{"id":"https://openalex.org/W3000992358","doi":"https://doi.org/10.1137/18m1213579","title":"Arbitrarily High-Order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models","display_name":"Arbitrarily High-Order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3000992358","doi":"https://doi.org/10.1137/18m1213579","mag":"3000992358"},"language":"en","primary_location":{"id":"doi:10.1137/18m1213579","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1213579","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","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/A5082208381","display_name":"Yuezheng Gong","orcid":"https://orcid.org/0000-0002-5381-7183"},"institutions":[{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuezheng Gong","raw_affiliation_strings":["College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;"],"raw_orcid":"https://orcid.org/0000-0002-5381-7183","affiliations":[{"raw_affiliation_string":"College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;","institution_ids":["https://openalex.org/I9842412"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5000892024","display_name":"Jia Zhao","orcid":"https://orcid.org/0000-0001-6021-0841"},"institutions":[{"id":"https://openalex.org/I121980950","display_name":"Utah State University","ror":"https://ror.org/00h6set76","country_code":"US","type":"education","lineage":["https://openalex.org/I121980950"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jia Zhao","raw_affiliation_strings":["Department of Mathematics & Statistics, Utah State University, Logan, UT 84322, USA;"],"raw_orcid":"https://orcid.org/0000-0001-6021-0841","affiliations":[{"raw_affiliation_string":"Department of Mathematics & Statistics, Utah State University, Logan, UT 84322, USA;","institution_ids":["https://openalex.org/I121980950"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035473719","display_name":"Qi Wang","orcid":"https://orcid.org/0000-0001-9928-2703"},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]},{"id":"https://openalex.org/I205237279","display_name":"Nankai University","ror":"https://ror.org/01y1kjr75","country_code":"CN","type":"education","lineage":["https://openalex.org/I205237279"]},{"id":"https://openalex.org/I4210139958","display_name":"Beijing Computational Science Research Center","ror":"https://ror.org/04tavf782","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4210139958"]}],"countries":["CN","US"],"is_corresponding":false,"raw_author_name":"Qi Wang","raw_affiliation_strings":["Beijing Computational Science Research Center, Beijing, China, 100193; Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA; School of Materials and Engineering,","Nankai University, Tianjin, China, 300084;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Beijing Computational Science Research Center, Beijing, China, 100193; Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA; School of Materials and Engineering,","institution_ids":["https://openalex.org/I155781252","https://openalex.org/I4210139958"]},{"raw_affiliation_string":"Nankai University, Tianjin, China, 300084;","institution_ids":["https://openalex.org/I205237279"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":7.6263,"has_fulltext":false,"cited_by_count":74,"citation_normalized_percentile":{"value":0.98885057,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"42","issue":"1","first_page":"B135","last_page":"B156"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9972000122070312,"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9972000122070312,"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/T10402","display_name":"Phase Equilibria and Thermodynamics","score":0.9955000281333923,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical 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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.991599977016449,"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/balanced-flow","display_name":"Balanced flow","score":0.8135415315628052},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.708888053894043},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6926718354225159},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.6666300892829895},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.6634964942932129},{"id":"https://openalex.org/keywords/flow","display_name":"Flow (mathematics)","score":0.5765806436538696},{"id":"https://openalex.org/keywords/dissipation","display_name":"Dissipation","score":0.574087917804718},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.518675684928894},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.4799007177352905},{"id":"https://openalex.org/keywords/collocation","display_name":"Collocation (remote