{"id":"https://openalex.org/W2084221711","doi":"https://doi.org/10.1137/070680965","title":"Numerical Tests of a Phase Field Model with Second Order Accuracy","display_name":"Numerical Tests of a Phase Field Model with Second Order Accuracy","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2084221711","doi":"https://doi.org/10.1137/070680965","mag":"2084221711"},"language":"en","primary_location":{"id":"doi:10.1137/070680965","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070680965","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Applied Mathematics","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/A5081095860","display_name":"Gunduz Caginalp","orcid":"https://orcid.org/0000-0003-4137-6680"},"institutions":[{"id":"https://openalex.org/I170201317","display_name":"University of Pittsburgh","ror":"https://ror.org/01an3r305","country_code":"US","type":"education","lineage":["https://openalex.org/I170201317"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Gunduz Caginalp","raw_affiliation_strings":["University of Pittsburgh"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Pittsburgh","institution_ids":["https://openalex.org/I170201317"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087971771","display_name":"Xinfu Chen","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xinfu Chen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5109038008","display_name":"Christof Eck","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Christof Eck","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5081095860"],"corresponding_institution_ids":["https://openalex.org/I170201317"],"apc_list":null,"apc_paid":null,"fwci":0.1751,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.54377283,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"68","issue":"6","first_page":"1518","last_page":"1534"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10670","display_name":"Aluminum Alloy Microstructure Properties","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace 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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9923999905586243,"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/computation","display_name":"Computation","score":0.6857749819755554},{"id":"https://openalex.org/keywords/limiting","display_name":"Limiting","score":0.6241601705551147},{"id":"https://openalex.org/keywords/phase","display_name":"Phase (matter)","score":0.6054462194442749},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.5805633664131165},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5482339859008789},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5279910564422607},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5270422101020813},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.49534764885902405},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.46572014689445496},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.44835197925567627},{"id":"https://openalex.org/keywords/phase-field-models","display_name":"Phase field models","score":0.4412805438041687},{"id":"https://openalex.org/keywords/phase-boundary","display_name":"Phase boundary","score":0.43288499116897583},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.34392470121383667},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3014488220214844},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10018002986907959},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.09390103816986084},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.08050909638404846}],"concepts":[{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.6857749819755554},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.6241601705551147},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.6054462194442749},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.5805633664131165},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5482339859008789},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5279910564422607},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5270422101020813},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.49534764885902405},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.46572014689445496},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.44835197925567627},{"id":"https://openalex.org/C2780568864","wikidata":"https://www.wikidata.org/wiki/Q1640482","display_name":"Phase field models","level":3,"score":0.4412805438041687},{"id":"https://openalex.org/C188324986","wikidata":"https://www.wikidata.org/wiki/Q7180938","display_name":"Phase boundary","level":3,"score":0.43288499116897583},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34392470121383667},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3014488220214844},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10018002986907959},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.09390103816986084},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.08050909638404846},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/070680965","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070680965","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Applied Mathematics","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":33,"referenced_works":["https://openalex.org/W592693156","https://openalex.org/W1599497244","https://openalex.org/W1782454978","https://openalex.org/W1969450068","https://openalex.org/W1975470728","https://openalex.org/W1976106537","https://openalex.org/W1977032047","https://openalex.org/W1981412090","https://openalex.org/W1990533693","https://openalex.org/W1999438643","https://openalex.org/W2003074491","https://openalex.org/W2009181446","https://openalex.org/W2018573482","https://openalex.org/W2019034989","https://openalex.org/W2020227380","https://openalex.org/W2029631131","https://openalex.org/W2037913112","https://openalex.org/W2041001120","https://openalex.org/W2056909920","https://openalex.org/W2061500684","https://openalex.org/W2079028266","https://openalex.org/W2086238793","https://openalex.org/W2132701476","https://openalex.org/W2140458275","https://openalex.org/W2142734831","https://openalex.org/W2156659007","https://openalex.org/W2162070653","https://openalex.org/W2169000983","https://openalex.org/W2327000728","https://openalex.org/W3022593705","https://openalex.org/W3028131295","https://openalex.org/W3085091986","https://openalex.org/W3203633735"],"related_works":["https://openalex.org/W2029404707","https://openalex.org/W4285325679","https://openalex.org/W1928239295","https://openalex.org/W4282982070","https://openalex.org/W2487963770","https://openalex.org/W4300274844","https://openalex.org/W155385625","https://openalex.org/W1985351605","https://openalex.org/W2133690079","https://openalex.org/W2111215486"],"abstract_inverted_index":{"Numerical":[0,96],"computations":[1,97],"are":[2],"performed":[3],"for":[4,11,46,117,133],"a":[5,99],"recently":[6],"derived":[7],"phase":[8,26,82,86],"field":[9,27,87],"model":[10,28,74,88,111],"the":[12,39,51,57,60,65,68,72,76,81,85,105,113,118,125,131],"interface":[13,42,70],"between":[14,67],"two":[15],"phases.":[16],"The":[17,137],"rigorous":[18],"results":[19],"indicate":[20],"that":[21],"solutions":[22,37,107,116],"to":[23,36],"this":[24,109],"new":[25,110],"should":[29],"converge":[30],"more":[31],"rapidly":[32],"than":[33,94],"traditional":[34],"ones":[35],"of":[38,50,59,71,80,90,108,130,139,143],"corresponding":[40],"sharp":[41,69],"(free":[43],"boundary)":[44],"formulation":[45],"sufficiently":[47,134],"small":[48,135],"values":[49],"approximation":[52],"parameter":[53],"$\\varepsilon":[54],"$":[55],"representing":[56],"thickness":[58],"interfacial":[61],"region.":[62],"In":[63],"particular,":[64],"distance":[66],"limiting":[73,119],"and":[75,123],"zero":[77],"level":[78],"set":[79],"function":[83],"in":[84,146],"is":[89],"order":[91,127],"$\\varepsilon^2$":[92],"rather":[93],"$\\varepsilon$.":[95,136],"within":[98],"three-dimensional":[100],"spherically":[101],"symmetric":[102],"setting":[103],"compare":[104],"computed":[106],"with":[112],"known":[114],"exact":[115],"free":[120],"boundary":[121],"problem":[122],"confirm":[124],"second":[126],"accuracy":[128],"predictions":[129],"theory":[132],"sets":[138],"parameters":[140],"include":[141],"those":[142],"succinonitrile":[144],"used":[145],"dendritic":[147],"experiments.":[148]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":5}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}
