Numerical schemes that conserve invariants have demonstrated superior performance in various contexts, and several unified methods have been developed for constructing such schemes. However, the mathematical properties of these schemes remain poorly understood, except in norm-preserving cases. This study introduces a novel analytical framework applicable to general energy-preserving schemes. The proposed framework is applied to Korteweg-de Vries (KdV)-type equations, establishing global existence and convergence estimates for the numerical solutions.

翻译：数值格式若能守恒不变量，已在多种情境中展现出优越性能，且已有多种统一方法用于构建此类格式。然而，除范数守恒情形外，这些格式的数学性质仍鲜为人知。本研究提出了一种适用于一般能量守恒格式的新型分析框架，并将其应用于Korteweg-de Vries (KdV)型方程，建立了数值解的全局存在性与收敛性估计。