{"id":"https://openalex.org/W2001033798","doi":"https://doi.org/10.1137/140959936","title":"The Picard Integral Formulation of Weighted Essentially Nonoscillatory Schemes","display_name":"The Picard Integral Formulation of Weighted Essentially Nonoscillatory Schemes","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2001033798","doi":"https://doi.org/10.1137/140959936","mag":"2001033798"},"language":"en","primary_location":{"id":"doi:10.1137/140959936","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140959936","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/A5083347493","display_name":"Andrew Christlieb","orcid":"https://orcid.org/0000-0002-5395-5455"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Andrew J. Christlieb","raw_affiliation_strings":["Computational Mathematics, Science and Engineering"],"affiliations":[{"raw_affiliation_string":"Computational Mathematics, Science and Engineering","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025498364","display_name":"Yaman G\u00fc\u00e7l\u00fc","orcid":"https://orcid.org/0000-0003-2619-5152"},"institutions":[{"id":"https://openalex.org/I4210139716","display_name":"Max Planck Institute for Plasma Physics","ror":"https://ror.org/03taest98","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210139716"]},{"id":"https://openalex.org/I149899117","display_name":"Max Planck Society","ror":"https://ror.org/01hhn8329","country_code":"DE","type":"funder","lineage":["https://openalex.org/I149899117"]},{"id":"https://openalex.org/I4210119692","display_name":"Numerical Method (China)","ror":"https://ror.org/02ns8zp42","country_code":"CN","type":"company","lineage":["https://openalex.org/I4210119692"]}],"countries":["CN","DE"],"is_corresponding":false,"raw_author_name":"Yaman G\u00fc\u00e7l\u00fc","raw_affiliation_strings":["Numerical Methods in Plasma Physics (NMPP), Max Planck Institute for Plasma Physics, Max Planck Society"],"affiliations":[{"raw_affiliation_string":"Numerical Methods in Plasma Physics (NMPP), Max Planck Institute for Plasma Physics, Max Planck Society","institution_ids":["https://openalex.org/I4210139716","https://openalex.org/I4210119692","https://openalex.org/I149899117"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5004529550","display_name":"David C. Seal","orcid":null},"institutions":[{"id":"https://openalex.org/I87216513","display_name":"Michigan State University","ror":"https://ror.org/05hs6h993","country_code":"US","type":"education","lineage":["https://openalex.org/I87216513"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David C. Seal","raw_affiliation_strings":["Michigan State University"],"affiliations":[{"raw_affiliation_string":"Michigan State University","institution_ids":["https://openalex.org/I87216513"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5083347493"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.0093,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.89965054,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"53","issue":"4","first_page":"1833","last_page":"1856"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9998000264167786,"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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9998000264167786,"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":0.9993000030517578,"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9986000061035156,"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":0.8834680318832397},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.7535810470581055},{"id":"https://openalex.org/keywords/taylor-series","display_name":"Taylor series","score":0.6116658449172974},{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.5964361429214478},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5713448524475098},{"id":"https://openalex.org/keywords/conservation-law","display_name":"Conservation law","score":0.523115336894989},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.5097288489341736},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4923642873764038},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.4792899489402771},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.42348340153694153},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.41358262300491333},{"id":"https://openalex.org/keywords/euler-method","display_name":"Euler method","score":0.410690039396286},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.3239174485206604}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8834680318832397},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7535810470581055},{"id":"https://openalex.org/C158946198","wikidata":"https://www.wikidata.org/wiki/Q131187","display_name":"Taylor series","level":2,"score":0.6116658449172974},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.5964361429214478},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5713448524475098},{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.523115336894989},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.5097288489341736},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4923642873764038},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4792899489402771},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.42348340153694153},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.41358262300491333},{"id":"https://openalex.org/C75380026","wikidata":"https://www.wikidata.org/wiki/Q868454","display_name":"Euler method","level":3,"score":0.410690039396286},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3239174485206604},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/140959936","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140959936","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"},{"id":"pmh:oai:pure.mpg.de:item_2083321","is_oa":false,"landing_page_url":"http://hdl.handle.net/11858/00-001M-0000-0028-8084-6","pdf_url":null,"source":{"id":"https://openalex.org/S4306400654","display_name":"MPG.PuRe (Max Planck Society)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I149899117","host_organization_name":"Max Planck Society","host_organization_lineage":["https://openalex.org/I149899117"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/15","score":0.6600000262260437,"display_name":"Life in Land"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":59,"referenced_works":["https://openalex.org/W83369956","https://openalex.org/W1513760541","https://openalex.org/W1555627060","https://openalex.org/W1886421468","https://openalex.org/W1967870607","https://openalex.org/W1969647398","https://openalex.org/W1971334156","https://openalex.org/W1973481688","https://openalex.org/W1976883414","https://openalex.org/W1977042465","https://openalex.org/W1983195644","https://openalex.org/W1984280619","https://openalex.org/W1986328299","https://openalex.org/W1988094638","https://openalex.org/W1988634891","https://openalex.org/W1989352162","https://openalex.org/W1993547144","https://openalex.org/W1999399767","https://openalex.org/W2001684614","https://openalex.org/W2003333105","https://openalex.org/W2006118548","https://openalex.org/W2006321195","https://openalex.org/W2008182598","https://openalex.org/W2010923956","https://openalex.org/W2014519896","https://openalex.org/W2015892502","https://openalex.org/W2019720983","https://openalex.org/W2020168202","https://openalex.org/W2027592715","https://openalex.org/W2033265834","https://openalex.org/W2038072397","https://openalex.org/W2039150507","https://openalex.org/W2042218545","https://openalex.org/W2045849310","https://openalex.org/W2048143037","https://openalex.org/W2048941652","https://openalex.org/W2052013840","https://openalex.org/W2054662916","https://openalex.org/W2057858153","https://openalex.org/W2068056613","https://openalex.org/W2068255553","https://openalex.org/W2069227042","https://openalex.org/W2071622123","https://openalex.org/W2072602885","https://openalex.org/W2079375216","https://openalex.org/W2082433079","https://openalex.org/W2087887080","https://openalex.org/W2093842327","https://openalex.org/W2102179892","https://openalex.org/W2107569990","https://openalex.org/W2112548197","https://openalex.org/W2128425522","https://openalex.org/W2157430242","https://openalex.org/W2162683902","https://openalex.org/W2163604103","https://openalex.org/W2170107618","https://openalex.org/W3010292040","https://openalex.org/W3125108654","https://openalex.org/W4245037559"],"related_works":["https://openalex.org/W2189596786","https://openalex.org/W47811045","https://openalex.org/W4382053489","https://openalex.org/W2089955609","https://openalex.org/W2039299714","https://openalex.org/W2466040413","https://openalex.org/W4383710307","https://openalex.org/W4307700932","https://openalex.org/W2290376096","https://openalex.org/W2887130266"],"abstract_inverted_index":{"High-order":[0],"temporal":[1],"discretizations":[2,52],"for":[3,224,251],"hyperbolic":[4,260],"conservation":[5,261],"laws":[6,262],"have":[7],"historically":[8],"been":[9],"formulated":[10],"as":[11,32,161],"either":[12],"a":[13,19,33,97,220],"method":[14,78],"of":[15,36,79,100,125,157,166,191,194,222],"lines":[16],"(MOL)":[17],"or":[18],"Lax--Wendroff":[20,51,149],"method.":[21,246,253],"In":[22,49,63,93,197],"the":[23,26,68,77,113,126,132,158,167,195,206,239,243,269],"MOL":[24,176],"viewpoint,":[25],"partial":[27],"differential":[28,38],"equation":[29],"is":[30,47,74,83,129,151,214],"treated":[31],"large":[34],"system":[35],"ordinary":[37],"equations":[39],"(ODEs),":[40],"where":[41],"an":[42],"ODE":[43],"tailored":[44],"time":[45,58,145,210],"integrator":[46],"applied.":[48],"contrast,":[50],"immediately":[53],"convert":[54],"Taylor":[55,88,155,164],"series":[56,156,165],"in":[57,263,272],"to":[59,85,112,131,163,183,219,259],"discrete":[60],"spatial":[61,192],"derivatives.":[62],"this":[64],"work,":[65],"we":[66,95,179,200,229],"propose":[67],"Picard":[69],"integral":[70],"formulation":[71],"(PIF),":[72],"which":[73,128],"based":[75,235,241],"on":[76,236,242],"modified":[80],"fluxes,":[81,127],"and":[82,89,216,238,265,268],"used":[84],"derive":[86],"new":[87,98],"Runge--Kutta":[90],"(RK)":[91],"methods.":[92,278],"particular,":[94],"construct":[96],"class":[99],"conservative":[101,121],"finite":[102],"difference":[103],"methods":[104],"by":[105,153],"applying":[106],"weighted":[107],"essentially":[108],"nonoscillatory":[109],"(WENO)":[110],"reconstructions":[111,137,182],"so-called":[114],"time-averaged":[115,184],"fluxes.":[116],"Our":[117],"schemes":[118,256],"are":[119,249,257,271],"automatically":[120],"under":[122],"any":[123],"modification":[124],"attributed":[130],"fact":[133],"that":[134],"classical":[135,175,244],"WENO":[136,181],"conserve":[138],"mass":[139],"when":[140],"coupled":[141],"with":[142,275],"forward":[143],"Euler":[144],"steps.":[146],"The":[147,170,212,254],"proposed":[148,255],"discretization":[150,172],"constructed":[152],"taking":[154,188],"flux":[159],"function":[160],"opposed":[162],"conserved":[168],"variables.":[169],"RK":[171,245],"differs":[173],"from":[174],"formulations":[177],"because":[178],"apply":[180],"fluxes":[185],"rather":[186],"than":[187],"linear":[189],"combinations":[190],"derivatives":[193],"flux.":[196],"both":[198],"cases,":[199],"only":[201],"need":[202],"one":[203,234,264],"projection":[204],"onto":[205],"characteristic":[207],"variables":[208],"per":[209],"step.":[211],"PIF":[213],"generic":[215],"lends":[217],"itself":[218],"multitude":[221],"options":[223],"further":[225],"investigation.":[226],"At":[227],"present,":[228],"present":[230],"two":[231,266],"canonical":[232],"examples:":[233],"Taylor,":[237],"other":[240],"Stability":[247],"analyses":[248],"presented":[250],"each":[252],"applied":[258],"dimensions":[267],"results":[270],"good":[273],"agreement":[274],"current":[276],"state-of-the-art":[277]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
