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Performing a careful validation in a battery of relevant test cases, we show that these schemes can accurately analyze the stability properties of stationary states in challenging problems such as phase transitions in collective behavior, generalized Euler--Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories.<\/jats:p>","DOI":"10.1137\/18m1230050","type":"journal-article","created":{"date-parts":[[2020,3,30]],"date-time":"2020-03-30T15:35:51Z","timestamp":1585582551000},"page":"502-541","source":"Crossref","is-referenced-by-count":11,"title":["Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations with General Free Energy"],"prefix":"10.1137","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8819-4660","authenticated-orcid":true,"given":"Jos\u00e9 A.","family":"Carrillo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Serafim","family":"Kalliadasis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8485-609X","authenticated-orcid":true,"given":"Sergio P.","family":"Perez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chi-Wang","family":"Shu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,3,30]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503431090"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.12.016"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/15M1043637"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1142\/S021820251550044X"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7930(94)90004-3"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"F. 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