{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T20:44:34Z","timestamp":1785357874205,"version":"3.55.0"},"reference-count":54,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2012,2,26]],"date-time":"2012-02-26T00:00:00Z","timestamp":1330214400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2012,8]]},"DOI":"10.1007\/s00211-012-0450-3","type":"journal-article","created":{"date-parts":[[2012,2,25]],"date-time":"2012-02-25T04:26:31Z","timestamp":1330143991000},"page":"753-779","source":"Crossref","is-referenced-by-count":33,"title":["An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier\u2013Stokes equations"],"prefix":"10.1007","volume":"121","author":[{"given":"Xiaoming","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2012,2,26]]},"reference":[{"issue":"3","key":"450_CR1","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1137\/0732037","volume":"32","author":"U.M. Ascher","year":"1995","unstructured":"Ascher U.M., Ruuth S.J., Wetton W.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797\u2013823 (1995)","journal-title":"SIAM J. Numer. Anal."},{"key":"450_CR2","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970623","volume-title":"Weak Convergence of Measures: Applications in Probability","author":"P. Billingsley","year":"1971","unstructured":"Billingsley P.: Weak Convergence of Measures: Applications in Probability. SIAM, Philadelphia (1971)"},{"key":"450_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-84108-8","volume-title":"Spectral Methods in Fluid Dynamics","author":"C. Canuto","year":"1988","unstructured":"Canuto C., Hussaini M.Y., Quarteroni A., Zang T.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)"},{"issue":"1","key":"450_CR4","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1137\/080713501","volume":"47","author":"W. Cheng","year":"2008","unstructured":"Cheng W., Wang X.: A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model. SIAM J. Numer. Anal. 47(1), 250\u2013270 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"450_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-8728-0","volume-title":"Vorticity and Turbulence","author":"A.J. Chorin","year":"1994","unstructured":"Chorin A.J.: Vorticity and Turbulence. Springer, New York (1994)"},{"key":"450_CR6","doi-asserted-by":"crossref","DOI":"10.7208\/chicago\/9780226764320.001.0001","volume-title":"Navier\u2013Stokes Equations","author":"P. Constantin","year":"1988","unstructured":"Constantin P., Foias C.: Navier\u2013Stokes Equations. The University of Chicago Press, Chicago (1988)"},{"key":"450_CR7","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/BF01396412","volume":"35","author":"M. Crouzeix","year":"1982","unstructured":"Crouzeix M.: Une methode multipas implicite-explicite pour l\u2019approximation des \u2019equations d\u2019evolution paraboliques. Numer. Math. 35, 257\u2013276 (1982)","journal-title":"Numer. Math."},{"key":"450_CR8","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511662829","volume-title":"Ergodicity for Infinite Dimensional Systems","author":"G. Da Prato","year":"1996","unstructured":"Da Prato G., Zabczyk J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)"},{"key":"450_CR9","doi-asserted-by":"crossref","unstructured":"E, W.: Stochastic hydrodynamics. Current Developments in Mathematics, 2000, pp. 109\u2013147. International Press, Somerville (2001)","DOI":"10.4310\/CDM.2000.v2000.n1.a2"},{"key":"450_CR10","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1088\/0951-7715\/4\/3\/001","volume":"4","author":"C. Foias","year":"1991","unstructured":"Foias C., Jolly M., Kevrekidis I.G., Titi E.S.: Dissipativity of numerical schemes. Nonlinearity 4, 591\u2013613 (1991)","journal-title":"Nonlinearity"},{"key":"450_CR11","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0375-9601(94)90926-1","volume":"186","author":"C. Foias","year":"1994","unstructured":"Foias C., Jolly M., Kevrekidis I.G., Titi E.S.: On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto\u2013Ivashinsky equation. Phys. Lett. A 186, 87\u201396 (1994)","journal-title":"Phys. Lett. A"},{"key":"450_CR12","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511546754","volume-title":"Navier\u2013Stokes equations and turbulence. Encyclopedia of Mathematics and its Applications","author":"C. Foias","year":"2001","unstructured":"Foias C., Manley O., Rosa R., Temam R.: Navier\u2013Stokes equations and turbulence. