Optimality and resonances in a class of compact finite difference schemes of high order
Provides theoretical convergence guarantees for a broad class of high-order compact finite difference schemes, addressing a foundational problem in numerical analysis.
The paper revisits compact finite difference schemes for the 1D homogeneous Dirichlet problem, constructing a family of arbitrary high-order schemes with an algebraic structure. It proves convergence for optimally efficient schemes via Padé approximants and shows that almost all schemes converge at the consistency order rate up to a logarithmic correction.
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{é} 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.