Projected Runge–Kutta methods for index 3 differential–algebraic equations near equilibria, periodic orbits and attracting sets

In the present paper, we analyse the geometric properties of projected Runge–Kutta methods for the solution of index 3 differential–algebraic equations in the Hessenberg form. We show that the geometric phase portrait is well reproduced under discretization in the vicinity of equilibria, periodic orbits or asymptotically stable invariant sets. The main tools are embedding techniques and an invariant manifold theorem which allow a reduction of the problem to the classical ordinary differential equation case.