Sequential regularization methods for nonlinear higher-index DAEs

Sequential regularization methods relate to a combination of stabilization methods and the usual penalty method for differential equations with algebraic equality constraints. This paper extends an earlier work [SIAM J. Numer. Anal., 33 (1996), pp. 1921--1940] to nonlinear problems and to differential algebraic equations (DAEs) with an index higher than 2. Rather than having one "winning" method, this is a class of methods from which a number of variants are singled out as being particularly effective methods in certain circumstances. We propose sequential regularization methods for index-2 and index-3 DAEs, both with and without constraint singularities. In the case of no constraint singularity we prove convergence results. Numerical experiments confirm our theoretical predictions and demonstrate the viability of the proposed methods. The examples include constrained multibody systems.