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.
Keywords: spectral methods, convergence rate, collocation projection, analytic Gevrey class
@article{M2AN_2007__41_1_95_0,
author = {Kalisch, Henrik and Raynaud, Xavier},
title = {On the rate of convergence of a collocation projection of the {KdV} equation},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {95--110},
year = {2007},
publisher = {EDP Sciences},
volume = {41},
number = {1},
doi = {10.1051/m2an:2007010},
mrnumber = {2323692},
zbl = {1129.65060},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an:2007010/}
}
TY - JOUR AU - Kalisch, Henrik AU - Raynaud, Xavier TI - On the rate of convergence of a collocation projection of the KdV equation JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 95 EP - 110 VL - 41 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an:2007010/ DO - 10.1051/m2an:2007010 LA - en ID - M2AN_2007__41_1_95_0 ER -
%0 Journal Article %A Kalisch, Henrik %A Raynaud, Xavier %T On the rate of convergence of a collocation projection of the KdV equation %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 95-110 %V 41 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an:2007010/ %R 10.1051/m2an:2007010 %G en %F M2AN_2007__41_1_95_0
Kalisch, Henrik; Raynaud, Xavier. On the rate of convergence of a collocation projection of the KdV equation. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 1, pp. 95-110. doi: 10.1051/m2an:2007010
[1] , and, Direct and inverse scattering on the line. Mathematical Surveys and Monographs 28, American Mathematical Society, Providence, RI (1988). | Zbl | MR
[2] and, Spatial analyticity for nonlinear waves. Math. Models Methods Appl. Sci. 13 (2003) 1-15. | Zbl
[3] , and, 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. | Zbl | Numdam
[4] , Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. GAFA 3 (1993) 107-156, 209-262. | Zbl
[5] , 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 | Numdam
[6] ,, and, Spectral Methods in Fluid Dynamics. Springer, Berlin (1988). | Zbl | MR
[7] ,,, and, Multilinear estimates for periodic KdV equations, and applications. J. Funct. Anal. 211 (2004) 173-218. | Zbl
[8] and, An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (1965) 297-301. | Zbl
[9] and, Regularity of solutions and the convergence of the Galerkin method in the Ginzburg-Landau equation. Numer. Funct. Anal. Optim. 14 (1993) 299-321. | Zbl
[10] and, Solitons: an introduction, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1989). | Zbl | MR
[11] and, Gevrey regularity for nonlinear analytic parabolic equations. Comm. Partial Differential Equations 23 (1998) 1-16. | Zbl
[12] and, Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Functional Anal. 87 (1989) 359-369. | Zbl
[13] and, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions. Diff. Integral Eq. 15 (2002) 1325-1334. | Zbl
[14] , Analyticity of solutions of the Korteweg-de Vries equation. SIAM J. Math. Anal. 22 (1991) 1738-1743. | Zbl
[15] , 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
[16] , Rapid convergence of a Galerkin projection of the KdV equation. C. R. Math. 341 (2005) 457-460. | Zbl
[17] and, Global well-posedness of KdV in . Duke Math. J. 7 135 (2006) 327-360. | Zbl
[18] and, Nonlinear evolution equations and analyticity I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 455-467. | Zbl | Numdam
[19] , and, A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9 (1996) 573-603. | Zbl
[20] and, 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
[21] and, Stability of the Fourier method. SIAM J. Numer. Anal. 16 (1979) 421-433. | Zbl
[22] and, Analyticity of solutions for a generalized Euler equation. J. Differential Equations 133 (1997) 321-339. | Zbl
[23] and, Error analysis for spectral approximation of the Korteweg-de Vries equation. RAIRO Modél. Math. Anal. Numér. 22 (1988) 499-529. | Zbl | Numdam
[24] , Spectral and pseudospectral methods for advection equations. Math. Comput. 35 (1980) 1081-1092. | Zbl
[25] , The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23 (1986) 1-10. | Zbl
[26] and, Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation. J. Comput. Phys. 55 (1984) 231-253. | Zbl
[27] , Sur un problème non linéaire. J. Math. Pures Appl. 48 (1969) 159-172. | Zbl
[28] , Linear and Nonlinear Waves. Wiley, New York (1974). | Zbl | MR
[29] and, Interaction of solutions in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15 (1965) 240-243.
Cité par Sources :
