This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the literature preserve fully discrete local conservation laws whose densities are either quadratic or a Hamiltonian. The approach generalizes to time integrators with more steps and conservation laws of other kinds; higher-dimensional PDEs can be treated by iterating the new strategy. We use the Boussinesq equation as a benchmark and introduce new families of schemes of order two and four that preserve three conservation laws. We show that the new technique is practicable for PDEs with three dependent variables, introducing as an example new families of second-order schemes for the potential Kadomtsev–Petviashvili equation.

本文介绍了一种新的符号数字策略，用于查找给定PDE的半离散化，该半离散化保留了多个本地保护定律。我们证明，对于一个空间维度，文献中的各种单步时间积分器保留了密度为二次方或哈密顿量的完全离散的局部守恒定律。该方法可以推广到具有更多步骤和其他守恒律的时间积分器。可以通过迭代新策略来处理高维PDE。我们使用Boussinesq方程作为基准，并介绍了保留三项守恒定律的新的二阶和四阶方案族。我们证明了这项新技术对于具有三个因变量的PDE是可行的，