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.

22 pages, 14 figures, 74 references. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/

References in corpus (1)

• Kinetic equation for a dense soliton gas

Cited by in corpus (11)

• Numerical simulation of a solitonic gas in KdV and KdV-BBM equations
• Macroscopic dynamics of incoherent soliton ensembles: soliton-gas kinetics and direct numerical modeling
• Two-soliton interaction in the framework of modified Korteweg - de Vries equation
• On the modelling of shallow turbidity flows
• Partial differential systems with nonlocal nonlinearities: Generation and solutions
• On the multi-symplectic structure of the Serre-Green-Naghdi equations
• On the multi-symplectic structure of Boussinesq-type systems. I: Derivation and mathematical properties
• On the multi-symplectic structure of Boussinesq-type systems. II: Geometric discretization
• Numerical study of the generalised Klein-Gordon equations
• Serre-type equations in deep water
• Multi-symplectic structure of fully-nonlinear weakly-dispersive internal gravity waves