{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,1]],"date-time":"2024-06-01T11:56:20Z","timestamp":1717242980942},"reference-count":25,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,1]]},"abstract":"Abstract<\/jats:title>\n We consider settings for which one needs to perform multiple flow simulations based on the Navier\u2013Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and\/or coefficients such as the viscosity. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.<\/jats:p>","DOI":"10.1515\/cmam-2017-0051","type":"journal-article","created":{"date-parts":[[2017,12,5]],"date-time":"2017-12-05T22:15:51Z","timestamp":1512512151000},"page":"681-701","source":"Crossref","is-referenced-by-count":19,"title":["A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems"],"prefix":"10.1515","volume":"19","author":[{"given":"Max","family":"Gunzburger","sequence":"first","affiliation":[{"name":"Department of Scientific Computing , Florida State University , Tallahassee , FL 32306-4120 , USA"}]},{"given":"Nan","family":"Jiang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics , Missouri University of Science and Technology , Rolla , MO 65409-0020 , USA"}]},{"given":"Zhu","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of South Carolina , Columbia , SC 29208 , USA"}]}],"member":"374","published-online":{"date-parts":[[2017,12,5]]},"reference":[{"key":"2023033110340757522_j_cmam-2017-0051_ref_001_w2aab3b7b1b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"L. C. Berselli,\nOn the large eddy simulation of the Taylor\u2013Green vortex,\nJ. Math. Fluid Mech. 7 (2005), no. 2, S164\u2013S191.","DOI":"10.1007\/s00021-005-0152-z"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_002_w2aab3b7b1b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_003_w2aab3b7b1b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"G. J. Fix, M. D. Gunzburger and J. S. Peterson,\nOn finite element approximations of problems having inhomogeneous essential boundary conditions,\nComput. Math. Appl. 9 (1983), 687\u2013700.","DOI":"10.1016\/0898-1221(83)90126-8"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_004_w2aab3b7b1b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"V. Girault and P.-A. Raviart,\nFinite Element Approximation of the Navier\u2013Stokes Equations,\nLecture Notes in Math. 749,\nSpringer, Berlin, 1979.","DOI":"10.1007\/BFb0063447"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_005_w2aab3b7b1b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"A. Green and G. Taylor,\nMechanism of the production of small eddies from larger ones,\nProc. Royal Soc. Lond. Ser. A Math. Phys. Sci. 158 (1937), 499\u2013521.","DOI":"10.1098\/rspa.1937.0036"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_006_w2aab3b7b1b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"J.-L. Guermond and L. Quartapelle,\nOn stability and convergence of projection methods based on pressure Poisson equation,\nInternat. J. Numer. Methods Fluids 26 (1998), no. 9, 1039\u20131053.","DOI":"10.1002\/(SICI)1097-0363(19980515)26:9<1039::AID-FLD675>3.0.CO;2-U"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_007_w2aab3b7b1b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"M. D. Gunzburger,\nFinite Element Methods for Viscous Incompressible Flows,\nComput. Sci. Sci. Computing,\nAcademic Press, Boston, 1989.","DOI":"10.1016\/B978-0-12-307350-1.50009-0"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_008_w2aab3b7b1b1b6b1ab2b1b8Aa","unstructured":"M. Gunzburger, N. Jiang and M. Schneier,\nA higher-order ensemble\/proper orthogonal decomposition method for the nonstationary Navier\u2013Stokes equations,\nInt. J. Numer. Anal. Model., to appear; https:\/\/arxiv.org\/abs\/1709.06422."},{"key":"2023033110340757522_j_cmam-2017-0051_ref_009_w2aab3b7b1b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"M. Gunzburger, N. Jiang and M. Schneier,\nAn ensemble-proper orthogonal decomposition method for the nonstationary Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 55 (2017), no. 1, 286\u2013304.","DOI":"10.1137\/16M1056444"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_010_w2aab3b7b1b1b6b1ab2b1c10Aa","unstructured":"M. Gunzburger, N. Jiang and Z. Wang,\nAn efficient algorithm for simulating ensembles of parameterized flow problems,\npreprint (2017), https:\/\/arxiv.org\/abs\/1705.09350."