{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:56:06Z","timestamp":1772294166682,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2014,7,15]],"date-time":"2014-07-15T00:00:00Z","timestamp":1405382400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2015,4]]},"DOI":"10.1007\/s00211-014-0652-y","type":"journal-article","created":{"date-parts":[[2014,7,14]],"date-time":"2014-07-14T03:57:54Z","timestamp":1405310274000},"page":"749-782","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the $$2\/3$$ 2 \/ 3 de-aliasing method"],"prefix":"10.1007","volume":"129","author":[{"given":"Claude","family":"Bardos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eitan","family":"Tadmor","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2014,7,15]]},"reference":[{"key":"652_CR1","unstructured":"Abarbanel, S., Gottlieb, D., Tadmor, E.: Spectral methods for discontinuous problems \u201cNumerical methods for fluid dynamics II\u201d. In: Morton, K.W., Baines, M.J. (eds.) Proceedings of the 1985 Conference on Numerical methods for fluid dynamics. Clarendon Press, Oxford, pp. 129\u2013153 (1986)"},{"key":"652_CR2","doi-asserted-by":"crossref","first-page":"2778","DOI":"10.1016\/j.jde.2009.08.012","volume":"247","author":"J Avrin","year":"2009","unstructured":"Avrin, J., Xiao, C.: Convergence of Galerkin solutions and continuous dependence on data in spectrally hyperviscous models of 3D turbulent flow. J. Differ. Equ. 247, 2778\u20132798 (2009)","journal-title":"J. Differ. Equ."},{"issue":"2","key":"652_CR3","doi-asserted-by":"crossref","first-page":"185","DOI":"10.3934\/dcdss.2010.3.185","volume":"3","author":"C Bardos","year":"2010","unstructured":"Bardos, C., Titi, E.: Loss of smoothness and energy conserving rough weak solutions for the 3D Euler equations. DCDS Ser. 5 3(2), 185\u2013197 (2010)","journal-title":"DCDS Ser. 5"},{"key":"652_CR4","unstructured":"Buckmaster, T.: Onsager\u2019s conjecture almost everywhere in time. arXiv:1304.1049 (2013)"},{"issue":"1","key":"652_CR5","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/s00205-008-0201-x","volume":"195","author":"C Lellis De","year":"2010","unstructured":"De Lellis, C., Sz\u00e9kelyhidi Jr, L.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225\u2013260 (2010)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"10","key":"652_CR6","doi-asserted-by":"crossref","first-page":"2981","DOI":"10.1090\/S0002-9939-97-03946-4","volume":"125","author":"G-Q Chen","year":"1997","unstructured":"Chen, G.-Q.: Remarks on DiPerna\u2019s paper \u201cConvergence of the viscosity method for isentropic gas dynamics\u201d. Proc. AMS 125(10), 2981\u20132986 (1997)","journal-title":"Proc. AMS"},{"issue":"4","key":"652_CR7","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1090\/S0273-0979-07-01184-6","volume":"44","author":"P Constantin","year":"2007","unstructured":"Constantin, P.: On the Euler equations of incompressible fluids. Bull. AMS 44(4), 603\u2013621 (2007)","journal-title":"Bull. AMS"},{"key":"652_CR8","unstructured":"Constantin, P., Weinan, E., Titi, E.S.: Onsager\u2019s conjecture on the energy conservation for solutions of Euler\u2019s equation. Commun. Math. Phys. 165, 207\u2013209 (1994)"},{"key":"652_CR9","doi-asserted-by":"crossref","unstructured":"Dafermos, C.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70(199), 167\u2013179","DOI":"10.1007\/BF00250353"},{"key":"652_CR10","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1090\/S0273-0979-2012-01376-9","volume":"49","author":"C Lellis De","year":"2012","unstructured":"De Lellis, C., Sz\u00e9kelyhidi, L.: The $$h$$ h -principle and the equations of fluid dynamics. Bull. Am. Math. Soc. 49, 347\u2013375 (2012)","journal-title":"Bull. Am. Math. Soc."},{"key":"652_CR11","doi-asserted-by":"crossref","unstructured":"DeVore, R., Lorentz, G.G.: Constructive Approximation, Springer Grundlehren, vol. 303 (1993)","DOI":"10.1007\/978-3-662-02888-9"},{"key":"652_CR12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01206047","volume":"91","author":"R DiPerna","year":"1983","unstructured":"DiPerna, R.