Structure preserving hybrid Finite Volume Finite Element method for compressible MHD
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913437270933504 |
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| author | Fambri, Francesco Sonnendrücker, Eric |
| author_facet | Fambri, Francesco Sonnendrücker, Eric |
| contents | In this manuscript we present a novel and efficient numerical method for the compressible viscous and resistive MHD equations for all Mach number regimes. The time-integration strategy is a semi-implicit splitting, combined with a hybrid finite-volume and finite-element (FE) discretization in space. The non-linear convection is solved by a robust explicit FV scheme, while the magneto-acoustic terms are treated implicitly in time. The resulting CFL stability condition depends only on the fluid velocity, and not on the Alfvénic and acoustic modes. The magneto-acoustic terms are discretized by compatible FE based on a continuous and a discrete de Rham complexes designed using Finite Element Exterior Calculus (FEEC). Thanks to the use of FEEC, energy stability, magnetic-helicity conservation and the divergence-free conditions can be preserved also at the discrete level. A very efficient splitting approach is used to separate the acoustic and the Alfvénic modes in such a fashion that the original symmetries of the PDE governing equations are preserved. In this way, the algorithm relies on the solution of linear, symmetric and positive-definite algebraic systems, that are very efficiently handled by the simple matrix-free conjugate-gradient method. The resulting algorithm showed to be robust and accurate in low and high Mach regimes even at large Courant numbers. Non-trivial tests are solved in one-, two- and three- space dimensions to confirm the robustness, accuracy, and the low-dissipative and conserving properties of the final algorithm. While the formulation of the method is very general, numerical results for a second-order accurate FV-FE scheme will be presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14141 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure preserving hybrid Finite Volume Finite Element method for compressible MHD Fambri, Francesco Sonnendrücker, Eric Numerical Analysis Instrumentation and Methods for Astrophysics Plasma Physics 65M60, 76M10, 85-08 (Primary) 65M12, 65M08, 76M12 (Secondary) G.1; J.2; F.2.1 In this manuscript we present a novel and efficient numerical method for the compressible viscous and resistive MHD equations for all Mach number regimes. The time-integration strategy is a semi-implicit splitting, combined with a hybrid finite-volume and finite-element (FE) discretization in space. The non-linear convection is solved by a robust explicit FV scheme, while the magneto-acoustic terms are treated implicitly in time. The resulting CFL stability condition depends only on the fluid velocity, and not on the Alfvénic and acoustic modes. The magneto-acoustic terms are discretized by compatible FE based on a continuous and a discrete de Rham complexes designed using Finite Element Exterior Calculus (FEEC). Thanks to the use of FEEC, energy stability, magnetic-helicity conservation and the divergence-free conditions can be preserved also at the discrete level. A very efficient splitting approach is used to separate the acoustic and the Alfvénic modes in such a fashion that the original symmetries of the PDE governing equations are preserved. In this way, the algorithm relies on the solution of linear, symmetric and positive-definite algebraic systems, that are very efficiently handled by the simple matrix-free conjugate-gradient method. The resulting algorithm showed to be robust and accurate in low and high Mach regimes even at large Courant numbers. Non-trivial tests are solved in one-, two- and three- space dimensions to confirm the robustness, accuracy, and the low-dissipative and conserving properties of the final algorithm. While the formulation of the method is very general, numerical results for a second-order accurate FV-FE scheme will be presented. |
| title | Structure preserving hybrid Finite Volume Finite Element method for compressible MHD |
| topic | Numerical Analysis Instrumentation and Methods for Astrophysics Plasma Physics 65M60, 76M10, 85-08 (Primary) 65M12, 65M08, 76M12 (Secondary) G.1; J.2; F.2.1 |
| url | https://arxiv.org/abs/2407.14141 |