{"id":"https://openalex.org/W4296849258","doi":"https://doi.org/10.3390/e24101336","title":"A Modular Grad-Div Stabilization Method for Time-Dependent Thermally Coupled MHD Equations","display_name":"A Modular Grad-Div Stabilization Method for Time-Dependent Thermally Coupled MHD Equations","publication_year":2022,"publication_date":"2022-09-22","ids":{"openalex":"https://openalex.org/W4296849258","doi":"https://doi.org/10.3390/e24101336","pmid":"https://pubmed.ncbi.nlm.nih.gov/37420356"},"language":"en","primary_location":{"id":"doi:10.3390/e24101336","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24101336","pdf_url":"https://www.mdpi.com/1099-4300/24/10/1336/pdf?version=1663848912","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/24/10/1336/pdf?version=1663848912","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016048648","display_name":"Xianzhu Li","orcid":null},"institutions":[{"id":"https://openalex.org/I96908189","display_name":"Xinjiang University","ror":"https://ror.org/059gw8r13","country_code":"CN","type":"education","lineage":["https://openalex.org/I96908189"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xianzhu Li","raw_affiliation_strings":["College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China","institution_ids":["https://openalex.org/I96908189"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5076978683","display_name":"Haiyan Su","orcid":"https://orcid.org/0000-0003-1722-8936"},"institutions":[{"id":"https://openalex.org/I96908189","display_name":"Xinjiang University","ror":"https://ror.org/059gw8r13","country_code":"CN","type":"education","lineage":["https://openalex.org/I96908189"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Haiyan Su","raw_affiliation_strings":["College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China","institution_ids":["https://openalex.org/I96908189"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5076978683"],"corresponding_institution_ids":["https://openalex.org/I96908189"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.4195,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.49608133,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"24","issue":"10","first_page":"1336","last_page":"1336"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9995999932289124,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9995999932289124,"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9993000030517578,"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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9990000128746033,"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/magnetohydrodynamic-drive","display_name":"Magnetohydrodynamic drive","score":0.8300681114196777},{"id":"https://openalex.org/keywords/magnetohydrodynamics","display_name":"Magnetohydrodynamics","score":0.7853220701217651},{"id":"https://openalex.org/keywords/modular-design","display_name":"Modular design","score":0.7522878050804138},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.7409031391143799},{"id":"https://openalex.org/keywords/divergence","display_name":"Divergence (linguistics)","score":0.6177785992622375},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5807934999465942},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5512599945068359},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.468520849943161},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.42232292890548706},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.38289427757263184},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1941957175731659},{"id":"https://openalex.org/keywords/plasma","display_name":"Plasma","score":0.12662655115127563}],"concepts":[{"id":"https://openalex.org/C42471609","wikidata":"https://www.wikidata.org/wiki/Q1779048","display_name":"Magnetohydrodynamic drive","level":4,"score":0.8300681114196777},{"id":"https://openalex.org/C31532427","wikidata":"https://www.wikidata.org/wiki/Q2549249","display_name":"Magnetohydrodynamics","level":3,"score":0.7853220701217651},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.7522878050804138},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.7409031391143799},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.6177785992622375},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5807934999465942},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5512599945068359},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.468520849943161},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42232292890548706},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.38289427757263184},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1941957175731659},{"id":"https://openalex.org/C82706917","wikidata":"https://www.wikidata.org/wiki/Q10251","display_name":"Plasma","level":2,"score":0.12662655115127563},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e24101336","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24101336","pdf_url":"https://www.mdpi.com/1099-4300/24/10/1336/pdf?version=1663848912","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:37420356","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/37420356","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:ba88aa4f1f6f49c399c049116fc04e69","is_oa":true,"landing_page_url":"https://doaj.org/article/ba88aa4f1f6f49c399c049116fc04e69","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 