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Hyper. Differential Equations"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:p> We study the Riemann problem for a new model on immiscible vertical two-phase flow under point injection. The point injection is modeled by a Dirac [Formula: see text]-source term as well as by a spatially discontinuous flux function, which defines two fluxes, one on each side of the discontinuity. The solutions comprise up to three wave groups: downward waves, a stationary shock and upward waves. Owning the interplay between the Dirac [Formula: see text]-source and the discontinuous flux, there is no standard entropy condition for the stationary shock (flux\u2019s connections). An entropy condition was deduced based on impinging characteristics and perturbation of the constant solution. This condition leads to shocks that do not satisfy the classical Lax\u2019s conditions \u2014 even for arbitrarily small shocks \u2014 and may also have no viscous profile. The Rankine\u2013Hugoniot condition \u2014 embedding the Dirac [Formula: see text]-source \u2014 and the entropy condition are geometrically represented by \u201cFlux Projections\u201d that support the analytical method proposed in this paper. We then obtain a [Formula: see text] analytic solution for all initial value problems. We verify the entropy condition using a Lagrangian\u2013Eulerian scheme recently introduced in the literature, which is based in the new concept of no-flow curves. Analytic and numerical solutions fit. <\/jats:p>","DOI":"10.1142\/s0219891624500012","type":"journal-article","created":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T06:00:49Z","timestamp":1718085649000},"page":"1-32","source":"Crossref","is-referenced-by-count":6,"title":["Riemann problem solutions for a balance law under Dirac-Delta source with a discontinuous flux"],"prefix":"10.1142","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1979-3082","authenticated-orcid":false,"given":"Eduardo","family":"Abreu","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific, Computing\u2013IMECC, University of Campinas\u2013UNICAMP, S\u00e9rgio Buarque de Holanda Street, 651, 13083-859 Campinas, S\u00e3o Paulo, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4252-3825","authenticated-orcid":false,"given":"Vitor","family":"Matos","sequence":"additional","affiliation":[{"name":"Centro de Matematica, Faculdade de Economia, Universidade do Porto, Rua Dr Roberto Frias, 4200-464 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6322-4103","authenticated-orcid":false,"given":"John","family":"P\u00e9rez","sequence":"additional","affiliation":[{"name":"ITM-Institucion Universitaria, Medellin, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0901-106X","authenticated-orcid":false,"given":"Panters","family":"Rodr\u00edguez-Berm\u00fadez","sequence":"additional","affiliation":[{"name":"Fluminense Federal University, Avenida dos Trabalhadores, 420, 27255-125 Volta Redonda, Rio de Janeiro, Brazil"}]}],"member":"219","published-online":{"date-parts":[[2024,6,7]]},"reference":[{"key":"S0219891624500012BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-023-10283-1"},{"key":"S0219891624500012BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-019-00923-4"},{"key":"S0219891624500012BIB003","doi-asserted-by":"publisher","DOI":"10.1137\/20M1320250"},{"key":"S0219891624500012BIB004","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2016.12.012"},{"key":"S0219891624500012BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-020-01392-w"},{"key":"S0219891624500012BIB006","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2018.12.019"},{"key":"S0219891624500012BIB007","doi-asserted-by":"publisher","DOI":"10.1137\/S003614290139562X"},{"key":"S0219891624500012BIB008","doi-asserted-by":"publisher","DOI":"10.1051\/cocv\/2020044"},{"key":"S0219891624500012BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-010-0389-4"},{"key":"S0219891624500012BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2014.08.002"},{"key":"S0219891624500012BIB011","doi-asserted-by":"publisher","DOI":"10.1007\/s00030-023-00857-9"},{"key":"S0219891624500012BIB012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1988-0964856-0"},{"key":"S0219891624500012BIB013","doi-asserted-by":"publisher","DOI":"10.2118\/942107-G"},{"key":"S0219891624500012BIB014","doi-asserted-by":"publisher","DOI":"10.1137\/07069314X"},{"key":"S0219891624500012BIB015","first-page":"1411","volume":"38","author":"Cheng S.","year":"2017","journal-title":"Acta Pet. 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