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On the stability of robust dynamical low-rank approximations for hyperbolic problems

Kusch, Jonas; Einkemmer, Lukas; Ceruti, Gianluca

Abstract:

The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a L2 stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then applying the DLRA. Based on this we propose a projector splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator has more favorable stability properties and explain why the projector splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. ... mehr


Volltext §
DOI: 10.5445/IR/1000135515
Veröffentlicht am 19.07.2021
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 07.2021
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000135515
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 23 S.
Serie CRC 1173 Preprint ; 2021/33
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter dynamical low-rank approximation, numerical stability, kinetic equations, uncertainty quantification, projector-splitting integrator, unconventional integrator
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