A Hamiltonian Approximation to Simulate Solitary Waves of the Korteweg - De Vries Equation

Given the Hamiltonian nature and conservation laws of the Korteweg-de Vries equation, the simulation of the solitary waves of this equation by numerical methods should be effected in such a way as to maintain the Hamiltonian nature of the problem. A semidiscrete finite element approximation of Petrov-Galerkin type, proposed by R. Winther, is analyzed here. It is shown that this approximation is a finite Hamiltonian system, and as a consequence, the energy integral /(h) = / I % + «J 1 dx «-jfM is exactly conserved by this method. In addition, there is a discussion of error estimates and superconvergence properties of the method, in which there is no perturbation term but instead a suitable choice of initial data. A single-step fully discrete scheme, and some numerical results, are presented. 1. The Hamiltonian nature and conservation laws In this paper, we shall consider the following problem for the Kortewegde Vries equation: ut 6uux + uxxx = 0, x G R, t > 0, (P) u(x+X,t) = u(x, t), u(x, 0) = u0(x) (a prescribed 1-periodic function). To study the Hamiltonian nature of problem (P), we introduce the following function space with / = [0, 1], H? = {vg Hm(I);v{i)(x+l) = v{i)(x), i = 0, 1, ... , m 1}, and the functional H(u)= f (y + "3J dx, Received March 24, 1989; revised December 7, 1989, May 22, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 65N15, 65N30.