# Convergence of Parareal for the Navier-Stokes equations depending on the Reynolds number

arXiv: https://arxiv.org/abs/1311.4588 · HTML: https://mlsanity.com/paper/1311.4588/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2013.

## Built on by (most cited first, 1 of 1)

- [Self-Refining Diffusion Samplers: Enabling Parallelization via Parareal Iterations](https://mlsanity.com/paper/2412.08292/) (2024)

## Abstract (arXiv)

The paper presents first a linear stability analysis for the time-parallel Parareal method, using an IMEX Euler as coarse and a Runge-Kutta-3 method as fine propagator, confirming that dominant imaginary eigenvalues negatively affect Parareal's convergence. This suggests that when Parareal is applied to the nonlinear Navier-Stokes equations, problems for small viscosities could arise. Numerical results for a driven cavity benchmark are presented, confirming that Parareal's convergence can indeed deteriorate as viscosity decreases and the flow becomes increasingly dominated by convection. The effect is found to strongly depend on the spatial resolution.