sensing)","score":0.46166497468948364},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.45630449056625366},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4372463822364807},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2558137774467468},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16177517175674438},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12077420949935913}],"concepts":[{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.8135415315628052},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.708888053894043},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6926718354225159},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.6666300892829895},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.6634964942932129},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.5765806436538696},{"id":"https://openalex.org/C135402231","wikidata":"https://www.wikidata.org/wiki/Q898440","display_name":"Dissipation","level":2,"score":0.574087917804718},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.518675684928894},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4799007177352905},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.46166497468948364},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.45630449056625366},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4372463822364807},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2558137774467468},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16177517175674438},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12077420949935913},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/18m1213579","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1213579","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:digitalcommons.usu.edu:mathsci_facpub-1406","is_oa":false,"landing_page_url":"https://digitalcommons.usu.edu/mathsci_facpub/266","pdf_url":null,"source":{"id":"https://openalex.org/S4377196327","display_name":"Digital Commons - USU (Utah State University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I121980950","host_organization_name":"Utah State University","host_organization_lineage":["https://openalex.org/I121980950"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics and Statistics Faculty Publications","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","score":0.8999999761581421,"display_name":"Affordable and clean energy"}],"awards":[{"id":"https://openalex.org/G1364899216","display_name":null,"funder_award_id":"U1530401","funder_id":"https://openalex.org/F4320336475","funder_display_name":"National Safety Academic Fund"},{"id":"https://openalex.org/G1420654066","display_name":null,"funder_award_id":"#91630207","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2687787774","display_name":null,"funder_award_id":"BK20180413","funder_id":"https://openalex.org/F4320322769","funder_display_name":"Natural Science Foundation of Jiangsu Province"},{"id":"https://openalex.org/G4165402629","display_name":"Collaborative Research: Computational Modeling of How Living Cells Utilize Liquid-Liquid Phase Separation to Organize Chemical Compartments","funder_award_id":"1816783","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4210119354","display_name":null,"funder_award_id":"BK20180413","funder_id":"https://openalex.org/F4320334982","funder_display_name":"Basic Research Program of Jiangsu Province"},{"id":"https://openalex.org/G4356423287","display_name":null,"funder_award_id":"OIA-1655740","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4542483587","display_name":"RII Track 1: Materials Assembly and Design Excellence in South Carolina: MADE in SC","funder_award_id":"1655740","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5017951845","display_name":null,"funder_award_id":"U1530401","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G581947329","display_name":null,"funder_award_id":"#NSAF-U1530401","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5968757890","display_name":"Collaborative Research:   Kinetic to Continuum Modeling of Active Anisotropic Fluids","funder_award_id":"1517347","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7084207421","display_name":"\u6d3b\u6027\u7684\u3001\u5404\u5411\u5f02\u6027\u6d41\u4f53\u7684\u6570\u5b66\u5efa\u6a21\u5206\u6790\u4e0e\u8ba1\u7b97","funder_award_id":"11571032","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7997164568","display_name":null,"funder_award_id":"U1530401","funder_id":"https://openalex.org/F4320335578","funder_display_name":"NSAF Joint Fund"},{"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"},{"id":"https://openalex.org/G8484640360","display_name":null,"funder_award_id":"11801269","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8954733464","display_name":"Collaborative Research: Computational Modeling of How Living Cells Utilize Liquid-Liquid Phase Separation to Organize Chemical Compartments","funder_award_id":"1815921","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321605","display_name":"Government