The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified equations with increasingly high frequencies which increase the regularity requirements on the analysis. In this paper, we consider the next order modified equations for the implicit midpoint rule applied to the semilinear wave equation to give a proof-of-concept of a new construction which works directly with the variational principle. We show that a carefully chosen change of coordinates yields a modified system which inherits its analytical properties from the original wave equation. Our method systematically exploits additional degrees of freedom by modifying the symplectic structure and the Hamiltonian together.

翻译：修正方程的构造是哈密顿系统辛积分数值后向误差分析的重要步骤。在偏微分方程背景下，标准构造方法会导致修正方程产生持续增长的高频分量，这提高了数值分析的规则性要求。本文以半线性波动方程的隐式中点格式为对象，提出一种直接基于变分原理的新型修正方程构造方法，并给出概念验证。研究表明，通过精心选择坐标变换，所构造的修正系统能够继承原始波动方程的解析特性。本方法通过同时修正辛结构与哈密顿量，系统地利用了额外的自由度资源。