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Wang, H. Tang and K. Wu,\nHigh-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms,\nJ. Comput. Phys. 519 2024, Article ID 113451]. The semi-discrete schemes are constructed based on the Harten-Lax-van Leer-Contact (HLLC) flux with modified solution states, along with suitable discretization and decomposition of the source terms. The fully-discrete schemes obtained by the explicit strong-stability-preserving Rung-Kutta time discretizations (including the forward Euler) can be proved to maintain the WB property for a given known hydrostatic equilibrium state. A rigorous PP analysis for the fully-discrete schemes is provided, based on several key properties of the HLLC flux and the admissible state set as well as the geometric quasilinearlization (GQL) approach\n[K. Wu and C.-W. Shu,\nGeometric quasilinearization framework for analysis and design of bound-preserving schemes,\nSIAM Rev. 65 2023, 4, 1031\u20131073] and\n[K. Wu and H. Tang,\nAdmissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations,\nMath. Models Methods Appl. Sci. 27 2017, 10, 1871\u20131928]\ntransforming complex nonlinear constraints in the admissible state set into simple linear ones. Based on several newly introduced properties of the HLLC flux, we may not decompose the high-order schemes into a convex combination of the \u201cfirst-order schemes\u201d, which permits us to skip the proof of the PP property for the first-order scheme and directly analyze the high-order schemes. Consequently, the present PP analysis is more simple and direct compared to that in Wang, Tang and Wu (2024).\nSeveral numerical experiments validate the high-order accuracy, WB and PP properties as well as the high resolution of the proposed schemes.<\/jats:p>","DOI":"10.1515\/cmam-2024-0198","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T21:00:13Z","timestamp":1748638813000},"page":"709-740","source":"Crossref","is-referenced-by-count":1,"title":["High-Order Accurate Structure-Preserving Finite Volume Scheme for Ten-Moment Gaussian Closure Equations with Source Terms: Positivity and Well-Balancedness"],"prefix":"10.1515","volume":"25","author":[{"given":"Zhihao","family":"Zhang","sequence":"first","affiliation":[{"name":"Center for Applied Physics and Technology, HEDPS and LMAM , School of Mathematical Sciences , 12465 Peking University , Beijing , 100871 , P. R. China"}]},{"given":"Jiangfu","family":"Wang","sequence":"additional","affiliation":[{"name":"Center for Applied Physics and Technology, HEDPS and LMAM , School of Mathematical Sciences , 12465 Peking University , Beijing , 100871 , P. R. China"}]},{"given":"Huazhong","family":"Tang","sequence":"additional","affiliation":[{"name":"Center for Applied Physics and Technology, HEDPS and LMAM , School of Mathematical Sciences , 12465 Peking University , Beijing , 100871 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"2025070217063922223_j_cmam-2024-0198_ref_001","doi-asserted-by":"crossref","unstructured":"E.  Audusse, F.  Bouchut, M.-O.  Bristeau, R.  Klein and B.  Perthame,\nA fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows,\nSIAM J. Sci. 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