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

Structure-Preserving Time Discretization of Port-Hamiltonian Systems via Discrete Gradient Pairs