A parallel-in-time isogeometric BEM for the 3D wave equation using B-spline linear multistep methods
Status: in preparation. Last update: July 2026
Abstract: Time-domain boundary integral equations (TDBIEs) provide integral representations of solutions to evolution problems in physics and engineering, such as acoustic scattering in isotropic, homogeneous media. Developing numerical methods for TDBIEs is a highly active research area featuring competing approaches such as the energetic Galerkin method, marching-on-in-time, and convolution quadrature (CQ). For spatial discretization, the combination of Isogeometric Analysis and the Boundary Element Method (IgA-BEM) has emerged as the leading high-order framework over the last decade.
Convolution quadrature methods transform TDBIEs to the Laplace domain, converting the time integral into a family of ordinary differential equations which are then discretized using linear multistep or Runge-Kutta methods. The TDBIE solution can be computed using Lubich’s classical forward-in-time approach, or newer parallel-in-time schemes. For the wave equation reformulated as a TDBIE, parallel-in-time schemes yield a decoupled set of Helmholtz problems. The combination of high-order IgA-BEM and Runge-Kutta convolution quadrature was recently explored in Kramer, Marussig, Schanz (2026).
In this work, we introduce a novel high-order parallel-in-time discretization for the 3D wave equation using linear multistep methods as Boundary Value Methods (BVM). Unlike Runge-Kutta CQ, which requires eigendecompositions at the timestep level, our BVM approach exhibits global diagonalizability, immediately yielding decoupled Helmholtz problems. To mitigate the growth of the condition number of the transformation matrix, we use symmetric B-spline linear multistep methods. Spatial discretization on multi-patch surfaces is achieved via isogeometric collocation, resulting in a fully spline-based scheme in both space and time. Numerical experiments show that B-spline BVMs can be significantly more accurate than state-of-the-art Runge-Kutta CQ for equivalent sets of decoupled Helmholtz problems.
I have given a talk on this topic at the IMSE 2026 conference in Matera, Italy. Slides are available for download.