Geometric numerical schemes for the KdV equation

Abstract

Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudo-spectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2013geometricnumerical,
  author = {Dutykh, D. and Fedele, F. and Chhay, M.},
  title = {Geometric numerical schemes for the KdV equation},
  journal = {Computational Mathematics and Mathematical Physics},
  year = {2013},
  volume = {53},
  number = {2},
  pages = {221--236},
  doi = {10.1134/S0965542513020103},
  abstract = {Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudo-spectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.},
  keywords = {Geometric numerical schemes, solitonic gas, Pseudo-spectral methods, Korteweg-de Vries equation, Nonlinear wave phenomena, KdV equation, Symplectic methods, wave turbulence, Multi-symplectic schemes, multi-symplectic scheme, symplectic scheme, Hamiltonian structures, Long-time dynamics}
}