The present work presents a stable POD-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh-B\'enard convection in a square geometry for three Rayleigh numbers: $10^4$ (steady state), $3\times 10^5$ (periodic), and $6 \times 10^6$ (chaotic). Stability is obtained through a particular (staggered-grid) full-order model (FOM) discretization that leads to a ROM that is pressure-free and has skew-symmetric (energy-conserving) convective terms. This yields long-time stable solutions without requiring stabilizing mechanisms, even outside the training data range. The ROM's stability is validated for the different test cases by investigating the Nusselt and Reynolds number time series and the mean and variance of the vertical temperature profile. In general, these quantities converge to the FOM when increasing the number of modes, and turn out to be a good measure of accuracy. However, for the chaotic case, convergence with increasing numbers of modes is relatively difficult and a high number of modes is required to resolve the low-energy structures that are important for the global dynamics.

翻译：本文针对方形区域内的二维瑞利-贝纳德对流，基于POD-Galerkin方法提出了一种稳定的降阶模型（ROM），覆盖三个瑞利数：$10^4$（稳态）、$3\times 10^5$（周期性）和$6 \times 10^6$（混沌）。通过采用特定的交错网格全阶模型（FOM）离散格式，所构建的ROM具有无压力特性及反对称（能量守恒）的对流项，从而无需引入稳定化机制即可实现长时间稳定解，甚至在训练数据范围之外亦能保持稳定性。通过分析努塞尔数和雷诺数时间序列以及垂直温度剖面的均值和方差，验证了ROM在不同算例中的稳定性。总体而言，随着模态数增加，这些量趋近于FOM结果，并成为衡量精度的重要指标。然而，对于混沌情况，随着模态数增加收敛相对困难，需要大量模态来解析对全局动力学至关重要但能量较低的结构。