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Time domain boundary integral equations and convolution quadrature for scattering by composite media (extended preprint)

Alexander Rieder, Francisco-Javier Sayas, Jens Markus Melenk

TL;DR

This work develops a fully discrete forward solver for acoustic scattering in media with piecewise constant parameters by marrying a Galerkin boundary element method in space with Runge-Kutta convolution quadrature in time. It proves well-posedness and derives rigorous a priori error estimates for both space and time discretizations, with convergence rates governed by the RK stage order $q$, classical order $p$, and data regularity through a parameter $\mu$. The analysis accommodates arbitrary numbers of subdomains and cross-points, extending the Costabel-Stephan framework to a time-domain, fully discrete setting, and is complemented by 2D numerical experiments demonstrating robust and high-order convergence on complex geometries. Overall, the paper delivers a theoretically sound and practically effective forward solver for transient scattering in composite media, with potential impact for seismic and geophysical applications.

Abstract

We consider acoustic scattering in heterogeneous media with piecewise constant wave number. The discretization is carried out using a Galerkin boundary element method in space and Runge-Kutta convolution quadrature in time. We prove well-posedness of the scheme and provide a priori estimates for the convergence in space and time.

Time domain boundary integral equations and convolution quadrature for scattering by composite media (extended preprint)

TL;DR

This work develops a fully discrete forward solver for acoustic scattering in media with piecewise constant parameters by marrying a Galerkin boundary element method in space with Runge-Kutta convolution quadrature in time. It proves well-posedness and derives rigorous a priori error estimates for both space and time discretizations, with convergence rates governed by the RK stage order , classical order , and data regularity through a parameter . The analysis accommodates arbitrary numbers of subdomains and cross-points, extending the Costabel-Stephan framework to a time-domain, fully discrete setting, and is complemented by 2D numerical experiments demonstrating robust and high-order convergence on complex geometries. Overall, the paper delivers a theoretically sound and practically effective forward solver for transient scattering in composite media, with potential impact for seismic and geophysical applications.

Abstract

We consider acoustic scattering in heterogeneous media with piecewise constant wave number. The discretization is carried out using a Galerkin boundary element method in space and Runge-Kutta convolution quadrature in time. We prove well-posedness of the scheme and provide a priori estimates for the convergence in space and time.

Paper Structure

This paper contains 18 sections, 24 theorems, 174 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model problem and notation
  3. The scattering problem
  4. A multiply overlapped wave problem
  5. A particular construction of the approximation spaces
  6. Towards an analyzable form
  7. Analysis
  8. A system of semidiscrete TDBIE
  9. Time discretization - Runge Kutta convolution quadrature
  10. The fully discrete method
  11. Some regularity theorems
  12. Convergence of the time discretization
  13. Convergence of the fully discrete scheme
  14. Particular geometric configurations
  15. Multiple homogeneous scatterers
  16. ...and 3 more sections

Key Result

Lemma 2.1

If $\boldsymbol u=(u_\ell)_{\ell=0}^L \in \mathcal{H}$ satisfies $\llbracket \boldsymbol{\gamma} \boldsymbol u \rrbracket\in \mathcal{Y}$ and $\boldsymbol{\gamma}^{\mathrm{ext}}{\boldsymbol u}\in \mathcal{Y}$, then the function $u:\mathbb{R}^d\to \mathbb{R}$ defined by $u|_{\Omega_\ell}:=u_\ell |_{

Figures (6)

  • Figure 2.1: Geometry of the problem and notation for subdomains
  • Figure 5.1: A cartoon representing two scatterers with inclusions, and the corresponding tree. The arrows point to the parenting (surrounding) domain and are tagged with the common intersection.
  • Figure 6.1: Comparison of discretization schemes
  • Figure 6.2: Convergence rates w.r.t. the spatial discretization
  • Figure 6.3: Evolution of Example \ref{['ex:complex_geometry']}
  • ...and 1 more figures

Theorems & Definitions (59)

  • Lemma 2.1: Restricting and gluing
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 49 more