vakonomic流体
Vakonomic Fluids
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中文总结 AI 辅助
该研究基于不可压缩欧拉方程的新离散化方法，从vakonomic视角出发，产生离散流体轨迹，其动力学为李 - 泊松且有离散对称性，借助低秩克莱布施变量的动量映射表示，使vakonomic流体在低分辨率下稳定一致，提升鲁棒性与物理真实性。
中文摘要 AI 辅助
我们基于不可压缩欧拉方程作为保体积微分同胚李群上的测地线方程的解释，引入了一种新颖的离散化方法。众所周知，通过离散的库普曼表示对微分同胚及其无穷小生成元进行编码，会对离散速度施加非完整约束，而对于这种约束的变分处理尚无共识。我们表明，采用vakonomic观点，与通常的拉格朗日 - 达朗贝尔观点相反，会产生在（子）黎曼流形上保持为测地线的离散流体轨迹。特别是，由此产生的vakonomic动力学是李 - 泊松的，其解具有离散重新标记对称性，从而导致卡西米尔不变量在机器精度上得到满足，以及开尔文环流定理的离散类似物。使用基于低秩克莱布施变量的有效动量映射表示，我们表明这些vakonomic流体即使在低网格分辨率下也能稳定且一致地表现，从长远来看会提高鲁棒性和物理真实性。
英文摘要
We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms. It is well known that encoding diffeomorphisms and their infinitesimal generators through a discretized Koopman representation places a nonholonomic constraint on discrete velocities, for which there is no consensus on a variational treatment. We show that taking the vakonomic perspective, as opposed to the usual perspective of Lagrange--d'Alembert, yields discrete fluid trajectories that remain geodesics on a (sub-)Riemannian manifold. In particular, the resulting vakonomic dynamics are Lie--Poisson and their solutions admit a discrete relabeling symmetry, leading to machine-precision satisfaction of Casimir invariants along with a discrete analogue of Kelvin's Circulation Theorem. Using an efficient momentum map representation based on low-rank Clebsch variables, we show that these vakonomic fluids behave stably and consistently even at low grid resolutions, leading to increased robustness and physical realism in the long term.
