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This model is formulated under the assumption of a variable cross-sectional area. A monotone numerical scheme to approximate solutions to this model is presented. The scheme is supported by three partial theoretical arguments. Firstly, it is proved that it satisfies an invariant-region property, i.e., the approximate volume fractions of the three phases, and their sum, stay between zero and one. Secondly, under the assumption of flow in a column with constant cross-sectional area it is shown that the scheme for the primary disperse phase converges to a suitably defined entropy solution. Thirdly, under the additional assumption of absence of flux discontinuities it is further demonstrated, by invoking arguments of compensated compactness, that the scheme for the secondary disperse phase converges to a weak solution of the corresponding conservation law. Numerical examples along with estimations of numerical error and convergence rates are presented for counter-current and co-current flows of the two disperse phases.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/nhm.2023006","type":"journal-article","created":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T13:28:06Z","timestamp":1669642086000},"page":"140-190","source":"Crossref","is-referenced-by-count":3,"title":["A difference scheme for a triangular system of conservation laws with discontinuous flux modeling three-phase flows"],"prefix":"10.3934","volume":"18","author":[{"given":"Raimund","family":"B\u00fcrger","sequence":"first","affiliation":[{"name":"CI<sup>2<\/sup>MA and Departamento de Ingenier\u00eda Matem\u00e1tica, Universidad de Concepci\u00f3n, Casilla 160-C, Concepci\u00f3n, Chile"}]},{"given":"Stefan","family":"Diehl","sequence":"additional","affiliation":[{"name":"Centre for Mathematical Sciences, Lund University, P.O. Box 118, S-221 00 Lund, Sweden"}]},{"given":"M. Carmen","family":"Mart\u00ed","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat de Val\u00e8ncia, Avda. Vicent Andr\u00e9s Estell\u00e9s s\/n, Burjassot, Val\u00e8ncia, Spain"}]},{"given":"Yolanda","family":"V\u00e1squez","sequence":"additional","affiliation":[{"name":"CI<sup>2<\/sup>MA and Departamento de Ingenier\u00eda Matem\u00e1tica, Universidad de Concepci\u00f3n, Casilla 160-C, Concepci\u00f3n, Chile"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2023006-1","doi-asserted-by":"publisher","unstructured":"Adimurthi, G. D. Veerappa Gowda, S. Mishra, Optimal entropy solutions for conservation laws with discontinuous flux, <i>J. Hyperbolic Differ. Equ.<\/i>, <b>2<\/b> (2005), 1\u201356. https:\/\/doi.org\/10.1142\/S0219891605000361","DOI":"10.1142\/S0219891605000361"},{"key":"key-10.3934\/nhm.2023006-2","doi-asserted-by":"publisher","unstructured":"B. Andreianov, K. H. Karlsen, N. 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