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On the rate of convergence of a collocation projection of the KdV equation. (English) Zbl 1129.65060
The authors consider the error analysis of a spectral collocation projection of the periodic Korteweg-de Vries (KdV) equation in analytic Gevrey classes and prove that, under appropriate assumptions on the initial data, the convergence of the numerical approximation is exponentially fast. This is in contrast to previous results that achieved spectral, or super-polynomial, convergence.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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References:
[1] R. Beals , P. Deift and C. Tomei , Direct and inverse scattering on the line . Mathematical Surveys and Monographs 28, American Mathematical Society, Providence, RI ( 1988 ). MR 954382 | Zbl 0679.34018 · Zbl 0679.34018
[2] J.L. Bona and Z. Grujić , Spatial analyticity for nonlinear waves . Math. Models Methods Appl. Sci. 13 ( 2003 ) 1 - 15 . Zbl 1137.35418 · Zbl 1137.35418 · doi:10.1142/S0218202503002532
[3] J.L. Bona , Z. Grujić and H. Kalisch , Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation . Ann. Inst. H. Poincaré, Anal. Non Linéaire 22 ( 2005 ) 783 - 797 . Numdam | Zbl 1095.35039 · Zbl 1095.35039 · doi:10.1016/j.anihpc.2004.12.004 · numdam:AIHPC_2005__22_6_783_0 · eudml:78678
[4] J. Bourgain , Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations . GAFA 3 ( 1993 ) 107 - 156 , 209 - 262 . Article | Zbl 0787.35098 · Zbl 0787.35098 · doi:10.1007/BF01895688 · eudml:58123
[5] J. Boussinesq , Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond . J. Math. Pures Appl. 17 (1872) 55 - 108 . Article | JFM 04.0493.04 · JFM 04.0493.04 · minidml.mathdoc.fr
[6] C. Canuto , M.Y. Hussaini , A. Quarteroni and T.A. Zang , Spectral Methods in Fluid Dynamics . Springer, Berlin ( 1988 ). MR 917480 | Zbl 0658.76001 · Zbl 0658.76001
[7] J. Colliander , M. Keel , G. Staffilani , H. Takaoka and T. Tao , Multilinear estimates for periodic KdV equations, and applications . J. Funct. Anal. 211 ( 2004 ) 173 - 218 . Zbl 1062.35109 · Zbl 1062.35109 · doi:10.1016/S0022-1236(03)00218-0
[8] J.M. Cooley and J.W. Tukey , An algorithm for the machine calculation of complex Fourier series . Math. Comp. 19 ( 1965 ) 297 - 301 . Zbl 0127.09002 · Zbl 0127.09002 · doi:10.2307/2003354
[9] A. Doelman and E.S. Titi , Regularity of solutions and the convergence of the Galerkin method in the Ginzburg-Landau equation . Numer. Funct. Anal. Optim. 14 ( 1993 ) 299 - 321 . Zbl 0792.35096 · Zbl 0792.35096 · doi:10.1080/01630569308816523
[10] P.G. Drazin and R.S. Johnson , Solitons: an introduction , Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge ( 1989 ). MR 985322 | Zbl 0661.35001 · Zbl 0661.35001
[11] A.B. Ferrari and E.S. Titi , Gevrey regularity for nonlinear analytic parabolic equations . Comm. Partial Differential Equations 23 ( 1998 ) 1 - 16 . Zbl 0907.35061 · Zbl 0907.35061 · doi:10.1080/03605309808821335
[12] C. Foias and R. Temam , Gevrey class regularity for the solutions of the Navier-Stokes equations . J. Functional Anal. 87 ( 1989 ) 359 - 369 . Zbl 0702.35203 · Zbl 0702.35203 · doi:10.1016/0022-1236(89)90015-3
[13] Z. Grujić and H. Kalisch , Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions . Diff. Integral Eq. 15 ( 2002 ) 1325 - 1334 . Zbl 1031.35124 · Zbl 1031.35124
