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Analysis and computation of a nonlinear Korteweg-de Vries system

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Abstract

In this work, we present an analytical and numerical study of the Korteweg-de Vries (KdV) equation on a bounded domain in the presence of a dissipation mechanism. We present results on the existence and uniqueness of strong solutions using the Faedo–Galerkin method and studying a regularized version of the KdV equation. We also analyze the influence of this dissipation mechanism on the system energy. We introduce a numerical method based on a finite element discretization in space using Hermite polynomials as basis functions and the Crank–Nicolson finite difference scheme in time. Error estimates in Sobolev space for both the semi and fully-discrete problems are presented. Numerical simulations are also included in order to illustrate the applicability of the method. They also show the influence of the dissipative mechanism on the energy of the system.

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Authors and Affiliations

  1. Departamento de Ciência da Computação, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Mauro A. Rincon & Daniel G. Alfaro Vigo

  2. Programa de Pós-graduação em Informática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Juliana C. Xavier

Authors
  1. Mauro A. Rincon
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  2. Juliana C. Xavier
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  3. Daniel G. Alfaro Vigo
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Corresponding author

Correspondence to Daniel G. Alfaro Vigo.

Additional information

Communicated by Jan Nordström.

M. A. Rincon research was partially supported by CNPq-Brazil.

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Rincon, M.A., Xavier, J.C. & Alfaro Vigo, D.G. Analysis and computation of a nonlinear Korteweg-de Vries system. Bit Numer Math 56, 1069–1099 (2016). https://doi.org/10.1007/s10543-015-0589-2

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  • DOI: https://doi.org/10.1007/s10543-015-0589-2

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