24, Iss 10, p 1336 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/24/10/1336/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e24101336","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy; Volume 24; Issue 10; Pages: 1336","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:9601374","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/9601374","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e24101336","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24101336","pdf_url":"https://www.mdpi.com/1099-4300/24/10/1336/pdf?version=1663848912","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.6100000143051147}],"awards":[{"id":"https://openalex.org/G2998767453","display_name":null,"funder_award_id":"12061076, 12126361","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4443476256","display_name":null,"funder_award_id":"12126361","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5848258319","display_name":null,"funder_award_id":"0 and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6729461843","display_name":"\u5927\u5bc6\u5ea6\u6bd4\u4e24\u76f8\u4e0d\u53ef\u538b\u78c1\u6d41\u4f53\u52a8\u529b\u5b66\u6269\u6563\u754c\u9762\u6a21\u578b\u53ca\u5176\u9ad8\u6548\u7b97\u6cd5\u7814\u7a76","funder_award_id":"12061076","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8045602365","display_name":null,"funder_award_id":"11701493","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"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4296849258.pdf","grobid_xml":"https://content.openalex.org/works/W4296849258.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W209265777","https://openalex.org/W1974461496","https://openalex.org/W1987393363","https://openalex.org/W2004423722","https://openalex.org/W2006454864","https://openalex.org/W2036207036","https://openalex.org/W2043605704","https://openalex.org/W2045478774","https://openalex.org/W2047430733","https://openalex.org/W2061423965","https://openalex.org/W2078857037","https://openalex.org/W2093225505","https://openalex.org/W2094009683","https://openalex.org/W2594247582","https://openalex.org/W2744050773","https://openalex.org/W2765295126","https://openalex.org/W2775518998","https://openalex.org/W2810151696","https://openalex.org/W2891023908","https://openalex.org/W2967609366","https://openalex.org/W2970105038","https://openalex.org/W2998280745","https://openalex.org/W3007244759","https://openalex.org/W3015401808","https://openalex.org/W3021359190","https://openalex.org/W3043688784","https://openalex.org/W3095793062","https://openalex.org/W3110353360","https://openalex.org/W4220716422","https://openalex.org/W4220970297","https://openalex.org/W4224442558","https://openalex.org/W4248450032","https://openalex.org/W4280622867","https://openalex.org/W4281747701","https://openalex.org/W4293230911","https://openalex.org/W4301283118","https://openalex.org/W6662436231","https://openalex.org/W6783975827","https://openalex.org/W6838539084"],"related_works":["https://openalex.org/W2386528229","https://openalex.org/W2049123749","https://openalex.org/W1482473826","https://openalex.org/W2072075229","https://openalex.org/W2028882311","https://openalex.org/W1973727899","https://openalex.org/W2096223544","https://openalex.org/W2031269569","https://openalex.org/W2914179537","https://openalex.org/W2771870683"],"abstract_inverted_index":{"In":[0,57],"this":[1,69],"paper,":[2],"we":[3,59],"consider":[4],"a":[5],"fully":[6],"discrete":[7],"modular":[8],"grad-div":[9,54,86],"stabilization":[10,55],"algorithm":[11,25,84],"for":[12,46],"time-dependent":[13],"thermally":[14],"coupled":[15],"magnetohydrodynamic":[16],"(MHD)":[17],"equations.":[18],"The":[19],"main":[20],"idea":[21],"of":[22,39,49,68],"the":[23,36,43,50,61,83],"proposed":[24],"is":[26],"to":[27,34],"add":[28],"an":[29],"extra":[30],"minimally":[31],"intrusive":[32],"module":[33],"penalize":[35],"divergence":[37],"errors":[38],"velocity":[40],"and":[41,53,64,77],"improve":[42],"computational":[44],"efficiency":[45],"increasing":[47],"values":[48],"Reynolds":[51],"number":[52],"parameters.":[56],"addition,":[58],"provide":[60],"unconditional":[62],"stability":[63],"optimal":[65],"convergence":[66],"analysis":[67],"algorithm.":[70],"Finally,":[71],"several":[72],"numerical":[73],"experiments":[74],"are":[75],"performed":[76],"further":[78],"indicated":[79],"these":[80],"advantages":[81],"over":[82],"without":[85],"stabilization.":[87]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}