# A stable parareal-like method for the second order wave equation

## TL;DR

This paper introduces a parallel-in-time iterative method for the second-order wave equation that combines coarse and fine propagators, utilizing a data-driven stabilization strategy to improve efficiency and accuracy.

## Contribution

It develops a stable parareal-like method with a data-driven coupling strategy for the wave equation, enhancing parallel efficiency and solution stability.

## Key findings

- Effective in stabilizing the solution process
- Demonstrated improved performance on Marmousi model
- Allows larger time steps with maintained accuracy

## Abstract

A new parallel-in-time iterative method is proposed for solving the homogeneous second-order wave equation. The new method involves a coarse scale propagator, allowing for larger time steps, and a fine scale propagator which fully resolves the medium using finer spatial grid and shorter time steps. The fine scale propagator is run in parallel for short time intervals. The two propagators are coupled in an iterative way that resembles the standard parareal method developed by Lions, Maday and Turinici. We present a data-driven strategy in which the computed data gathered from each iteration are re-used to stabilize the coupling by minimizing the energy residual of the fine and coarse propagated solutions. An example of Marmousi model is provided to demonstrate the performance of the proposed method.