A null infinity layer for wave scattering

TL;DR

The paper introduces a null infinity layer method for solving wave scattering problems on unbounded domains without truncation, enabling accurate treatment of radiative fields at infinity and applicability to variable coefficients.

Contribution

It develops a novel null infinity layer technique that maps unbounded domains to bounded ones, avoiding the need for local Green functions and handling complex wave equations.

Findings

Efficiently solves Helmholtz equations with variable coefficients.

Accurately captures radiative fields at infinity.

Demonstrates effectiveness in 1D and 2D numerical examples.

Abstract

We show how to solve time-harmonic wave scattering problems on unbounded domains without truncation. The technique, first developed in numerical relativity for time-domain wave equations, maps the unbounded domain to a bounded domain and scales out the known oscillatory decay towards infinity. We design a null infinity layer that corresponds to the infinite exterior domain and restricts the transformations to an annular domain. The method does not require the local Green function. Therefore we can use it to solve Helmholtz equations with variable coefficients and certain nonlinear source terms. The method's main advantages are the exact treatment of the local boundary and access to radiative fields at infinity. The freedom in the transformations allows us to choose parameters adapted to high-frequency wave propagation in the exterior domain. We demonstrate the efficiency of the technique in one- and two-dimensional numerical examples.