Goal-Oriented Time Adaptivity for Linear Port-Hamiltonian Differential-Algebraic Equations of Index~1
For researchers modeling physical systems with port-Hamiltonian DAEs, this work provides a novel approach to enforce energy balance in discrete approximations, though it is incremental as it extends existing adaptive techniques to a specific structure.
The authors propose a time-adaptive method for port-Hamiltonian differential-algebraic equations that controls energy balance violation using dual-weighted residual error estimators. Simulations on electrical circuits demonstrate the method's efficacy.
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.