A non-iterative domain decomposition time integrator for linear wave equations

TL;DR

This paper introduces a non-iterative, parallelizable domain decomposition method for linear wave equations that achieves high accuracy and larger time steps by combining implicit and prediction steps, with proven convergence.

Contribution

It adapts existing methods from parabolic problems to wave equations, enabling non-iterative parallel computation with second-order accuracy and convergence guarantees.

Findings

Method achieves second-order accuracy in time.

Allows larger time steps compared to explicit schemes.

Numerical experiments confirm theoretical convergence.

Abstract

We propose and analyze a non-iterative domain decomposition integrator for the linear acoustic wave equation. The core idea is to combine an implicit Crank-Nicolson step on spatial subdomains with a local prediction step at the subdomain interfaces. This enables parallelization across space while advancing sequentially in time, without requiring iterations at each time step. The method is similar to the methods from Blum, Lisky and Rannacher (1992) or Dawson and Dupont (1992), which have been designed for parabolic problems. Our approach adapts them to the case of the wave equation in a fully discrete setting, using linear finite elements with mass lumping. Compared to explicit schemes, our method permits significantly larger time steps and retains high accuracy. We prove that the resulting method achieves second-order accuracy in time and global convergence of order $\mathcal{O}(h + \tau^2)$ under a CFL-type condition, which depends on the overlap width between subdomains. We conclude with numerical experiments which confirm the theoretical results.