{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:45:06Z","timestamp":1776786306791,"version":"3.51.2"},"reference-count":28,"publisher":"American Mathematical Society (AMS)","issue":"259","license":[{"start":{"date-parts":[[2008,1,25]],"date-time":"2008-01-25T00:00:00Z","timestamp":1201219200000},"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>\n                    We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case single conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these classes of equations based on a two-step approach. In the first step, an interface connection vector is used to define infinite classes of entropy solutions. We show that each of these classes of solutions is stable in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This allows for the possibility of choosing one of these classes of solutions based on the physics of the problem. In the second step, we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient. This method leads to optimal entropy solutions that are consistent with physically observed solutions in two-phase flows in heterogeneous porous media. Another central aim of this paper is to develop suitable numerical schemes for these equations. We develop and analyze a set of Godunov type finite volume methods that are based on exact solutions of the corresponding Riemann problem. Numerical experiments are shown comparing the performance of these schemes on a set of test problems.\n                  <\/p>","DOI":"10.1090\/s0025-5718-07-01960-6","type":"journal-article","created":{"date-parts":[[2008,3,21]],"date-time":"2008-03-21T10:50:03Z","timestamp":1206096603000},"page":"1219-1242","source":"Crossref","is-referenced-by-count":16,"title":["Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function"],"prefix":"10.1090","volume":"76","author":[{"family":"Adimurthi","sequence":"first","affiliation":[]},{"given":"Siddhartha","family":"Mishra","sequence":"additional","affiliation":[]},{"given":"G.","family":"Gowda","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2007,1,25]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1215\/kjm\/1250283740","article-title":"Conservation law with discontinuous flux","volume":"43","author":"Adimurthi","year":"2003","journal-title":"J. Math. Kyoto Univ.","ISSN":"https:\/\/id.crossref.org\/issn\/0023-608X","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1137\/S003614290139562X","article-title":"Godunov-type methods for conservation laws with a flux function discontinuous in space","volume":"42","author":"Adimurthi","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1142\/S0219891605000622","article-title":"Optimal entropy solutions for conservation laws with discontinuous flux-functions","volume":"2","author":"Adimurthi","year":"2005","journal-title":"J. Hyperbolic Differ. Equ.","ISSN":"https:\/\/id.crossref.org\/issn\/0219-8916","issn-type":"print"},{"key":"4","unstructured":"Adimurthi, Siddhartha Mishra and G. D. Veerappa Gowda, Conservation laws with flux function discontinuous in the space variable - II, Discontinuous convex-concave type fluxes and generalised entropy solutions, To appear in J. Comp. Appl. Math."},{"key":"5","unstructured":"Adimurthi, Siddhartha Mishra and G. D. Veerappa Gowda, Conservation laws with flux function discontinuous in the space variable - III, The general case, Preprint."},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/s00211-003-0503-8","article-title":"Well-posedness in \ud835\udc35\ud835\udc49_{\ud835\udc61} and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units","volume":"97","author":"B\u00fcrger, R.","year":"2004","journal-title":"Numer. 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