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。为在时间离散化中精确解析该能量平衡，我们提出一种基于后验误差估计的自适应网格细化技术。误差估计量的计算涉及伴随敏感度的求解。为将该伴随方程解释为时间反向方程，我们证明了其对偶变量的分段弱可微性。进而，利用端口-哈密顿动力学的耗散特性，我们以块雅可比方法的思想提出一种可并行化的伴随系统近似方案，以高效计算误差指示器。通过数值实验验证所提方案的性能，结果表明：与均匀细化及传统步长控制时间步进方法相比，该方法产生的能量平衡违例更小。