The Numerical Solution of Higher Iindex Differential/Algebraic Equations by Implicit Methods

Abstract

This paper studies the order, stability, and convergence properties of implicit Runge–Kutta (IRK) methods applied to differential/algebraic systems with index greater than one. These methods do not in general attain the same order of accuracy for higher index differential/algebraic systems as they do for index 1 systems or for purely differential systems. Necessary and sufficient conditions on the method coefficients are derived to ensure that the local and global errors of the method attain a given order of accuracy for higher index linear constant coefficient systems. IRK methods applied to nonlinear semi-explicit index 2 systems are studied, and a sufficient set of conditions is derived which ensures that a method is accurate to a given order for these systems. Finally, some numerical experiments are presented that illustrate the theoretical results and demonstrate the effects of roundoff' errors on the solution.