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1\u201330 (1983)","journal-title":"Commun. Math. Phys."},{"key":"652_CR13","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/0167-2789(94)90117-1","volume":"78","author":"GL Eyink","year":"1994","unstructured":"Eyink, G.L.: Energy dissipation without viscosity in ideal hydrodynamics, I. Fourier analysis and local energy transfer. Phys. D 78, 222\u2013240 (1994)","journal-title":"Phys. D"},{"issue":"1","key":"652_CR14","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s002110050019","volume":"67","author":"J Goodman","year":"1994","unstructured":"Goodman, J., Hou, T., Tadmor, E.: On the stability of the unsmoothed Fourier method for hyperbolic equations. Numer. Math. 67(1), 93\u2013129 (1994)","journal-title":"Numer. Math."},{"key":"652_CR15","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., Orzag, S.: Numerical Analysis of Spectral Methods : Theory and Applications. SIAM, Philadelphia (1977)"},{"key":"652_CR16","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1051\/m2an:2003060","volume":"37","author":"J-L Guermond","year":"2003","unstructured":"Guermond, J.-L., Prudhomme, S.: Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows. Math. Model. Numer. Anal. 37, 893\u2013908 (2003)","journal-title":"Math. Model. Numer. Anal."},{"key":"652_CR17","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.jcp.2007.04.014","volume":"226","author":"TY Hou","year":"2007","unstructured":"Hou, T.Y., Li, R.: Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379\u2013397 (2007)","journal-title":"J. Comput. Phys."},{"key":"652_CR18","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1006\/jcph.2000.6552","volume":"163","author":"G-S Karamanos","year":"2000","unstructured":"Karamanos, G.-S., Karniadakis, G.E.: A spectral vanishing viscosity method for large eddy-simulations. J. Comput. Phys. 163, 22\u201350 (2000)","journal-title":"J. Comput. Phys."},{"issue":"7","key":"652_CR19","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1063\/1.858849","volume":"5","author":"RM Kerr","year":"1993","unstructured":"Kerr, R.M.: Evidence for a singularity of the three dimensional, incompressible Euler equations. Phys. Fluids 5(7), 1725\u20131746 (1993)","journal-title":"Phys. Fluids"},{"key":"652_CR20","doi-asserted-by":"crossref","first-page":"075103","DOI":"10.1063\/1.1905183","volume":"17","author":"RM Kerr","year":"2005","unstructured":"Kerr, R.M.: Velocity and scaling of collapsing Euler vortices. Phys. Fluids 17, 075103\u2013114 (2005)","journal-title":"Phys. Fluids"},{"key":"652_CR21","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1016\/0167-2789(89)90151-6","volume":"37","author":"RM Kerr","year":"1989","unstructured":"Kerr, R.M., Hussain, F.: Simulation of vortex reconnection. Phys. D 37, 474 (1989)","journal-title":"Phys. D"},{"key":"652_CR22","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1111\/j.2153-3490.1972.tb01547.x","volume":"24","author":"H-O Kreiss","year":"1972","unstructured":"Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199\u2013215 (1972)","journal-title":"Tellus"},{"key":"652_CR23","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/0167-2789(86)90185-5","volume":"18","author":"PD Lax","year":"1986","unstructured":"Lax, P.D.: On dispersive difference schemes. Phys. D 18, 250\u2013254 (1986)","journal-title":"Phys. D"},{"key":"652_CR24","unstructured":"Lions, P.L.: Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, Oxford (1996)"},{"key":"652_CR25","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/S0294-1449(00)00113-X","volume":"17","author":"M Lopes Filho","year":"2000","unstructured":"Lopes Filho, M., Nussenzveig, H.J., Tadmor, E.: Approximate solutions of the incompressible Euler equations with no concentrations. Annales De L\u2019institut Henri Poincare (c) Non Linear Analysis 17, 371\u2013412 (2000)","journal-title":"Annales De L\u2019institut Henri Poincare (c) Non Linear Analysis"},{"key":"652_CR26","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1090\/S0025-5718-1978-0501995-4","volume":"32","author":"A Majda","year":"1978","unstructured":"Majda, A., McDonough, J.M., Osher, S.: The Fourier method for non-smooth initial data. Math. Comput. 