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In this paper, we revisit the old problem of compact finite differenceapproximations of the homogeneous Dirichlet problem in dimension 1. We design alarge and natural set of schemes of arbitrary high order, and we equip this setwith an algebraic structure. We give some general criteria of convergence andwe 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 mostefficient schemes and we prove their convergence. On the other hand, we usediophantine approximation theory to prove that almost all of these schemes areconvergent at the same rate as the consistency order, up to some logarithmiccorrection.

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