Numerical methods for the simulation of transient systems with structure-preserving properties are known to exhibit greater accuracy and physical reliability, in particular over long durations. These schemes are often built on powerful geometric ideas for broad classes of problems, such as Hamiltonian or reversible systems. However, there remain difficulties in devising timestepping schemes that conserve non-quadratic invariants or dissipation laws. In this work, we propose an approach for the construction of timestepping schemes that preserve dissipation laws and conserve multiple general invariants, via finite elements in time and the systematic introduction of auxiliary variables. The approach generalises several existing ideas in the literature, including Gauss methods, the framework of Cohen & Hairer, and the energy- and helicity-conserving scheme of Rebholz. We demonstrate the ideas by devising novel arbitrary-order schemes that conserve to machine precision all known invariants of Hamiltonian ODEs, including the Kepler and Kovalevskaya problems, and arbitrary-order schemes for the compressible Navier-Stokes equations that conserve mass, momentum, and energy, and provably possess non-decreasing entropy.

翻译：具有结构保持特性的瞬态系统数值模拟方法，因其在长时间尺度上展现出更高的精度与物理可靠性而备受关注。这类方法通常建立在适用于广泛问题类别（如哈密顿系统或可逆系统）的几何理论基础上。然而，在构造能够保持非二次不变量或耗散律的时间步进格式方面仍存在困难。本研究提出一种通过时间有限元法及系统引入辅助变量来构造时间步进格式的方法，该格式能够保持耗散律并同时守恒多个广义不变量。该方法推广了文献中现有的若干思想，包括高斯方法、Cohen与Hairer的理论框架，以及Rebholz的能量与螺旋度守恒格式。我们通过构造新型任意阶格式来验证该思想：针对哈密顿常微分方程（包括开普勒问题与Kovalevskaya问题）的格式能以机器精度守恒所有已知不变量；针对可压缩Navier-Stokes方程的任意阶格式能够守恒质量、动量与能量，并严格证明其熵具有非递减特性。