# High order surface radiation conditions for time-harmonic waves in exterior domains

## TL;DR

This paper introduces a new family of high-order on-surface radiation conditions for accurately approximating outgoing solutions to the Helmholtz equation in exterior domains, enhancing computational methods for wave problems.

## Contribution

It develops a systematic procedure to implement high-order pseudo-differential symbols for surface radiation conditions, improving solution accuracy in exterior Helmholtz problems.

## Key findings

- Numerical results demonstrate improved accuracy of the proposed method.
- The approach effectively handles Dirichlet and Neumann boundary conditions.
- Potential extensions suggest broader applicability in wave simulations.

## Abstract

We formulate a new family of high order on-surface radiation conditions to approximate the outgoing solution to the Helmholtz equation in exterior domains. Motivated by the pseudo-differential expansion of the Dirichlet-to-Neumann operator developed by Antoine et al. (J. Math. Anal. Appl. 229:184-211, 1999), we design a systematic procedure to apply pseudo-differential symbols of arbitrarily high order. Numerical results are presented to illustrate the performance of the proposed method for solving both the Dirichlet and the Neumann boundary value problems. Possible improvements and extensions are also discussed.