{"id":"https://openalex.org/W3008597509","doi":"https://doi.org/10.1137/19m1289157","title":"On Energy Stable, Maximum-Principle Preserving, Second-Order BDF Scheme with Variable Steps for the Allen--Cahn Equation","display_name":"On Energy Stable, Maximum-Principle Preserving, Second-Order BDF Scheme with Variable Steps for the Allen--Cahn Equation","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3008597509","doi":"https://doi.org/10.1137/19m1289157","mag":"3008597509"},"language":"en","primary_location":{"id":"doi:10.1137/19m1289157","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1289157","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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2003.00421","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5052651480","display_name":"Hong-lin Liao","orcid":"https://orcid.org/0000-0003-0777-6832"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Hong-lin Liao","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5058967726","display_name":"Tao Tang","orcid":"https://orcid.org/0000-0003-2919-2774"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tao Tang","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5042861452","display_name":"Tao Zhou","orcid":"https://orcid.org/0000-0002-6507-8912"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tao Zhou","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5052651480"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3256,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.48656298,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"58","issue":"4","first_page":"2294","last_page":"2314"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9997000098228455,"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":0.9997000098228455,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9994000196456909,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9966999888420105,"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/allen\u2013cahn-equation","display_name":"Allen\u2013Cahn equation","score":0.9191920757293701},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7983438968658447},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.643638014793396},{"id":"https://openalex.org/keywords/backward-differentiation-formula","display_name":"Backward differentiation formula","score":0.6248795986175537},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.5398854613304138},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.507782518863678},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5000486373901367},{"id":"https://openalex.org/keywords/approx","display_name":"Approx","score":0.4980752468109131},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4760848581790924},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4737720787525177},{"id":"https://openalex.org/keywords/scheme","display_name":"Scheme (mathematics)","score":0.44683530926704407},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.4387034773826599},{"id":"https://openalex.org/keywords/work","display_name":"Work (physics)","score":0.4369202256202698},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.4286729097366333},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3959053158760071},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3593929409980774},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1602783501148224},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1486654281616211},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.12672919034957886},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.12503013014793396},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.1080491840839386},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09317436814308167},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08809566497802734}],"concepts":[{"id":"https://openalex.org/C2777166216","wikidata":"https://www.wikidata.org/wiki/Q4732089","display_name":"Allen\u2013Cahn equation","level":2,"score":0.9191920757293701},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7983438968658447},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.643638014793396},{"id":"https://openalex.org/C152522140","wikidata":"https://www.wikidata.org/wiki/Q795746","display_name":"Backward differentiation formula","level":5,"score":0.6248795986175537},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.5398854613304138},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.507782518863678},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5000486373901367},{"id":"https://openalex.org/C2777894999","wikidata":"https://www.wikidata.org/wiki/Q4781758","display_name":"Approx","level":2,"score":0.4980752468109131},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4760848581790924},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4737720787525177},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.44683530926704407},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4387034773826599},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.4369202256202698},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.4286729097366333},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3959053158760071},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3593929409980774},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1602783501148224},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1486654281616211},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.12672919034957886},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.12503013014793396},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.1080491840839386},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09317436814308167},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08809566497802734},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C2126413","wikidata":"https://www.wikidata.org/wiki/Q1779479","display_name":"Collocation method","level":4,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/19m1289157","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1289157","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:arXiv.org:2003.00421","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2003.00421","pdf_url":"https://arxiv.org/pdf/2003.00421","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:3008597509","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2003.00421.