{"id":"https://openalex.org/W4391782442","doi":"https://doi.org/10.1007/s11075-024-01770-7","title":"On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters","display_name":"On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters","publication_year":2024,"publication_date":"2024-02-13","ids":{"openalex":"https://openalex.org/W4391782442","doi":"https://doi.org/10.1007/s11075-024-01770-7"},"language":"en","primary_location":{"id":"doi:10.1007/s11075-024-01770-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-024-01770-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-024-01770-7.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s11075-024-01770-7.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027471730","display_name":"Thomas Izgin","orcid":"https://orcid.org/0000-0003-3235-210X"},"institutions":[{"id":"https://openalex.org/I106157433","display_name":"University of Kassel","ror":"https://ror.org/04zc7p361","country_code":"DE","type":"education","lineage":["https://openalex.org/I106157433"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Thomas Izgin","raw_affiliation_strings":["Institute of Mathematics, University of Kassel, Untere K\u00f6nigsstr. 86, 34117, Kassel, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, University of Kassel, Untere K\u00f6nigsstr. 86, 34117, Kassel, Germany","institution_ids":["https://openalex.org/I106157433"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5054696906","display_name":"Stefan Kopecz","orcid":"https://orcid.org/0000-0003-0593-450X"},"institutions":[{"id":"https://openalex.org/I106157433","display_name":"University of Kassel","ror":"https://ror.org/04zc7p361","country_code":"DE","type":"education","lineage":["https://openalex.org/I106157433"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Stefan Kopecz","raw_affiliation_strings":["Institute of Mathematics, University of Kassel, Untere K\u00f6nigsstr. 86, 34117, Kassel, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, University of Kassel, Untere K\u00f6nigsstr. 86, 34117, Kassel, Germany","institution_ids":["https://openalex.org/I106157433"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109294577","display_name":"Andreas Meister","orcid":null},"institutions":[{"id":"https://openalex.org/I106157433","display_name":"University of Kassel","ror":"https://ror.org/04zc7p361","country_code":"DE","type":"education","lineage":["https://openalex.org/I106157433"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Andreas Meister","raw_affiliation_strings":["Institute of Mathematics, University of Kassel, Untere K\u00f6nigsstr. 86, 34117, Kassel, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, University of Kassel, Untere K\u00f6nigsstr. 86, 34117, Kassel, Germany","institution_ids":["https://openalex.org/I106157433"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5092829211","display_name":"Amandine Schilling","orcid":null},"institutions":[{"id":"https://openalex.org/I106157433","display_name":"University of Kassel","ror":"https://ror.org/04zc7p361","country_code":"DE","type":"education","lineage":["https://openalex.org/I106157433"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Amandine Schilling","raw_affiliation_strings":["Institute of Mathematics, University of Kassel, Untere K\u00f6nigsstr. 86, 34117, Kassel, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, University of Kassel, Untere K\u00f6nigsstr. 86, 34117, Kassel, Germany","institution_ids":["https://openalex.org/I106157433"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5054696906"],"corresponding_institution_ids":["https://openalex.org/I106157433"],"apc_list":{"value":2990,"currency":"USD","value_usd":2990},"apc_paid":{"value":2990,"currency":"USD","value_usd":2990},"fwci":1.0624,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.68280224,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":97},"biblio":{"volume":"96","issue":"3","first_page":"1221","last_page":"1242"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","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"}},"topics":[{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","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"}},{"id":"https://openalex.org/T10597","display_name":"Nuclear reactor physics and engineering","score":0.983299970626831,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"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/spurious-relationship","display_name":"Spurious relationship","score":0.8581579923629761},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7974742650985718},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.6895443201065063},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6026105284690857},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4514899253845215},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3876579701900482},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.21727442741394043},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.15615680813789368},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09332120418548584}],"concepts":[{"id":"https://openalex.org/C97256817","wikidata":"https://www.wikidata.org/wiki/Q1462316","display_name":"Spurious relationship","level":2,"score":0.8581579923629761},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7974742650985718},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.6895443201065063},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6026105284690857},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4514899253845215},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3876579701900482},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.21727442741394043},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.15615680813789368},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09332120418548584},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s11075-024-01770-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-024-01770-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-024-01770-7.