of Jiangsu Province","ror":"https://ror.org/004svx814"},{"id":"https://openalex.org/F4320322769","display_name":"Natural Science Foundation of Jiangsu Province","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334982","display_name":"Basic Research Program of Jiangsu Province","ror":null},{"id":"https://openalex.org/F4320335578","display_name":"NSAF Joint Fund","ror":null},{"id":"https://openalex.org/F4320336475","display_name":"National Safety Academic Fund","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":48,"referenced_works":["https://openalex.org/W1626544264","https://openalex.org/W1920690606","https://openalex.org/W1976156829","https://openalex.org/W1994870183","https://openalex.org/W2013307001","https://openalex.org/W2025043986","https://openalex.org/W2025944355","https://openalex.org/W2029449803","https://openalex.org/W2030350543","https://openalex.org/W2050969603","https://openalex.org/W2104346351","https://openalex.org/W2112804981","https://openalex.org/W2118152112","https://openalex.org/W2119682008","https://openalex.org/W2121954553","https://openalex.org/W2136727992","https://openalex.org/W2152024595","https://openalex.org/W2152501918","https://openalex.org/W2153986075","https://openalex.org/W2168653054","https://openalex.org/W2309796220","https://openalex.org/W2328649268","https://openalex.org/W2394745169","https://openalex.org/W2408542077","https://openalex.org/W2473369144","https://openalex.org/W2519077923","https://openalex.org/W2521393946","https://openalex.org/W2542505516","https://openalex.org/W2562132457","https://openalex.org/W2587203146","https://openalex.org/W2588054282","https://openalex.org/W2626341890","https://openalex.org/W2735222073","https://openalex.org/W2735843500","https://openalex.org/W2766149883","https://openalex.org/W2787186442","https://openalex.org/W2789891565","https://openalex.org/W2790418049","https://openalex.org/W2796453292","https://openalex.org/W2802568736","https://openalex.org/W2913950299","https://openalex.org/W2962689056","https://openalex.org/W2962815619","https://openalex.org/W2962906715","https://openalex.org/W2963333235","https://openalex.org/W2963533073","https://openalex.org/W2992498009","https://openalex.org/W4244360209"],"related_works":["https://openalex.org/W2370840338","https://openalex.org/W2347486132","https://openalex.org/W2388641108","https://openalex.org/W2316789606","https://openalex.org/W2350340797","https://openalex.org/W2368317224","https://openalex.org/W4293224283","https://openalex.org/W2950501077","https://openalex.org/W2370651559","https://openalex.org/W4385473842"],"abstract_inverted_index":{"We":[0],"present":[1],"a":[2,41,53,71,98],"systematic":[3],"approach":[4],"to":[5,74,92,119,136,166],"developing":[6],"arbitrarily":[7,104],"high-order,":[8],"unconditionally":[9,121],"energy":[10,22,27,45,61,122],"stable":[11,62],"numerical":[12,63,80,143,154],"schemes":[13,126,147],"for":[14,107],"thermodynamically":[15,169],"consistent":[16,170],"gradient":[17,33,108,132],"flow":[18,34,109,133],"models":[19,134],"that":[20,160],"satisfy":[21],"dissipation":[23],"laws.":[24],"Utilizing":[25],"the":[26,32,49,66,76,83,88,94],"quadratization":[28],"method,":[29],"we":[30,56,69,86],"formulate":[31],"model":[35],"into":[36],"an":[37,103],"equivalent":[38,50,95],"form":[39,51,96],"with":[40,52,97],"corresponding":[42],"quadratic":[43,54,99],"free":[44],"functional.":[46],"Based":[47],"on":[48],"energy,":[55,100],"propose":[57],"two":[58],"classes":[59],"of":[60,78],"approximations.":[64],"In":[65,82],"first":[67],"approach,":[68,85],"use":[70],"prediction-correction":[72],"strategy":[73],"improve":[75],"accuracy":[77,139],"linear":[79],"schemes.":[81],"second":[84],"adopt":[87],"Gaussian":[89],"collocation":[90],"method":[91],"discretize":[93],"arriving":[101],"at":[102],"high-order":[105],"scheme":[106],"models.":[110,172],"Schemes":[111],"derived":[112],"using":[113],"both":[114],"approaches":[115],"are":[116,127,148,156],"proved":[117],"rigorously":[118],"be":[120,163],"stable.":[123],"The":[124],"proposed":[125],"then":[128],"implemented":[129],"in":[130],"four":[131],"numerically":[135],"demonstrate":[137],"their":[138],"and":[140],"effectiveness.":[141],"Detailed":[142],"comparisons":[144],"among":[145],"these":[146],"carried":[149],"out":[150],"as":[151],"well.":[152],"These":[153],"strategies":[155],"rather":[157],"general":[158],"so":[159],"they":[161],"can":[162],"readily":[164],"generalized":[165],"solve":[167],"any":[168],"PDE":[171]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":13},{"year":2024,"cited_by_count":9},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":14},{"year":2021,"cited_by_count":15},{"year":2020,"cited_by_count":9},{"year":2019,"cited_by_count":1}],"updated_date":"2026-08-06T08:24:18.245995","created_date":"2025-10-10T00:00:00"}