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

Published online by Cambridge University Press:  26 April 2007

Henrik Kalisch  and
Xavier Raynaud
Henrik Kalisch
Affiliation:
Department of Mathematics, University of Bergen, 5008 Bergen, Norway. [email protected]
Xavier Raynaud
Affiliation:
Department of Mathematics, NTNU, 7491 Trondheim, Norway. [email protected]
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Abstract

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

Keywords

Spectral methodsconvergence ratecollocation projectionanalytic Gevrey class.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 1 , January 2007 , pp. 95 - 110
Copyright
© EDP Sciences, SMAI, 2007

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References

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).
Bona, J.L. and Grujić, Z., Spatial analyticity for nonlinear waves. Math. Models Methods Appl. Sci. 13 (2003) 1–15. CrossRef
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.
Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. GAFA 3 (1993) 107–156, 209–262.
Boussinesq, J., 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.
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics. Springer, Berlin (1988).
Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., Multilinear estimates for periodic KdV equations, and applications. J. Funct. Anal. 211 (2004) 173–218. CrossRef
Cooley, J.M. and Tukey, J.W., An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (1965) 297–301. CrossRef
Doelman, A. and Titi, E.S., Regularity of solutions and the convergence of the Galerkin method in the Ginzburg-Landau equation. Numer. Funct. Anal. Optim. 14 (1993) 299–321. CrossRef
P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1989).
Ferrari, A.B. and Titi, E.S., Gevrey regularity for nonlinear analytic parabolic equations. Comm. Partial Differential Equations 23 (1998) 1–16. CrossRef
Foias, C. and Temam, R., Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Functional Anal. 87 (1989) 359–369. CrossRef
Grujić, Z. and Kalisch, H., Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions. Diff. Integral Eq. 15 (2002) 1325–1334.
Hayashi, N., Analyticity of solutions of the Korteweg-de Vries equation. SIAM J. Math. Anal. 22 (1991) 1738–1743. CrossRef
Hayashi, N., Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and Szegö spaces on a sector. Duke Math. J. 62 (1991) 575–591. CrossRef
Kalisch, H., Rapid convergence of a Galerkin projection of the KdV equation. C. R. Math. 341 (2005) 457–460. CrossRef
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.
T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 455–467.
Kenig, C.E, Ponce, G. and Vega, L., A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9 (1996) 573–603. CrossRef
Korteweg, D.J. and de Vries, G., 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. CrossRef
Kreiss, H.-O. and Oliger, J., Stability of the Fourier method. SIAM J. Numer. Anal. 16 (1979) 421–433. CrossRef
Levermore, C.D. and Oliver, M., Analyticity of solutions for a generalized Euler equation. J. Differential Equations 133 (1997) 321–339. CrossRef
Maday, Y. and Quarteroni, A., Error analysis for spectral approximation of the Korteweg-de Vries equation. RAIRO Modél. Math. Anal. Numér. 22 (1988) 499–529. CrossRef
Pasciak, J.E., Spectral and pseudospectral methods for advection equations. Math. Comput. 35 (1980) 1081–1092.
Tadmor, E., The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23 (1986) 1–10. CrossRef
Taha, T. and Ablowitz, M., Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation. J. Comput. Phys. 55 (1984) 231–253. CrossRef
Temam, R., Sur un problème non linéaire. J. Math. Pures Appl. 48 (1969) 159–172.
G.B. Whitham, Linear and Nonlinear Waves. Wiley, New York (1974).
Zabusky, N.J. and Kruskal, M.D., Interaction of solutions in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15 (1965) 240–243. CrossRef