A Multi-Symplectic Compact Method for the Two-Component Camassa–Holm Equation with Singular Solutions**Supported by the National Natural Science Foundation of China under Grant Nos 11571366 and 11501570, the Open Foundation of State Key Laboratory of High Performance Computing of China, the Research Fund of the National University of Defense Technology under Grant No JC15-02-02, and the Fund from HPCL.

Abstract

The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.