{"id":"https://openalex.org/W4406417941","doi":"https://doi.org/10.1090/mcom/4067","title":"Global-in-time energy stability analysis for a second-order accurate exponential time differencing Runge\u2013Kutta scheme for the phase field crystal equation","display_name":"Global-in-time energy stability analysis for a second-order accurate exponential time differencing Runge\u2013Kutta scheme for the phase field crystal equation","publication_year":2025,"publication_date":"2025-01-15","ids":{"openalex":"https://openalex.org/W4406417941","doi":"https://doi.org/10.1090/mcom/4067"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/4067","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/4067","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"preprint","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/A5035541867","display_name":"Xiao Li","orcid":"https://orcid.org/0000-0002-3553-9677"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Xiao Li","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081360062","display_name":"Zhonghua Qiao","orcid":"https://orcid.org/0000-0002-0553-1404"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhonghua Qiao","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100417045","display_name":"Cheng Wang","orcid":"https://orcid.org/0000-0003-4220-8080"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cheng Wang","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5039725019","display_name":"Nan Zheng","orcid":"https://orcid.org/0000-0003-2512-7879"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nan Zheng","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5035541867"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":12.68365718,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.94577577,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"95","issue":"358","first_page":"803","last_page":"831"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9864000082015991,"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":0.9864000082015991,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/runge\u2013kutta-methods","display_name":"Runge\u2013Kutta methods","score":0.6133530139923096},{"id":"https://openalex.org/keywords/phase","display_name":"Phase (matter)","score":0.5767845511436462},{"id":"https://openalex.org/keywords/scheme","display_name":"Scheme (mathematics)","score":0.5298433303833008},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.5241979956626892},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.4753420352935791},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.4689309895038605},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4674687385559082},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.45537739992141724},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.40228021144866943},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3853626847267151},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3739126920700073},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3530808985233307},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3031288981437683},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.14792656898498535},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.13515335321426392},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.11423274874687195},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.06888660788536072}],"concepts":[{"id":"https://openalex.org/C181582579","wikidata":"https://www.wikidata.org/wiki/Q725944","display_name":"Runge\u2013Kutta methods","level":3,"score":0.6133530139923096},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.5767845511436462},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.5298433303833008},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.5241979956626892},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4753420352935791},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.4689309895038605},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4674687385559082},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.45537739992141724},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40228021144866943},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3853626847267151},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3739126920700073},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3530808985233307},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3031288981437683},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.14792656898498535},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.13515335321426392},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.11423274874687195},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.06888660788536072},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/mcom/4067","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/4067","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.8500000238418579,"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":56,"referenced_works":["https://openalex.org/W429183865","https://openalex.org/W1597278668","https://openalex.org/W1913288047","https://openalex.org/W1971518097","https://openalex.org/W1979546093","https://openalex.org/W1980158383","https://openalex.org/W1999107555","https://openalex.org/W2000883601","https://openalex.org/W2005859257","https://openalex.org/W2029449803","https://openalex.org/W2031651409","https://openalex.org/W2049165066","https://openalex.org/W2050969603","https://openalex.org/W2057808716","https://openalex.org/W2072801225","https://openalex.org/W2077001143","https://openalex.org/W2083604306","https://openalex.org/W2088291843","https://openalex.org/W2088535056","https://openalex.org/W2090704787","https://openalex.org/W2090738965","https://openalex.org/W2094884329","https://openalex.org/W2127166951","https://openalex.org/W2167347425","https://openalex.org/W2167886272","https://openalex.org/W2184266932","https://openalex.org/W2314914323","https://openalex.org/W2323645990","https://openalex.org/W2501163367","https://openalex.org/W2501507403","https://openalex.org/W2521393946","https://openalex.org/W2526274843","https://openalex.org/W2553175992","https://openalex.org/W2588054282","https://openalex.org/W2593432127","https://openalex.org/W2626341890","https://openalex.org/W2725599832","https://openalex.org/W2735222073","https://openalex.org/W2790424815","https://openalex.org/W2911401788","https://openalex.org/W2913677680","https://openalex.org/W2922030862","https://openalex.org/W2954429508","https://openalex.org/W2954504234","https://openalex.org/W3038878809","https://openalex.org/W3087775895","https://openalex.org/W3096953690","https://openalex.org/W3115671004","https://openalex.org/W4206766080","https://openalex.org/W4211077197","https://openalex.org/W4297724997","https://openalex.org/W4319781739","https://openalex.org/W4391532981","https://openalex.org/W4391617410","https://openalex.org/W4392161959","https://openalex.org/W4402271415"],"related_works":["https://openalex.org/W4206768127","https://openalex.org/W4213270568","https://openalex.org/W4301315183","https://openalex.org/W2157764110","https://openalex.org/W2083551655","https://openalex.org/W4244999877","https://openalex.org/W2121498459","https://openalex.org/W2069270950","https://openalex.org/W2082250026","https://openalex.org/W4224102766"],"abstract_inverted_index":{"The":[0,228],"global-in-time":[1,62,195,210],"energy":[2,47,63,77,176,196,211,226],"estimate":[3,64,114,145,197],"is":[4,119,198,231],"derived":[5],"for":[6,65,80,214,236],"the":[7,18,33,46,66,71,76,89,116,120,124,147,152,156,162,165,175,179,185,194,203,223,243],"second-order":[8,216],"accurate":[9],"exponential":[10],"time":[11,91,158,181],"differencing":[12],"Runge\u2013Kutta":[13,239],"(ETDRK2)":[14],"numerical":[15,68,117,153,218,240],"scheme":[16,69,219],"to":[17,70,206,233,242],"phase":[19],"field":[20],"crystal":[21,29],"(PFC)":[22],"equation,":[23],"a":[24,52,61,95,209,215],"sixth-order":[25],"parabolic":[26],"equation":[27,73],"modeling":[28],"evolution.":[30],"To":[31],"recover":[32],"value":[34,163],"of":[35,115,151,164,222],"stabilization":[36,166],"constant,":[37],"some":[38],"local-in-time":[39],"convergence":[40],"analysis":[41,213],"has":[42],"been":[43],"reported,":[44],"and":[45,160],"stability":[48,212],"becomes":[49],"available":[50,235],"over":[51],"fixed":[53],"final":[54,82],"time.":[55,83],"In":[56],"this":[57],"work,":[58],"we":[59],"develop":[60],"ETDRK2":[67],"PFC":[72],"by":[74,192],"showing":[75],"dissipation":[78,177],"property":[79],"any":[81],"An":[84],"<italic>a":[85],"priori</italic>":[86],"assumption":[87],"at":[88,155,178],"previous":[90],"step,":[92,159,182],"combined":[93],"with":[94],"single-step":[96],"<inline-formula":[97,128],"content-type=\"math/mathml\">":[98,129],"<mml:math":[99,130],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[100,131],"alttext=\"upper":[101,132],"H":[102,133],"squared\">":[103,134],"<mml:semantics>":[104,135],"<mml:msup>":[105,136],"<mml:mi>H</mml:mi>":[106,137],"<mml:mn>2</mml:mn>":[107,138],"</mml:msup>":[108,139],"<mml:annotation":[109,140],"encoding=\"application/x-tex\">H^2</mml:annotation>":[110,141],"</mml:semantics>":[111,142],"</mml:math>":[112,143],"</inline-formula>":[113,144],"solution,":[118],"key":[121],"point":[122],"in":[123,220],"analysis.":[125],"Such":[126],"an":[127],"recovers":[146],"maximum":[148],"norm":[149],"bound":[150],"solution":[154],"next":[157,180],"then":[161,193],"parameter":[167],"can":[168,188],"be":[169,189,234],"theoretically":[170,207],"justified.":[171],"This":[172,200],"justification":[173],"ensures":[174],"so":[183],"that":[184],"mathematical":[186],"induction":[187],"effectively":[190],"applied,":[191],"accomplished.":[199],"paper":[201],"represents":[202],"first":[204],"effort":[205],"establish":[208],"stabilized":[217],"terms":[221],"original":[224],"free":[225],"functional.":[227],"presented":[229],"methodology":[230],"expected":[232],"many":[237],"other":[238],"schemes":[241],"gradient":[244],"flow":[245],"equations.":[246]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":7}],"updated_date":"2026-02-15T07:36:51.736424","created_date":"2025-10-10T00:00:00"}