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2001)"},{"key":"450_CR13","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0022-1236(89)90015-3","volume":"97","author":"C. Foias","year":"1989","unstructured":"Foias C., Temam R.: Gevrey class regularity for the solutions of the Navier\u2013Stokes equations. J. Funct. Anal. 97, 359\u2013369 (1989)","journal-title":"J. Funct. Anal."},{"key":"450_CR14","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139170666","volume-title":"Turbulence: The Legacy of A.N. Kolmogorov","author":"U. Frisch","year":"1995","unstructured":"Frisch U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge (1995)"},{"key":"450_CR15","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1090\/S0025-5718-1989-0969488-5","volume":"53","author":"T. Geveci","year":"1989","unstructured":"Geveci T.: On the convergence of a time discretization scheme for the Navier\u2013Stokes equations. Math. Comput. 53, 43\u201353 (1989)","journal-title":"Math. Comput."},{"key":"450_CR16","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970425","volume-title":"Numerical Analysis of Spectral Methods, Theory and Applications","author":"D. Gottlieb","year":"1977","unstructured":"Gottlieb D., Orszag S.A.: Numerical Analysis of Spectral Methods, Theory and Applications. SIAM, Philadelphia (1977)"},{"key":"450_CR17","doi-asserted-by":"crossref","unstructured":"Gottlieb, S., Tone, F., Wang, C., Wang, X., Wirosoetisno, D.: Long time stability of a classical efficient scheme for two dimensional Navier\u2013Stokes equations. SIAM J. Numer. Anal. 50(1) 126\u2013150 (2012) . http:\/\/dx.doi.org\/10.1137\/110834901","DOI":"10.1137\/110834901"},{"key":"450_CR18","unstructured":"Gottlieb, S., Wang, C.: High order long time stable schemes for 2D Navier\u2013Stokes system in a periodic box. In preparation (2011)"},{"issue":"6","key":"450_CR19","doi-asserted-by":"crossref","first-page":"2050","DOI":"10.1214\/08-AOP392","volume":"36","author":"M. Hairer","year":"2008","unstructured":"Hairer M., Mattingly J.C.: Spectral gaps in Wasserstein distances and the 2D stochastic Navier\u2013Stokes equations. Ann. Probab. 36(6), 2050\u20132091 (2008)","journal-title":"Ann. Probab."},{"key":"450_CR20","volume-title":"Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems","author":"E. Hairer","year":"2002","unstructured":"Hairer E., Wanner G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (2002)","edition":"2"},{"key":"450_CR21","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1090\/S0025-5718-1988-0917820-X","volume":"50","author":"J.K. Hale","year":"1988","unstructured":"Hale J.K., Lin X.B., Raugel G.: Upper semicontinuity of attractors for approximations of semigroups and partial differential equations. Math. Comput. 50, 89\u2013123 (1988)","journal-title":"Math. Comput."},{"issue":"11","key":"450_CR22","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1090\/S0025-5718-1995-1297470-2","volume":"64","author":"A.T. Hill","year":"1995","unstructured":"Hill A.T., S\u00fcli E.: Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations. Math. Comput. 64(11), 1097\u20131122 (1995)","journal-title":"Math. Comput."},{"key":"450_CR23","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1093\/imanum\/20.4.633","volume":"20","author":"A.T. Hill","year":"2000","unstructured":"Hill A.T., S\u00fcli E.: Approximation of the global attractor for the incompressible Navier\u2013Stokes equations. IMA J. Numer. Anal. 20, 633\u2013667 (2000)","journal-title":"IMA J. Numer. Anal."},{"key":"450_CR24","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/BF01047708","volume":"3","author":"M. Jolly","year":"1991","unstructured":"Jolly M., Kevrekidis I., Titi E.S.: Preserving dissipation in approximate inertial forms for the Kuramoto\u2013Sivashinsky equation. J. Dyn. Differ. Equ. 3, 179\u2013197 (1991)","journal-title":"J. Dyn. Differ. Equ."},{"key":"450_CR25","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1093\/imanum\/22.4.577","volume":"22","author":"N. Ju","year":"2002","unstructured":"Ju N.: On the global stability of a temporal discretization scheme for the Navier\u2013Stokes equations. IMA J. Numer. Anal. 22, 577\u2013597 (2002)","journal-title":"IMA J. Numer. Anal."},{"key":"450_CR26","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1016\/0021-9991(91)90007-8","volume":"97","author":"G. Karniadakis","year":"1991","unstructured":"Karniadakis G., Israeli M., Orszag S.: High-order splitting methods for the incompressible Navier\u2013Stokes equations. J. Comput. Phys. 97, 414\u2013443 (1991)","journal-title":"J. Comput. Phys."