{"id":"https://openalex.org/W7213439798","doi":"https://doi.org/10.48550/arxiv.2609.16863","title":"Mixed precision solvers for the all-at-once Runge--Kutta discretization of the heat equation","display_name":"Mixed precision solvers for the all-at-once Runge--Kutta discretization of the heat equation","publication_year":2026,"publication_date":"2026-09-15","ids":{"openalex":"https://openalex.org/W7213439798","doi":"https://doi.org/10.48550/arxiv.2609.16863"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.16863","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16863","pdf_url":"https://arxiv.org/pdf/2609.16863","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.16863","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5074896686","display_name":"Santolo Leveque","orcid":"https://orcid.org/0000-0001-8167-088X","observed_orcids":["https://orcid.org/0000-0001-8167-088X"]},"institutions":[{"id":"https://openalex.org/I21250087","display_name":"Charles University","ror":"https://ror.org/024d6js02","country_code":"CZ","type":"education","lineage":["https://openalex.org/I21250087"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Leveque, Santolo","raw_affiliation_strings":[", Department of Numerical Mathematics , Faculty of Mathematics and Physics , Charles Uni- versity , 186 75 , Praha 8 , Czech Republic"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":", Department of Numerical Mathematics , Faculty of Mathematics and Physics , Charles Uni- versity , 186 75 , Praha 8 , Czech Republic","institution_ids":["https://openalex.org/I21250087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5123176635","display_name":"Luca Bergamaschi","orcid":"https://orcid.org/0000-0001-8273-9674","observed_orcids":["https://orcid.org/0000-0001-8273-9674"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bergamaschi, Luca","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5151905067","display_name":"\u00c1ngeles Mart\u00ednez","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mart\u00ednez, \u00c1ngeles","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5133624810","display_name":"Erin Carson","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I21250087","display_name":"Charles University","ror":"https://ror.org/024d6js02","country_code":"CZ","type":"education","lineage":["https://openalex.org/I21250087"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Carson, Erin","raw_affiliation_strings":[", Department of Numerical Mathematics , Faculty of Mathematics and Physics , Charles Uni- versity , 186 75 , Praha 8 , Czech Republic"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":", Department of Numerical Mathematics , Faculty of Mathematics and Physics , Charles Uni- versity , 186 75 , Praha 8 , Czech Republic","institution_ids":["https://openalex.org/I21250087"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9843999743461609,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9843999743461609,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"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":0.003800000064074993,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.003100000089034438,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mixed-precision","display_name":"mixed precision","score":0.9300000071525574},{"id":"https://openalex.org/keywords/heat-equation","display_name":"heat equation","score":0.8629999756813049},{"id":"https://openalex.org/keywords/runge-kutta-integration","display_name":"Runge-Kutta time integration","score":0.8019999861717224},{"id":"https://openalex.org/keywords/preconditioned-iterative-solvers","display_name":"preconditioned iterative solvers","score":0.8930000066757202},{"id":"https://openalex.org/keywords/sparse-linear-systems","display_name":"sparse linear systems","score":0.953000009059906},{"id":"https://openalex.org/keywords/parallel-preconditioners","display_name":"parallel preconditioning","score":0.8420000076293945},{"id":"https://openalex.org/keywords/cpu-time-reduction","display_name":"CPU time reduction","score":0.6079999804496765}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.8335999846458435},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.728600025177002},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.6714000105857849},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5644999742507935},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.5471000075340271},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5187000036239624},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.47850000858306885},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4185999929904938},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.38679999113082886},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.37860000133514404},{"id":"https://openalex.org/C77346573","wikidata":"https://www.wikidata.org/wiki/Q484637","display_name":"Linear equation","level":2,"score":0.3671000003814697},{"id":"https://openalex.org/C94523830","wikidata":"https://www.wikidata.org/wiki/Q11203","display_name":"System of linear equations","level":2,"score":0.36039999127388},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34610000252723694},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3280999958515167},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.3260999917984009},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.28769999742507935},{"id":"https://openalex.org/C126148662","wikidata":"https://www.wikidata.org/wiki/Q3890355","display_name":"Discretization error","level":3,"score":0.2865000069141388},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.26820001006126404},{"id":"https://openalex.org/C3018824978","wikidata":"https://www.wikidata.org/wiki/Q2894891","display_name":"Error analysis","level":2,"score":0.263700008392334},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.25290000438690186}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.16863","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16863","pdf_url":"https://arxiv.org/pdf/2609.16863","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},{"id":"doi:10.48550/arxiv.2609.16863","doi":"https://doi.org/10.48550/arxiv.2609.16863","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.16863","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.16863","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16863","pdf_url":"https://arxiv.org/pdf/2609.16863","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1321919621","display_name":null,"funder_award_id":"UNCE24/SCI/005","funder_id":"https://openalex.org/F4320309755","funder_display_name":"Univerzita Karlova v Praze"},{"id":"https://openalex.org/G8130583117","display_name":null,"funder_award_id":"101075632","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320309755","display_name":"Univerzita Karlova v Praze","ror":"https://ror.org/024d6js02"},{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213439798.pdf","grobid_xml":"https://content.openalex.org/works/W7213439798.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7168092361","https://openalex.org/W4415752784","https://openalex.org/W3179588297","https://openalex.org/W4416419340","https://openalex.org/W3213194041","https://openalex.org/W2323200980","https://openalex.org/W3116961995","https://openalex.org/W4411955994","https://openalex.org/W4312122728","https://openalex.org/W7162339782"],"abstract_inverted_index":{"We":[0],"study":[1],"the":[2,8,12,42,46,76,79,91,94,108,112],"effect":[3],"of":[4,11,49,93,107,111],"mixed":[5,85,100,113],"precision":[6,86,114],"on":[7],"numerical":[9,43,105],"integration":[10],"heat":[13],"equation":[14],"discretized":[15],"with":[16],"a":[17,31,60,84,119,125],"Runge--Kutta":[18,47],"method":[19,64],"in":[20,30,69,99,118,128],"time.":[21,130],"A":[22],"full":[23],"space-time":[24],"discretization":[25],"is":[26,56,81],"applied,":[27],"which":[28],"results":[29,89],"very":[32],"large":[33],"and":[34,45],"sparse":[35],"linear":[36,54],"system":[37,55],"to":[38,73,124],"be":[39,67],"solved":[40,57],"for":[41],"approximations":[44],"stages":[48],"all":[50],"time":[51],"steps.":[52],"The":[53],"by":[58],"applying":[59],"suitable":[61],"preconditioned":[62],"iterative":[63],"that":[65],"can":[66],"run":[68],"parallel.":[70],"In":[71],"order":[72],"speed":[74],"up":[75,123],"solution":[77],"process,":[78],"preconditioner":[80],"applied":[82,98,117],"using":[83],"framework.":[87],"Sequential":[88],"show":[90],"robustness":[92],"preconditioner,":[95],"even":[96],"when":[97,116],"precision.":[101],"Finally,":[102],"we":[103],"present":[104],"evidence":[106],"improved":[109],"performance":[110],"strategy":[115],"parallel":[120],"environment,":[121],"achieving":[122],"50%":[126],"reduction":[127],"CPU":[129]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-17T00:00:00"}