提出新数值方法，精准模拟物理系统能量变化。
Discrete gradient methods for port-Hamiltonian differential-algebraic equations
- 用离散梯度法处理非线性物理系统方程，保持能量结构。
- 针对半显式微分代数方程设计新算法，精度高且稳定。
- 适合多体动力学等物理仿真，尤其关注能量守恒场景。
离散梯度方法是动态系统时间离散的强大工具，因其对总能量形式的无关性而保持结构特性。本文研究其在非线性端口-哈密顿微分代数方程（port-Hamiltonian differential-algebraic equations）中的应用，该类方程源自多领域物理系统的端口与能量建模。我们提出一种专为半显式微分代数方程设计的新数值格式，并通过离散梯度对与狄拉克耗散结构的概念拓展至更一般情形。此外，分析了系统变换下的行为，证明在适当假设下，端口-哈密顿微分代数方程可表示为参数化的端口-哈密顿半显式系统与一个无结构方程的组合。最后，将方法应用于多体系统动力学，展示其数值效果，验证了该方法在保持系统结构和能量特性方面的有效性。
原文摘要 · Abstract (English)
Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient methods to the system class of nonlinear port-Hamiltonian differential-algebraic equations - as they emerge from the port- and energy-based modeling of physical systems in various domains. We introduce a novel numerical scheme tailored for semi-explicit differential-algebraic equations and further address more general settings using the concepts of discrete gradient pairs and Dirac-dissipative structures. Additionally, the behavior under system transformations is investigated and we demonstrate that under suitable assumptions port-Hamiltonian differential-algebraic equations admit a representation which consists of a parametrized port-Hamiltonian semi-explicit system and an unstructured equation. Finally, we present the application to multibody system dynamics and discuss numerical results to demonstrate the capabilities of our approach.
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