32, 1041 (1978)","journal-title":"Math. Comput."},{"key":"652_CR27","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1002\/cpa.3160310403","volume":"31","author":"M Mock","year":"1978","unstructured":"Mock, M., Lax, P.D.: The computation of discontinuous solutions of linear hyperbolic equations. Commun. Pure Appl. Math. 31, 423\u2013430 (1978)","journal-title":"Commun. Pure Appl. Math."},{"key":"652_CR28","unstructured":"Murat, F.: Compacit\u2019e par compensation. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 489\u2013507 (1978)"},{"key":"652_CR29","unstructured":"von Neumann, J.: Proposed and analysis of a new numerical method in the treatment of hydrodynamical shock problem, vol. VI, Collected Works, pp. 361\u2013379. Pergamon, London (1963)"},{"key":"652_CR30","doi-asserted-by":"crossref","unstructured":"Ohlsson, J., Schlatter, P., Fischer, P.F., Henningson, D.S.: Stabilization of the spectral-element method in turbulent flow simulation, ICOSAHOM (2010)","DOI":"10.1007\/978-3-642-15337-2_43"},{"key":"652_CR31","doi-asserted-by":"crossref","unstructured":"Onsager, L.: Statistical hydrodynamics, Nuovo Cimento (9), 6 Supplemento, 2(Convegno Internazionale di Meccanica Statistica), pp. 279\u2013287 (1949)","DOI":"10.1007\/BF02780991"},{"key":"652_CR32","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1002\/sapm1972513253","volume":"51","author":"S Orszag","year":"1972","unstructured":"Orszag, S.: Comparison of pseudospectral and spectral approximations. Stud. Appl. Math. 51, 253\u2013259 (1972)","journal-title":"Stud. Appl. Math."},{"key":"652_CR33","doi-asserted-by":"crossref","unstructured":"Panov, E.: Uniqueness of the solution of the Cauchy problem for a first-order quasilinear equation with an admissible strictly convex entropy. Mat. Zametki 55(5), 116\u2013129 (1994) [translation in Math. Notes 55(5\u20136), 517\u2013525 (1994)]","DOI":"10.1007\/BF02110380"},{"issue":"3\u20134","key":"652_CR34","first-page":"261","volume":"22","author":"R Pasquetti","year":"2007","unstructured":"Pasquetti, R., Severac, E., Serre, E., Bontoux, P., Schafer, M.: From stratified wakes to rotor-stator flows by an SVV-LES method. Theor. Commput. Fluid Dyn. 22(3\u20134), 261\u2013273 (2007)","journal-title":"Theor. Commput. Fluid Dyn."},{"key":"652_CR35","doi-asserted-by":"crossref","unstructured":"Pereira, R.M., Nguyen van yen, R., Farge, M., Schneider, K.: Wavelet methods to eliminate resonances in the Galerkin-truncated Burgers and Euler equations. Phys. Rev. E 87, 033017 (2013)","DOI":"10.1103\/PhysRevE.87.033017"},{"key":"652_CR36","first-page":"016301","volume":"84","author":"SS Ray","year":"2011","unstructured":"Ray, S.S., Frisch, U., Nazarenko, S., Matsumuto, T.: Resonance phenomenon for the Galerkin-truncated Burgers and Euler equations. Phys. Rev. E 84, 016301 (2011)","journal-title":"Phys. Rev. E"},{"key":"652_CR37","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/BF02921318","volume":"3","author":"V Scheffer","year":"1993","unstructured":"Scheffer, V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3, 343\u2013401 (1993)","journal-title":"J. Geom. Anal."},{"key":"652_CR38","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1016\/j.jcp.2007.05.023","volume":"226","author":"E Severac","year":"2007","unstructured":"Severac, E., Serre, E.: A spectral vanishing viscosity for the LES of turbulent flows within rotating cavities. J. Comput. Phys. 226, 1234\u20131255 (2007)","journal-title":"J. Comput. Phys."},{"key":"652_CR39","doi-asserted-by":"crossref","first-page":"1261","DOI":"10.1002\/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6","volume":"50","author":"A Shnirelman","year":"1997","unstructured":"Shnirelman, A.: On the nonuniqueness of weak solutions of the Euler equations. Commun. Pure Appl. Math. 50, 1261\u20131286 (1997)","journal-title":"Commun. Pure Appl. Math."