{"id":"https://openalex.org/W4308404550","doi":"https://doi.org/10.1515/cmam-2022-0002","title":"High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations","display_name":"High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations","publication_year":2022,"publication_date":"2022-11-07","ids":{"openalex":"https://openalex.org/W4308404550","doi":"https://doi.org/10.1515/cmam-2022-0002"},"language":"en","primary_location":{"id":"doi:10.1515/cmam-2022-0002","is_oa":false,"landing_page_url":"https://doi.org/10.1515/cmam-2022-0002","pdf_url":null,"source":{"id":"https://openalex.org/S2765013998","display_name":"Computational Methods in Applied Mathematics","issn_l":"1609-4840","issn":["1609-4840","1609-9389"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Methods in 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/A5100376387","display_name":"Zhi Wang","orcid":"https://orcid.org/0000-0002-2167-830X"},"institutions":[{"id":"https://openalex.org/I21642278","display_name":"Ningxia University","ror":"https://ror.org/04j7b2v61","country_code":"CN","type":"education","lineage":["https://openalex.org/I21642278"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhi Wang","raw_affiliation_strings":["Institute of Applied Mathematics and Mechanics , Ningxia University , Yinchuan , P. R. China"],"affiliations":[{"raw_affiliation_string":"Institute of Applied Mathematics and Mechanics , Ningxia University , Yinchuan , P. R. China","institution_ids":["https://openalex.org/I21642278"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101767197","display_name":"Yongbin Ge","orcid":"https://orcid.org/0000-0001-9622-3134"},"institutions":[{"id":"https://openalex.org/I21642278","display_name":"Ningxia University","ror":"https://ror.org/04j7b2v61","country_code":"CN","type":"education","lineage":["https://openalex.org/I21642278"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yongbin Ge","raw_affiliation_strings":["Institute of Applied Mathematics and Mechanics , Ningxia University , Yinchuan , P. R. China"],"affiliations":[{"raw_affiliation_string":"Institute of Applied Mathematics and Mechanics , Ningxia University , Yinchuan , P. R. China","institution_ids":["https://openalex.org/I21642278"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101501154","display_name":"Hai\u2010Wei Sun","orcid":"https://orcid.org/0000-0002-5507-6083"},"institutions":[{"id":"https://openalex.org/I204512498","display_name":"University of Macau","ror":"https://ror.org/01r4q9n85","country_code":"MO","type":"education","lineage":["https://openalex.org/I204512498"]}],"countries":["MO"],"is_corresponding":false,"raw_author_name":"Hai-Wei Sun","raw_affiliation_strings":["Department of Mathematics , University of Macau , Macau , Macao"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Macau , Macau , Macao","institution_ids":["https://openalex.org/I204512498"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5101767197"],"corresponding_institution_ids":["https://openalex.org/I21642278"],"apc_list":null,"apc_paid":null,"fwci":0.1847,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.48931992,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":98},"biblio":{"volume":"23","issue":"2","first_page":"491","last_page":"516"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/compact-finite-difference","display_name":"Compact finite difference","score":0.8942018747329712},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7898629903793335},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.7202272415161133},{"id":"https://openalex.org/keywords/truncation-error","display_name":"Truncation error","score":0.670508623123169},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5745278596878052},{"id":"https://openalex.org/keywords/truncation","display_name":"Truncation (statistics)","score":0.5642473697662354},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.553034782409668},{"id":"https://openalex.org/keywords/finite-difference-coefficient","display_name":"Finite difference coefficient","score":0.5449915528297424},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5172244310379028},{"id":"https://openalex.org/keywords/finite-difference","display_name":"Finite difference","score":0.5107440948486328},{"id":"https://openalex.org/keywords/helmholtz-free-energy","display_name":"Helmholtz free energy","score":0.5041543245315552},{"id":"https://openalex.org/keywords/finite-difference-method","display_name":"Finite difference method","score":0.5031306147575378},{"id":"https://openalex.org/keywords/laplace-transform","display_name":"Laplace transform","score":0.4927532970905304},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.48973870277404785},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4885343313217163},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.46012070775032043},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.42414993047714233},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.23116835951805115},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.18488633632659912},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.0942571759223938},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08174559473991394}],"concepts":[{"id":"https://openalex.org/C2780829570","wikidata":"https://www.wikidata.org/wiki/Q25113057","display_name":"Compact finite difference","level":3,"score":0.8942018747329712},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7898629903793335},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.7202272415161133},{"id":"https://openalex.org/C104942944","wikidata":"https://www.wikidata.org/wiki/Q3434686","display_name":"Truncation error","level":2,"score":0.670508623123169},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5745278596878052},{"id":"https://openalex.org/C106195933","wikidata":"https://www.wikidata.org/wiki/Q7847935","display_name":"Truncation (statistics)","level":2,"score":0.5642473697662354},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.553034782409668},{"id":"https://openalex.org/C108268861","wikidata":"https://www.wikidata.org/wiki/Q1747602","display_name":"Finite difference coefficient","level":4,"score":0.5449915528297424},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5172244310379028},{"id":"https://openalex.org/C181330731","wikidata":"https://www.wikidata.org/wiki/Q2068418","display_name":"Finite