On the rate of convergence of a collocation projection of the KdV equation

Kalisch, Henrik ; Raynaud, Xavier

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007), p. 95-110 / Harvested from Numdam

Résumé

Based on estimates for the KdV equation in analytic Gevrey classes, a spectral collocation approximation of the KdV equation is proved to converge exponentially fast.

Détail

Publié le : 2007-01-01

MR 2323692 | Zbl 1129.65060

DOI : https://doi.org/10.1051/m2an:2007010
Classification:  35Q53,  65M12,  65M70

Bibliographie

[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

[2] J.L. Bona and Z. Grujić, Spatial analyticity for nonlinear waves. Math. Models Methods Appl. Sci. 13 (2003) 1-15. | Zbl 1137.35418

[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

[4] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. GAFA 3 (1993) 107-156, 209-262. | Zbl 0787.35098

[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. | JFM 04.0493.04

[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

[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

[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

[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

[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

[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

[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

[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

[14] N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation. SIAM J. Math. Anal. 22 (1991) 1738-1743. | Zbl 0742.35056

[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. | Zbl 0729.35119

[16] H. Kalisch, Rapid convergence of a Galerkin projection of the KdV equation. C. R. Math. 341 (2005) 457-460. | Zbl 1081.65539

[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. | Zbl 1106.35081

[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

[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

[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

[21] H.-O. Kreiss and J. Oliger, Stability of the Fourier method. SIAM J. Numer. Anal. 16 (1979) 421-433. | Zbl 0419.65076

[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

[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

[24] J.E. Pasciak, Spectral and pseudospectral methods for advection equations. Math. Comput. 35 (1980) 1081-1092. | Zbl 0448.65071

[25] E. Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23 (1986) 1-10. | Zbl 0613.65017

[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

[27] R. Temam, Sur un problème non linéaire. J. Math. Pures Appl. 48 (1969) 159-172. | Zbl 0187.03902

[28] G.B. Whitham, Linear and Nonlinear Waves. Wiley, New York (1974). | MR 483954 | 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.