# Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

## TL;DR

This paper analyzes the discretization of Navier-Stokes equations, focusing on the numerical difficulties caused by system strangeness and how common time-stepping schemes address these issues.

## Contribution

It provides a detailed analysis of the semi-discrete and fully discrete Navier-Stokes equations, highlighting the role of strangeness and the effectiveness of standard time-integration schemes.

## Key findings

- Semi-discrete Navier-Stokes equations form a nonlinear DAE with strangeness.
- Fully discrete schemes mitigate strangeness by removing it through time-stepping.
- Numerical examples confirm the theoretical analysis.

## Abstract

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.