On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods

TL;DR

This paper explores how conformal maps, specifically polynomial adjustments to the sinh map, can accelerate the convergence of the trapezoidal rule and Sinc methods, especially for integrals with singularities, using adaptive techniques like Sinc-Padé approximants.

Contribution

It introduces a polynomial-adjusted conformal map approach to improve convergence of numerical methods handling singularities, including adaptive strategies for unknown singularities.

Findings

Conformal maps significantly improve convergence rates.

Adaptive methods perform well with unknown singularities.

High-accuracy solutions achieved for complex integrals.

Abstract

We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the $\sinh$ map, and allows the treatment of a finite number of singularities in the complex plane. In the case where locations are unknown, the so-called Sinc-Pad\'e approximants are used to provide approximate results. This adaptive method is shown to have almost the same convergence properties. We use the conformal maps to generate high accuracy solutions to several challenging integrals, nonlinear waves, and multidimensional integrals.