Petrov–Galerkin Dynamical Low Rank Approximation: SUPG stabilisation of advection-dominated problems

Abstract

We propose a novel framework of generalised Petrov–Galerkin Dynamical Low Rank (DLR) Approximations 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 Streamline 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.