[14] N. Hayashi , Analyticity of solutions of the Korteweg-de Vries equation . SIAM J. Math. Anal. 22 ( 1991 ) 1738 - 1743 . Zbl 0742.35056 · Zbl 0742.35056 · doi:10.1137/0522107
[15] N. Hayashi , Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and Szegö spaces on a sector . Duke Math. J. 62 ( 1991 ) 575 - 591 . Article | Zbl 0729.35119 · Zbl 0729.35119 · doi:10.1215/S0012-7094-91-06224-1 · minidml.mathdoc.fr
[16] H. Kalisch , Rapid convergence of a Galerkin projection of the KdV equation . C. R. Math. 341 ( 2005 ) 457 - 460 . Zbl 1081.65539 · Zbl 1081.65539 · doi:10.1016/j.crma.2005.09.006
[17] T. Kappeler and P. Topalov , Global well-posedness of KdV in \(H^{-1} (\mathbb{T},\mathbb{R})\) . Duke Math. J. 7 135 ( 2006 ) 327 - 360 . Article | Zbl 1106.35081 · Zbl 1106.35081 · doi:10.1215/S0012-7094-06-13524-X · minidml.mathdoc.fr
[18] T. Kato and K. Masuda , Nonlinear evolution equations and analyticity I . Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 ( 1986 ) 455 - 467 . Numdam | Zbl 0622.35066 · Zbl 0622.35066 · numdam:AIHPC_1986__3_6_455_0 · eudml:78123
[19] C.E Kenig , G. Ponce and L. Vega , A bilinear estimate with applications to the KdV equation . J. Amer. Math. Soc. 9 ( 1996 ) 573 - 603 . Zbl 0848.35114 · Zbl 0848.35114 · doi:10.1090/S0894-0347-96-00200-7
[20] D.J. Korteweg and G. de Vries , On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave . Philos. Mag. 39 (1895) 422 - 443 . JFM 26.0881.02 · JFM 26.0881.02
[21] H.-O. Kreiss and J. Oliger , Stability of the Fourier method . SIAM J. Numer. Anal. 16 ( 1979 ) 421 - 433 . Zbl 0419.65076 · Zbl 0419.65076 · doi:10.1137/0716035
[22] C.D. Levermore and M. Oliver , Analyticity of solutions for a generalized Euler equation . J. Differential Equations 133 ( 1997 ) 321 - 339 . Zbl 0876.35090 · Zbl 0876.35090 · doi:10.1006/jdeq.1996.3200
[23] Y. Maday and A. Quarteroni , Error analysis for spectral approximation of the Korteweg-de Vries equation . RAIRO Modél. Math. Anal. Numér. 22 ( 1988 ) 499 - 529 . Numdam | Zbl 0647.65082 · Zbl 0647.65082 · eudml:193540
[24] J.E. Pasciak , Spectral and pseudospectral methods for advection equations . Math. Comput. 35 ( 1980 ) 1081 - 1092 . Zbl 0448.65071 · Zbl 0448.65071 · doi:10.2307/2006376
[25] E. Tadmor , The exponential accuracy of Fourier and Chebyshev differencing methods . SIAM J. Numer. Anal. 23 ( 1986 ) 1 - 10 . Zbl 0613.65017 · Zbl 0613.65017 · doi:10.1137/0723001
[26] T. Taha and M. Ablowitz , Analytical and numerical aspects of certain nonlinear evolution equations . III. Numerical, Korteweg-de Vries equation. J. Comput. Phys. 55 ( 1984 ) 231 - 253 . Zbl 0541.65083 · Zbl 0541.65083 · doi:10.1016/0021-9991(84)90004-4
[27] R. Temam , Sur un problème non linéaire . J. Math. Pures Appl. 48 ( 1969 ) 159 - 172 . Zbl 0187.03902 · Zbl 0187.03902
[28] G.B. Whitham , Linear and Nonlinear Waves . Wiley, New York ( 1974 ). MR 483954 | Zbl 0373.76001 · Zbl 0373.76001
[29] N.J. Zabusky and M.D. Kruskal , Interaction of solutions in a collisionless plasma and the recurrence of initial states . Phys. Rev. Lett. 15 ( 1965 ) 240 - 243 . · Zbl 1201.35174 · doi:10.1103/PhysRevLett.15.240 · staff.ustc.edu.cn
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