{"id":"https://openalex.org/W2054748390","doi":"https://doi.org/10.1090/s0025-5718-99-01159-x","title":"Stability and \ud835\udc35-convergence properties of multistep Runge-Kutta methods","display_name":"Stability and \ud835\udc35-convergence properties of multistep Runge-Kutta methods","publication_year":1999,"publication_date":"1999-08-17","ids":{"openalex":"https://openalex.org/W2054748390","doi":"https://doi.org/10.1090/s0025-5718-99-01159-x","mag":"2054748390"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-99-01159-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01159-x","pdf_url":"https://www.ams.org/mcom/2000-69-232/S0025-5718-99-01159-X/S0025-5718-99-01159-X.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":null,"license_id":null,"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":"bronze","oa_url":"https://www.ams.org/mcom/2000-69-232/S0025-5718-99-01159-X/S0025-5718-99-01159-X.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009101042","display_name":"Shoufu Li","orcid":null},"institutions":[{"id":"https://openalex.org/I4610292","display_name":"Xiangtan University","ror":"https://ror.org/00xsfaz62","country_code":"CN","type":"education","lineage":["https://openalex.org/I4610292"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Shoufu Li","raw_affiliation_strings":["Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan Province, People's Republic of China","Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan Province, People\u2019s Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan Province, People's Republic of China","institution_ids":["https://openalex.org/I4610292"]},{"raw_affiliation_string":"Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan Province, People\u2019s Republic of China","institution_ids":["https://openalex.org/I4610292"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5009101042"],"corresponding_institution_ids":["https://openalex.org/I4610292"],"apc_list":null,"apc_paid":null,"fwci":1.5048,"has_fulltext":true,"cited_by_count":25,"citation_normalized_percentile":{"value":0.81688466,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"69","issue":"232","first_page":"1481","last_page":"1504"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9998999834060669,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/runge\u2013kutta-methods","display_name":"Runge\u2013Kutta methods","score":0.9088890552520752},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8617287874221802},{"id":"https://openalex.org/keywords/linear-multistep-method","display_name":"Linear multistep method","score":0.800809919834137},{"id":"https://openalex.org/keywords/numerical-methods-for-ordinary-differential-equations","display_name":"Numerical methods for ordinary differential equations","score":0.5651116967201233},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5625616312026978},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5576229691505432},{"id":"https://openalex.org/keywords/gauss\u2013seidel-method","display_name":"Gauss\u2013Seidel method","score":0.5040930509567261},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4633292555809021},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.42978599667549133},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.41578811407089233},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3939124643802643},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.27022701501846313},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.19542324542999268},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.14458316564559937},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.14193159341812134},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.09430795907974243},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06475299596786499},{"id":"https://openalex.org/keywords/collocation-method","display_name":"Collocation method","score":0.06223723292350769},{"id":"https://openalex.org/keywords/differential-algebraic-equation","display_name":"Differential algebraic equation","score":0.05416440963745117}],"concepts":[{"id":"https://openalex.org/C181582579","wikidata":"https://www.wikidata.org/wiki/Q725944","display_name":"Runge\u2013Kutta methods","level":3,"score":0.9088890552520752},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8617287874221802},{"id":"https://openalex.org/C65889100","wikidata":"https://www.wikidata.org/wiki/Q1462003","display_name":"Linear multistep method","level":5,"score":0.800809919834137},{"id":"https://openalex.org/C1164432","wikidata":"https://www.wikidata.org/wiki/Q541182","display_name":"Numerical methods for ordinary differential equations","level":5,"score":0.5651116967201233},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5625616312026978},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5576229691505432},{"id":"https://openalex.org/C3828260","wikidata":"https://www.wikidata.org/wiki/Q1069090","display_name":"Gauss\u2013Seidel