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

TL;DR

This work introduces a generalised Petrov-Galerkin Dynamical Low Rank (PG-DLR) framework for random time-dependent PDEs, enabling standard FEM stabilisation techniques such as SUPG to be incorporated directly into the online evolution of the low-rank DO/DLR basis. By formulating the dynamics as an oblique projection onto the tangent space of the low-rank manifold and carefully designing a skewing operator $\mathcal{H}$ (often separable as $\mathcal{H}_1\otimes\mathcal{H}_2$), the authors derive evolution equations for the deterministic and stochastic modes and implement time-stepping schemes that preserve stability properties. The SUPG-stabilised DLR variant is analysed for norm-stability under implicit Euler and semi-implicit integrators, with coercivity-based bounds and practical dt/Δt constraints; numerical experiments on rotating bodies and boundary-layer-type problems validate the stabilising effect and demonstrate reduced oscillations compared to standard DLR without offline stabilization. The framework generalises to a broad class of finite element stabilisations and offers an online-stabilised ROM approach for uncertain, time-dependent PDEs that obviates costly offline stabilization while preserving accuracy and stability in advection-dominated regimes.

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.

Figures (5)

• Figure 1: Realisation at $t=0$ with parameters $\mathbf{y} = (0.05, -0.63, 0.67)$.
• Figure 2: Numerical solution for same parameters as in Figure \ref{['fig:initial-cond']} obtained via standard DLR or SUPG-DLR at $t=2\pi$.
• Figure 3: (a) Comparison of $u_{\mathrm{DLR}}^{\mathrm{SUPG}}$ and $u_{\mathrm{DLR}}^{\mathrm{standard}}$ to proxy solution, each colour corresponds to a realisation. (b) Oscillations in realisations of $u_{\mathrm{DLR}}^{\mathrm{SUPG}}$ and $u_{\mathrm{DLR}}^{\mathrm{standard}}$.
• Figure 4: Boundary of the domain, separated in $\partial D_1$ (black) and $\partial D_2$ (grey).
• Figure 5: Evolution of parametrised solution for $\mathbf{y} = (1.6 \cdot 10^{-4} , 1, -0.33, -0.77)$.

Theorems & Definitions (22)

• Lemma 2.1
• Remark 1
• Remark 2
• Remark 3
• Proposition 3.1
• proof
• Lemma 3.1
• proof
• Theorem 3.1
• proof