An Energy-Stable Discontinuous Galerkin Method for the Compressible Navier--Stokes--Allen--Cahn System
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917045399977984 |
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| author | Ostrowski, Lukas Rohde, Christian |
| author_facet | Ostrowski, Lukas Rohde, Christian |
| contents | We consider a Navier--Stokes--Allen--Cahn (NSAC) system that governs the compressible motion of a viscous, immiscible two-phase fluid at constant temperature. Weak solutions of the NSAC system dissipate an appropriate energy functional. Based on an equivalent re-formulation of the NSAC system we propose a fully-discrete discontinuous Galerkin (dG) discretization that is mass-conservative, energy-stable, and provides higher-order accuracy in space and second-order accuracy in time. The approach relies on the approach in \cite{Giesselmann2015a} and a special splitting discretization of the derivatives of the free energy function within the Crank-Nicolson time-stepping. Numerical experiments confirm the analytical statements and show the applicability of the approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_23289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Energy-Stable Discontinuous Galerkin Method for the Compressible Navier--Stokes--Allen--Cahn System Ostrowski, Lukas Rohde, Christian Numerical Analysis 65M60 76T10 35Q30 We consider a Navier--Stokes--Allen--Cahn (NSAC) system that governs the compressible motion of a viscous, immiscible two-phase fluid at constant temperature. Weak solutions of the NSAC system dissipate an appropriate energy functional. Based on an equivalent re-formulation of the NSAC system we propose a fully-discrete discontinuous Galerkin (dG) discretization that is mass-conservative, energy-stable, and provides higher-order accuracy in space and second-order accuracy in time. The approach relies on the approach in \cite{Giesselmann2015a} and a special splitting discretization of the derivatives of the free energy function within the Crank-Nicolson time-stepping. Numerical experiments confirm the analytical statements and show the applicability of the approach. |
| title | An Energy-Stable Discontinuous Galerkin Method for the Compressible Navier--Stokes--Allen--Cahn System |
| topic | Numerical Analysis 65M60 76T10 35Q30 |
| url | https://arxiv.org/abs/2510.23289 |