{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T06:02:07Z","timestamp":1762408927605,"version":"build-2065373602"},"reference-count":0,"publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Port. Math."],"published-print":{"date-parts":[[2013,7,5]]},"abstract":"<jats:p>\n                    We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in\n                    <jats:inline-formula>\n                      <jats:tex-math>L^2<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    ), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.\n                  <\/jats:p>","DOI":"10.4171\/pm\/1924","type":"journal-article","created":{"date-parts":[[2013,7,5]],"date-time":"2013-07-05T17:45:01Z","timestamp":1373046301000},"page":"23-50","source":"Crossref","is-referenced-by-count":6,"title":["Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data"],"prefix":"10.4171","volume":"70","author":[{"given":"Paulo","family":"Amorim","sequence":"first","affiliation":[{"name":"Universidade de Lisboa, Portugal"}]},{"given":"M\u00e1rio","family":"Figueira","sequence":"additional","affiliation":[{"name":"Universidade de Lisboa, Portugal"}]}],"member":"2673","container-title":["Portugaliae Mathematica"],"original-title":[],"link":[{"URL":"http:\/\/www.ems-ph.org\/fulltext\/10.4171\/PM\/1924","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T12:05:26Z","timestamp":1761825926000},"score":1,"resource":{"primary":{"URL":"https:\/\/ems.press\/doi\/10.4171\/pm\/1924"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,7,5]]},"references-count":0,"journal-issue":{"issue":"1"},"URL":"https:\/\/doi.org\/10.4171\/pm\/1924","relation":{},"ISSN":["0032-5155","1662-2758"],"issn-type":[{"type":"print","value":"0032-5155"},{"type":"electronic","value":"1662-2758"}],"subject":[],"published":{"date-parts":[[2013,7,5]]}}}