Goal-Oriented Time Adaptivity for Linear Port-Hamiltonian Differential-Algebraic Equations of Index~1

TL;DR

This paper introduces a time adaptivity method for port-Hamiltonian differential-algebraic equations of index 1, leveraging energy structure to improve simulation accuracy through a posteriori error control.

Contribution

It develops a novel energy balance control approach using dual weighted residual methods tailored for port-Hamiltonian systems with efficient error estimation.

Findings

Effective energy balance control demonstrated in electrical circuit simulations.

The method achieves accurate adaptive time-stepping by exploiting system dissipativity.

Error estimators are efficiently computed using a dissipativity-exploiting block-Jacobi approximation.

Abstract

Port-Hamiltonian systems provide a highly-structured framework for modeling of physical systems. By definition, they encode a balance equation relating energy changes to supplied and dissipated energy. Capturing this energy balance in discrete approximations is a fundamental challenge and often has been achieved by designing particular schemes such as discrete gradient methods. In this work, we propose an approach that controls the energy balance violation for port-Hamiltonian differential algebraic equations via time adaptivity using a posteriori grid refinement techniques based on the dual weighted residual method. In particular, we show how one may leverage the port-Hamiltonian structure to efficiently compute the error estimators using a dissipativity-exploiting block-Jacobi approximation. We illustrate the efficacy of the method by means of simulations of electrical circuit models.