{"id":"https://openalex.org/W2964203639","doi":"https://doi.org/10.1137/130920046","title":"A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and Its Application to the Nonlinear Schr\u00f6dinger Equation","display_name":"A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and Its Application to the Nonlinear Schr\u00f6dinger Equation","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2964203639","doi":"https://doi.org/10.1137/130920046","mag":"2964203639"},"language":"en","primary_location":{"id":"doi:10.1137/130920046","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130920046","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/A5044039585","display_name":"Ronald M. Caplan","orcid":"https://orcid.org/0000-0002-2633-4290"},"institutions":[{"id":"https://openalex.org/I26538001","display_name":"San Diego State University","ror":"https://ror.org/0264fdx42","country_code":"US","type":"education","lineage":["https://openalex.org/I26538001"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"R. M. Caplan","raw_affiliation_strings":["Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182-7720, USA","Nonlinear Dynamical System Group \u2020 , Computational Science Research Center \u2021 , and"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182-7720, USA","institution_ids":["https://openalex.org/I26538001"]},{"raw_affiliation_string":"Nonlinear Dynamical System Group \u2020 , Computational Science Research Center \u2021 , and","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030089337","display_name":"R. Carretero-Gonz\u00e1lez","orcid":"https://orcid.org/0000-0002-7995-7491"},"institutions":[{"id":"https://openalex.org/I26538001","display_name":"San Diego State University","ror":"https://ror.org/0264fdx42","country_code":"US","type":"education","lineage":["https://openalex.org/I26538001"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"R. Carretero-Gonz\u00e1lez","raw_affiliation_strings":["Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182-7720, USA","Nonlinear Dynamical System Group \u2020 , Computational Science Research Center \u2021 , and"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182-7720, USA","institution_ids":["https://openalex.org/I26538001"]},{"raw_affiliation_string":"Nonlinear Dynamical System Group \u2020 , Computational Science Research Center \u2021 , and","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5044039585"],"corresponding_institution_ids":["https://openalex.org/I26538001"],"apc_list":null,"apc_paid":null,"fwci":1.2771,"has_fulltext":false,"cited_by_count":26,"citation_normalized_percentile":{"value":0.83873067,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"36","issue":"1","first_page":"A1","last_page":"A19"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9994999766349792,"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":0.9994999766349792,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9993000030517578,"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/T10514","display_name":"Numerical methods in engineering","score":0.9882000088691711,"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/mathematics","display_name":"Mathematics","score":0.7543004155158997},{"id":"https://openalex.org/keywords/dirichlet-boundary-condition","display_name":"Dirichlet boundary condition","score":0.7271440625190735},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.7229224443435669},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.7138969898223877},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.6511285305023193},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.6020951867103577},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5244855284690857},{"id":"https://openalex.org/keywords/mixed-boundary-condition","display_name":"Mixed boundary condition","score":0.4725648760795593},{"id":"https://openalex.org/keywords/dirichlet-problem","display_name":"Dirichlet problem","score":0.45298370718955994},{"id":"https://openalex.org/keywords/first-order-partial-differential-equation","display_name":"First-order partial differential equation","score":0.4413470923900604},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1889728307723999}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7543004155158997},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.7271440625190735},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.7229224443435669},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.7138969898223877},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6511285305023193},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.6020951867103577},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5244855284690857},{"id":"https://openalex.org/C42045870","wikidata":"https://www.wikidata.org/wiki/Q2992268","display_name":"Mixed boundary condition","level":3,"score":0.4725648760795593},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.45298370718955994},{"id":"https://openalex.org/C64057670","wikidata":"https://www.wikidata.org/wiki/Q4381442","display_name":"First-order partial differential equation","level":3,"score":0.4413470923900604},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1889728307723999},{"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:10.1137/130920046","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130920046","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8372609528","display_name":"Modeling, Analysis, Computation and Experiments of Two-Component Bose-Einstein Condensates","funder_award_id":"0806762","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8643518142","display_name":null,"funder_award_id":"DMS-0806762","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/F4320309507","display_name":"San Diego State University","ror":"https://ror.org/0264fdx42"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":36,"referenced_works":["https://openalex.org/W11286117","https://openalex.org/W322174869","https://openalex.org/W1558122728","https://openalex.org/W1965855805","https://openalex.org/W2003017040","https://openalex.org/W2009837750","https://openalex.org/W2017463281","https://openalex.org/W2029830616","https://openalex.org/W2030876368","https://openalex.org/W2034276104","https://openalex.org/W2042893663","https://openalex.org/W2046600110","https://openalex.org/W2049904442","https://openalex.org/W2050338742","https://openalex.org/W2054739989","https://openalex.org/W2057427839","https://openalex.org/W2058203215","https://openalex.org/W2064178994","https://openalex.org/W2066939711","https://openalex.org/W2067352798","https://openalex.org/W2075696551","https://openalex.org/W2089291942","https://openalex.org/W2103617362","https://openalex.org/W2105097288","https://openalex.org/W2106343087","https://openalex.org/W2112700629","https://openalex.org/W2116239971","https://openalex.org/W2118860521","https://openalex.org/W2121498459","https://openalex.org/W2140846764","https://openalex.org/W2151428129","https://openalex.org/W2153631870","https://openalex.org/W2161675745","https://openalex.org/W2171483571","https://openalex.org/W3101835573","https://openalex.org/W4233987090"],"related_works":["https://openalex.org/W2359206539","https://openalex.org/W2479314534","https://openalex.org/W2388355944","https://openalex.org/W2563073486","https://openalex.org/W2485938319","https://openalex.org/W1991459776","https://openalex.org/W3021320825","https://openalex.org/W4238318050","https://openalex.org/W2072438171","https://openalex.org/W4232257209"],"abstract_inverted_index":{"An":[0],"easy":[1],"to":[2,65,74,84,98],"implement":[3],"modulus-squared":[4,30],"Dirichlet":[5],"(MSD)":[6],"boundary":[7,25,67,82,107],"condition":[8,26,83],"is":[9,39,54,91],"formulated":[10],"for":[11],"numerical":[12,94],"simulations":[13,85,95],"of":[14,32,79,86],"time-dependent":[15],"complex":[16,56],"partial":[17],"differential":[18],"equations":[19],"in":[20],"multidimensional":[21],"settings.":[22],"The":[23],"MSD":[24,81],"approximates":[27],"a":[28,66],"constant":[29],"value":[31],"the":[33,36,55,59,70,75,80,87],"solution":[34],"at":[35],"boundaries":[37],"and":[38,58,62,69,93,102],"defined":[40],"as":[41],"$\\frac{\\partial":[42],"\\Psi}{\\partial":[43,49],"t}|_b":[44],"\\approx":[45],"i\\,\\mbox{Im}[":[46],"\\frac{1}{\\Psi_{b-1}}":[47],"\\frac{\\partial":[48],"t}|_{b-1}":[50],"]\\,\\Psi_b,$":[51],"where":[52],"$\\Psi$":[53],"field":[57],"subscripts":[60],"$b$":[61],"$b-1$":[63],"refer":[64],"point":[68,73],"closest":[71],"interior":[72],"boundary,":[76],"respectively.":[77],"Application":[78],"nonlinear":[88],"Schr\u00f6dinger":[89],"equation":[90],"shown,":[92],"are":[96],"performed":[97],"demonstrate":[99],"its":[100],"usefulness":[101],"advantages":[103],"over":[104],"other":[105],"simple":[106],"conditions.":[108]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}