Convergence of generalized- α $\boldsymbol{\alpha}$ time integration for nonlinear systems with stiff potential forces

Abstract

Generalized- $\alpha$ time integration schemes, originally developed for application in structural dynamics, are increasingly popular throughout many branches of multibody system simulation. Their simple implementation and the opportunity to control the numerical dissipation make them highly appealing for use in broad fields of application. Initially introduced for the solution of linear ordinary differential equations, there have been several extensions to nonlinear structural dynamics and constrained multibody systems in various formulations. In the present paper, we consider the application to systems with very stiff potential forces (singular singularly perturbed systems) whose solution approaches in the limit case that of a constrained system (index-3 differential–algebraic equation). We give a convergence analysis comparing the highly oscillatory solutions of the stiff system to those of the associated constrained one and show that the classical second order convergence result holds for position coordinates as well as for appropriately projected errors on the velocity level. The theoretical results are verified by numerical experiments for a simple test example.