Two high order energy conserving variational limit integral schemes for the Korteweg–de Vries–Benjamin–Bona–Mahony equation

In this paper, we propose two high-order energy-conserving variational limit integral schemes to solve the Korteweg–de Vries–Benjamin–Bona–Mahony equation, coupled with an implicit midpoint time discretization. The two proposed fully discrete schemes can be shown to preserve the discrete counterparts of both mass and energy of the continuous solution. The existence and uniqueness of the solution, the a priori estimate, and the unconditional stability of the two fully discrete schemes are proved. The optimal convergence rate of the two fully discrete schemes is obtained at the order of O(τ2 + h4) in both discrete L2 norm and L∞ norm. Our numerical experiments demonstrate optimal rates of convergence O(τ2 + h4) as well as the mass- and energy-conserving properties and show that the errors of the numerical solutions of the two fully discrete schemes do not grow significantly in the long-time test due to the energy-conserving property. A numerical experiment is provided to show that the proposed schemes can effectively simulate the collision of two solitary waves.