Discretization of inherent ODEs and the geometric integration of DAEs with symmetries

TL;DR

This paper introduces a discretization approach for DAEs based on their inherent ODEs, enabling the use of geometric integration schemes that preserve symmetries and geometric properties of the original system.

Contribution

It demonstrates how to construct inherent ODEs for DAEs with symmetries that inherit their geometric properties, facilitating structure-preserving numerical integration.

Findings

Inherent ODEs can be constructed to inherit DAE symmetries.

Geometric integration schemes can be applied to inherent ODEs.

Preservation of geometric properties in numerical flow.

Abstract

Discretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.