# A parallel-in-time approach for wave-type PDEs

## TL;DR

This paper introduces a parallel-in-time explicit extrapolation method optimized for wave-type PDEs, enhancing stability and efficiency by shaping the stability domain and enabling high-order solutions on multi-core systems.

## Contribution

It develops a novel extrapolation-based solver that maintains accuracy while expanding the stability domain for wave PDEs, enabling efficient parallel computation.

## Key findings

- Achieves larger stable time steps for wave PDEs.
- Demonstrates high-order accuracy with parallel efficiency.
- Outperforms traditional ODE solvers in speed on multi-core systems.

## Abstract

Numerical solutions to wave-type PDEs utilizing method-of-lines require the ODE solver's stability domain to include a large stretch of the imaginary axis surrounding the origin. We show here that extrapolation based solvers of Gragg-Bulirsch-Stoer (GBS) type can meet this requirement. Extrapolation methods utilize several independent time stepping sequences, making them highly suited for parallel execution. Traditional extrapolation schemes use all time stepping sequences to maximize the method's order of accuracy. The present method instead maintains a desired order of accuracy while employing additional time stepping sequences to shape the resulting stability domain. We optimize the extrapolation coefficients to maximize the stability domain's imaginary axis coverage. This yields a family of explicit schemes that approaches maximal time step size for wave propagation problems. On a computer with several cores we achieve both high order and fast time to solution compared with traditional ODE integrators.