{"doi":"10.1002/num.20010","title":"Uniform stability of spectral nonlinear Galerkin methods","abstract":"<jats:title>Abstract</jats:title><jats:p>This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier‐Stokes equations with some appropriate assumption of the data (λ, <jats:italic>u</jats:italic><jats:sub>0</jats:sub>, <jats:italic>f</jats:italic>). If the backward Euler scheme with the semi‐implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δ<jats:italic>t</jats:italic> ≤ (2/λλ<jats:sub>1</jats:sub>). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δ<jats:italic>t = O</jats:italic>(λ<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/tex2gif-stack-1.gif\" xlink:title=\"urn:x-wiley:0749159X:media:NUM20010:tex2gif-stack-1\" />) and Δ<jats:italic>t = O</jats:italic>(λ<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/tex2gif-stack-2.gif\" xlink:title=\"urn:x-wiley:0749159X:media:NUM20010:tex2gif-stack-2\" />), respectively, where λ<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/tex2gif-stack-3.gif\" xlink:title=\"urn:x-wiley:0749159X:media:NUM20010:tex2gif-stack-3\" /> ≤ λ<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/tex2gif-stack-4.gif\" xlink:title=\"urn:x-wiley:0749159X:media:NUM20010:tex2gif-stack-4\" />, which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004</jats:p>","journal":"Numerical Methods for Partial Differential Equations","year":2004,"id":24003,"datarank":0.4191060547299307,"base_score":1.6094379124341003,"endowment":1.6094379124341003,"self_citation_contribution":0.24141568686511508,"citation_network_contribution":0.17769036786481557,"self_endowment_contribution":0.24141568686511508,"citer_contribution":0.17769036786481557,"corpus_percentile":null,"corpus_rank":null,"citation_count":4,"citer_count":4,"citers_with_citation_signal":4,"citers_with_endowment":4,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":null,"is_data_producer":false,"deposit_databanks":null,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":null,"fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":145351,"name":"Kaitai Li","orcid":null,"position":1,"is_corresponding":false},{"id":145352,"name":"Chunshan Zhao","orcid":null,"position":2,"is_corresponding":false},{"id":142563,"name":"Yinnian He","orcid":null,"position":0,"is_corresponding":false}],"reference_count":0,"raw_metadata":{"has_enrichment":true,"base_score":1.6094379124341003,"endowment":1.6094379124341003,"datacite_reuse_total":0,"file_count":0,"downloads":0,"views":0,"has_version_chain":false,"is_dataset":false,"is_oa":false,"pmid":"24523987","pmcid":null,"openalex_id":"https://openalex.org/W2096756136","authors":[],"funders":[],"total_grants":0,"fwci":0.0,"citation_percentile":0.16509274,"influential_citations":0,"citation_trend":[{"year":2012,"count":1},{"year":2013,"count":1},{"year":2019,"count":1}],"oa_status":"closed","license":"http://onlinelibrary.wiley.com/termsAndConditions#vor","oa_locations":[{"url":"https://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2Fnum.20010","host_type":"publisher"},{"url":"https://onlinelibrary.wiley.com/doi/pdf/10.1002/num.20010","host_type":"publisher"},{"url":"https://doi.org/10.1002/num.20010","host_type":"journal"}],"fields_of_study":["Advanced Numerical Methods in Computational Mathematics","Differential Equations and Numerical Methods","Advanced Mathematical Modeling in Engineering","Mathematics","Engineering"],"mesh_terms":[],"keywords":["Mathematics","Galerkin method","Discretization","Nonlinear system","Spectral method","Backward Euler method","Euler's formula","Mathematical analysis","Discontinuous Galerkin method","Stability (learning theory)","Applied mathematics","Partial differential equation","Euler equations","Spectral element method","Finite element method","Physics","Computer science"],"sdg_mappings":[],"linked_datasets":[],"clinical_trials":[],"software_tools":[],"database_accessions":[],"source":"live","citation_network_status":"fetched"},"created_at":"2026-06-07T21:10:25.271360Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}