High-order robust basis-update & Galerkin integrators for dynamical low-rank approximation
Abstract
Dynamical low-rank approximation has become a widely used numerical method in diverse disciplines. Its main idea is to represent the matrix or tensor-valued solution to a time-dependent differential equation as a low-rank factorization. The evolution of the factorization leads to highly stiff dynamics and requires the derivation of novel time integration methods that are not prone to this stiffness. A promising family of integrators are basis-update \& Galerkin (BUG) integrators as they enable implicit time integration and structure--preservation. However, current BUG integrators are limited to second--order accuracy while general-order BUG integrators are designed as projections of explicit time integration methods, thus severely limiting their use. In this work, we propose general-order BUG integrators that do not rely on an explicit time integration scheme while requiring a smaller number of basis functions to achieve high-order accuracy. We prove a general order error bound for the proposed augmented and parallel BUG integrators and demonstrate their behaviour for a series of stiff and non-stiff numerical benchmarks in which they significantly outperform previous BUG versions.
Related
- HIGH-ORDER DISCRETE DIFFERENTIAL AND INTEGRAL CALCULUS AND GALERKIN VARIATIONAL INTEGRATORS
- High-Order BUG Dynamical Low-Rank Integrators Based on Explicit Runge-Kutta Methods.
- Robust High-Order Projector-Splitting Integrators
- Flux-Matched Poisson Iteration for Generated Jacobian Equations: Local Convergence and Spectral Galerkin Approximation
- Galerkin Approximation of the Fractional Sobolev Constant
BibTeX
@misc{hauck2026highorder,
title = {{High-order robust basis-update & Galerkin integrators for dynamical low-rank approximation}},
author = {Cory D. Hauck and Jonas Kusch and Steffen Schotthöfer},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27749},
eprint = {2608.27749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27749},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - High-order robust basis-update & Galerkin integrators for dynamical low-rank approximation
AU - Cory D. Hauck
AU - Jonas Kusch
AU - Steffen Schotthöfer
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27749
UR - https://arxiv.org/abs/2608.27749
AB - Dynamical low-rank approximation has become a widely used numerical method in diverse disciplines. Its main idea is to represent the matrix or tensor-valued solution to a time-dependent differential equation as a low-rank factorization. The evolution of the factorization leads to highly stiff dynamics and requires the derivation of novel time integration methods that are not prone to this stiffness. A promising family of integrators are basis-update \& Galerkin (BUG) integrators as they enable implicit time integration and structure--preservation. However, current BUG integrators are limited to second--order accuracy while general-order BUG integrators are designed as projections of explicit time integration methods, thus severely limiting their use. In this work, we propose general-order BUG integrators that do not rely on an explicit time integration scheme while requiring a smaller number of basis functions to achieve high-order accuracy. We prove a general order error bound for the proposed augmented and parallel BUG integrators and demonstrate their behaviour for a series of stiff and non-stiff numerical benchmarks in which they significantly outperform previous BUG versions.
ER -