Behavior of Runge-Kutta discretizations near equilibria of index 2 differential algebraic systems (Q1612473)

The author studies the behavior of Runge-Kutta discretizations applied to index 2 differential algebraic systems near hyperbolic equilibria. The analysis is restricted to projected and half-explicit Runge-Kutta methods, in order to avoid drift phenomena in longtime behavior. The main results are very interesting. It is shown that near hyperbolic equilibria the discrete Runge-Kutta time \(h\)-map restricted to its invariant manifold is locally homeomorphic to the continuous time \(h\)-flow. Therefore the qualitative geometric properties of the considered differential algebraic equations are preserved under discretization with projected and half-explicit Runge-Kutta methods.