Convergence of Galerkin Approximations for the Korteweg-de Vries Equation,

Standard Galerkin approximations, using smooth splines on a uniform mesh, to 1-periodic solutions of the Korteweg-de Vries equation are analyzed. Optimal rate of convergence estimates are obtained for both semidiscrete and second order in time fully discrete schemes. At each time level, the resulting system of nonlinear equations can be solved by Newton's method. It is shown that if a proper extrapolation is used as a starting value, then only one step of the Newton iteration is required.

[1]  R. Temam.  Navier-Stokes Equations , 1977 .

[2]  Jerry L. Bona,et al.  Solutions of the Korteweg-de Vries equation in frac-tional order Sobolev space , 1976 .

[3]  Ragnar Winther,et al.  A conservative finite element method for the Korteweg-de Vries equation , 1980 .

[4]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[5]  J. Bona,et al.  The initial-value problem for the Korteweg-de Vries equation , 1975, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[6]  Osamu Inoue,et al.  Fourier expansion solution of the Korteweg-de Vries equation , 1980 .

[7]  N. Zabusky,et al.  Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States , 1965 .

[8]  Bengt Fornberg,et al.  A numerical and theoretical study of certain nonlinear wave phenomena , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[9]  A. C. Vliegenthart,et al.  On finite-difference methods for the Korteweg-de Vries equation , 1971 .

[10]  M. E. Alexander,et al.  Galerkin methods applied to some model equations for non-linear dispersive waves , 1979 .

[11]  H. Schamel,et al.  The application of the spectral method to nonlinear wave propagation , 1976 .

[12]  J. Ll. Morris,et al.  A Hopscotch method for the Korteweg-de-Vries equation , 1976 .

[13]  Vidar Thomée,et al.  Convergence Estimates for Galerkin Methods for Variable Coefficient Initial Value Problems , 1974 .

[14]  Peter D. Lax,et al.  Almost Periodic Solutions of the KdV Equation , 1976 .

[15]  Lars B. Wahlbin.  A Dissipative Galerkin Method for the Numerical Solution of First Order Hyperbolic Equations , 1974 .