zbMATH — the first resource for mathematics

Rapid convergence of a Galerkin projection of the KdV equation. (English) Zbl 1081.65539
Summary: It is shown that a Fourier-Galerkin approximation of the Korteweg-de Vries equation with periodic boundary conditions converges exponentially fast if the initial data can be continued analytically to a strip about the real axis.

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI
[1] Bona, J.L.; Grujić, Z., Spatial analyticity for nonlinear waves, Math. models methods appl. sci., 13, 1-15, (2003)
[2] 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, 55-108, (1872) · JFM 04.0493.04
[3] Foias, C.; Temam, R., Gevrey class regularity for the solutions of the navier – stokes equations, J. funct. anal., 87, 359-369, (1989) · Zbl 0702.35203
[4] Hayashi, N., Analyticity of solutions of the korteweg – de Vries equation, SIAM J. math. anal., 22, 1738-1743, (1991) · Zbl 0742.35056
[5] T. Kappeler, P. Topalov, Global well-posedness of KdV in \(H^{−1}(\mathbb{T}, \mathbb{R})\), Preprint, University of Zürich · Zbl 1101.35367
[6] Kato, T.; Masuda, K., Nonlinear evolution equations and analyticity I, Ann. inst. H. Poincaré anal. non linéaire, 3, 455-467, (1986) · Zbl 0622.35066
[7] Kenig, C.E.; Ponce, G.; Vega, L., A bilinear estimate with applications to the KdV equation, J. amer. math. soc., 9, 573-603, (1996) · Zbl 0848.35114
[8] Korteweg, D.J.; 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, 422-443, (1895) · JFM 26.0881.02
[9] Maday, Y.; Quarteroni, A., Error analysis for spectral approximation of the korteweg – de Vries equation, RAIRO modél. math. anal., 22, 499-529, (1988) · Zbl 0647.65082
[10] Taha, T.; Ablowitz, M., Analytical and numerical aspects of certain nonlinear evolution equations. III. numerical, korteweg – de Vries equation, J. comput. phys., 55, 231-253, (1984) · Zbl 0541.65083
[11] Temam, R., Sur un problème non linéaire, J. math. pures appl., 48, 159-172, (1969) · Zbl 0187.03902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.