Filtered interpolation for solving Prandtl's integro-differential equations

TL;DR

This paper introduces a filtered interpolation collocation-quadrature method for solving Prandtl's equations, achieving optimal convergence rates without increased computational cost, and providing stability and efficiency proofs.

Contribution

It presents a novel filtered interpolation approach that improves convergence and stability in solving Prandtl-type equations, outperforming classical methods without extra computational effort.

Findings

Achieves optimal convergence rates in uniform norms

Provides a stable, efficient algorithm based on a 2-bandwidth linear system

Proves condition numbers tend to a finite limit as system size grows

Abstract

In order to solve Prandtl-type equations we propose a collocation-quadrature method based on VP filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Holder - Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation nodes, we succeed in reproducing the optimal convergence rates of the L2 case by cutting off the typical log factor which seemed inevitable dealing with uniform norms. Such an improvement does not require a greater computational effort. In particular we propose a fast algorithm based on the solution of a simple 2-bandwidth linear system and prove that, as its dimension tends to infinity, the sequence of the condition numbers (in any natural matrix norm) tends to a finite limit.