# Space-time domain solutions of the wave equation by a non-singular boundary integral method and Fourier transform

## TL;DR

This paper presents a high-precision, stable boundary integral method combined with Fourier transform to efficiently solve the space-time wave equation for complex scattering problems, accurately capturing multiple scattering effects.

## Contribution

It introduces a non-singular boundary integral approach for Helmholtz problems and applies Fourier transform for efficient space-time wave solutions, improving stability and accuracy.

## Key findings

- Exact enforcement of radiation boundary conditions
- High numerical stability for small degrees of freedom
- Efficient response analysis to various incident pulses

## Abstract

The general space-time evolution of the scattering of an incident acoustic plane wave pulse by an arbitrary configuration of targets is treated by employing a recently developed non-singular boundary integral method to solve the Helmholtz equation in the frequency domain from which the fast Fourier transform is used to obtain the full space-time solution of the wave equation. The non-singular boundary integral solution can enforce the radiation boundary condition at infinity exactly and can account for multiple scattering effects at all spacings between scatterers without adverse effects on the numerical precision. More generally, the absence of singular kernels in the non-singular integral equation confers high numerical stability and precision for smaller numbers of degrees of freedom. The use of fast Fourier transform to obtain the time dependence is not constrained to discrete time steps and is particularly efficient for studying the response to different incident pulses by the same configuration of scatterers. The precision that can be attained using a smaller number of Fourier components is also quantified.