This paper proposes a new class of arbitarily high-order conservative numerical schemes for the generalized Korteweg-de Vries (KdV) equation. This approach is based on the scalar auxiliary variable (SAV) method. The equation is reformulated into an equivalent system by introducing a scalar auxiliary variable, and the energy is reformulated into a sum of two quadratic terms. Therefore, the quadratic preserving Runge-Kutta method will preserve both the mass and the reformulated energy in the discrete time flow. With the Fourier pseudo-spectral spatial discretization, the scheme conserves the first and third invariant quantities (momentum and energy) exactly in the fully discrete sense. The discrete mass possesses the precision of the spectral accuracy.

翻译：本文为通用的 Korteweg-de Vries (KdV) 等式提出了一个新的等级,即高等级保守的保守数字方案。这个办法以 scalar 辅助变量 (SAV) 方法为基础。该等式通过引入一个 scal 辅助变量重新改制为等效系统,能源则重新改制为两个二次条件之和。因此,龙格-库塔四边保护方法将在离散时间流中保留质量和重订能量。随着Fourier 伪光谱空间分解,该等式方案在完全离散的意义上保存了第一和第二个变异数量(流动和能源)。离散质量具有光谱精度的精确度。