# Parareal Algorithm Implementation and Simulation in Julia

## TL;DR

This paper implements the parareal algorithm in Julia, demonstrating its flexibility with different integrators and providing a simulation tool for performance analysis and educational purposes.

## Contribution

The paper provides a complete Julia implementation of the parareal algorithm with customizable integrators and a simulation environment for testing and educational use.

## Key findings

- Implementation supports various integrators like Euler and Runge-Kutta.
- Simulation tool aids in understanding and analyzing algorithm performance.
- Demonstrates the algorithm's applicability to general first-order initial-value problems.

## Abstract

We present a full implementation of the parareal algorithm---an integration technique to solve differential equations in parallel---in the Julia programming language for a fully general, first-order, initial-value problem. We provide a brief overview of Julia---a concurrent programming language for scientific computing. Our implementation of the parareal algorithm accepts both coarse and fine integrators as functional arguments. We use Euler's method and another Runge-Kutta integration technique as the integrators in our experiments. We also present a simulation of the algorithm for purposes of pedagogy and as a tool for investigating the performance of the algorithm.