A Hamiltonian approximation to simulate solitary waves of the Korteweg-de Vries equation

Huang, Ming You

doi:10.1090/S0025-5718-1991-1068815-X

A Hamiltonian approximation to simulate solitary waves of the Korteweg-de Vries equation
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by Ming You Huang PDF

Math. Comp. 56 (1991), 607-620 Request permission

Abstract:

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 \[ I(u) = \int _0^1 {\left ( {\frac {{u_x^2}}{2} + {u^3}} \right )\;dx} \] 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.

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