{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T11:39:20Z","timestamp":1778326760927,"version":"3.51.4"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"274","license":[{"start":{"date-parts":[[2011,9,17]],"date-time":"2011-09-17T00:00:00Z","timestamp":1316217600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We provide a new analytical approach to operator splitting for equations of the type\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript t Baseline equals upper A u plus upper B left-parenthesis u right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u_t=Au+B(u)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a linear operator and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\">\n                        <mml:semantics>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is quadratic. A particular example is the Korteweg\u2013de Vries (KdV) equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript t Baseline minus u u Subscript x Baseline plus u Subscript x x x Baseline equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mi>x<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mi>x<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u_t-u u_x+u_{xxx}=0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2010-02402-0","type":"journal-article","created":{"date-parts":[[2010,12,29]],"date-time":"2010-12-29T14:39:24Z","timestamp":1293633564000},"page":"821-846","source":"Crossref","is-referenced-by-count":68,"title":["Operator splitting for the KdV equation"],"prefix":"10.1090","volume":"80","author":[{"given":"Helge","family":"Holden","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kenneth","family":"Karlsen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nils","family":"Risebro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Terence","family":"Tao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,9,17]]},"reference":[{"key":"1","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","volume-title":"A primer of nonlinear analysis","volume":"34","author":"Ambrosetti, Antonio","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521485738"},{"issue":"1287","key":"2","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1098\/rsta.1975.0035","article-title":"The initial-value problem for the Korteweg-de Vries equation","volume":"278","author":"Bona, J. L.","year":"1975","journal-title":"Philos. Trans. Roy. Soc. London Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0080-4614","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1006\/jcph.1999.6273","article-title":"Operator splitting methods for generalized Korteweg-de Vries equations","volume":"153","author":"Holden, Helge","year":"1999","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"H. Holden, K. H. Karlsen, K.-A. Lie, and N. H. Risebro. Splitting for Partial Differential Equations with Rough Solutions. European Math. Soc. Publishing House, Z\u00fcrich, 2010.","DOI":"10.4171\/078"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"323","DOI":"10.2307\/2939277","article-title":"Well-posedness of the initial value problem for the Korteweg-de Vries equation","volume":"4","author":"Kenig, Carlos E.","year":"1991","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"},{"key":"6","series-title":"Universitext","isbn-type":"print","volume-title":"Introduction to nonlinear dispersive equations","author":"Linares, Felipe","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387848983"},{"issue":"264","key":"7","doi-asserted-by":"publisher","first-page":"2141","DOI":"10.1090\/S0025-5718-08-02101-7","article-title":"On splitting methods for Schr\u00f6dinger-Poisson and cubic nonlinear Schr\u00f6dinger equations","volume":"77","author":"Lubich, Christian","year":"2008","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","series-title":"CBMS Regional Conference Series in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/cbms\/106","volume-title":"Nonlinear dispersive equations","volume":"106","author":"Tao, Terence","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/0821841432"},{"key":"9","unstructured":"F. Tappert. Numerical solutions of the Korteweg\u2013de Vries equation and its generalizations by the split-step Fourier method. In: (A. C. Newell, editor) Nonlinear Wave Motion, Amer. Math. Soc., 1974, pp. 215\u2013216."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-274\/S0025-5718-2010-02402-0\/S0025-5718-2010-02402-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-274\/S0025-5718-2010-02402-0\/S0025-5718-2010-02402-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:46:46Z","timestamp":1776790006000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-274\/S0025-5718-2010-02402-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9,17]]},"references-count":9,"journal-issue":{"issue":"274","published-print":{"date-parts":[[2011,4]]}},"alternative-id":["S0025-5718-2010-02402-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2010-02402-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2010,9,17]]}}}