{"id":"https://openalex.org/W2026736496","doi":"https://doi.org/10.1137/s0895479897316588","title":"Iterative Solution of the Helmholtz Equation by a Second-Order Method","display_name":"Iterative Solution of the Helmholtz Equation by a Second-Order Method","publication_year":1999,"publication_date":"1999-01-01","ids":{"openalex":"https://openalex.org/W2026736496","doi":"https://doi.org/10.1137/s0895479897316588","mag":"2026736496"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479897316588","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479897316588","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Matrix Analysis and Applications","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/A5056523037","display_name":"Kurt Otto","orcid":null},"institutions":[{"id":"https://openalex.org/I123387679","display_name":"Uppsala University","ror":"https://ror.org/048a87296","country_code":"SE","type":"education","lineage":["https://openalex.org/I123387679"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"Kurt Otto","raw_affiliation_strings":["Department of Scienti c Computing, Uppsala University, Box 120, S-751 04 Uppsala, Sweden"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Scienti c Computing, Uppsala University, Box 120, S-751 04 Uppsala, Sweden","institution_ids":["https://openalex.org/I123387679"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072928406","display_name":"Elisabeth Larsson","orcid":"https://orcid.org/0000-0003-1154-9587"},"institutions":[{"id":"https://openalex.org/I123387679","display_name":"Uppsala University","ror":"https://ror.org/048a87296","country_code":"SE","type":"education","lineage":["https://openalex.org/I123387679"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"Elisabeth Larsson","raw_affiliation_strings":["Department of Scienti c Computing, Uppsala University, Box 120, S-751 04 Uppsala, Sweden"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Scienti c Computing, Uppsala University, Box 120, S-751 04 Uppsala, Sweden","institution_ids":["https://openalex.org/I123387679"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.2296,"has_fulltext":false,"cited_by_count":44,"citation_normalized_percentile":{"value":0.93756878,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"21","issue":"1","first_page":"209","last_page":"229"},"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/T10514","display_name":"Numerical methods in engineering","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/preconditioner","display_name":"Preconditioner","score":0.9571224451065063},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7863119840621948},{"id":"https://openalex.org/keywords/krylov-subspace","display_name":"Krylov subspace","score":0.6304274201393127},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.5934014916419983},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5577954053878784},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.5277503132820129},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.5215047001838684},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4915352761745453},{"id":"https://openalex.org/keywords/generalized-minimal-residual-method","display_name":"Generalized minimal residual method","score":0.47911810874938965},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4574841558933258},{"id":"https://openalex.org/keywords/gaussian-elimination","display_name":"Gaussian elimination","score":0.4281516671180725},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.40088656544685364},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.3126509189605713},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.28575563430786133},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.205888569355011},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06508392095565796}],"concepts":[{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.9571224451065063},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7863119840621948},{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.6304274201393127},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.5934014916419983},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5577954053878784},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.5277503132820129},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.5215047001838684},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4915352761745453},{"id":"https://openalex.org/C155332342","wikidata":"https://www.wikidata.org/wiki/Q1432976","display_name":"Generalized minimal residual method","level":3,"score":0.47911810874938965},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4574841558933258},{"id":"https://openalex.org/C126312332","wikidata":"https://www.wikidata.org/wiki/Q2658","display_name":"Gaussian elimination","level":3,"score":0.4281516671180725},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.40088656544685364},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.3126509189605713},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.28575563430786133},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.205888569355011},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06508392095565796},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/s0895479897316588","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479897316588","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.22.8928","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.8928","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.umd.edu/Library/TRs/CS-TR-3727/CS-TR-3727.ps.Z","raw_type":"text"},{"id":"pmh:oai:drum.lib.umd.edu:1903/866","is_oa":false,"landing_page_url":"http://hdl.handle.net/1903/866","pdf_url":null,"source":{"id":"https://openalex.org/S4306402644","display_name":"Digital Repository at the University of Maryland (University of Maryland College Park)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I66946132","host_organization_name":"University of Maryland, College Park","host_organization_lineage":["https://openalex.org/I66946132"],"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":"Technical Report"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W18468166","https://openalex.org/W574635522","https://openalex.org/W584890378","https://openalex.org/W1560263223","https://openalex.org/W1647103664","https://openalex.org/W1973808722","https://openalex.org/W1976538983","https://openalex.org/W1982772190","https://openalex.org/W1986293636","https://openalex.org/W1990619251","https://openalex.org/W2000537069","https://openalex.org/W2005041367","https://openalex.org/W2005733061","https://openalex.org/W2007657616","https://openalex.org/W2032231305","https://openalex.org/W2036155965","https://openalex.org/W2037048184","https://openalex.org/W2037092148","https://openalex.org/W2048269627","https://openalex.org/W2056978740","https://openalex.org/W2064733018","https://openalex.org/W2069595221","https://openalex.org/W2071897054","https://openalex.org/W2078272972","https://openalex.org/W2078498319","https://openalex.org/W2082761676","https://openalex.org/W2085067463","https://openalex.org/W2100384730","https://openalex.org/W2134572726","https://openalex.org/W2140153041","https://openalex.org/W3003821699","https://openalex.org/W3111897257"],"related_works":["https://openalex.org/W2042846224","https://openalex.org/W3157581606","https://openalex.org/W2113830538","https://openalex.org/W2358905935","https://openalex.org/W2942315554","https://openalex.org/W2804461272","https://openalex.org/W2040391781","https://openalex.org/W2382988560","https://openalex.org/W2024149451","https://openalex.org/W4322732133"],"abstract_inverted_index":{"The":[0,15,29,93],"numerical":[1],"solution":[2],"of":[3,21,45,51,89],"the":[4,19,49,87,113,117,130,133,142],"Helmholtz":[5,75],"equation":[6],"subject":[7],"to":[8],"nonlocal":[9],"radiation":[10],"boundary":[11],"conditions":[12],"is":[13,18,31,58,65,91,97],"studied.":[14],"specific":[16],"problem":[17,30],"propagation":[20],"hydroacoustic":[22],"waves":[23],"in":[24,41,119],"a":[25,34,42,53,62,72,100,138],"two-dimensional":[26],"curvilinear":[27],"duct.":[28],"discretized":[32],"with":[33,99,108,125],"second-order":[35],"accurate":[36],"finite":[37],"difference":[38],"method,":[39],"resulting":[40],"linear":[43],"system":[44,50],"equations.":[46],"To":[47],"solve":[48],"equations,":[52],"preconditioned":[54,114],"Krylov":[55],"subspace":[56],"method":[57],"employed.":[59],"We":[60],"construct":[61],"preconditioner":[63,96,136,144],"that":[64,86],"based":[66],"on":[67],"fast":[68,74,94,134],"transforms":[69],"and":[70,106],"yields":[71,137],"direct":[73],"solver":[76],"for":[77,82],"rectangular":[78],"domains.":[79],"Numerical":[80],"experiments":[81],"curved":[83],"ducts":[84],"demonstrate":[85],"rate":[88],"convergence":[90],"high.":[92],"transform":[95,135],"compared":[98,124],"symmetric":[101],"successive":[102],"over-relaxation":[103],"(SSOR)":[104],"preconditioner,":[105],"also":[107],"band":[109,126,148],"Gaussian":[110,127,149],"elimination.":[111,128,150],"For":[112],"iterative":[115],"methods,":[116],"gains":[118],"storage":[120],"requirement":[121],"are":[122],"large":[123],"Regarding":[129],"arithmetic":[131],"complexity,":[132],"significant":[139],"gain,":[140],"whereas":[141],"SSOR":[143],"performs":[145],"worse":[146],"than":[147]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":4}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}