In this paper, a conservative compact difference scheme for the generalized Kawahara equation is constructed based on the scalar auxiliary variable (SAV) approach. The discrete conservative laws of mass and Hamiltonian energy and boundedness estimates are studied in detail. The error estimates in discrete $L^{\infty}$-norm and $L^2$-norm of the presented scheme are analyzed by using the discrete energy method. We give an efficiently algorithm of the presented scheme which only needs to solve two decoupled equations.

In this paper, a fourth-order accurate conservative finite difference scheme for solving the coupled nonlinear Schr\”odinger (CNLS) equations is proposed. Conservation of the discrete energy and masses, priori estimates, existence and uniqueness of numerical solutions, convergence with second-order in time and fourth-order in space as well as stability of the present scheme are proved by discrete energy method. A convergent iterative method for the present scheme is developed. Numerical experiments are given to support the theoretical analysis.