This work develops an energy-based discontinuous Galerkin (EDG) method for the nonlinear Schr\"odinger equation with the wave operator. The focus of the study is on the energy-conserving or energy-dissipating behavior of the method with some simple mesh-independent numerical fluxes we designed. We establish error estimates in the energy norm that require careful selection of a test function for the auxiliary equation involving the time derivative of the displacement variable. A critical part of the convergence analysis is to establish the L2 error bounds for the time derivative of the approximation error in the displacement variable by using the equation that determines its mean value. Using a specially chosen test function, we show that one can create a linear system for the time evolution of the unknowns even when dealing with nonlinear properties in the original problem. Extensive numerical experiments are provided to demonstrate the optimal convergence of the scheme in the L2 norm with our choices of the numerical flux.

翻译：本文针对含波动算子的非线性薛定谔方程，提出了一种基于能量的间断伽辽金（EDG）方法。研究重点在于该方法在我们设计的若干简单网格无关数值通量作用下的能量守恒或能量耗散特性。我们建立了能量范数下的误差估计，该估计需要对辅助方程中涉及位移变量时间导数的测试函数进行精心选取。收敛性分析的关键环节是通过确定位移变量平均值的方程，建立近似误差时间导数的L2误差界。通过使用特殊选取的测试函数，我们证明即使处理原问题中的非线性特性时，仍可构建未知量时间演化的线性系统。大量数值实验表明，在我们选择的数值通量方案下，该方法在L2范数下具有最优收敛性。