Projected collocation for higher-order higher-index differential-algebraic equations

Abstract

Higher-order, higher-index Hessenberg systems of initial and boundary value differential-algebraic equations (DAEs) are considered. These types of systems arise in a variety of applications, including multibody systems. We extend a class of recently introduced projected implicit Runge-Kutta methods and define a new class of projected piecewise polynomial collocation methods for the solution of these problems. Stabilizing reformulations are considered as well, and a new projected invariant method is proposed. The higher-order ODE part of the DAE is collocated directly by a piecewise polynomial. A projection modification helps restore all the properties of stability and superconvergence which a corresponding collocation method for an ODE possesses. Higher-order collocation at Radau points is recommended for initial-value problems. The projection methods appear to be particularly promising for the solution of DAE boundary value problems, where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations, like collocation at Gauss points. Previously defined symmetric methods have severe limitations when applied to these problems, including instability, oscillation and loss of accuracy; the new methods overcome these difficulties. For higher-index problems we consider reformulation methods of stabilizing index reduction. We propose new methods of projected invariants to handle particularly tough higher-index problems. The advantages offered by these methods are demonstrated numerically.