{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T10:44:02Z","timestamp":1709203442447},"reference-count":19,"publisher":"University of Zielona G\u00f3ra, Poland","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2007,10,1]]},"abstract":"<jats:title>A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation<\/jats:title><jats:p>This article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1 + 1)-dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be said<jats:italic>optimal<\/jats:italic>in the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter \u03b5 which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge in<jats:italic>L<\/jats:italic><jats:sup>\u221e<\/jats:sup>towards the exact ones as \u03b5 and \u03b4<jats:italic>t<\/jats:italic>tend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.<\/jats:p>","DOI":"10.2478\/v10006-007-0029-9","type":"journal-article","created":{"date-parts":[[2007,10,12]],"date-time":"2007-10-12T14:22:52Z","timestamp":1192198972000},"page":"351-359","source":"Crossref","is-referenced-by-count":2,"title":["A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation"],"prefix":"10.61822","volume":"17","author":[{"given":"Martin","family":"Pinto","sequence":"first","affiliation":[]}],"member":"37438","reference":[{"key":"1","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1137\/S0036142902410775","article-title":"Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system","volume":"1","author":"N Besse","year":"2004","journal-title":"SIAM Journal on Numerical Analysis"},{"key":"2","first-page":"437","author":"N. Besse","year":"2001","journal-title":"An adaptive numerical method for the Vlasov equation based on a multiresolution analysis"},{"key":"3","unstructured":"Campos Pinto M. (2005): <i>D\u00e9veloppement et analyse de sch\u00e9mas adaptatifs pour les \u00e9quations de transport.<\/i> Ph.D. thesis (in French), Universit\u00e9 Pierre et Marie Curie, Paris."},{"key":"4","unstructured":"Campos Pinto M. (2007): <i>P<sub>1<\/sub> interpolation in the plane and functions of bounded total curvature.<\/i> (in preparation)."},{"key":"5","first-page":"43","volume":"7","author":"M. Campos Pinto","year":"2005","journal-title":"Adaptive numerical resolution of the Vlasov equation"},{"key":"6","doi-asserted-by":"crossref","unstructured":"Campos Pinto M. and Mehrenberger M. 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