We propose a novel framework of generalised Petrov-Galerkin Dynamical Low Rank Approximations (DLR) in the context of random PDEs. It builds on the standard Dynamical Low Rank Approximations in their Dynamically Orthogonal formulation. It allows to seamlessly build-in many standard and well-studied stabilisation techniques that can be framed as either generalised Galerkin methods, or Petrov-Galerkin methods. The framework is subsequently applied to the case of Streamine Upwind/Petrov Galerkin (SUPG) stabilisation of advection-dominated problems with small stochastic perturbations of the transport field. The norm-stability properties of two time discretisations are analysed. Numerical experiments confirm that the stabilising properties of the SUPG method naturally carry over to the DLR framework.

翻译：本文针对随机偏微分方程，提出了一种广义 Petrov-Galerkin 动力学低秩逼近（DLR）的新框架。该框架建立在标准动力学低秩逼近的动态正交表述基础之上，能够无缝融入许多标准且经过充分研究的稳定化技术，这些技术可被表述为广义 Galerkin 方法或 Petrov-Galerkin 方法。随后，该框架被应用于对流主导问题的流线迎风/Petrov-Galerkin（SUPG）稳定化，其中输运场存在小的随机扰动。本文分析了两种时间离散格式的范数稳定性。数值实验证实，SUPG 方法的稳定化性质自然地延续到了 DLR 框架中。