A non-iterative domain decomposition time integrator combined with discontinuous Galerkin space discretizations for acoustic wave equations

TL;DR

The paper addresses efficient, accurate time integration for acoustic wave propagation in heterogeneous media by introducing a non-iterative domain decomposition method that couples overlapping DG spatial discretizations with a local Crank-Nicolson time integrator and a local prediction step. The approach supports higher-order polynomial approximations and heterogeneous material parameters, overcoming limitations of mass-lumped linear elements. Numerical experiments demonstrate that the domain splitting method achieves accuracy comparable to global time integration while offering parallelization and reduced global communication, with detailed implementation and performance insights. The work provides public code and a practical framework for scalable simulations of acoustic waves in complex media.

Abstract

We propose a novel non-iterative domain decomposition time integrator for acoustic wave equations using a discontinuous Galerkin discretization in space. It is based on a local Crank-Nicolson approximation combined with a suitable local prediction step in time. In contrast to earlier work using linear continuous finite elements with mass lumping, the proposed approach enables higher-order approximations and also heterogeneous material parameters in a natural way.

Figures (9)

• Figure 1: Extension $\mathcal{N}_2(\widehat{\Omega})$ by $\ell = 2$ layers of a subdomain $\widehat{\Omega} \subset \Omega$ colored in dark blue.
• Figure 2: Overlapping subdomain $\Omega_{i}^{\ell}$ (left, dark and light red area) and prediction domain $\Omega_{\Gamma_{},i}^{\ell}$ (right, yellow area). The interface $\Gamma_{i}^{\ell}$ is shown in dark red in both pictures.
• Figure 3: Example for a weighted undirected communication graph for $\mathcal{I} = 6$ subdomains. Weights correspond to the number of values, which need to get exchanged between two subdomains.
• Figure 4: For each local mesh we build a 'dofmap', which maps the local DoFs to the global DoFs in $\mathcal{T}_h$. Here the index denotes the local DoF, while the value stores the global DoF. Note, that the size of the map is only related to the local mesh.
• Figure 5: Example grid on $[0.8] \times [0,4]$ for the prism example with quite coarse $h$. Red area $\kappa(x) = \kappa_i$, white area $\kappa = \kappa_o$. Dirichlet boundary $\Gamma_D$ (green) and Neumann boundary $\Gamma_N$ (blue). For simulations much finer $h$ were used.