{"id":"https://openalex.org/W2956284904","doi":"https://doi.org/10.1137/18m121931x","title":"Uniformly Accurate Nested Picard Iterative Integrators for the Dirac Equation in the Nonrelativistic Limit Regime","display_name":"Uniformly Accurate Nested Picard Iterative Integrators for the Dirac Equation in the Nonrelativistic Limit Regime","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2956284904","doi":"https://doi.org/10.1137/18m121931x","mag":"2956284904"},"language":"en","primary_location":{"id":"doi:10.1137/18m121931x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m121931x","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/A5013651928","display_name":"Yongyong Cai","orcid":"https://orcid.org/0000-0002-3890-0878"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yongyong Cai","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5100322912","display_name":"Yan Wang","orcid":"https://orcid.org/0000-0003-1778-5417"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yan Wang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.375,"has_fulltext":false,"cited_by_count":30,"citation_normalized_percentile":{"value":0.95207668,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"57","issue":"4","first_page":"1602","last_page":"1624"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":1.0,"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":1.0,"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9970999956130981,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8069243431091309},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5898630619049072},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5600543022155762},{"id":"https://openalex.org/keywords/dirac","display_name":"Dirac (video compression format)","score":0.5118622183799744},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.4978675842285156},{"id":"https://openalex.org/keywords/fast-fourier-transform","display_name":"Fast Fourier transform","score":0.474334180355072},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.4304753541946411},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.42740651965141296},{"id":"https://openalex.org/keywords/oscillation","display_name":"Oscillation (cell signaling)","score":0.42720627784729004},{"id":"https://openalex.org/keywords/dirac-equation","display_name":"Dirac equation","score":0.42594707012176514},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.4124314785003662},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.3851264417171478},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3712958097457886},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2164817452430725},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.21537381410598755},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.08076432347297668}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8069243431091309},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5898630619049072},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5600543022155762},{"id":"https://openalex.org/C61039578","wikidata":"https://www.wikidata.org/wiki/Q604279","display_name":"Dirac (video compression format)","level":3,"score":0.5118622183799744},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4978675842285156},{"id":"https://openalex.org/C75172450","wikidata":"https://www.wikidata.org/wiki/Q623950","display_name":"Fast Fourier transform","level":2,"score":0.474334180355072},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.4304753541946411},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.42740651965141296},{"id":"https://openalex.org/C2778439541","wikidata":"https://www.wikidata.org/wiki/Q7106412","display_name":"Oscillation (cell signaling)","level":2,"score":0.42720627784729004},{"id":"https://openalex.org/C82601208","wikidata":"https://www.wikidata.org/wiki/Q272621","display_name":"Dirac