Numerical study of long-time Camassa–Holm solution behavior for soliton transport

In this paper a three-step solution scheme is employed to numerically explore the long-time solution behavior of the Camassa–Holm equation. In the present u−P−αu−P−α formation, we conduct modified equation analysis to eliminate several leading discretization error terms and perform Fourier analysis for minimizing the wave-like type of error. A three-point seventh-order spatially accurate combined compact upwind scheme is developed for the approximation of first-order derivative term. For the purpose of retaining Hamiltonian and multi-symplectic geometric structures in the non-dissipative Camassa–Holm equation, the adopted time integrator conserves symplecticity. Another main emphasis of this study is to numerically shed light on the scenario of the soliton transport.