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s11075-024-01770-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-024-01770-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-024-01770-7.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320327735","display_name":"Universit\u00e4t Kassel","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4391782442.pdf","grobid_xml":"https://content.openalex.org/works/W4391782442.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W1661414203","https://openalex.org/W1902819800","https://openalex.org/W1980276182","https://openalex.org/W2060312991","https://openalex.org/W2073540225","https://openalex.org/W2085800969","https://openalex.org/W2274604370","https://openalex.org/W2572911505","https://openalex.org/W2588894268","https://openalex.org/W2603908784","https://openalex.org/W2896501110","https://openalex.org/W2902813631","https://openalex.org/W2945798956","https://openalex.org/W2995135341","https://openalex.org/W3033142563","https://openalex.org/W3116265717","https://openalex.org/W3216639823","https://openalex.org/W4220769286","https://openalex.org/W4285047878","https://openalex.org/W4309325534","https://openalex.org/W4315631935","https://openalex.org/W4360877427","https://openalex.org/W4378509412","https://openalex.org/W4380318450","https://openalex.org/W4385075801"],"related_works":["https://openalex.org/W2162899405","https://openalex.org/W3113091479","https://openalex.org/W941090075","https://openalex.org/W2044987316","https://openalex.org/W3134374554","https://openalex.org/W2237480245","https://openalex.org/W2519167559","https://openalex.org/W2075065631","https://openalex.org/W4288358396","https://openalex.org/W4311248832"],"abstract_inverted_index":{"Abstract":[0],"Recently,":[1,98],"a":[2,25,123],"stability":[3,12,19,81,106,135,182,191],"theory":[4,20],"has":[5],"been":[6],"developed":[7],"to":[8,32,67,322,326],"study":[9],"the":[10,39,46,50,56,61,68,92,104,108,140,151,181,255,282,286,298,311,314,323],"linear":[11,80,96,105,134,190],"of":[13,28,41,49,107,125,130,153,285,310],"modified":[14],"Patankar\u2013Runge\u2013Kutta":[15],"(MPRK)":[16],"schemes.":[17],"This":[18],"provides":[21],"sufficient":[22],"conditions":[23],"for":[24,38,94,179,195,290,297,308],"fixed":[26],"point":[27],"an":[29,42],"MPRK":[30,43],"scheme":[31,44,110],"be":[33,319],"stable":[34,253,278],"as":[35,37,90],"well":[36],"convergence":[40],"towards":[45,281],"steady":[47,69,283,324],"state":[48,284,325],"corresponding":[51],"initial":[52,62,287,315],"value":[53,63,288],"problem,":[54],"whereas":[55],"main":[57],"assumption":[58],"is":[59,64,91,111,119,122,136,192,212,259,305],"that":[60,78,103,120,133,171,214,307],"sufficiently":[65,260],"close":[66,320],"state.":[70],"Initially,":[71],"numerical":[72],"experiments":[73],"in":[74,115],"several":[75],"publications":[76],"indicated":[77],"these":[79,174],"properties":[82,183],"are":[83,143,177,184,251,277],"not":[84,193],"only":[85,113,185,252],"local":[86,114,189],"but":[87],"even":[88,172],"global,":[89],"case":[93],"general":[95],"methods.":[97],"however,":[99],"it":[100,211,304],"was":[101],"discovered":[102],"MPDeC(8)":[109,131],"indeed":[112,137],"nature.":[116],"Our":[117],"conjecture":[118],"this":[121,147,188,328],"result":[124],"negative":[126,166],"Runge\u2013Kutta":[127,207],"(RK)":[128],"parameters":[129,142,168],"and":[132,169,279],"global":[138],"if":[139,254],"RK":[141,167],"nonnegative.":[144],"To":[145],"support":[146],"conjecture,":[148],"we":[149],"examine":[150],"family":[152],"MPRK22(":[154,196,215],"$$\\varvec{\\alpha":[155],"}$$":[156],"<mml:math":[157,199,218,227,240,268],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[158,200,219,228,241,269],"<mml:mrow>":[159,229,242,270],"<mml:mi>\u03b1</mml:mi>":[160,201,220,232,246,271],"</mml:mrow>":[161,235,249,275],"</mml:math>":[162,202,221,236,250,276],")":[163,203,222],"methods":[164,175,178],"with":[165,205,224,264],"show":[170],"among":[173],"there":[176],"which":[180],"local.":[186],"However,":[187],"observed":[194],"$$\\alpha":[197,216,265],"$$":[198,217],"schemes":[204,223,263],"nonnegative":[206],"parameters.":[208],"In":[209],"particular,":[210],"shown":[213,306],"$$0&lt;\\alpha":[225],"&lt;0.5$$":[226],"<mml:mn>0</mml:mn>":[230,248],"<mml:mo>&lt;</mml:mo>":[231,233,245,247],"<mml:mn>0.5</mml:mn>":[234,244,274],"or":[237],"$$-0.5&lt;\\alpha":[238],"&lt;0$$":[239],"<mml:mo>-</mml:mo>":[243,273],"time":[256,292],"step":[257,293],"size":[258],"small.":[261],"But":[262],"\\le":[266],"-0.5$$":[267],"<mml:mo>\u2264</mml:mo>":[272],"converge":[280],"problems":[289],"all":[291],"sizes,":[294],"at":[295],"least":[296],"test":[299],"problem":[300],"under":[301],"consideration.":[302],"Furthermore,":[303],"some":[309],"latter":[312],"systems,":[313],"values":[316],"must":[317],"actually":[318],"enough":[321],"guarantee":[327],"result.":[329]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1}],"updated_date":"2026-08-18T07:49:30.821534","created_date":"2025-10-10T00:00:00"}