{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T19:42:52Z","timestamp":1793130172649,"version":"4.5.0"},"reference-count":15,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We establish a convergence rate of the renormalized meshfree numerical scheme applied to scalar conservation laws. Renormalization is a tool introduced in order to eliminate the smoothed particle hydrodynamics (SPH) lack of consistency. A conservative scheme, the weak renormalized scheme, is derived from the general conservation laws weak formulation and is time discretized by using finite volume techniques. Because of the new form of the derivative approximation, the convergence proof of the classical SPH method cannot be applied. Thus, we use a Kruskov technique to obtain an $L^1$ norm comparison between the approximate solution and the solution of a regularized conservation law, the pseudoviscous problem.<\/jats:p>","DOI":"10.1137\/s003614290444739x","type":"journal-article","created":{"date-parts":[[2008,4,18]],"date-time":"2008-04-18T18:06:43Z","timestamp":1208542003000},"page":"1935-1964","source":"Crossref","is-referenced-by-count":44,"title":["Renormalized Meshfree Schemes II: Convergence for Scalar Conservation Laws"],"prefix":"10.1137","volume":"46","author":[{"given":"Nathalie","family":"Lanson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Paul","family":"Vila","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,18]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0732036"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996307119"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-98-02204-1"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0732032"},{"key":"R5","unstructured":"R. Eymard, T. Gallouet, and R. Herbin,\n                      The finite volume method\n                      , in Handbook of Numerical Analysis, Ph. Ciarlet and J. L. Lions eds., North-Holland, Amsterdam, 2000."},{"key":"R6","unstructured":"E. Godlewski and P. A. Raviart,\n                      Hyperbolic systems of conservation laws\n                      , SMAI Math. Appl., 3\/4 (1991)."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1070\/SM1970v010n02ABEH002156"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(76)90046-X"},{"key":"R9","unstructured":"N. Lanson,\n                      Etude des M\u00e9thodes Particulaires Renormalis\u00e9es. Applications aux Prob\u00e8mes de Dynamique Rapide\n                      , Ph.D. thesis, INSA-Universit\u00e9 Paul Sabatier, Toulouse, France, 2001; also available online from www.math.uwaterloo.ca\/~\u02dcnlanson\/thesis1.pdf."},{"key":"R10","unstructured":"N. Lanson and J. P. Vila,\n                      Renormalized meshfree schemes: Consistency, stability and hybrid methods for conservation laws\n                      , SIAM J. Numer. Anal., submitted."},{"key":"R11","unstructured":"A. Leroux,\n                      Approximation de Quelques Probl\u00e8mes Hyperboliques non Lin\u00e9aires\n                      , Ph.D. thesis, Universit\u00e9 de Rennes, France, 1979."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0721016"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"J. M\u00e1lek, J. Necas, M. Rokyta, and M. Ruzicka,\n                      Weak and Measure Valued Solutions to Evolutionary PDE's\n                      , Applied Mathematics and Mathematical Computation 13, Chapman and Hall, London, 1996.","DOI":"10.1007\/978-1-4899-6824-1"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1984-0758189-X"},{"key":"R15","unstructured":"J. P. Vila,\n                      Weighted particle-finite volume hybrid schemes\n                      , in Finite Volume for Complex Applications, Benkhaldoun-Vilsmeir, eds., HERMES, London, 1996."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S003614290444739X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:11Z","timestamp":1787325251000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S003614290444739X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":15,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/S003614290444739X"],"URL":"https:\/\/doi.org\/10.1137\/s003614290444739x","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}