},{"issue":"4","key":"450_CR27","doi-asserted-by":"crossref","first-page":"1653","DOI":"10.1137\/080725349","volume":"41","author":"N. Kim","year":"2009","unstructured":"Kim N.: Large friction limit and the inviscid limit of 2D Navier\u2013Stokes equations under Navier friction condition. SIAM J. Math. Anal. 41(4), 1653\u20131663 (2009)","journal-title":"SIAM J. Math. Anal."},{"key":"450_CR28","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s002090000130","volume":"235","author":"H. Kozono","year":"2000","unstructured":"Kozono H., Taniuchi Y.: Bilinear estimates in BMO and the Navier\u2013Stokes equations. Math. Z. 235, 173\u2013194 (2000)","journal-title":"Math. Z."},{"issue":"2","key":"450_CR29","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1137\/0726019","volume":"26","author":"S. Larsson","year":"1989","unstructured":"Larsson S.: The long-time behavior of finite-element approximations of solutions to semilinear parabolic problems. SIAM J. Numer. Anal. 26(2), 348\u2013365 (1989)","journal-title":"SIAM J. Numer. Anal."},{"key":"450_CR30","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-4286-4","volume-title":"Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics","author":"A. Lasota","year":"1994","unstructured":"Lasota A., Mackey M.C.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd edn. Springer, New York (1994)","edition":"2"},{"key":"450_CR31","volume-title":"Functional Analysis","author":"P.D. Lax","year":"2002","unstructured":"Lax P.D.: Functional Analysis. Wiley, New York (2002)"},{"key":"450_CR32","volume-title":"Simulating Hamiltonian Dynamics","author":"B. Leimkuhler","year":"2004","unstructured":"Leimkuhler B., Reich S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004)"},{"key":"450_CR33","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511616778","volume-title":"Nonlinear Dynamics and Statistical Theory for Basic Geophysical Flows","author":"A.J. Majda","year":"2006","unstructured":"Majda A.J., Wang X.: Nonlinear Dynamics and Statistical Theory for Basic Geophysical Flows. Cambridge University Press, Cambridge (2006)"},{"key":"450_CR34","doi-asserted-by":"crossref","unstructured":"Marion, M., Temam, R.M.: Navier\u2013Stokes equations: Theory and approximation. In: Handbook of Numerical Analysis, vol. VI, pp. 96\u2013102 (1998)","DOI":"10.1016\/S1570-8659(98)80010-0"},{"key":"450_CR35","unstructured":"Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence. English edn. updated, augmented and rev. by the authors. MIT Press, Cambridge (1975)"},{"key":"450_CR36","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-6557-1","volume-title":"Spectral Methods for Incompressible Viscous Flow","author":"R. Peyret","year":"2002","unstructured":"Peyret R.: Spectral Methods for Incompressible Viscous Flow. Springer, New York (2002)"},{"issue":"11&12","key":"450_CR37","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1080\/01630568908816354","volume":"10","author":"J. Shen","year":"1989","unstructured":"Shen J.: Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations. Numer. Funct. Anal. Optim. 10(11&12), 1213\u20131234 (1989)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"450_CR38","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1080\/00036819008839963","volume":"38","author":"J. Shen","year":"1990","unstructured":"Shen J.: Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods. Appl. Anal. 38, 201\u2013229 (1990)","journal-title":"Appl. Anal."},{"key":"450_CR39","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1017\/CBO9781107360198.012","volume-title":"Foundations of Computational Mathematics","author":"H. Sigurgeirsson","year":"2001","unstructured":"Sigurgeirsson H., Stuart A.M.: Statistics from computations. In: Devore, R., Iserles, A., Suli, E. (eds) Foundations of Computational Mathematics, pp. 323\u2013344. Cambridge University Press, Cambridge (2001)"},{"key":"450_CR40","volume-title":"Dynamical Systems and Numerical Analysis","author":"A.M. Stuart","year":"1996","unstructured":"Stuart A.M., Humphries A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996)"},{"key":"450_CR41","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1006\/jcph.1993.1142","volume":"107","author":"C.K.W. Tam","year":"1993","unstructured":"Tam C.K.W., Webb J.C.: Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107, 262\u2013281 (1993)","journal-title":"J. Comput. Phys."