difference","level":2,"score":0.5107440948486328},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.5041543245315552},{"id":"https://openalex.org/C205951836","wikidata":"https://www.wikidata.org/wiki/Q1147751","display_name":"Finite difference method","level":2,"score":0.5031306147575378},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.4927532970905304},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.48973870277404785},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4885343313217163},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.46012070775032043},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.42414993047714233},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.23116835951805115},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.18488633632659912},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0942571759223938},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08174559473991394},{"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"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/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C144468803","wikidata":"https://www.wikidata.org/wiki/Q16965966","display_name":"Mixed finite element method","level":3,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1515/cmam-2022-0002","is_oa":false,"landing_page_url":"https://doi.org/10.1515/cmam-2022-0002","pdf_url":null,"source":{"id":"https://openalex.org/S2765013998","display_name":"Computational Methods in Applied Mathematics","issn_l":"1609-4840","issn":["1609-4840","1609-9389"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Methods in Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1232573025","display_name":null,"funder_award_id":"11902170","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G307924310","display_name":null,"funder_award_id":"12161067","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4397907366","display_name":null,"funder_award_id":"11961054","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4773438052","display_name":null,"funder_award_id":"2018BEE03007","funder_id":"https://openalex.org/F4320334010","funder_display_name":"Key Research and Development Program of Ningxia"},{"id":"https://openalex.org/G4782656278","display_name":null,"funder_award_id":"2020AAC03059","funder_id":"https://openalex.org/F4320322877","funder_display_name":"Natural Science Foundation of Ningxia Province"},{"id":"https://openalex.org/G6652897409","display_name":null,"funder_award_id":"11772165","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G779617455","display_name":null,"funder_award_id":"2022AAC02023","funder_id":"https://openalex.org/F4320322877","funder_display_name":"Natural Science Foundation of Ningxia Province"},{"id":"https://openalex.org/G7959072416","display_name":null,"funder_award_id":"MYRG2020-00224-FST","funder_id":"https://openalex.org/F4320322841","funder_display_name":"Universidade de Macau"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322841","display_name":"Universidade de Macau","ror":"https://ror.org/01r4q9n85"},{"id":"https://openalex.org/F4320322877","display_name":"Natural Science Foundation of Ningxia Province","ror":null},{"id":"https://openalex.org/F4320334010","display_name":"Key Research and Development Program of Ningxia","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":41,"referenced_works":["https://openalex.org/W1488435683","https://openalex.org/W1964822953","https://openalex.org/W1966763308","https://openalex.org/W1969041915","https://openalex.org/W1987066751","https://openalex.org/W1991290104","https://openalex.org/W1995616111","https://openalex.org/W2002710898","https://openalex.org/W2023769913","https://openalex.org/W2030729182","https://openalex.org/W2039057361","https://openalex.org/W2042897750","https://openalex.org/W2047463395","https://openalex.org/W2056361319","https://openalex.org/W2057073736","https://openalex.org/W2070875961","https://openalex.org/W2071881281","https://openalex.org/W2072912212","https://openalex.org/W2100276724","https://openalex.org/W2101615520","https://openalex.org/W2108332067","https://openalex.org/W2150300407","https://openalex.org/W2164639651","https://openalex.org/W2171890171","https://openalex.org/W2340739500","https://openalex.org/W2401178568","https://openalex.org/W2509466404","https://openalex.org/W2522648897","https://openalex.org/W2569928605","https://openalex.org/W2605452465","https://openalex.org/W2609596110","https://openalex.org/W2677589427","https://openalex.org/W2783325651","https://openalex.org/W2903420666","https://openalex.org/W2944280986","https://openalex.org/W2986318811","https://openalex.org/W3037545212","https://openalex.org/W4206097460","https://openalex.org/W4283641523","https://openalex.org/W6600344744","https://openalex.org/W6600459194"],"related_works":["https://openalex.org/W1990789772","https://openalex.org/W2155459193","https://openalex.org/W2340830474","https://openalex.org/W2042176734","https://openalex.org/W2161704621","https://openalex.org/W2007426143","https://openalex.org/W4308404550","https://openalex.org/W2389032076","https://openalex.org/W2133620072","https://openalex.org/W1987066751"],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2],"paper,":[3],"the":[4,22,28,35,40,44,52,56,62,71,75,91,96,107],"sixth-order":[5,23,45,76,83],"compact":[6,24,46,77,84],"finite":[7],"difference":[8,25,47],"schemes":[9,48,109],"for":[10,27],"solving":[11],"two-dimensional":[12],"(2D)":[13],"and":[14,66,93],"three-dimensional":[15],"(3D)":[16],"Helmholtz":[17],"equations":[18],"are":[19,31,49,100,110],"proposed.":[20,50],"Firstly,":[21],"operators":[26],"second-order":[29],"derivatives":[30],"applied":[32],"to":[33,79],"approximate":[34],"Laplace":[36],"operator.":[37],"Meanwhile,":[38],"with":[39],"original":[41],"differential":[42],"equation,":[43],"However,":[51],"truncation":[53,72],"errors":[54],"of":[55,74,95],"proposed":[57],"scheme":[58,78,85],"obviously":[59],"depend":[60],"on":[61],"unknowns,":[63],"source":[64],"function":[65],"wavenumber.":[67],"Thus,":[68],"we":[69],"correct":[70],"error":[73],"obtain":[80],"an":[81],"improved":[82,98,108],"that":[86,106],"is":[87],"more":[88,111],"accurate.":[89,112],"Theoretically,":[90],"convergence":[92],"stability":[94],"present":[97],"method":[99],"proved.":[101],"Finally,":[102],"numerical":[103],"tests":[104],"verify":[105]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2}],"updated_date":"2026-04-07T14:57:38.498316","created_date":"2025-10-10T00:00:00"}