and has a clear and well‐known analogy in control theory. In particular, it is not necessary to use BDF methods for the solution of high index problems; indeed, “nonstiff” methods may be used for part of the system provided that the state‐space form is nonstiff. We illustrate the techniques and demonstrate the results with a simplified multibody model of a truck.

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A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of k − 1 or k parameters. This construction includes all methods of maximal order (p = k for stiff, and p = k + 1 for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software, although variable order has not yet been included. Mathematics subject classification: 65L06, 65L05, 65L80