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.

在本文中，我们为带波算子的非线性薛定谔方程开发了两类线性高阶保守数值格式。基于时间降阶方法和标量辅助变量技术，我们将原始模型转换为等效系统，其中能量被修改为二次形式。为了构造线性高阶保守方案，我们首先采用外推策略推导出线性化的偏微分方程系统，该系统高精度地逼近变换模型并继承了修正的能量守恒定律。然后我们及时采用辛Runge-Kutta方法得到一类线性高阶能量守恒方案。这种数值策略为开发可以简单实现的任意高阶线性结构保持算法提供了一个范例。为了补充新的线性方案，提出了预测校正方法以获得另一类能量保持算法。此外，三角伪谱方法应用于空间离散化以匹配时间精度的顺序。我们提供了充足的数值结果来确认所提出方案的收敛性、准确性和守恒性。将三角伪谱方法应用于空间离散化以匹配时间上的精度顺序。我们提供了充足的数值结果来确认所提出方案的收敛性、准确性和守恒性。将三角伪谱方法应用于空间离散化以匹配时间上的精度顺序。我们提供了充足的数值结果来确认所提出方案的收敛性、准确性和守恒性。