{"title":"Asymptotic-Preserving and Energy Stable Dynamical Low-Rank Approximation","authors":"Lukas Einkemmer, Jingwei Hu, Jonas Kusch","doi":"10.1137/23m1547603","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 73-92, February 2024. <br/> Abstract. Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical methods for DLRA need to be carefully constructed to guarantee stability while preserving crucial properties of the original problem. Important physical effects that one likes to preserve with DLRA include capturing the diffusion limit in the high-scattering regimes as well as dissipating energy. In this work we propose and analyze a dynamical low-rank method based on the “unconventional” basis update & Galerkin step integrator. We show that this method is asymptotic preserving, i.e., it captures the diffusion limit, and energy stable under a CFL condition. The derived CFL condition captures the transition from the hyperbolic to the parabolic regime when approaching the diffusion limit.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.8000,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1547603","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 73-92, February 2024. Abstract. Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical methods for DLRA need to be carefully constructed to guarantee stability while preserving crucial properties of the original problem. Important physical effects that one likes to preserve with DLRA include capturing the diffusion limit in the high-scattering regimes as well as dissipating energy. In this work we propose and analyze a dynamical low-rank method based on the “unconventional” basis update & Galerkin step integrator. We show that this method is asymptotic preserving, i.e., it captures the diffusion limit, and energy stable under a CFL condition. The derived CFL condition captures the transition from the hyperbolic to the parabolic regime when approaching the diffusion limit.
期刊介绍:
SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.