Many conservative partial differential equations such as the Korteweg-de Vries (KdV) equation, and the nonlinear Schr\"{o}dinger equations, the Klein-Gordon equation have more than one invariant functionals. In this paper, we propose the definition of the discrete variational derivative, based on which, a novel semi-analytical multiple invariants-preserving integrator for the conservative partial differential equations is constructed by projection technique. The proposed integrators are shown to have the same order of accuracy as the underlying integrators. For applications, some concrete mass-momentum-energy-preserving integrators are derived for the KdV equation.

翻译：许多保守型偏微分方程，例如 Korteweg-de Vries (KdV) 方程、非线性薛定谔方程以及 Klein-Gordon 方程，具有多个不变量泛函。本文提出了离散变分导数的定义，并在此基础上，通过投影技术构造了一种新型的半解析多不变量保持保守型偏微分方程积分器。研究表明，所提出的积分器与底层积分器具有相同的精度阶。在应用方面，针对 KdV 方程推导了一些具体的质量-动量-能量保持积分器。