A Nystrom flavored Calder\'on Calculus of order three for two dimensional waves

TL;DR

This paper introduces a third-order accurate discretization method for the Calderón Calculus related to the Helmholtz equation in 2D, tested across various integral equations and time-domain approximations.

Contribution

It presents a comprehensive, high-order discretization approach for all elements of the Calderón Calculus in 2D Helmholtz problems, including both frequency and time domain methods.

Findings

Achieves third-order accuracy for all variables

Successfully tests across multiple integral equation formulations

Demonstrates effectiveness in both frequency and time domain scenarios

Abstract

In this paper we present and test a full discretization of all elements of the Calder\'on Calculus (layer potentials and integral operators) for the Helmholtz equation in smooth closed curves in the plane. The resulting integral equations provide approximations of order three for all variables involved. Test are shown for a wide array of direct, indirect and combined field integral equation at fixed frequency and for a Convolution Quadrature based approximation in the time domain.