SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2459-2483, December 2024.
Abstract. This work develops an energy-based discontinuous Galerkin (EDG) method for the nonlinear Schrödinger 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 weak formulation for the auxiliary equation involving the time derivative of the displacement variable. A critical part of the convergence analysis is to establish the [math] 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 special weak formulation, 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. Numerical experiments were performed to demonstrate the optimal convergence of the scheme in the [math] norm. These experiments involved specific choices of numerical fluxes combined with specific choices of approximation spaces.

中文翻译：

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一种基于能量的不连续伽辽金方法，用于具有波算子的非线性薛定谔方程


SIAM 数值分析杂志，第 62 卷，第 6 期，第 2459-2483 页，2024 年 12 月。

抽象。这项工作开发了一种基于能量的间断伽辽金 （EDG） 方法，用于具有波运算符的非线性薛定谔方程。该研究的重点是该方法的能量守恒或能量耗散行为，我们设计了一些简单的与网格无关的数值通量。我们在能量范数中建立了误差估计，这需要为涉及位移变量的时间导数的辅助方程仔细选择一个弱公式。收敛分析的一个关键部分是通过使用确定位移变量平均值的方程来建立位移变量中近似误差的时间导数的 [数学] 误差边界。使用特殊的弱公式，我们表明，即使在处理原始问题中的非线性性质时，也可以为未知数的时间演变创建一个线性系统。进行了数值实验，以证明该方案在 [数学] 范数中的最佳收敛性。这些实验涉及数值通量的特定选择以及近似空间的特定选择。