Discrete pseudo-differential operators and applications to numerical schemes

TL;DR

This paper introduces a class of discrete pseudo-differential operators that mimic continuous properties, enabling improved analysis of numerical schemes for differential equations, including error estimates and preconditioning techniques.

Contribution

It defines a new class of discrete pseudo-differential operators with properties similar to continuous ones, and applies this framework to analyze and improve numerical methods for differential equations.

Findings

Discrete operators mimic continuous pseudo-differential properties

Error estimates for splitting methods show no derivative loss in some cases

Preconditioners inspired by normal form analysis improve numerical solutions

Abstract

We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the fundamental property that the commutator of two discrete operators gains one order of regularity. We show that standard differential operators acting on periodic functions, finite difference operators and fully discrete pseudo-spectral methods fall into this class of discrete pseudo-differential operators. 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.