{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:44:18Z","timestamp":1787334258732,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>\n                    We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point is subject to a constraint: it must stay inside a closed set K with boundary of class C\n                    <jats:sup>3<\/jats:sup>\n                    . We assume that, at impact, the tangential component of the impulsion is conserved, while its normal coordinate is reflected and multiplied by a given coefficient of restitution $e\\in[0,1]$: the mechanically relevant notion of orthogonality is defined in terms of the local metric for the impulsions (local cotangent metric). We define a numerical scheme which enables us to approximate the solutions of the Cauchy problem: this is a generalization of the scheme presented in the companion paper [L. Paoli and M. Schatzman, SIAM J. Numer. Anal., 40 (2002), pp. 702--733]. We prove the convergence of this numerical scheme to a solution, which also yields an existence result. Without any a priori estimates, the convergence and the existence are local; with some a priori estimates, the convergence and the existence are proved on intervals depending exclusively on these estimates. The technique of proof uses a localization of the scheme close to the boundary of K; this idea is classical for a differential system studied in the framework of flows of a vector field. It is much more difficult to implement here because finite differences schemes are only approximately local: straightening the boundary creates quadratic terms which cause all the difficulties of the proof.\n                  <\/jats:p>","DOI":"10.1137\/s003614290037873x","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"734-768","source":"Crossref","is-referenced-by-count":66,"title":["A Numerical Scheme for Impact Problems II: The Multidimensional Case"],"prefix":"10.1137","volume":"40","author":[{"given":"Laetitia","family":"Paoli","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michelle","family":"Schatzman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","first-page":"953","volume":"327","author":"Ballard P.","year":"1999","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. IIb"},{"key":"R2","first-page":"271","volume":"29","author":"Bressan Aldo","year":"1959","journal-title":"Rend. Sem. Mat. Univ. Padova"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(83)90035-9"},{"key":"R4","first-page":"87","volume":"1","author":"Carriero Michele","year":"1982","journal-title":"Boll. Un. Mat. Ital. A (6)"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(98)80193-6"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/S0997-7538(98)80007-7"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"M. Panet, L. Paoli, and M. Schatzman,\n                      Theoretical and numerical study for a model of vibrations with unilateral constraints\n                      , in Contact Mechanics, M. Raous, M. Jean, and J.\u2010J. Moreau, eds., Plenum Press, New York, 1995, pp. 457\u2013464.","DOI":"10.1007\/978-1-4615-1983-6_63"},{"key":"R8","unstructured":"L. Paoli,\n                      Analyse num\u00e9rique de vibrations avec contraintes unilat\u00e9rales\n                      , Ph.D. thesis, Universit\u00e9 Claude Bernard\u2014Lyon 1, Lyon, France, 1993."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142900378728"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1993270606731"},{"key":"R11","first-page":"211","volume":"317","author":"Paoli Laetitia","year":"1993","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"key":"R12","unstructured":"L. Paoli and M. Schatzman,\n                      Dynamics of an impacting bar: a model and numerics\n                      , in Proceedings of the European Congress on Computational Mechanics, W. Wunderlich, ed., CD\u2010ROM, Munich, 1999."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(85)90105-6"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01769220"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(91)90150-8"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(78)90022-6"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/S0895-7177(98)00104-6"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1115\/1.2788865"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S003614290037873X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:55:30Z","timestamp":1787327730000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S003614290037873X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2002,1]]}},"alternative-id":["10.1137\/S003614290037873X"],"URL":"https:\/\/doi.org\/10.1137\/s003614290037873x","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1]]}}}