# Efficient approximation of flow problems with multiple scales in time

## TL;DR

This paper introduces a multiscale approximation scheme for time-dependent flow problems like Navier-Stokes, achieving significant computational speed-ups while maintaining accuracy through averaging and advanced discretization techniques.

## Contribution

It develops a first-order averaging scheme and a second-order discretization for multiscale Navier-Stokes flows, enabling efficient simulations without resolving all fast-scale details.

## Key findings

- Achieves speed-ups up to 1:10000 compared to resolved simulations.
- Demonstrates convergence and accuracy through numerical examples.
- Provides a complete error analysis for the simplified ODE system.

## Abstract

In this article we address flow problems that carry a multiscale character in time. In particular we consider the Navier-Stokes flow in a channel on a fast scale that influences the movement of the boundary which undergoes a deformation on a slow scale in time. We derive an averaging scheme that is of first order with respect to the ratio of time-scales $\epsilon$. In order to cope with the problem of unknown initial data for the fast scale problem, we assume near-periodicity in time. Moreover, we construct a second-order accurate time discretisation scheme and derive a complete error analysis for a corresponding simplified ODE system. The resulting multiscale scheme does not ask for the continuous simulation of the fast scale variable and shows powerful speed-ups up to 1:10000 compared to a resolved simulation. Finally, we present some numerical examples for the full Navier-Stokes system to illustrate the convergence and performance of the approach.