High-order energy-preserving compact difference schemes for the improved Boussinesq equation

In this paper, some efficient energy-preserving schemes for solving the improved Boussinesq (IBq) equation are presented and discussed. Firstly, a scalar auxiliary variable is introduced to transform the Hamiltonian functional into a quadratic form, and the original IBq equation is written as an equivalent system. Then, the space variable is approximated by sixth-order compact finite difference method and the time direction is discretized making use of the Crank–Nicolson (C–N) scheme, Leap–Frog (L–F) scheme and second-order backward differential formula (BDF). The important thing is that a stabilized energy-preserving L–F scheme and an energy-preserving BDF scheme in the recursive sense are devised; the solvability, stability, and the conservation properties are proved. Finally, numerical examples are presented to illustrate the effectiveness of the proposed schemes.