},{"key":"2023033110340757522_j_cmam-2017-0051_ref_011_w2aab3b7b1b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"Y. He,\nTwo-level method based on finite element and Crank\u2013Nicolson extrapolation for the time-dependent Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 41 (2003), no. 4, 1263\u20131285.","DOI":"10.1137\/S0036142901385659"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_012_w2aab3b7b1b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"Y. He,\nThe Euler implicit\/explicit scheme for the 2D time-dependent Navier\u2013Stokes equations with smooth or non-smooth initial data,\nMath. Comp. 77 (2008), no. 264, 2097\u20132124.","DOI":"10.1090\/S0025-5718-08-02127-3"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_013_w2aab3b7b1b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"Y. He and W. Sun,\nStability and convergence of the Crank\u2013Nicolson\/Adams\u2013Bashforth scheme for the time-dependent Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 45 (2007), no. 2, 837\u2013869.","DOI":"10.1137\/050639910"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_014_w2aab3b7b1b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"J. G. Heywood and R. Rannacher,\nFinite-element approximation of the nonstationary Navier\u2013Stokes problem. IV. Error analysis for second-order time discretization,\nSIAM J. Numer. Anal. 27 (1990), no. 2, 353\u2013384.","DOI":"10.1137\/0727022"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_015_w2aab3b7b1b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"N. Jiang,\nA higher order ensemble simulation algorithm for fluid flows,\nJ. Sci. Comput. 64 (2015), no. 1, 264\u2013288.","DOI":"10.1007\/s10915-014-9932-z"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_016_w2aab3b7b1b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"N. Jiang,\nA second-order ensemble method based on a blended backward differentiation formula timestepping scheme for time-dependent Navier\u2013Stokes equations,\nNumer. Methods Partial Differential Equations 33 (2017), no. 1, 34\u201361.","DOI":"10.1002\/num.22070"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_017_w2aab3b7b1b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"N. Jiang, S. Kaya and W. Layton,\nAnalysis of model variance for ensemble based turbulence modeling,\nComput. Methods Appl. Math. 15 (2015), no. 2, 173\u2013188.","DOI":"10.1515\/cmam-2014-0029"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_018_w2aab3b7b1b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"N. Jiang and W. Layton,\nAn algorithm for fast calculation of flow ensembles,\nInt. J. Uncertain. Quantif. 4 (2014), no. 4, 273\u2013301.","DOI":"10.1615\/Int.J.UncertaintyQuantification.2014007691"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_019_w2aab3b7b1b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"N. Jiang and W. Layton,\nNumerical analysis of two ensemble eddy viscosity numerical regularizations of fluid motion,\nNumer. Methods Partial Differential Equations 31 (2015), no. 3, 630\u2013651.","DOI":"10.1002\/num.21908"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_020_w2aab3b7b1b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"H. Johnston and J.-G. Liu,\nAccurate, stable and efficient Navier\u2013Stokes solvers based on explicit treatment of the pressure term,\nJ. Comput. Phys. 199 (2004), no. 1, 221\u2013259.","DOI":"10.1016\/j.jcp.2004.02.009"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_021_w2aab3b7b1b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"W. Layton,\nIntroduction to the Numerical Analysis of Incompressible Viscous Flows,\nComput. Sci. Eng. 6,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2008.","DOI":"10.1137\/1.9780898718904"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_022_w2aab3b7b1b1b6b1ab2b1c22Aa","doi-asserted-by":"crossref","unstructured":"M. Marion and R. Temam,\nNavier\u2013Stokes equations: Theory and approximation,\nHandbook of Numerical Analysis. VI,\nNorth-Holland, Amsterdam (1998), 503\u2013688.","DOI":"10.1016\/S1570-8659(98)80010-0"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_023_w2aab3b7b1b1b6b1ab2b1c23Aa","doi-asserted-by":"crossref","unstructured":"M. Mohebujjaman and L. G. Rebholz,\nAn efficient algorithm for computation of MHD flow ensembles,\nComput. Methods Appl. Math. 17 (2017), no. 1, 121\u2013137.","DOI":"10.1515\/cmam-2016-0033"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_024_w2aab3b7b1b1b6b1ab2b1c24Aa","doi-asserted-by":"crossref","unstructured":"A. Takhirov, M. Neda and J. Waters,\nTime relaxation algorithm for flow ensembles,\nNumer. Methods Partial Differential Equations 32 (2016), no. 3, 757\u2013777.","DOI":"10.1002\/num.22024"},{"key":"2023033110340757522_j_cmam-2017-0051_ref_025_w2aab3b7b1b1b6b1ab2b1c25Aa","doi-asserted-by":"crossref","unstructured":"F. Tone,\nError analysis for a second order scheme for the Navier\u2013Stokes equations,\nAppl. Numer. Math. 50 (2004), no. 1, 93\u2013119.","DOI":"10.1016\/j.apnum.2003.12.003"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/19\/3\/article-p681.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0051\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0051\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:07:41Z","timestamp":1680264461000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0051\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,5]]},"references-count":25,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,6,27]]},"published-print":{"date-parts":[[2019,7,1]]}},"alternative-id":["10.1515\/cmam-2017-0051"],"URL":"http:\/\/dx.doi.org\/10.1515\/cmam-2017-0051","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,12,5]]}}}
