{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T18:10:58Z","timestamp":1778436658663,"version":"3.51.4"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,6,6]],"date-time":"2007-06-06T00:00:00Z","timestamp":1181088000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We propose a numerical scheme to approximate the weak solutions of the 10-moment Gaussian closure. The moment Gaussian closure for gas dynamics is governed by a conservative hyperbolic system supplemented by entropy inequalities whose solutions satisfy positiveness of density and tensorial pressure. We consider a Suliciu-type relaxation numerical scheme to approximate the solutions. These methods are proved to satisfy all the expected positiveness properties and all the discrete entropy inequalities. The scheme is illustrated by several numerical experiments.<\/p>","DOI":"10.1090\/s0025-5718-06-01860-6","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"1809-1831","source":"Crossref","is-referenced-by-count":36,"title":["Numerical approximations of the 10-moment Gaussian closure"],"prefix":"10.1090","volume":"75","author":[{"given":"Christophe","family":"Berthon","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,6,6]]},"reference":[{"issue":"1-2","key":"1","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1080\/00036819608840453","article-title":"Convergence of relaxation schemes for conservation laws","volume":"61","author":"Aregba-Driollet, Denise","year":"1996","journal-title":"Appl. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-6811","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/s00211-004-0558-1","article-title":"A relaxation method for two-phase flow models with hydrodynamic closure law","volume":"99","author":"Baudin, Micha\u00ebl","year":"2005","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/j.crma.2004.10.008","article-title":"In\u00e9galit\u00e9s d\u2019entropie pour un sch\u00e9ma de relaxation","volume":"340","author":"Berthon, Christophe","year":"2005","journal-title":"C. R. Math. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/1631-073X","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"C. Berthon, B. Braconnier, and B. Nkonga, Numerical approximation of a degenerated non conservative multifluid model: relaxation scheme, Int. J. Numer. Methods Fluids, 48 (2005), No.1, 85\u201390.","DOI":"10.1002\/fld.933"},{"key":"5","doi-asserted-by":"crossref","unstructured":"C. Berthon, M. Breuss, and M.-O. Titeux, A relaxation scheme for the approximation of the pressureless Euler equations, Numer. Methods Partial Diff. Eq., 22 (2006), 484\u2013505.","DOI":"10.1002\/num.20108"},{"key":"6","unstructured":"C. Berthon and B. Dubroca, work in preparation."},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"623","DOI":"10.1007\/s00211-002-0426-9","article-title":"Entropy satisfying flux vector splittings and kinetic BGK models","volume":"94","author":"Bouchut, F.","year":"2003","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"8","unstructured":"C. Chalons and F. Coquel, Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes, preprint."},{"issue":"6","key":"9","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1002\/cpa.3160470602","article-title":"Hyperbolic conservation laws with stiff relaxation terms and entropy","volume":"47","author":"Chen, Gui Qiang","year":"1994","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"key":"10","isbn-type":"print","first-page":"179","article-title":"Some new Godunov and relaxation methods for two-phase flow problems","author":"Coquel, F.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0306466015"},{"issue":"6","key":"11","doi-asserted-by":"publisher","first-page":"2223","DOI":"10.1137\/S0036142997318528","article-title":"Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics","volume":"35","author":"Coquel, Fr\u00e9d\u00e9ric","year":"1998","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"12","unstructured":"B. Dubroca, M. Tchong, P. Charrier, V.T. Tikhonchuk, and P.-M. Morreeuw, Magnetic field generation in plasma due to anisotropic laser heating, J. Phys. Plasma, accepted."},{"key":"13","unstructured":"T. I. Gombosi, C. P. T. Groth, P. L. Roe, and S. L. Brown, 35-Moment closure for rarefied gases: Derivation, transport equations, and wave structure, preprint."},{"key":"14","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0713-9","volume-title":"Numerical approximation of hyperbolic systems of conservation laws","volume":"118","author":"Godlewski, Edwige","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0387945296"},{"key":"15","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1002\/cpa.3160020403","article-title":"On the kinetic theory of rarefied gases","volume":"2","author":"Grad, Harold","year":"1949","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1137\/1025002","article-title":"On upstream differencing and Godunov-type schemes for hyperbolic conservation laws","volume":"25","author":"Harten, Amiram","year":"1983","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"3","key":"17","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1002\/cpa.3160480303","article-title":"The relaxation schemes for systems of conservation laws in arbitrary space dimensions","volume":"48","author":"Jin, Shi","year":"1995","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"key":"18","series-title":"Cambridge Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511791253","volume-title":"Finite volume methods for hyperbolic problems","author":"LeVeque, Randall J.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0521810876"},{"issue":"5-6","key":"19","doi-asserted-by":"publisher","first-page":"1021","DOI":"10.1007\/BF02179552","article-title":"Moment closure hierarchies for kinetic theories","volume":"83","author":"Levermore, C. David","year":"1996","journal-title":"J. Statist. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4715","issn-type":"print"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1137\/S0036139996299236","article-title":"The Gaussian moment closure for gas dynamics","volume":"59","author":"Levermore, C. David","year":"1999","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"995","DOI":"10.1137\/S1064827502402120","article-title":"Comparison of several difference schemes on 1D and 2D test problems for the Euler equations","volume":"25","author":"Liska, Richard","year":"2003","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"22","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/BF01210707","article-title":"Hyperbolic conservation laws with relaxation","volume":"108","author":"Liu, Tai-Ping","year":"1987","journal-title":"Comm. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3616","issn-type":"print"},{"issue":"8","key":"23","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1002\/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3","article-title":"Convergence to equilibrium for the relaxation approximations of conservation laws","volume":"49","author":"Natalini, Roberto","year":"1996","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"key":"24","doi-asserted-by":"crossref","unstructured":"W.F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. Comput. Phys., 72 (1987), 78\u2013120.","DOI":"10.1016\/0021-9991(87)90074-X"},{"key":"25","doi-asserted-by":"crossref","unstructured":"I. Suliciu, Energy estimates in rate-type thermo-viscoplasticity, Int. J. Plast., 14 (1998), 227\u2013244.","DOI":"10.1016\/S0749-6419(97)00053-3"},{"key":"26","series-title":"Pure and Applied Mathematics","volume-title":"Linear and nonlinear waves","author":"Whitham, G. B.","year":"1974"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01860-6\/S0025-5718-06-01860-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01860-6\/S0025-5718-06-01860-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:40:50Z","timestamp":1776782450000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01860-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,6,6]]},"references-count":26,"journal-issue":{"issue":"256","published-print":{"date-parts":[[2006,10]]}},"alternative-id":["S0025-5718-06-01860-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01860-6","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2006,6,6]]}}}