In this paper, we propose to improve the stabilized POD-ROM introduced in [48] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we propose a three-stage stabilizing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This approach mainly consists in three ingredients: (1) the addition of a “streamline diffusion” stabilization term to the governing projected equations, (2) the modification of the correlation matrix defining the POD modes associated to the advection stabilization term, and (3) an a-posteriori stabilization scheme. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases.

在本文中，我们提出对[48]中介绍的稳定的POD-ROM进行改进，以处理以对流为主的对流扩散反应方程的数值模拟。特别是，我们提出了一个三阶段稳定策略，当考虑极低的扩散系数时，即在强对流为主的情况下，将非常有用。此方法主要包含三个要素：（1）在控制投影方程式中添加“流线扩散”稳定项，（2）修改定义与对流稳定项相关的POD模式的相关矩阵，以及（3） ）后验稳定方案。进行了数值研究，以讨论该新方法在处理强对流占主导地位的情况下的准确性和性能。