Geometric Properties of Runge--Kutta Discretizations for Index 2 Differential Algebraic Equations

We analyze Runge--Kutta discretizations applied to index 2 differential algebraic equations (DAEs). The asymptotic features of the numerical and the exact solutions are compared. It is shown that Runge--Kutta methods satisfying the first order constraint condition of the DAE correctly reproduce the geometric properties of the continuous system. The proof combines embedding techniques of index 2 DAEs and ordinary differential equations (ODEs) with some invariant manifolds results of Nipp and Stoffer [Attractive Invariant Manifolds for Maps, SAM Research Report 92-11, ETH, Zurich, Switzerland, 1992]. The results support the favorable behavior of these Runge--Kutta methods applied to index 2 DAEs for $t \ge 0$.