---
abstract: |
    We consider the efficient numerical solution of the three-dimensional wave equation with Neumann boundary conditions via time-domain boundary integral equations. A space-time Galerkin method with $C^\infty$-smooth, compactly supported basis functions in time and piecewise polynomial basis functions in space is employed. We discuss the structure of the system matrix and its efficient parallel assembly. Different preconditioning strategies for the solution of the arising systems with block Hessenberg matrices are proposed and investigated numerically. Furthermore, a C++ implementation parallelized by OpenMP and MPI in shared and distributed memory, respectively, is presented. The code is part of the boundary element library BEM4I. Results of numerical experiments including convergence and scalability tests up to a thousand cores on a cluster are provided. The presented implementation shows good parallel scalability of the system matrix assembly. Moreover, the proposed algebraic preconditioner in combination with the FGMRES solver leads to a significant reduction of the computational time.

    [**AMS subject classifications.** ]{}35L05, 65N38, 65R20, 65F08
author:
- 'A. Veit[^1]'
- 'M. Merta[^2]'
- 'J. Zapletal[^3]'
- 'D. Lukáš[^4]'
bibliography:
- 'merta\_bib.bib'
title: 'Efficient Solution of Time-Domain Boundary Integral Equations Arising in Sound-Hard Scattering'
---

Introduction
============

Integral Formulation of the Wave Equation
=========================================

Numerical Discretization
========================

Solution of the linear system {#Sec:SolLinSys}
=============================

Implementation {#Sec:Implementation}
==============

Numerical experiments {#Sec:NumExp}
=====================

Conclusion
==========

We considered the numerical modelling of time-dependent, three-dimensional sound-hard scattering phenomena in unbounded domains. We used time-domain boundary integral equations in combination with space-time Galerkin methods to solve the arising problems numerically. The use of $C^\infty$-smooth and compactly supported temporal basis functions simplifies the generation of the system matrix and as a consequence allows the use of higher order approximation schemes in time which significantly improves the accuracy of the method. Due to the overlap of the basis functions in time the arising linear systems that need to be solved admit a block Hessenberg structure. We examined the performance of a restarted GMRES solver preconditioned by deflation and proposed an algebraic preconditioner that exploits the block Hessenberg structure of the matrix. Various examples show that both solvers require a considerably lower number of iterations than a conventional GMRES method. Especially the second approach is promising since the overall solution time could be significantly reduced. A rigorous analysis and a possible parallel implementation of the preconditioner, however, will be considered in the future.

We furthermore presented a parallel implementation of the scheme that is based on the boundary element library BEM4I. It exploits the special structure of the system matrix to reduce computational complexity and memory requirements. It uses OpenMP and MPI for parallelization in shared and distributed memory. Numerical experiments show good parallel scalability and efficiency of the computations in a distributed memory architecture with up to $1024$ cores.

Acknowledgement {#acknowledgement .unnumbered}
===============

The first author gratefully acknowledges the support given by the Swiss National Science Foundation (SNF), No. P2ZHP2\_148705. Latter authors were supported by the IT4Innovations Centre of Excellence project (CZ.1.05/ 1.1.00/02.0070), funded by the European Regional Development Fund and the national budget of the Czech Republic via the Research and Development for Innovations Operational Programme, as well as Czech Ministry of Education, Youth and Sports via the project Large Research, Development and Innovations Infrastructures (LM2011033). The second and third authors were supported by VŠB-TU Ostrava under the grant SGS SP2015/160. The last author was supported by VŠB-TU Ostrava under the grant SGS SP2015/100.

[^1]: aveit@uchicago.edu, Department of Computer Science, The University of Chicago, 1100 E. 58th Street, Chicago, IL 60637 US

[^2]: michal.merta@vsb.cz, VŠB-Technical University of Ostrava, 17. listopadu 2172/15, 708 33 Ostrava, Czech Republic

[^3]: (jan.zapletal@vsb.cz), VŠB-Technical University of Ostrava, 17. listopadu 2172/15, 708 33 Ostrava, Czech Republic

[^4]: (dalibor.lukas@vsb.cz), VŠB-Technical University of Ostrava, 17. listopadu 2172/15, 708 33 Ostrava, Czech Republic
