二维Euler方程的稀疏随机模型降阶
Sparse-stochastic model reduction for 2D Euler equations
摘要：
二维欧拉方程是一组简单但丰富的非线性偏微分方程，描述了理想无粘流体的演化，其中一维可以忽略不计。数值求解这些方程可能是非常苛刻的。在过去的几十年中，已经开发了几种技术来获得快速和准确的模拟。在本文中，我们提出了一种新的方法，它结合了最近的发展，随机模型的减少和保守的半离散的欧拉方程。特别是，从Zeitlin模型的2球，我们推导出大尺度的动力学减少，我们关闭的方程确定性或一个合适的随机项。数值实验表明，在初始湍流状态后，小尺度对大尺度的影响可以忽略不计，即使不同模式之间存在非零能量转移。
英文摘要：
The 2D Euler equations are a simple but rich set of non-linear PDEs that describe the evolution of an ideal inviscid fluid, for which one dimension is negligible. Solving numerically these equations can be extremely demanding. Several techniques to obtain fast and accurate simulations have been developed during the last decades. In this paper, we present a novel approach which combines recent developments in the stochastic model reduction and conservative semi-discretization of the Euler equations. In particular, starting from the Zeitlin model on the 2-sphere, we derive reduced dynamics for large scales and we close the equations either deterministically or with a suitable stochastic term. Numerical experiments show that, after an initial turbulent regime, the influence of small scales to large scales is negligible, even though a non-zero transfer of energy among different modes is present.
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