equation","level":2,"score":0.42594707012176514},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.4124314785003662},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3851264417171478},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3712958097457886},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2164817452430725},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.21537381410598755},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.08076432347297668},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C186453547","wikidata":"https://www.wikidata.org/wiki/Q2126","display_name":"Neutrino","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/18m121931x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m121931x","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":[{"id":"https://openalex.org/G1196839600","display_name":null,"funder_award_id":"91630204","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6284513310","display_name":null,"funder_award_id":"11771036","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G814881854","display_name":null,"funder_award_id":"U-1530401","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W1965471588","https://openalex.org/W1967057381","https://openalex.org/W1982686824","https://openalex.org/W1990853264","https://openalex.org/W1999077739","https://openalex.org/W2004974504","https://openalex.org/W2012997730","https://openalex.org/W2013807320","https://openalex.org/W2036516029","https://openalex.org/W2045505006","https://openalex.org/W2047270251","https://openalex.org/W2050338742","https://openalex.org/W2056470162","https://openalex.org/W2064726339","https://openalex.org/W2083382031","https://openalex.org/W2086585878","https://openalex.org/W2093993931","https://openalex.org/W2100447495","https://openalex.org/W2101962226","https://openalex.org/W2124382907","https://openalex.org/W2143076561","https://openalex.org/W2143097120","https://openalex.org/W2187257405","https://openalex.org/W2318376939","https://openalex.org/W2618966431","https://openalex.org/W2792515126","https://openalex.org/W2963174244","https://openalex.org/W2963600323","https://openalex.org/W2964199329","https://openalex.org/W3098636032","https://openalex.org/W3100405256","https://openalex.org/W3101835573","https://openalex.org/W3123381822","https://openalex.org/W4211077197","https://openalex.org/W4232654152"],"related_works":["https://openalex.org/W2349221428","https://openalex.org/W1564485841","https://openalex.org/W4287219806","https://openalex.org/W4244688541","https://openalex.org/W105939817","https://openalex.org/W3100754759","https://openalex.org/W1919901593","https://openalex.org/W2951283098","https://openalex.org/W105150783","https://openalex.org/W2326064537"],"abstract_inverted_index":{"This":[0],"paper":[1],"is":[2,30,92,150,169,174],"devoted":[3],"to":[4,37,97,110,116,128,191],"the":[5,18,22,38,43,60,64,71,98,102,112,117,122,141,154,162,170,175,182,185,193,197,203,210],"construction":[6,72],"and":[7,42,53,82,105,135,153,178,201,212],"analysis":[8],"of":[9,40,73,101,140,184],"uniformly":[10,79],"accurate":[11,205],"nested":[12,94,118],"Picard":[13,95,119],"iterative":[14,120],"integrators":[15,109],"(NPI)":[16],"for":[17,196,209],"Dirac":[19,103],"equation":[20,44,104],"in":[21,51,56,85,131],"nonrelativistic":[23],"limit":[24],"regime.":[25],"In":[26],"this":[27],"regime,":[28],"there":[29],"a":[31],"dimensionless":[32],"parameter":[33],"$\\varepsilon\\in(0,1]$":[34],"inversely":[35],"proportional":[36],"speed":[39],"light":[41],"admits":[45],"propagating":[46],"waves":[47],"with":[48,133],"$O(1)$":[49],"wavelength":[50,55],"space":[52],"$O(\\varepsilon^2)$":[54],"time.":[57],"To":[58],"overcome":[59],"difficulty":[61],"induced":[62],"by":[63],"temporal":[65,113],"$\\varepsilon$":[66],"dependent":[67],"oscillation,":[68],"we":[69],"present":[70],"several":[74],"NPI":[75,123,144,199,214],"methods":[76,215],"which":[77],"are":[78,158,189],"first-,":[80],"second-,":[81],"third-order":[83,213],"convergent":[84],"time":[86,132,176],"w.r.t.":[87],"$\\varepsilon$.":[88],"The":[89,138],"general":[90],"idea":[91],"applying":[93],"iterations":[96],"integral":[99],"form":[100],"using":[106],"exponential":[107],"wave":[108],"approximate":[111],"integrals.":[114],"Thanks":[115],"idea,":[121],"method":[124,145,164,200],"can":[125],"be":[126],"extended":[127],"arbitrary":[129],"higher-order":[130],"optimal":[134],"uniform":[136,204],"accuracy.":[137],"implementation":[139],"second-order":[142,198],"in-time":[143],"via":[146],"Fourier":[147],"pseudospectral":[148],"discretization":[149],"clearly":[151],"demonstrated,":[152],"corresponding":[155],"error":[156,194],"bounds":[157],"rigorously":[159],"established":[160],"through":[161],"energy":[163],"as":[165,216],"$h^{m_0}+\\tau^2$,":[166],"where":[167],"$h$":[168],"mesh":[171],"size,":[172],"$\\tau$":[173],"step,":[177],"$m_0$":[179],"depends":[180],"on":[181],"regularity":[183],"solution.":[186],"Numerical":[187],"results":[188],"reported":[190],"confirm":[192],"estimates":[195],"show":[202],"properties":[206],"(w.r.t.":[207],"$\\varepsilon$)":[208],"first-":[211],"well.":[217]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2025-10-10T00:00:00"}