Discrete Pseudo-differential Operators and Applications to Numerical Schemes

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-dif...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 25; no. 2; pp. 587 - 630
Main Authors: Faou, Erwan, Grébert, Benoît
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2025
Springer Nature B.V
Springer Verlag
Subjects:
Applications of Mathematics
Canonical forms
Commutators
Computer Science
Differential calculus
Differential equations
Discretization
Economics
Estimates
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Operators (mathematics)
Regularity
Standard error
Splitting methods
65T50
Pseudo-differential operators
65M06
Numerical methods for discrete and fast Fourier transforms
35S05
Finite difference methods
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary: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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-024-09645-y