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A general framework for the numerical approximation of evolution problems ispresented that allows to preserve exactly an underlying Hamiltonian- orgradient structure. The approach relies on rewriting the evolution problem in aparticular form that complies with the underlying geometric structure. TheGalerkin approximation of a corresponding variational formulation in space thenautomatically preserves this structure which allows to deduce importantproperties for appropriate discretization schemes including projection basedmodel order reduction. We further show that the underlying structure ispreserved also under time discretization by a Petrov-Galerkin approach. Thepresented framework is rather general and allows the numerical approximation ofa wide range of applications, including nonlinear partial differentialequations and port-Hamiltonian systems. Some examples will be discussed forillustration of our theoretical results and connections to other discretizationapproaches will be revealed.

Energy stable Galerkin approximation of Hamiltonian and gradient systems