},{"key":"450_CR42","first-page":"219","volume":"262","author":"R. Temam","year":"1966","unstructured":"Temam R.: Sur l\u2019approximation des solutions des \u00e9quations de Navier\u2013Stokes. C.R. Acad. Sci. Paris Serie A 262, 219\u2013221 (1966)","journal-title":"C.R. Acad. Sci. Paris Serie A"},{"key":"450_CR43","unstructured":"Temam, R.M.: Navier\u2013Stokes equations and nonlinear functional analysis. In: CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1983)"},{"key":"450_CR44","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0645-3","volume-title":"Infinite Dimensional Dynamical Systems in Mechanics and Physics","author":"R.M. Temam","year":"1997","unstructured":"Temam R.M.: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)","edition":"2"},{"key":"450_CR45","doi-asserted-by":"crossref","unstructured":"Tone, F., Wang, X.: Approximation of the stationary statistical properties of the dynamical systems generated by the two-dimensional Rayleigh-Benard convection problem. Anal. Appl. (2010, accepted)","DOI":"10.1142\/S0219530511001935"},{"issue":"1","key":"450_CR46","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1137\/040618527","volume":"44","author":"F. Tone","year":"2006","unstructured":"Tone F., Wirosoetisno D.: On the long-time stability of the implicit Euler scheme for the two-dimensional Navier\u2013Stokes equations. SIAM J. Numer. Anal. 44(1), 29\u201340 (2006)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"450_CR47","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1137\/040603802","volume":"4","author":"P.F. Tupper","year":"2005","unstructured":"Tupper P.F.: Ergodicity and the numerical simulation of Hamiltonian systems. SIAM J. Appl. Dyn. Syst. 4(3), 563\u2013587 (2005)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"450_CR48","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1137\/0717025","volume":"17","author":"J. Varah","year":"1980","unstructured":"Varah J.: Stability restrictions on second-order, three level finite difference schemes for parabolic equations. SIAM J. Numer. Anal. 17, 300\u2013309 (1980)","journal-title":"SIAM J. Numer. Anal."},{"key":"450_CR49","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-1423-0","volume-title":"Mathematical Problems of Statistical Hydromechanics","author":"M.I. Vishik","year":"1988","unstructured":"Vishik M.I., Fursikov A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer, Dordrecht (1988)"},{"issue":"6","key":"450_CR50","first-page":"789","volume":"61","author":"X. Wang","year":"2008","unstructured":"Wang X.: Stationary statistical properties of Rayleigh-B\u00e9nard convection at large Prandtl number. CPAM 61(6), 789\u2013815 (2008)","journal-title":"CPAM"},{"issue":"1\/2","key":"450_CR51","doi-asserted-by":"crossref","first-page":"521","DOI":"10.3934\/dcds.2009.23.521","volume":"23","author":"X. Wang","year":"2009","unstructured":"Wang X.: Upper semi-continuity of stationary statistical properties of dissipative systems. Dedicated to Prof. Li Ta-Tsien on the occasion of his 70th birthday. Discrete Contin. Dyn. Syst. A 23(1\/2), 521\u2013540 (2009). doi: 10.3934\/dcds.2009.23.521","journal-title":"Discrete Contin. Dyn. Syst. A"},{"key":"450_CR52","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1090\/S0025-5718-09-02256-X","volume":"79","author":"X. Wang","year":"2010","unstructured":"Wang X.: Approximation of stationary statistical properties of dissipative dynamical systems: time discretization. Math. Comput. 79, 259\u2013280 (2010)","journal-title":"Math. Comput."},{"key":"450_CR53","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/0022-247X(69)90156-5","volume":"26","author":"H.C. Wente","year":"1960","unstructured":"Wente H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318\u2013344 (1960)","journal-title":"J. Math. Anal. Appl."},{"issue":"4","key":"450_CR54","doi-asserted-by":"crossref","first-page":"1451","DOI":"10.1137\/S0036142993248092","volume":"33","author":"Y. Yan","year":"1996","unstructured":"Yan Y.: Attractors and error estimates for discretizations of incompressible Navier\u2013Stokes equations. SIAM J. Numer. Anal. 33(4), 1451\u20131472 (1996)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-012-0450-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-012-0450-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-012-0450-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,21]],"date-time":"2025-03-21T19:05:28Z","timestamp":1742583928000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-012-0450-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2,26]]},"references-count":54,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2012,8]]}},"alternative-id":["450"],"URL":"https:\/\/doi.org\/10.1007\/s00211-012-0450-3","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,2,26]]}}}