Port-Hamiltonian systems provide an energy-based modeling paradigm for dynamical input-state-output systems. At their core, they fulfill an energy balance relating stored, dissipated and supplied energy. To accurately resolve this energy balance in time discretizations, we propose an adaptive grid refinement technique based on a posteriori error estimation. The evaluation of the error estimator includes the computation of adjoint sensitivities. To interpret this adjoint equation as a backwards-in-time equation, we show piecewise weak differentiability of the dual variable. Then, leveraging dissipativity of the port-Hamiltonian dynamics, we present a parallelizable approximation of the underlying adjoint system in the spirit of a block-Jacobi method to efficiently compute error indicators. We illustrate the performance of the proposed scheme by means of numerical experiments showing that it yields a smaller violation of the energy balance when compared to uniform refinements and traditional step-size controlled time stepping.

翻译：端口-哈密顿系统为动态输入-状态-输出系统提供了一种基于能量的建模范式。其核心在于满足存储能量、耗散能量与供给能量之间的能量平衡关系。为在时间离散过程中精确解析该能量平衡，本文提出了一种基于后验误差估计的自适应网格细化技术。误差估计量的评估涉及伴随灵敏度的计算。为将伴随方程解释为时间反向方程，我们证明了对偶变量的分段弱可微性。进而利用端口-哈密顿动力学耗散性，提出一种基于块雅可比方法精神的伴随系统可并行近似方案，以高效计算误差指标。数值实验表明，所提方法相较于均匀细化与传统步长控制时间推进方案，能够更小程度地违反能量平衡关系。