Optimality and resonances in a class of compact finite difference schemes of high order

TL;DR

This paper develops a broad class of high-order compact finite difference schemes for 1D Dirichlet problems, analyzing their optimality and convergence properties using algebraic, Padé, and Diophantine approximation theories.

Contribution

It introduces a large, structured set of high-order schemes and establishes their convergence and optimality using advanced mathematical tools.

Findings

Constructed high-order schemes with proven convergence.

Identified most efficient schemes for each order of consistency.

Showed almost all schemes converge at the optimal rate.

Abstract

In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an algebraic structure. We give some general criteria of convergence and we apply them to obtain two new results. On the one hand, we use Pad{\'e} approximant theory to construct, for each given order of consistency, the most efficient schemes and we prove their convergence. On the other hand, we use diophantine approximation theory to prove that almost all of these schemes are convergent at the same rate as the consistency order, up to some logarithmic correction.