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We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.<\/jats:p>","DOI":"10.1515\/jnma-2023-0065","type":"journal-article","created":{"date-parts":[[2024,3,25]],"date-time":"2024-03-25T12:41:17Z","timestamp":1711370477000},"page":"275-299","source":"Crossref","is-referenced-by-count":1,"title":["A fully well-balanced hydrodynamic reconstruction"],"prefix":"10.1515","volume":"32","author":[{"given":"Christophe","family":"Berthon","sequence":"first","affiliation":[{"name":"Universit\u00e9 de Nantes, CNRS UMR 6629, Laboratoire de Math\u00e9matiques Jean Leray , 2 rue de la Houssini\u00e8re, BP 92208 , Nantes , France"}]},{"given":"Victor","family":"Michel-Dansac","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Strasbourg , CNRS, Inria, IRMA , Strasbourg , France"}]}],"member":"374","published-online":{"date-parts":[[2024,3,25]]},"reference":[{"key":"2024100712341967360_j_jnma-2023-0065_ref_001","unstructured":"M. Abramowitz and I. A. 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