We consider a class of discrete operators introduced by O. Chodosh, acting on infinite sequences and mimicking standard properties of pseudo-differential operators. By using a new approach, we extend this class to finite or periodic sequences, allowing a general representation of discrete pseudo-differential operators obtained by finite differences approximations and easily transferred to time discretizations. In particular we can define the notion of order and regularity, and we recover the fundamental property, well known in pseudo-differential calculus, that the commutator of two discrete operators gains one order of regularity. As examples of practical applications, we revisit standard error estimates for the convergence of splitting methods, obtaining in some Hamiltonian cases no loss of derivative in the error estimates, in particular for discretizations of general waves and/or water-waves equations. Moreover, we give an example of preconditioner constructions inspired by normal form analysis to deal with the similar question for more general cases.

我们考虑 O. Chodosh 引入的一类离散算子，作用于无限序列并模仿伪微分算子的标准属性。通过使用一种新方法，我们将此类扩展到有限序列或周期序列，从而允许通过有限差分近似获得离散伪微分算子的一般表示，并轻松转移到时间离散化。特别是，我们可以定义阶数和正则性的概念，并且恢复伪微分学中众所周知的基本属性，即两个离散算子的交换子获得一阶正则性。作为实际应用的示例，我们重新审视分裂方法收敛的标准误差估计，在某些哈密顿情况下获得误差估计中没有导数损失，特别是对于一般波浪和/或水波方程的离散化。此外，我们给出了一个受范式分析启发的预处理器构造示例，以处理更一般情况下的类似问题。