Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914483438354432 |
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| author | Giesselmann, Jan Öffner, Philipp Sauerborn, Robert |
| author_facet | Giesselmann, Jan Öffner, Philipp Sauerborn, Robert |
| contents | We propose a concept of dissipative weak (DW) solutions for the Navier-Stokes-Korteweg (NSK) system and prove conditional convergence of a structure-preserving finite volume scheme towards such a solution. DW solutions provide a generalized solution concept in computational fluid dynamics and have recently attracted significant attention. They provide an extension of the famous Lax Equivalence Theorem to nonlinear problems, i.e. consistency and stability of a numerical scheme imply convergence. Our work builds on recent advances where convergence towards DW solutions of structure-preserving schemes has been established for the Euler and Navier-Stokes equations. Indeed, we prove convergence of a recently proposed FV scheme by leveraging its conservation and dissipation properties as well as its consistency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16110 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions Giesselmann, Jan Öffner, Philipp Sauerborn, Robert Numerical Analysis 76M12, 65M08, 65M12, 35Q35, 35G50 We propose a concept of dissipative weak (DW) solutions for the Navier-Stokes-Korteweg (NSK) system and prove conditional convergence of a structure-preserving finite volume scheme towards such a solution. DW solutions provide a generalized solution concept in computational fluid dynamics and have recently attracted significant attention. They provide an extension of the famous Lax Equivalence Theorem to nonlinear problems, i.e. consistency and stability of a numerical scheme imply convergence. Our work builds on recent advances where convergence towards DW solutions of structure-preserving schemes has been established for the Euler and Navier-Stokes equations. Indeed, we prove convergence of a recently proposed FV scheme by leveraging its conservation and dissipation properties as well as its consistency. |
| title | Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions |
| topic | Numerical Analysis 76M12, 65M08, 65M12, 35Q35, 35G50 |
| url | https://arxiv.org/abs/2604.16110 |