{"id":"https://openalex.org/W2029383777","doi":"https://doi.org/10.1137/100819333","title":"Crank--Nicolson Schemes for Optimal Control Problems with Evolution Equations","display_name":"Crank--Nicolson Schemes for Optimal Control Problems with Evolution Equations","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2029383777","doi":"https://doi.org/10.1137/100819333","mag":"2029383777"},"language":"en","primary_location":{"id":"doi:10.1137/100819333","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100819333","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069305279","display_name":"Thomas Apel","orcid":"https://orcid.org/0000-0003-3642-3956"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Thomas Apel","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5050384421","display_name":"Thomas G. Flaig","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thomas G. Flaig","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5069305279"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.9754,"has_fulltext":false,"cited_by_count":49,"citation_normalized_percentile":{"value":0.96517663,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"50","issue":"3","first_page":"1484","last_page":"1512"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9997000098228455,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9990000128746033,"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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9930999875068665,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8500880599021912},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.7468684315681458},{"id":"https://openalex.org/keywords/crank\u2013nicolson-method","display_name":"Crank\u2013Nicolson method","score":0.5825517773628235},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5451587438583374},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5074049830436707},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.49786853790283203},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.4975486099720001},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.4736807644367218},{"id":"https://openalex.org/keywords/parabolic-partial-differential-equation","display_name":"Parabolic partial differential equation","score":0.4248164892196655},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2959648370742798},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.2829921841621399},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2499805986881256}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8500880599021912},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7468684315681458},{"id":"https://openalex.org/C180931078","wikidata":"https://www.wikidata.org/wiki/Q588725","display_name":"Crank\u2013Nicolson method","level":3,"score":0.5825517773628235},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5451587438583374},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5074049830436707},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.49786853790283203},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.4975486099720001},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4736807644367218},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.4248164892196655},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2959648370742798},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.2829921841621399},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2499805986881256}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/100819333","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100819333","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W657076157","https://openalex.org/W1480679962","https://openalex.org/W1492326914","https://openalex.org/W1544770838","https://openalex.org/W1979080874","https://openalex.org/W1979159785","https://openalex.org/W1992056335","https://openalex.org/W1998967184","https://openalex.org/W2000039609","https://openalex.org/W2008621392","https://openalex.org/W2018816843","https://openalex.org/W2031291057","https://openalex.org/W2035273811","https://openalex.org/W2040574660","https://openalex.org/W2060546108","https://openalex.org/W2063164826","https://openalex.org/W2069623243","https://openalex.org/W2082621526","https://openalex.org/W2083310254","https://openalex.org/W2113020599","https://openalex.org/W2135320821","https://openalex.org/W2152848099","https://openalex.org/W2727339451","https://openalex.org/W3021722416","https://openalex.org/W3165030955","https://openalex.org/W3204708751","https://openalex.org/W4300184951"],"related_works":["https://openalex.org/W1597420886","https://openalex.org/W3049665010","https://openalex.org/W2028690217","https://openalex.org/W1850237974","https://openalex.org/W2020915824","https://openalex.org/W2090855536","https://openalex.org/W2985697654","https://openalex.org/W1973273292","https://openalex.org/W4246241682","https://openalex.org/W1977346014"],"abstract_inverted_index":{"Crank--Nicolson":[0,34],"methods":[1,56],"are":[2],"often":[3],"used":[4],"for":[5,13,27,40,94,105],"the":[6,66,77,92,109],"simulation":[7],"of":[8,25,54,68,72,76],"initial":[9],"boundary":[10],"value":[11],"problems":[12,31],"parabolic":[14,28],"partial":[15],"differential":[16],"equations.":[17],"In":[18],"this":[19,106],"paper":[20],"we":[21,90],"present":[22],"a":[23,62,84,95],"family":[24],"discretizations":[26,39],"optimal":[29],"control":[30],"based":[32],"on":[33],"schemes":[35,78,93],"with":[36,87,115],"different":[37],"time":[38,97,110],"state":[41,45],"$y$":[42],"and":[43,50,100],"adjoint":[44],"$p$":[46],"so":[47],"that":[48],"discretization":[49],"optimization":[51],"commute.":[52],"One":[53],"these":[55],"can":[57],"also":[58,80],"be":[59,81],"explained":[60],"as":[61,83],"St\u00f6rmer--Verlet":[63],"scheme":[64],"in":[65],"context":[67],"geometric":[69],"numerical":[70],"integration":[71],"Hamiltonian":[73],"systems.":[74],"Two":[75],"may":[79],"obtained":[82],"Galerkin":[85],"method":[86],"quadrature.":[88],"Further":[89],"investigate":[91],"variable":[96],"step":[98,111],"size":[99,112],"prove":[101],"second":[102],"order":[103],"convergence":[104],"case":[107],"if":[108],"is":[113],"chosen":[114],"respect":[116],"to":[117],"an":[118],"arbitrary,":[119],"sufficiently":[120],"smooth":[121],"mesh":[122],"generating":[123],"function.":[124]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2026-03-28T08:17:26.163206","created_date":"2025-10-10T00:00:00"}