An energy-preserving finite difference scheme with fourth-order accuracy for the generalized Camassa–Holm equation

Abstract

In this article, an energy-preserving finite difference scheme for solving the generalized Camassa–Holm (gCH) equation with the dual-power law nonlinearities is proposed. We first show that the solution of the initial–boundary-value gCH equation is unique and continuously dependent on the initial condition, then we construct a linear energy-preserving difference scheme for the gCH equation. The proposed difference scheme is three-level implicit, and the numerical convergence order is O(τ2+h4). The energy conservation, unique solvability, convergence and stability of the finite difference scheme are rigorously proved by using the discrete energy method. Finally, some numerical examples show that the proposed numerical scheme is efficient and reliable.