Geometric numerical schemes for the KdV equation

TL;DR

This paper demonstrates that geometric discretization schemes for the KdV equation are as accurate and robust as spectral methods for long-term simulations, offering advantages for modeling nonlinear wave phenomena.

Contribution

It shows that geometric schemes effectively preserve Hamiltonian structures, matching the accuracy of spectral methods in long-time KdV simulations.

Findings

Geometric schemes are robust and accurate for long-time KdV dynamics.

They outperform traditional methods in modeling complex nonlinear waves.

Geometric discretizations preserve Hamiltonian structures at the discrete level.

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.