{"id":"https://openalex.org/W2009657287","doi":"https://doi.org/10.1090/s0025-5718-2010-02402-0","title":"Operator splitting for the KdV equation","display_name":"Operator splitting for the KdV equation","publication_year":2010,"publication_date":"2010-09-17","ids":{"openalex":"https://openalex.org/W2009657287","doi":"https://doi.org/10.1090/s0025-5718-2010-02402-0","mag":"2009657287"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2010-02402-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2010-02402-0","pdf_url":"https://www.ams.org/mcom/2011-80-274/S0025-5718-2010-02402-0/S0025-5718-2010-02402-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.ams.org/mcom/2011-80-274/S0025-5718-2010-02402-0/S0025-5718-2010-02402-0.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037337641","display_name":"Helge Holden","orcid":"https://orcid.org/0000-0002-8564-0343"},"institutions":[{"id":"https://openalex.org/I184942183","display_name":"University of Oslo","ror":"https://ror.org/01xtthb56","country_code":"NO","type":"education","lineage":["https://openalex.org/I184942183"]},{"id":"https://openalex.org/I204778367","display_name":"Norwegian University of Science and Technology","ror":"https://ror.org/05xg72x27","country_code":"NO","type":"education","lineage":["https://openalex.org/I204778367"]}],"countries":["NO"],"is_corresponding":false,"raw_author_name":"Helge Holden","raw_affiliation_strings":["Department of Mathematical Sciences, Norwegian University of Science and Technology, NO\u00e2\u0080\u00937491 Trondheim, Norway and Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO\u00e2\u0080\u00930316 Oslo, Norway","Department of Mathematical Sciences, Norwegian University of Science and Technology, NO\u20137491 Trondheim, Norway, and Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO\u20130316 Oslo, Norway"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Norwegian University of Science and Technology, NO\u00e2\u0080\u00937491 Trondheim, Norway and Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO\u00e2\u0080\u00930316 Oslo, Norway","institution_ids":["https://openalex.org/I184942183","https://openalex.org/I204778367"]},{"raw_affiliation_string":"Department of Mathematical Sciences, Norwegian University of Science and Technology, NO\u20137491 Trondheim, Norway, and Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO\u20130316 Oslo, Norway","institution_ids":["https://openalex.org/I184942183","https://openalex.org/I204778367"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034008502","display_name":"Kenneth H. Karlsen","orcid":"https://orcid.org/0000-0003-3375-2084"},"institutions":[{"id":"https://openalex.org/I184942183","display_name":"University of Oslo","ror":"https://ror.org/01xtthb56","country_code":"NO","type":"education","lineage":["https://openalex.org/I184942183"]}],"countries":["NO"],"is_corresponding":false,"raw_author_name":"Kenneth Karlsen","raw_affiliation_strings":["Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO\u00e2\u0080\u00930316 Oslo, Norway","Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO\u00e2\u0080\u00930316 Oslo, Norway","institution_ids":["https://openalex.org/I184942183"]},{"raw_affiliation_string":"Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway","institution_ids":["https://openalex.org/I184942183"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081764581","display_name":"Nils Henrik Risebro","orcid":"https://orcid.org/0000-0003-4069-8282"},"institutions":[{"id":"https://openalex.org/I184942183","display_name":"University of Oslo","ror":"https://ror.org/01xtthb56","country_code":"NO","type":"education","lineage":["https://openalex.org/I184942183"]}],"countries":["NO"],"is_corresponding":false,"raw_author_name":"Nils Risebro","raw_affiliation_strings":["Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO\u00e2\u0080\u00930316 Oslo, Norway","Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO\u00e2\u0080\u00930316 Oslo, Norway","institution_ids":["https://openalex.org/I184942183"]},{"raw_affiliation_string":"Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway","institution_ids":["https://openalex.org/I184942183"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066728260","display_name":"Terence Tao","orcid":"https://orcid.org/0000-0002-0140-7641"},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Terence Tao","raw_affiliation_strings":["Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555","Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095\u20101555"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555","institution_ids":["https://openalex.org/I161318765"]},{"raw_affiliation_string":"Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095\u20101555","institution_ids":["https://openalex.org/I161318765"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.768,"has_fulltext":true,"cited_by_count":92,"citation_normalized_percentile":{"value":0.97990044,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"80","issue":"274","first_page":"821","last_page":"846"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9944000244140625,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/korteweg\u2013de-vries-equation","display_name":"Korteweg\u2013de Vries equation","score":0.9417411088943481},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8976186513900757},{"id":"https://openalex.org/keywords/operator-splitting","display_name":"Operator splitting","score":0.803257167339325},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.6902157664299011},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.5822623372077942},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.45580625534057617},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4276677072048187},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.42202863097190857},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.34719187021255493},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.15648731589317322},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07954749464988708},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06216415762901306},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.04642316699028015},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.04422086477279663}],"concepts":[{"id":"https://openalex.org/C146630112","wikidata":"https://www.wikidata.org/wiki/Q601796","display_name":"Korteweg\u2013de