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

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

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...

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

We consider the numerical solution of systems of semi-explicit index 2 differential-algebraic equations (DAEs) by methods based on RungeKutta (RK) coefficients. For nonstiffly accurate RK coefficients, such as Gauss and Radau IA coefficients, the standard application of implicit RK methods is generally not superconvergent. To reestablish superconvergence projected RK methods and partitioned RK ...

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...

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...