In this paper, a novel high-order, mass and energy-conserving scheme is proposed for the regularized logarithmic Schr\"{o}dinger equation(RLogSE). Based on the idea of the supplementary variable method (SVM), we firstly reformulate the original system into an equivalent form by introducing two supplementary variables, and the resulting SVM reformulation is then discretized by applying a high-order prediction-correction scheme in time and a Fourier pseudo-spectral method in space, respectively. The newly developed scheme can produce numerical solutions along which the mass and original energy are precisely conserved, as is the case with the analytical solution. Additionally, it is extremely efficient in the sense that only requires solving a constant-coefficient linear systems plus two algebraic equations, which can be efficiently solved by the Newton iteration at every time step. Numerical experiments are presented to confirm the accuracy and structure-preserving properties of the new scheme.

翻译：本文针对正则化对数薛定谔方程(RLogSE)提出了一种新颖的高阶、守恒质量和能量的数值格式。基于辅助变量法(SVM)的思想，我们首先通过引入两个辅助变量将原系统重构为等价形式，随后分别采用时间方向的高阶预测校正格式与空间方向的傅里叶伪谱法对所得SVM重构系统进行离散。新构建的格式能够生成严格保持质量和原始能量的数值解，与解析解的性质一致。此外，该格式具有极高的计算效率，因为每个时间步仅需求解一个常系数线性方程组和两个代数方程（可通过牛顿迭代高效求解）。数值实验验证了新格式的精度与保结构特性。