不可压缩流体的保结构离散化
Structure-Preserving Discretization of Incompressible Fluids
摘要：
多年来，欧拉流体的几何性质已经被清楚地识别和广泛研究，最终以流体动力学的拉格朗日和哈密顿描述为高潮，其中配置空间被定义为体积保持的非同构，开尔文循环定理被视为与流体力学的粒子重新标记对称性相关的诺特定理的结果。然而，流体力学的计算方法在很大程度上是从数值分析的角度推导出来的，很少考虑到结构的保留而设计，并且经常遭受虚假的数值伪影，如能量和循环漂移。与此相反，本文几何推导出离散的运动方程的流体动力学的第一原则，在一个纯粹的欧拉形式。我们的方法近似组的体积保持同构使用一个有限维李群，和相关的离散欧拉方程来自非完整约束的变分原理。由此产生的离散运动方程产生一个结构保持时间积分器具有良好的长期能量行为和一个确切的离散开尔文循环定理。
英文摘要：
The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed as a consequence of Noether's theorem associated with the particle relabeling symmetry of fluid mechanics. However computational approaches to fluid mechanics have been largely derived from a numerical-analytic point of view, and are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts such as energy and circulation drift. In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin's circulation theorem holds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