},{"key":"652_CR40","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.jcp.2003.08.021","volume":"194","author":"S Sirisup","year":"2004","unstructured":"Sirisup, S., Karniadakis, G.E.: A spectral viscosity method for correcting the long-term behavior of POD models. J. Comput. Phys. 194, 92\u2013116 (2004)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"652_CR41","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/BF02091780","volume":"10","author":"C-W Shu","year":"1995","unstructured":"Shu, C.-W., Wong, P.S.: A note on the accuracy of spectral method applied to nonlinear conservation laws. J. Sci. Comput. 10(3), 357\u2013369 (1995)","journal-title":"J. Sci. Comput."},{"key":"652_CR42","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1137\/1029110","volume":"29","author":"E Tadmor","year":"1987","unstructured":"Tadmor, E.: Stability analysis of finite-difference, pseudospectral and Fourier\u2013Galerkin approximations for time-dependent problems. SIAM Rev. 29, 525\u2013555 (1987)","journal-title":"SIAM Rev."},{"key":"652_CR43","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1137\/0726003","volume":"26","author":"E Tadmor","year":"1989","unstructured":"Tadmor, E.: Convergence of spectral methods for nonlinear conservation laws. SINUM 26, 30\u201344 (1989)","journal-title":"SINUM"},{"key":"652_CR44","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1090\/S0025-5718-1993-1153170-9","volume":"60","author":"E Tadmor","year":"1993","unstructured":"Tadmor, E.: Total variation and error estimates for spectral viscosity approximations. Math. Comput. 60, 245\u2013256 (1993)","journal-title":"Math. Comput."},{"key":"652_CR45","doi-asserted-by":"crossref","unstructured":"Tadmor, E.: Super viscosity and spectral approximations of nonlinear conservation laws. In: Baines, J., Morton, K.W. (eds.) Numerical Methods for Fluid Dynamics IV, Proceedings of the 1992 Conference on Numerical Methods for Fluid Dynamics, pp. 69\u201382. Clarendon Press, Oxford (1993)","DOI":"10.1093\/oso\/9780198536963.003.0005"},{"key":"652_CR46","unstructured":"Tadmor, E.: Spectral methods for hyperbolic problems, Methods numeriques d\u2019ordre eleve pour les ondes en regime transitoire, Lecture notes delivered at Ecole des Ondes, INRIA\u2014Rocquencourt (1994)"},{"issue":"2","key":"652_CR47","doi-asserted-by":"crossref","first-page":"317","DOI":"10.4310\/CMS.2004.v2.n2.a9","volume":"2","author":"E Tadmor","year":"2004","unstructured":"Tadmor, E.: Burgers\u2019 equation with vanishing hyper-viscosity. Commun. Math. Sci. 2(2), 317\u2013324 (2004)","journal-title":"Commun. Math. Sci."},{"key":"652_CR48","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1017\/S0962492906320016","volume":"16","author":"E Tadmor","year":"2007","unstructured":"Tadmor, E.: Filters, mollifiers and the computation of the Gibbs phenomenon. Acta Numer. 16, 305\u2013378 (2007)","journal-title":"Acta Numer."},{"key":"652_CR49","unstructured":"Tartar, L.: Compensated compactness and applications to partial differential equations. In: Knopps, R.J. (ed.) Research Notes in Mathematics 39, Nonlinear Analysis and Mechanics, Heriott-Watt Symposium, vol. 4. Pittman Press, pp. 136\u2013211 (1979)"},{"key":"652_CR50","unstructured":"Tartar, L.: The compensated compactness method for a scalar hyperbolic equation, Carnegie Mellon Univ. Lecture notes, pp. 87\u201320 (1987)"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0652-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-014-0652-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0652-y","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T23:21:29Z","timestamp":1717024889000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-014-0652-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,7,15]]},"references-count":50,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,4]]}},"alternative-id":["652"],"URL":"https:\/\/doi.org\/10.1007\/s00211-014-0652-y","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,7,15]]}}}