A multi-symplectic numerical integrator for the two-component Camassa–Holm equation

Abstract

A new multi-symplectic formulation of the two-component Camassa-Holm equation (2CH) is presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. Furthermore, this scheme preserves exactly two discrete versions of the Casimir functions of 2CH. Numerical experiments show that the proposed numerical scheme has good conservation properties.

Keywords:

• Two-component Camassa–Holm equation
• Hamiltonian PDE
• Casimir function
• Numerical discretisation
• Multi-symplectic formulation
• Multi-symplectic schemes
• Euler box scheme

2010 Mathematics Subject Classification:

• 35Q51
• 35Q53
• 37K05
• 37K10
• 37M15
• 65M06
• 65M99
• 65P10