Vries equation","level":3,"score":0.9417411088943481},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8976186513900757},{"id":"https://openalex.org/C2987355758","wikidata":"https://www.wikidata.org/wiki/Q25037251","display_name":"Operator splitting","level":2,"score":0.803257167339325},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.6902157664299011},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5822623372077942},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.45580625534057617},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4276677072048187},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.42202863097190857},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.34719187021255493},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.15648731589317322},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07954749464988708},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06216415762901306},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.04642316699028015},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.04422086477279663},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/s0025-5718-2010-02402-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2010-02402-0","pdf_url":"https://www.ams.org/mcom/2011-80-274/S0025-5718-2010-02402-0/S0025-5718-2010-02402-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.230.1880","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.230.1880","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.ntnu.no/conservation/2009/038.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.249.238","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.249.238","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0906.4902v1.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-2010-02402-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2010-02402-0","pdf_url":"https://www.ams.org/mcom/2011-80-274/S0025-5718-2010-02402-0/S0025-5718-2010-02402-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7201218755","display_name":"Global Behaviour of Critical Nonlinear PDE","funder_award_id":"0649473","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306142","display_name":"John D. and Catherine T. MacArthur Foundation","ror":"https://ror.org/00dxczh48"},{"id":"https://openalex.org/F4320323299","display_name":"Norges Forskningsr\u00e5d","ror":"https://ror.org/00epmv149"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2009657287.pdf","grobid_xml":"https://content.openalex.org/works/W2009657287.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W438964906","https://openalex.org/W1506032630","https://openalex.org/W1547431580","https://openalex.org/W1559694286","https://openalex.org/W2023174645","https://openalex.org/W2071742362","https://openalex.org/W2092400970","https://openalex.org/W2158738915","https://openalex.org/W4236581885"],"related_works":["https://openalex.org/W2027202866","https://openalex.org/W3149920197","https://openalex.org/W4293569346","https://openalex.org/W2397056152","https://openalex.org/W2949885877","https://openalex.org/W2950645300","https://openalex.org/W3144491984","https://openalex.org/W2009657287","https://openalex.org/W2954949794","https://openalex.org/W412354254"],"abstract_inverted_index":{"We":[0,155],"provide":[1],"a":[2,70],"new":[3],"analytical":[4],"approach":[5],"to":[6],"operator":[7,72],"splitting":[8,162],"for":[9],"equations":[10],"of":[11],"the":[12,93,158,166,170],"type":[13],"<inline-formula":[14,56,74,98],"content-type=\"math/mathml\">":[15,57,75,99],"<mml:math":[16,58,76,100],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[17,59,77,101],"alttext=\"u":[18,102],"Subscript":[19,103,109,114],"t":[20,104],"Baseline":[21,105,111,118],"equals":[22,119],"upper":[23,27],"A":[24,89],"u":[25,30,107,108,113],"plus":[26,112],"B":[28],"left-parenthesis":[29],"right-parenthesis\">":[31],"<mml:semantics>":[32,62,80,121],"<mml:mrow>":[33,122],"<mml:msub>":[34,123,131,136],"<mml:mi>u</mml:mi>":[35,40,45,124,130,132,137],"<mml:mi>t</mml:mi>":[36,125],"</mml:msub>":[37,126,134,144],"<mml:mo>=</mml:mo>":[38,145],"<mml:mi>A</mml:mi>":[39,63],"<mml:mo>+</mml:mo>":[41,135],"<mml:mi>B</mml:mi>":[42,81],"<mml:mo":[43,46],"stretchy=\"false\">(</mml:mo>":[44],"stretchy=\"false\">)</mml:mo>":[47],"</mml:mrow>":[48,143,147],"<mml:annotation":[49,64,82,148],"encoding=\"application/x-tex\">u_t=Au+B(u)</mml:annotation>":[50],"</mml:semantics>":[51,66,84,151],"</mml:math>":[52,67,85,152],"</inline-formula>":[53,68,86,153],",":[54],"where":[55],"alttext=\"upper":[60,78],"A\">":[61],"encoding=\"application/x-tex\">A</mml:annotation>":[65],"is":[69,87,92],"linear":[71],"and":[73,160],"B\">":[79],"encoding=\"application/x-tex\">B</mml:annotation>":[83],"quadratic.":[88],"particular":[90],"example":[91],"Korteweg\u2013de":[94],"Vries":[95],"(KdV)":[96],"equation":[97],"minus":[106],"x":[110,115,116,117],"0\">":[120],"<mml:mo>":[127],"\u2212":[128],"</mml:mo>":[129],"<mml:mi>x</mml:mi>":[133,140,141,142],"<mml:mrow":[138],"class=\"MJX-TeXAtom-ORD\">":[139],"<mml:mn>0</mml:mn>":[146],"encoding=\"application/x-tex\">u_t-u":[149],"u_x+u_{xxx}=0</mml:annotation>":[150],".":[154],"show":[156],"that":[157],"Godunov":[159],"Strang":[161],"methods":[163],"converge":[164],"with":[165],"expected":[167],"rates":[168],"if":[169],"initial":[171],"data":[172],"are":[173],"sufficiently":[174],"regular.":[175]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":7},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":14},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":10},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":6}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
