Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917223729201152 |
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| author | Giesselmann, Jan Keim, Jens Leotta, Fabio Rohde, Christian |
| author_facet | Giesselmann, Jan Keim, Jens Leotta, Fabio Rohde, Christian |
| contents | The Navier-Stokes-Cahn-Hilliard (NSCH) system governs the diffuse-interface dynamics of two incompressible and immiscible fluids. We consider a relaxation approximation of the NSCH system that is composed by a system of first-order hyperbolic balance laws and second-order elliptic operators. We prove first that the solutions of an initial boundary value problem for the approximation recover the limiting NSCH system for vanishing relaxation parameters. To cope with the singular limit we exploit the fact that the approximate solutions dissipate an almost quadratic energy, and employ the relative entropy-framework. In the second part of the work we provide numerical evidence for the analytical results, even in flow regimes not covered by the assumptions needed for the theoretical results. Using a novel marker-and-cell conservative finite-difference approach for both the approximation and the limit system, we are able to compute physically relevant interfacial flow problems including Ostwald ripening and high-velocity flow. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18463 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System Giesselmann, Jan Keim, Jens Leotta, Fabio Rohde, Christian Numerical Analysis Analysis of PDEs 35L03, 35M11, 65M12, 35Q30 The Navier-Stokes-Cahn-Hilliard (NSCH) system governs the diffuse-interface dynamics of two incompressible and immiscible fluids. We consider a relaxation approximation of the NSCH system that is composed by a system of first-order hyperbolic balance laws and second-order elliptic operators. We prove first that the solutions of an initial boundary value problem for the approximation recover the limiting NSCH system for vanishing relaxation parameters. To cope with the singular limit we exploit the fact that the approximate solutions dissipate an almost quadratic energy, and employ the relative entropy-framework. In the second part of the work we provide numerical evidence for the analytical results, even in flow regimes not covered by the assumptions needed for the theoretical results. Using a novel marker-and-cell conservative finite-difference approach for both the approximation and the limit system, we are able to compute physically relevant interfacial flow problems including Ostwald ripening and high-velocity flow. |
| title | Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System |
| topic | Numerical Analysis Analysis of PDEs 35L03, 35M11, 65M12, 35Q30 |
| url | https://arxiv.org/abs/2601.18463 |