Specialized Runge-Kutta methods for index 2 differential-algebraic equations

Lagrange-d’Alembert SPARK Integrators for Nonholonomic Lagrangian Systems

Lagrangian systems with ideal nonholonomic constraints can be expressed as implicit index 2 differential-algebraic equations (DAEs) and can be derived from the Lagrange-d’Alembert principle. We define a new nonholonomically constrained discrete Lagrange-d’Alembert principle based on a discrete Lagrange-d’Alembert principle for forced Lagrangian systems. Nonholonomic constraints are considered a...

Projected Runge-kutta Methods for Differential Algebraic Equations of Index 3

In the present paper we introduce a new class of methods, Projected RungeKutta methods, for the solution of index 3 differential algebraic equations (DAEs) in Hessenberg form. The methods admit the integration of index 3 DAEs without any drift effects. This makes them particularly well suited for long term integration. Finally, implemented on the basis of the Radau5 code, the projected Runge-Ku...

Efficient Runge-Kutta integrators for index-2 differential algebraic equations

In seeking suitable Runge-Kutta methods for differential algebraic equations, we consider singly-implicit methods to which are appended diagonally-implicit stages. Methods of this type are either similar to those of Butcher and Cash or else allow for the importation of a final derivative from a previous step. For these two classes, with up to three additional diagonallyimplicit stages, we deriv...

Solution of Index 2 Implicit Differential-algebraic Equations by Lobatto Runge-kutta Methods

We consider the numerical solution of systems of index 2 implicit differential-algebraic equations (DAEs) by a class of super partitioned additive Runge-Kutta (SPARK) methods. The families of Lobatto IIIA-B-C-C-D methods are included. We show super-convergence of optimal order 2s−2 for the s-stage Lobatto families provided the constraints are treated in a particular way which strongly relies on...

Invariant manifolds in differential algebraic equations of index 3 and in their Runge-Kutta discretizations