In this paper, we develop a new class of high-order energy-preserving schemes for the Korteweg-de Vries equation based on the quadratic auxiliary variable technique, which can conserve the original energy of the system. By introducing a quadratic auxiliary variable, the original system is reformulated into an equivalent form with a modified quadratic energy, where the way of the introduced variable naturally produces a quadratic invariant of the new system. A class of Runge-Kutta methods satisfying the symplectic condition is applied to discretize the reformulated model in time, arriving at arbitrarily high-order schemes, which naturally conserve the modified quadratic energy and the produced quadratic invariant. Under the consistent initial condition, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In order to match the high order precision of time, the Fourier pseudo-spectral method is employed for spatial discretization, resulting in fully discrete energy-preserving schemes. To solve the proposed nonlinear schemes effectively, we present a very efficient practically-structure-preserving iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed schemes. This new class of numerical strategies is rather general so that they can be readily generalized for any conservative systems with a polynomial energy.

翻译：在本文中,我们为Korteweg-de Vries 方程式开发了新型高阶节能计划,它以二次辅助可变技术为基础,可以保护系统原有的能量。通过引入二次辅助变量,原系统被重新改制为等效形式,使用经修改的二次能源,采用引入变量的方式自然产生新系统的二次变异。一种满足共振条件的龙格-库塔方法,用来及时分解重塑模型,达到任意的高级计划,自然保存经修改的二次能量和生成的二次变异。在相同的初始条件下,拟议方法被严格地证明能够维持Korteweweg-de Vries方程式的原始节能法。为了与时间的高度定序一致,四面伪光方法可用于空间分解,导致完全离散的节能计划。为了有效解决拟议的非线性计划,我们提出了一个非常高效的近乎常规的二次变异体能源和生成的二次变异体能源计划。在现实的轨道上实现了一种非常保守的常规的常规的计算方法,因此也能够大大地维持了预期的定序。