We discuss structure-preserving time discretization for nonlinear port-Hamiltonian systems with state-dependent mass matrix. Such systems occur, for instance, in the context of structure-preserving nonlinear model order reduction for port-Hamiltonian systems and, in this context, structure-preserving time discretization is crucial for preserving some of the properties of the time-continuous reduced-order model. For this purpose, we introduce a new class of time discretization schemes which is based on so-called discrete gradient pairs and leads to an exact power balance on the time-discrete level. Moreover, for the special case of a pointwise symmetric and positive definite mass matrix, we present an explicit construction of a discrete gradient pair. Finally, we illustrate the theoretical findings by means of a numerical example, where the time-continuous system is a nonlinear reduced-order model for an advection-diffusion problem.

翻译：本文讨论具有状态依赖质量矩阵的非线性端口哈密顿系统的保结构时间离散化方法。此类系统常见于端口哈密顿系统的保结构非线性模型降阶中，而保结构时间离散化对于保持时间连续降阶模型的某些特性至关重要。为此，我们引入一类基于所谓离散梯度对的新型时间离散格式，该格式在时间离散层面实现精确的功率平衡。进一步，针对逐点对称正定质量矩阵的特殊情形，我们给出离散梯度对的显式构造方法。最后，通过以时间连续系统为对流扩散问题非线性降阶模型的数值算例，验证了理论结果。