Linear High-Order Energy-Preserving Schemes for the Nonlinear Schrödinger Equation with Wave Operator Using the Scalar Auxiliary Variable Approach

In this paper, we develop two classes of linear high-order conservative numerical schemes for the nonlinear Schrödinger equation with wave operator. Based on the method of order reduction in time and the scalar auxiliary variable technique, we transform the original model into an equivalent system, where the energy is modified as a quadratic form. To construct linear high-order conservative schemes, we first adopt the extrapolation strategy to derive a linearized PDE system, which approximates the transformed model with high precision and inherits the modified energy conservation law. Then we employ the symplectic Runge–Kutta method in time to arrive at a class of linear high-order energy-preserving schemes. This numerical strategy presents a paradigm for developing arbitrarily high-order linear structure-preserving algorithms which could be implemented simply. In order to complement the new linear schemes, the prediction-correction method is presented to obtain another class of energy-preserving algorithms. Furthermore, the trigonometric pseudo-spectral method is applied for the spatial discretization to match the order of accuracy in time. We provide ample numerical results to confirm the convergence, accuracy and conservation property of the proposed schemes.