Splitting Techniques for DAEs with port-Hamiltonian Applications

TL;DR

This paper develops operator-splitting techniques for port-Hamiltonian differential-algebraic equations, ensuring structure preservation and energy conservation, with proven convergence and validated on electric circuit benchmarks.

Contribution

It introduces two novel splitting strategies for DAEs within the port-Hamiltonian framework, maintaining algebraic constraints and energy properties, with convergence guarantees.

Findings

Splitting schemes achieve ODE-like convergence rates for coupled index-1 DAEs.

Energy-preserving decompositions are effective for port-Hamiltonian systems.

Validated on electric circuit benchmarks demonstrating practical applicability.

Abstract

In the simulation of differential-algebraic equations (DAEs), it is essential to employ numerical schemes that take into account the inherent structure and maintain explicit or hidden algebraic constraints without altering them. This paper focuses on operator-splitting techniques for coupled systems and aims at preserving the structure in the port-Hamiltonian framework. The study explores two decomposition strategies: one considering the underlying coupled subsystem structure and the other addressing energy-associated properties such as conservation and dissipation. We show that for coupled index-$1$ DAEs with and without private index-2 variables, the splitting schemes on top of a dimension-reducing decomposition achieve the same convergence rate as in the case of ordinary differential equations. Additionally, we discuss an energy-associated decomposition for index-1 pH-DAEs and introduce generalized Cayley transforms to uphold energy conservation. The effectiveness of both strategies is evaluated using port-Hamiltonian benchmark examples from electric circuits.