欧拉方程的高阶Mori-Zwanzig模型
Higher order Mori-Zwanzig models for the Euler equations
AI总结：
本文基于Mori-Zwanzig形式，通过正交动力学演化算子的泰勒展开构造了欧拉方程的高阶模型，并数值模拟了零阶、一阶和二阶模型，在Taylor-Green涡问题中能量衰减率与理论估计吻合。
摘要：
在最近的一篇论文\cite{CHSS06}中，提出并分析了欧拉方程的一个无限长记忆模型（t模型）。该模型可以通过在Mori-Zwanzig形式中保留记忆被积函数泰勒展开的零阶项来推导。本文提出了一系列同样基于Mori-Zwanzig形式的欧拉方程模型。这些模型源于对另一个算子——出现在记忆被积函数中的正交动力学演化算子——的泰勒展开。我们构造了零阶、一阶和二阶模型，并进行了数值模拟。欧拉方程中非线性的形式、所用投影算子的特殊性质以及任何投影算子的一般性质，都可以用来促进更高阶模型的递归计算。我们使用这些模型计算了Taylor-Green涡问题的能量衰减率。结果与理论估计吻合良好。能量衰减似乎以“波”的形式组织，即快速衰减和慢速衰减交替出现。我们的结果证实了\cite{CHSS06}中的假设，即通过少数低波数模式对三维欧拉方程进行建模应包含长记忆。
英文摘要：
In a recent paper \cite{CHSS06}, an infinitely long memory model (the t-model) for the Euler equations was presented and analyzed. The model can be derived by keeping the zeroth order term in a Taylor expansion of the memory integrand in the Mori-Zwanzig formalism. We present here a collection of models for the Euler equations which are based also on the Mori-Zwanzig formalism. The models arise from a Taylor expansion of a different operator, the orthogonal dynamics evolution operator, which appears in the memory integrand. The zero, first and second order models are constructed and simulated numerically. The form of the nonlinearity in the Euler equations, the special properties of the projection operator used and the general properties of any projection operator can be exploited to facilitate the recursive calculation of even higher order models. We use our models to compute the rate of energy decay for the Taylor-Green vortex problem. The results are in good agreement with the theoretical estimates. The energy decay appears to be organized in "waves" of activity, i.e. alternating periods of fast and slow decay. Our results corroborate the assumption in \cite{CHSS06}, that the modeling of the 3D Euler equations by a few low wavenumber modes should include a long memory.
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