{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:17:23Z","timestamp":1776863843260,"version":"3.51.2"},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"261","license":[{"start":{"date-parts":[[2008,6,18]],"date-time":"2008-06-18T00:00:00Z","timestamp":1213747200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper we present some classes of high-order semi-Lagran- gian schemes for solving the periodic one-dimensional Vlasov-Poisson system in phase-space on uniform grids. We prove that the distribution function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis t comma x comma v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(t,x,v)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the electric field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis t comma x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E(t,x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    converge in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm with a rate of\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis normal upper Delta t squared plus h Superscript m plus 1 Baseline plus StartFraction h Superscript m plus 1 Baseline Over normal upper Delta t EndFraction right-parenthesis comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:msup>\n                                <mml:mi>t<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msup>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>h<\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi>m<\/mml:mi>\n                                  <mml:mo>+<\/mml:mo>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:msup>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mfrac>\n                                <mml:msup>\n                                  <mml:mi>h<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mrow>\n                                  <mml:mi mathvariant=\"normal\">\n                                    \u0394\n                                    \n                                  <\/mml:mi>\n                                  <mml:mi>t<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mfrac>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {O}\\left (\\Delta t^2 +h^{m+1}+ \\frac {h^{m+1}}{\\Delta t}\\right ),<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the degree of the polynomial reconstruction, and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Delta t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are respectively the time and the phase-space discretization parameters.\n                  <\/p>","DOI":"10.1090\/s0025-5718-07-01912-6","type":"journal-article","created":{"date-parts":[[2007,10,29]],"date-time":"2007-10-29T06:30:33Z","timestamp":1193639433000},"page":"93-123","source":"Crossref","is-referenced-by-count":43,"title":["Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov\u2013Poisson 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Besse, Etude math\u00e9matique et num\u00e9rique de l\u2019\u00e9quation de Vlasov non lin\u00e9aire sur des maillages non structur\u00e9s de l\u2019espace des phases Ph.D. thesis of Institut de Recherche Math\u00e9matique Avanc\u00e9e, IRMA, Universit\u00e9 Louis Pasteur, Strasbourg, 2003."},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1137\/S0036142902410775","article-title":"Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system","volume":"42","author":"Besse, Nicolas","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"7","unstructured":"N. Besse, Convergence of a high-order semi-Lagrangian scheme with propagation of gradients for the Vlasov-Poisson system, submitted."},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1016\/S0021-9991(03)00318-8","article-title":"Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space","volume":"191","author":"Besse, N.","year":"2003","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendr\u00fccker, Adaptive numerical method for the Vlasov equation based on a multiresolution analysis, In F. Brezzi, A. Buffa, S. Escorsaro, and A. 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