Construction of weak solutions to compressible Navier–Stokes equations with general inflow/outflow boundary conditions via a numerical approximation
The construction of weak solutions to compressible Navier–Stokes equations via a numerical method (including a rigorous proof of the convergence) is in a short supply, and so far, available only for one sole numerical scheme suggested in Karper (Numer Math, 125(3):441–510, 2013) for the no slip boun...
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Veröffentlicht in: | Numerische Mathematik 2021-12, Vol.149 (4), p.717-778 |
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description | The construction of weak solutions to compressible Navier–Stokes equations via a numerical method (including a rigorous proof of the convergence) is in a short supply, and so far, available only for one sole numerical scheme suggested in Karper (Numer Math, 125(3):441–510, 2013) for the no slip boundary conditions and the isentropic pressure with adiabatic coefficient
γ
>
3
. Here we consider the same problem for the general non zero inflow–outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper. |
doi_str_mv | 10.1007/s00211-021-01237-0 |
format | Article |
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γ
>
3
. Here we consider the same problem for the general non zero inflow–outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper.</description><identifier>ISSN: 0029-599X</identifier><identifier>EISSN: 0945-3245</identifier><identifier>DOI: 10.1007/s00211-021-01237-0</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Boundary conditions ; Compressibility ; Differential geometry ; Fluid flow ; Fourier analysis ; Harmonic analysis ; Inflow ; Mathematical and Computational Engineering ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Navier-Stokes equations ; Numerical Analysis ; Numerical and Computational Physics ; Numerical methods ; Outflow ; Simulation ; Theoretical</subject><ispartof>Numerische Mathematik, 2021-12, Vol.149 (4), p.717-778</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-d4a7ac8a8b5ad6ea7edfc1e73c9d0dbc234abb6ca33bc02aee01965631a08bba3</citedby><cites>FETCH-LOGICAL-c319t-d4a7ac8a8b5ad6ea7edfc1e73c9d0dbc234abb6ca33bc02aee01965631a08bba3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00211-021-01237-0$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00211-021-01237-0$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Kwon, Young-Sam</creatorcontrib><creatorcontrib>Novotný, Antonin</creatorcontrib><title>Construction of weak solutions to compressible Navier–Stokes equations with general inflow/outflow boundary conditions via a numerical approximation</title><title>Numerische Mathematik</title><addtitle>Numer. Math</addtitle><description>The construction of weak solutions to compressible Navier–Stokes equations via a numerical method (including a rigorous proof of the convergence) is in a short supply, and so far, available only for one sole numerical scheme suggested in Karper (Numer Math, 125(3):441–510, 2013) for the no slip boundary conditions and the isentropic pressure with adiabatic coefficient
γ
>
3
. Here we consider the same problem for the general non zero inflow–outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper.</description><subject>Boundary conditions</subject><subject>Compressibility</subject><subject>Differential geometry</subject><subject>Fluid flow</subject><subject>Fourier analysis</subject><subject>Harmonic analysis</subject><subject>Inflow</subject><subject>Mathematical and Computational Engineering</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Navier-Stokes equations</subject><subject>Numerical Analysis</subject><subject>Numerical and Computational Physics</subject><subject>Numerical methods</subject><subject>Outflow</subject><subject>Simulation</subject><subject>Theoretical</subject><issn>0029-599X</issn><issn>0945-3245</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kMtKBDEQRRtR8PkDrgKuWyudfkwvZfAFogsV3IVKulqjPcmYdDvOzn8Q_EC_xIwtuHNTVRTn3ipukuxzOOQA1VEAyDhPY0mBZ6JKYS3ZgjovUpHlxXqcIavToq7vN5PtEJ4AeFXmfCv5nDobej_o3jjLXMsWhM8suG5YLQLrHdNuNvcUglEdsSt8NeS_3j9uevdMgdHLgCO5MP0jeyBLHjtmbNu5xZEb-lVnyg22Qb-MXrYxI_9qkCGzw4y80VGC87l3b2b2Y7ebbLTYBdr77TvJ3enJ7fQ8vbw-u5geX6Za8LpPmxwr1BOcqAKbkrCiptWcKqHrBhqlM5GjUqVGIZSGDImA12VRCo4wUQrFTnIw-sbbLwOFXj65wdt4UmYliLwoCl5FKhsp7V0Inlo59_FRv5Qc5Cp_OeYvY5E_-UuIIjGKQoTtA_k_639U3xijj8k</recordid><startdate>20211201</startdate><enddate>20211201</enddate><creator>Kwon, Young-Sam</creator><creator>Novotný, Antonin</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20211201</creationdate><title>Construction of weak solutions to compressible Navier–Stokes equations with general inflow/outflow boundary conditions via a numerical approximation</title><author>Kwon, Young-Sam ; Novotný, Antonin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-d4a7ac8a8b5ad6ea7edfc1e73c9d0dbc234abb6ca33bc02aee01965631a08bba3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Boundary conditions</topic><topic>Compressibility</topic><topic>Differential geometry</topic><topic>Fluid flow</topic><topic>Fourier analysis</topic><topic>Harmonic analysis</topic><topic>Inflow</topic><topic>Mathematical and Computational Engineering</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Navier-Stokes equations</topic><topic>Numerical Analysis</topic><topic>Numerical and Computational Physics</topic><topic>Numerical methods</topic><topic>Outflow</topic><topic>Simulation</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kwon, Young-Sam</creatorcontrib><creatorcontrib>Novotný, Antonin</creatorcontrib><collection>CrossRef</collection><jtitle>Numerische Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kwon, Young-Sam</au><au>Novotný, Antonin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Construction of weak solutions to compressible Navier–Stokes equations with general inflow/outflow boundary conditions via a numerical approximation</atitle><jtitle>Numerische Mathematik</jtitle><stitle>Numer. 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γ
>
3
. Here we consider the same problem for the general non zero inflow–outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00211-021-01237-0</doi><tpages>62</tpages></addata></record> |
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subjects | Boundary conditions Compressibility Differential geometry Fluid flow Fourier analysis Harmonic analysis Inflow Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Navier-Stokes equations Numerical Analysis Numerical and Computational Physics Numerical methods Outflow Simulation Theoretical |
title | Construction of weak solutions to compressible Navier–Stokes equations with general inflow/outflow boundary conditions via a numerical approximation |
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