method","level":3,"score":0.5040930509567261},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4633292555809021},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.42978599667549133},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.41578811407089233},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3939124643802643},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.27022701501846313},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.19542324542999268},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.14458316564559937},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.14193159341812134},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.09430795907974243},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06475299596786499},{"id":"https://openalex.org/C2126413","wikidata":"https://www.wikidata.org/wiki/Q1779479","display_name":"Collocation method","level":4,"score":0.06223723292350769},{"id":"https://openalex.org/C186219872","wikidata":"https://www.wikidata.org/wiki/Q955889","display_name":"Differential algebraic equation","level":4,"score":0.05416440963745117},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/s0025-5718-99-01159-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01159-x","pdf_url":"https://www.ams.org/mcom/2000-69-232/S0025-5718-99-01159-X/S0025-5718-99-01159-X.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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-99-01159-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01159-x","pdf_url":"https://www.ams.org/mcom/2000-69-232/S0025-5718-99-01159-X/S0025-5718-99-01159-X.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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2054748390.pdf","grobid_xml":"https://content.openalex.org/works/W2054748390.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W144420692","https://openalex.org/W1588914840","https://openalex.org/W1600945564","https://openalex.org/W1964106719","https://openalex.org/W1969218214","https://openalex.org/W1975522669","https://openalex.org/W1977542511","https://openalex.org/W1995504856","https://openalex.org/W2009724929","https://openalex.org/W2011331110","https://openalex.org/W2021970348","https://openalex.org/W2032085008","https://openalex.org/W2049344980","https://openalex.org/W2063738015","https://openalex.org/W2064369834","https://openalex.org/W2068766408","https://openalex.org/W2082526528","https://openalex.org/W2376015929","https://openalex.org/W2540328989","https://openalex.org/W4293515545"],"related_works":["https://openalex.org/W832825236","https://openalex.org/W2378531610","https://openalex.org/W2136451125","https://openalex.org/W56640444","https://openalex.org/W1579385565","https://openalex.org/W2001563191","https://openalex.org/W2065445194","https://openalex.org/W2360969022","https://openalex.org/W2381493582","https://openalex.org/W2528836882"],"abstract_inverted_index":{"This":[0],"paper":[1,162],"continues":[2],"earlier":[3],"work":[4],"by":[5,83,124],"the":[6,10,20,121,141,152,160,169],"same":[7],"author":[8],"concerning":[9],"stability":[11],"and":[12,37,94,131,145],"B-convergence":[13],"properties":[14],"of":[15,23,34,85,89,133,140,154],"multistep":[16,42,109,155],"Runge-Kutta":[17,43,110,147,156,173],"methods":[18,111,119,157],"for":[19,40],"numerical":[21],"solution":[22],"nonlinear":[24],"stiff":[25],"initial-value":[26],"problems":[27],"in":[28,114,126,159,165],"a":[29,41,115],"Hilbert":[30],"space.":[31],"A":[32],"series":[33],"sufficient":[35],"conditions":[36,39],"necessary":[38],"method":[44],"to":[45,168],"be":[46,136],"algebraically":[47,92],"stable,":[48,50],"diagonally":[49],"<inline-formula":[51,67,95],"content-type=\"math/mathml\">":[52,68,96],"<mml:math":[53,69,97],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[54,70,98],"alttext=\"upper":[55,71,99],"B\">":[56,72,100],"<mml:semantics>":[57,73,101],"<mml:mi>B</mml:mi>":[58,74,102],"<mml:annotation":[59,75,103],"encoding=\"application/x-tex\">B</mml:annotation>":[60,76,104],"</mml:semantics>":[61,77,105],"</mml:math>":[62,78,106],"</inline-formula>":[63,79,107],"-":[64],"or":[65],"optimally":[66],"-convergent":[80,108],"are":[81,112,163],"established,":[82],"means":[84],"which":[86],"six":[87],"classes":[88,153],"high":[90],"order":[91],"stable":[93],"constructed":[113,123,158],"unified":[116],"pattern.":[117],"These":[118],"include":[120],"class":[122],"Burrage":[125],"1987":[127],"as":[128,138],"special":[129],"case,":[130],"most":[132],"them":[134],"can":[135],"regarded":[137],"extension":[139],"Gauss,":[142],"RadauIA,":[143],"RadauIIA":[144],"LobattoIIIC":[146],"methods.":[148],"We":[149],"find":[150],"that":[151],"present":[161],"superior":[164],"many":[166],"respects":[167],"corresponding":[170],"existing":[171],"one-step":[172],"schemes.":[174]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
