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Mathematics of Computation

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

Author: Ming You Huang
Journal: Math. Comp. 56 (1991), 607-620
MSC: Primary 65M60; Secondary 35Q53, 76B15, 76B25
MathSciNet review: 1068815
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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

$\displaystyle 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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Keywords: Korteweg-de Vries equation, finite element method, Hamiltonian approximation, superconvergence
Article copyright: © Copyright 1991 American Mathematical Society

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