A general framework for the numerical approximation of evolution problems is presented that allows to preserve an underlying dissipative Hamiltonian or gradient structure exactly. The approach relies on rewriting the evolution problem in a particular form that complies with the underlying geometric structure. The Galerkin approximation of a corresponding variational formulation in space then automatically preserves this structure which allows to deduce important properties for appropriate discretization schemes including projection based model order reduction. We further show that the underlying structure is preserved also under time discretization by a Petrov–Galerkin approach. The presented framework is rather general and allows the numerical approximation of a wide range of applications, including nonlinear partial differential equations and port-Hamiltonian systems. Some examples will be discussed for illustration of our theoretical results, and connections to other discretization approaches will be highlighted.

中文翻译：

---

哈密​​顿量和梯度系统的能量稳定逼近

提出了演化问题数值近似的通用框架，该框架允许精确地保留底层的耗散哈密顿量或梯度结构。该方法依赖于以符合底层几何结构的特定形式重写演化问题。然后，空间中相应变分公式的Galerkin近似会自动保留此结构，从而可以推断出适当的离散化方案（包括基于投影的模型阶数减少）的重要属性。我们进一步表明，通过彼得罗夫-加勒金方法在时间离散下也保留了基础结构。提出的框架相当笼统，可以对各种应用进行数值近似，包括非线性偏微分方程和Port-Hamiltonian系统。我们将讨论一些例子来说明我们的理论结果，并着重说明与其他离散化方法的联系。