{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:31:00Z","timestamp":1787329860894,"version":"build-2736575974"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the \u201cnonlocal pair-interaction model,\u201d which was recently introduced by Du, Huang, and LeFloch [ SIAM J. Numer. Anal., 55 (2017), pp. 2465--2489]. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and converges towards a weak solution. In fact, in contrast to local conservation laws, our second-order reconstruction-based method converges towards the unique entropy solution provided that the nonlocal interaction kernel satisfies a certain growth condition near zero. Furthermore, as the nonlocal horizon parameter in our method approaches zero we recover a well-known second-order method for local conservation laws. In addition, we answer several questions from the paper by Du, Huang, and LeFloch [ SIAM J. Numer. Anal., 55 (2017), pp. 2465--2489] concerning regularity of solutions. In particular, we prove that any discontinuity present in a weak solution must be stationary and that if the interaction kernel satisfies a certain growth condition, then weak solutions are unique. We present a series of numerical experiments in which we investigate the accuracy of our second-order scheme, demonstrate shock formation in the nonlocal pair-interaction model, and examine how the regularity of the solution depends on the choice of flux function.<\/jats:p>","DOI":"10.1137\/20m1360979","type":"journal-article","created":{"date-parts":[[2021,5,4]],"date-time":"2021-05-04T11:42:55Z","timestamp":1620128575000},"page":"1167-1194","source":"Crossref","is-referenced-by-count":4,"title":["Second-Order Accurate TVD Numerical Methods for Nonlocal Nonlinear Conservation Laws"],"prefix":"10.1137","volume":"59","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8528-5786","authenticated-orcid":true,"given":"Ulrik S.","family":"Fjordholm","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8058-7881","authenticated-orcid":true,"given":"Adrian M.","family":"Ruf","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,5,4]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-019-01375-8"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1980-0551288-3"},{"key":"atypb3","first-page":"1","volume":"37","author":"Q.","year":"2017","journal-title":"J. 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