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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.08913 |
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| _version_ | 1866911102894342144 |
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| author | Yan, Ruifang Cao, Huihui Wu, Kailiang |
| author_facet | Yan, Ruifang Cao, Huihui Wu, Kailiang |
| contents | This paper proposes a numerical method, termed PosDiv-CDG, that provably preserves both positivity and the globally divergence-free (DF) condition at arbitrarily high order in multiple dimensions. It resolves the fundamental structural incompatibility between standard positivity-preserving limiters and global DF enforcement in the central discontinuous Galerkin (CDG) framework. The method integrates a novel positivity-limiting strategy, a modified dissipation mechanism guided by convex decomposition, and an auxiliary evolution equation for the magnetic field, which are designed based on rigorous theoretical analysis. Notably, we provide a rigorous proof of positivity preservation for the updated cell averages under an explicit CFL-type condition. The proof leverages the geometric quasi-linearization (GQL) technique, which reformulates the nonlinear positivity constraint into an equivalent linear form. This enables the derivation of flux-based inequalities and technical estimates under the global DF constraint. To suppress nonphysical oscillations near shocks, we develop a compact, non-intrusive convex-oscillation-suppressing (COS) procedure based on the entropy function. The COS process acts only on non-magnetic variables, avoids costly characteristic decomposition, and maintains both the globally DF property and high-order accuracy. Several challenging experiments -- including low plasma-beta MHD jets with Mach numbers up to 1,000,000 -- demonstrate the proposed method robustness, high-order accuracy, non-oscillatory behavior, and its ability to preserve both positivity and globally DF structures under extreme conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Provably positivity-preserving, globally divergence-free central DG methods for ideal MHD system Yan, Ruifang Cao, Huihui Wu, Kailiang Numerical Analysis This paper proposes a numerical method, termed PosDiv-CDG, that provably preserves both positivity and the globally divergence-free (DF) condition at arbitrarily high order in multiple dimensions. It resolves the fundamental structural incompatibility between standard positivity-preserving limiters and global DF enforcement in the central discontinuous Galerkin (CDG) framework. The method integrates a novel positivity-limiting strategy, a modified dissipation mechanism guided by convex decomposition, and an auxiliary evolution equation for the magnetic field, which are designed based on rigorous theoretical analysis. Notably, we provide a rigorous proof of positivity preservation for the updated cell averages under an explicit CFL-type condition. The proof leverages the geometric quasi-linearization (GQL) technique, which reformulates the nonlinear positivity constraint into an equivalent linear form. This enables the derivation of flux-based inequalities and technical estimates under the global DF constraint. To suppress nonphysical oscillations near shocks, we develop a compact, non-intrusive convex-oscillation-suppressing (COS) procedure based on the entropy function. The COS process acts only on non-magnetic variables, avoids costly characteristic decomposition, and maintains both the globally DF property and high-order accuracy. Several challenging experiments -- including low plasma-beta MHD jets with Mach numbers up to 1,000,000 -- demonstrate the proposed method robustness, high-order accuracy, non-oscillatory behavior, and its ability to preserve both positivity and globally DF structures under extreme conditions. |
| title | Provably positivity-preserving, globally divergence-free central DG methods for ideal MHD system |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2508.08913 |