{"title":"An energy-stable variable-step L1 scheme for time-fractional Navier–Stokes equations","authors":"Ruimin Gao , Dongfang Li , Yaoda Li , Yajun Yin","doi":"10.1016/j.physd.2024.134264","DOIUrl":null,"url":null,"abstract":"<div><p>We propose a structure-preserving scheme and its error analysis for time-fractional Navier–Stokes equations (TFNSEs) with periodic boundary conditions. The equations are first rewritten as an equivalent system by eliminating the pressure explicitly. Then, the spatial and temporal discretization are done by the Fourier spectral method and variable-step L1 scheme, respectively. It is proved that the fully-discrete scheme is energy-stable and divergence-free. The energy is an asymptotically compatible one since it recovers the classical energy when <span><math><mrow><mi>α</mi><mo>→</mo><mn>1</mn></mrow></math></span>. Moreover, optimal error estimates are presented very technically by the obtained boundedness of the numerical solutions and some Sobolev inequalities. To our knowledge, they are the first results of the construction and analysis of structure-preserving schemes for TFNSEs. Several interesting numerical examples are given to confirm the theoretical results at last.</p></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"467 ","pages":"Article 134264"},"PeriodicalIF":2.7000,"publicationDate":"2024-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S016727892400215X","RegionNum":3,"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
We propose a structure-preserving scheme and its error analysis for time-fractional Navier–Stokes equations (TFNSEs) with periodic boundary conditions. The equations are first rewritten as an equivalent system by eliminating the pressure explicitly. Then, the spatial and temporal discretization are done by the Fourier spectral method and variable-step L1 scheme, respectively. It is proved that the fully-discrete scheme is energy-stable and divergence-free. The energy is an asymptotically compatible one since it recovers the classical energy when . Moreover, optimal error estimates are presented very technically by the obtained boundedness of the numerical solutions and some Sobolev inequalities. To our knowledge, they are the first results of the construction and analysis of structure-preserving schemes for TFNSEs. Several interesting numerical examples are given to confirm the theoretical results at last.
期刊介绍:
Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.