Convergence of Lobatto-type Runge-Kutta methods for partitioned differential-algebraic systems of index 2

TL;DR

This paper introduces a numerical scheme based on Lobatto-type Runge-Kutta methods for solving partitioned index 2 differential-algebraic equations, with applications in nonholonomic mechanics and related fields.

Contribution

It establishes the convergence and order of a new class of Lobatto-type Runge-Kutta methods tailored for partitioned DAEs of index 2.

Findings

Proves the order of convergence for the proposed methods.

Addresses the interaction of two different coefficient sets in the scheme.

Provides a theoretical foundation for numerical solutions of nonholonomic systems.

Abstract

In this paper we propose a numerical scheme for partitioned systems of index 2 DAEs, such as those arising from nonholonomic mechanical problems and prove the order of a certain class of Runge-Kutta methods we call of Lobatto-type. The study of nonholonomic systems has recently shown a new interest in that theory and also in its relation to the new developments in control theory, subriemannian geometry, robotics, etc. The proofs and general outline of the paper follow a similar procedure of the one by L.O. Jay in the non-partitioned setting, but we tackle the issue of having two different sets of coefficients in interaction.