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

TL;DR

This paper introduces a new Petrov-Galerkin dynamical low rank approximation framework that incorporates stabilization techniques like SUPG for advection-dominated stochastic PDEs, ensuring stability and effective numerical performance.

Contribution

It develops a generalized DLR framework integrating stabilization methods such as SUPG, extending the applicability to stochastic advection-dominated problems with stability analysis.

Findings

SUPG stabilization effectively carried over to DLR framework

Norm-stability of time discretizations confirmed

Numerical experiments demonstrate improved stability and accuracy

Abstract

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.