Error Estimates of High-order Conservative Schemes for the Generalized Nonlinear Schrödinger Equation in One Dimension

Abstract

In this paper, we use the high-order difference operator Dμ to compensate the central difference operator . Then, we discretize time by the time-midpoint method. Fully discrete schemes are obtained for the generalized nonlinear Schrödinger equation. For the high-order methods, conservation of mass and energy can be proved theoretically in the discrete sense. Then, to establish the optimal error estimates without any requirement on the grid ratio, we adopt the cut-off function technique. The error estimates of the schemes are proved to be of O(τ2 + h2(μ+1)) with time step size τ and mesh size h. Some numerical experiments are given to verify the accuracy order. In addition, the dynamics of the generalized nonlinear Schrödinger equation in one dimension are simulated.