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.2003.00421","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2003.00421","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2003.00421","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2003.00421","pdf_url":"https://arxiv.org/pdf/2003.00421","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[{"score":0.7300000190734863,"display_name":"Affordable and clean energy","id":"https://metadata.un.org/sdg/7"}],"awards":[{"id":"https://openalex.org/G1231421488","display_name":null,"funder_award_id":"under","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G1536835442","display_name":null,"funder_award_id":"No. TZ2018001","funder_id":"https://openalex.org/F4320336025","funder_display_name":"Science Challenge Project"},{"id":"https://openalex.org/G2621865199","display_name":null,"funder_award_id":"11731006","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3582785677","display_name":null,"funder_award_id":"TZ201800","funder_id":"https://openalex.org/F4320336025","funder_display_name":"Science Challenge Project"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5311346577","display_name":null,"funder_award_id":"2018001","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G544407669","display_name":null,"funder_award_id":"1008-56SYAH18037","funder_id":"https://openalex.org/F4320322438","funder_display_name":"Nanjing University of Aeronautics and Astronautics"},{"id":"https://openalex.org/G5921281487","display_name":null,"funder_award_id":"number","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5935691477","display_name":null,"funder_award_id":"11571351","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6361732797","display_name":"Insufficient Sleep and Dietary Choices: An Ecologically Valid Examination of the Decision Foundations of Eating Behaviors when Sleep Restricted.","funder_award_id":"2018001","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6967058303","display_name":null,"funder_award_id":"11688101","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7105951358","display_name":null,"funder_award_id":"11822111","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G729797197","display_name":null,"funder_award_id":"11731006","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7608752429","display_name":null,"funder_award_id":"Talent","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8512854826","display_name":null,"funder_award_id":"TZ2018001","funder_id":"https://openalex.org/F4320336025","funder_display_name":"Science Challenge Project"},{"id":"https://openalex.org/G8536560856","display_name":null,"funder_award_id":"TZ2018001","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"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/F4320322438","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794"},{"id":"https://openalex.org/F4320335892","display_name":"Youth Innovation Promotion Association","ror":null},{"id":"https://openalex.org/F4320336025","display_name":"Science Challenge Project","ror":null},{"id":"https://openalex.org/F4320336876","display_name":"National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences","ror":"https://ror.org/00s97k668"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W1512153219","https://openalex.org/W1605821011","https://openalex.org/W1968790851","https://openalex.org/W1973118344","https://openalex.org/W1977638711","https://openalex.org/W2030940697","https://openalex.org/W2031440284","https://openalex.org/W2037963783","https://openalex.org/W2041001120","https://openalex.org/W2043017073","https://openalex.org/W2046528357","https://openalex.org/W2050969603","https://openalex.org/W2058080316","https://openalex.org/W2082621526","https://openalex.org/W2090708196","https://openalex.org/W2112804981","https://openalex.org/W2119682008","https://openalex.org/W2168653054","https://openalex.org/W2328649268","https://openalex.org/W2400894298","https://openalex.org/W2586463154","https://openalex.org/W2589656637","https://openalex.org/W2593562401","https://openalex.org/W2794616828","https://openalex.org/W2799821673","https://openalex.org/W2911401788","https://openalex.org/W2914074116","https://openalex.org/W2917228799","https://openalex.org/W2942783709","https://openalex.org/W2990575151","https://openalex.org/W3018009055","https://openalex.org/W3106441075"],"related_works":["https://openalex.org/W2069608005","https://openalex.org/W3119350078","https://openalex.org/W2734544481","https://openalex.org/W3093318084","https://openalex.org/W3199928648","https://openalex.org/W2963632047","https://openalex.org/W2950235712","https://openalex.org/W2368199140","https://openalex.org/W2301659188","https://openalex.org/W2786736789","https://openalex.org/W2460690371","https://openalex.org/W2606453112","https://openalex.org/W3099925677","https://openalex.org/W2950913345","https://openalex.org/W2432922329","https://openalex.org/W2163438972","https://openalex.org/W2513678179","https://openalex.org/W2357415541","https://openalex.org/W1976445014","https://openalex.org/W2104575663"],"abstract_inverted_index":{"In":[0],"this":[1],"work,":[2],"we":[3,44],"investigate":[4],"the":[5,15,21,29,47,50,56,60,77,89],"two-step":[6],"backward":[7],"differentiation":[8],"formula":[9],"(BDF2)":[10],"with":[11],"nonuniform":[12,22],"grids":[13],"for":[14,46],"Allen--Cahn":[16],"equation.":[17],"We":[18],"show":[19],"that":[20],"BDF2":[23,57],"scheme":[24,58],"is":[25,80],"energy":[26],"stable":[27],"under":[28,59],"time-step":[30,61,69],"ratio":[31,62],"restriction":[32,63],"$r_k:=\\tau_k/\\tau_{k-1}<(3+\\sqrt{17})/2\\approx3.561.$":[33],"Moreover,":[34],"by":[35],"developing":[36],"a":[37,67],"novel":[38],"kernel":[39],"recombination":[40],"and":[41,66],"complementary":[42],"technique,":[43],"show,":[45],"first":[48],"time,":[49],"discrete":[51],"maximum":[52,78],"bound":[53],"principle":[54],"of":[55,74],"$r_k<1+\\sqrt{2}\\approx":[64],"2.414$":[65],"practical":[68],"constraint.":[70],"The":[71],"second-order":[72],"rate":[73],"convergence":[75],"in":[76],"norm":[79],"also":[81],"presented.":[82],"Numerical":[83],"experiments":[84],"are":[85],"provided":[86],"to":[87],"support":[88],"theoretical":[90],"findings.":[91]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-04-16T08:26:57.006410","created_date":"2025-10-10T00:00:00"}
