{"id":"https://openalex.org/W2056971052","doi":"https://doi.org/10.1137/s0036142900381734","title":"A Wigner-Measure Analysis of the Dufort--Frankel Scheme for the Schr\u00f6dinger Equation","display_name":"A Wigner-Measure Analysis of the Dufort--Frankel Scheme for the Schr\u00f6dinger Equation","publication_year":2002,"publication_date":"2002-01-01","ids":{"openalex":"https://openalex.org/W2056971052","doi":"https://doi.org/10.1137/s0036142900381734","mag":"2056971052"},"language":"en","primary_location":{"id":"doi:10.1137/s0036142900381734","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036142900381734","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/A5110260067","display_name":"Peter A. Markowich","orcid":"https://orcid.org/0000-0002-3704-1821"},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Peter A. Markowich","raw_affiliation_strings":["Univ. of Vienna#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Univ. of Vienna#TAB#","institution_ids":["https://openalex.org/I129774422"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056631958","display_name":"Paola Pietra","orcid":"https://orcid.org/0000-0002-5967-6983"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Paola Pietra","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005241309","display_name":"Carsten Pohl","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Carsten Pohl","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5035204764","display_name":"Hans Peter Stimming","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hans Peter Stimming","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.6593,"has_fulltext":false,"cited_by_count":51,"citation_normalized_percentile":{"value":0.95620438,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"40","issue":"4","first_page":"1281","last_page":"1310"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.9994999766349792,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9994999766349792,"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":0.9980999827384949,"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":0.9975000023841858,"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/semiclassical-physics","display_name":"Semiclassical physics","score":0.9200273156166077},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8326019644737244},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.633122980594635},{"id":"https://openalex.org/keywords/schr\u00f6dinger-equation","display_name":"Schr\u00f6dinger equation","score":0.6024901270866394},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5974929928779602},{"id":"https://openalex.org/keywords/planck-constant","display_name":"Planck constant","score":0.5405120253562927},{"id":"https://openalex.org/keywords/observable","display_name":"Observable","score":0.5374119877815247},{"id":"https://openalex.org/keywords/scheme","display_name":"Scheme (mathematics)","score":0.5070147514343262},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4524073898792267},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37146613001823425},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.3515104055404663},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.2500113546848297},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1349797546863556},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.08875823020935059}],"concepts":[{"id":"https://openalex.org/C535169671","wikidata":"https://www.wikidata.org/wiki/Q17080006","display_name":"Semiclassical physics","level":3,"score":0.9200273156166077},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8326019644737244},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.633122980594635},{"id":"https://openalex.org/C63036615","wikidata":"https://www.wikidata.org/wiki/Q165498","display_name":"Schr\u00f6dinger equation","level":2,"score":0.6024901270866394},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5974929928779602},{"id":"https://openalex.org/C197111296","wikidata":"https://www.wikidata.org/wiki/Q122894","display_name":"Planck constant","level":3,"score":0.5405120253562927},{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.5374119877815247},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.5070147514343262},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4524073898792267},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37146613001823425},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3515104055404663},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.2500113546848297},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1349797546863556},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.08875823020935059},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s0036142900381734","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036142900381734","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":[{"score":0.4000000059604645,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W1419708213","https://openalex.org/W1532154335","https://openalex.org/W1983797262","https://openalex.org/W1989121669","https://openalex.org/W1997844687","https://openalex.org/W2015683572","https://openalex.org/W2063727277","https://openalex.org/W2072482841","https://openalex.org/W2073339836","https://openalex.org/W2074306248","https://openalex.org/W2083916485","https://openalex.org/W2367355694","https://openalex.org/W3165898261"],"related_works":["https://openalex.org/W1648886000","https://openalex.org/W2044808828","https://openalex.org/W2963575582","https://openalex.org/W2917602225","https://openalex.org/W2011800114","https://openalex.org/W2767017392","https://openalex.org/W1981619409","https://openalex.org/W2058390952","https://openalex.org/W2046913111","https://openalex.org/W1762747973"],"abstract_inverted_index":{"We":[0],"apply":[1],"Wigner":[2],"transform":[3],"techniques":[4],"to":[5,18,42,63,95,101,106],"the":[6,9,14,19,23,26,47,59,98,103,111],"analysis":[7],"of":[8,22,28,56],"Dufort--Frankel":[10,104],"difference":[11,108],"scheme":[12,24,105],"for":[13,53,110],"Schr\u00f6dinger":[15],"equation":[16],"and":[17,76,80,100],"continuous":[20],"analogue":[21],"in":[25,68],"case":[27],"a":[29],"small":[30],"(scaled)":[31],"Planck":[32,60],"constant":[33,61],"(semiclassical":[34],"regime).":[35],"In":[36],"this":[37,69],"way":[38],"we":[39],"are":[40,93],"able":[41],"obtain":[43],"sharp":[44],"conditions":[45],"on":[46,74,84],"spatial-temporal":[48],"grid":[49],"which":[50],"guarantee":[51],"convergence":[52],"average":[54],"values":[55],"observables":[57],"as":[58],"tends":[62],"zero.":[64],"The":[65],"theory":[66,99],"developed":[67],"paper":[70],"is":[71],"not":[72,82,87],"based":[73],"local":[75],"global":[77],"error":[78],"estimates":[79],"does":[81],"depend":[83],"whether":[85],"or":[86],"caustics":[88],"develop.":[89],"Numerical":[90],"test":[91],"examples":[92],"presented":[94],"help":[96],"interpret":[97],"compare":[102],"other":[107],"schemes":[109],"Schr{\u00f6dinger":[112],"equation.":[113]},"counts_by_year":[{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
