{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:18:56Z","timestamp":1776795536457,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"281","license":[{"start":{"date-parts":[[2013,6,12]],"date-time":"2013-06-12T00:00:00Z","timestamp":1370995200000},"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 u u Subscript x\">\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>u<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mi>x<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u_t=Au+u u_x<\/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 differential operator such that the equation is well-posed. Particular examples include the viscous Burgers equation, the Korteweg\u2013de Vries (KdV) equation, the Benney\u2013Lin equation, and the Kawahara equation. We show that the Strang splitting method converges with the expected rate if the initial data are sufficiently regular. In particular, for the KdV equation we obtain second-order convergence in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript r\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for initial data in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript r plus 5\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>r<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>5<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^{r+5}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with arbitrary\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r greater-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r\\ge 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-2012-02624-x","type":"journal-article","created":{"date-parts":[[2012,6,12]],"date-time":"2012-06-12T09:08:01Z","timestamp":1339492081000},"page":"173-185","source":"Crossref","is-referenced-by-count":64,"title":["Operator splitting for partial differential equations with Burgers nonlinearity"],"prefix":"10.1090","volume":"82","author":[{"given":"Helge","family":"Holden","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christian","family":"Lubich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nils","family":"Risebro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2012,6,12]]},"reference":[{"key":"1","unstructured":"A. Ambrosetti and G. Prodi. A Primer of Nonlinear Analysis. Cambridge UP, Cambridge, 1995."},{"key":"2","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1002\/sapm1966451150","article-title":"Long waves on liquid films","volume":"45","author":"Benney, D. J.","year":"1966","journal-title":"J. Math. and Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-1421","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1006\/jmaa.1997.5438","article-title":"On the Benney-Lin and Kawahara equations","volume":"211","author":"Biagioni, H. A.","year":"1997","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"1287","key":"4","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"},{"key":"5","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Solving ordinary differential equations. I","volume":"8","author":"Hairer, E.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540566708","edition":"2"},{"key":"6","series-title":"EMS Series of Lectures in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.4171\/078","volume-title":"Splitting methods for partial differential equations with rough solutions","author":"Holden, Helge","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783037190784"},{"issue":"1","key":"7","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"},{"issue":"274","key":"8","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1090\/S0025-5718-2010-02402-0","article-title":"Operator splitting for the KdV equation","volume":"80","author":"Holden, Helge","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"9","unstructured":"H. Holden, K. H. Karlsen, T. Karper. Operator splitting for two-dimensional incompressible fluid equations. Math. Comp., to appear."},{"issue":"4","key":"10","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1023\/A:1022396519656","article-title":"Error bounds for exponential operator splittings","volume":"40","author":"Jahnke, Tobias","year":"2000","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"6","key":"11","doi-asserted-by":"publisher","first-page":"2448","DOI":"10.1016\/j.jde.2008.10.027","article-title":"Well-posedness for the fifth-order shallow water equations","volume":"246","author":"Jia, Yueling","year":"2009","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"key":"12","doi-asserted-by":"crossref","unstructured":"T. Kawahara. Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33:260\u2013264 (1975).","DOI":"10.1143\/JPSJ.33.260"},{"issue":"2","key":"13","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":"14","volume-title":"{\\cyr Line\\u{i}} {\\cyr nye i kvaziline\\u{i}} {\\cyr nye uravneniya parabolicheskogo tipa}","author":"Lady\u017eenskaja, O. A.","year":"1967"},{"key":"15","doi-asserted-by":"crossref","unstructured":"S. P. Lin. Finite amplitude side-band stability of a viscous film. J. Fluid. Mech. 63:417\u2013429 (1974).","DOI":"10.1017\/S0022112074001704"},{"key":"16","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":"17","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":"18","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":"19","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\/2013-82-281\/S0025-5718-2012-02624-X\/S0025-5718-2012-02624-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2013-82-281\/S0025-5718-2012-02624-X\/S0025-5718-2012-02624-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:24:47Z","timestamp":1776792287000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2013-82-281\/S0025-5718-2012-02624-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,6,12]]},"references-count":19,"journal-issue":{"issue":"281","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["S0025-5718-2012-02624-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2012-02624-x","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":[[2012,6,12]]}}}