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

## 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 discretisation. 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 analysing 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.

## Keywords

DAEs Difference-algebraic equations Navier-Stokes equations Strangeness index## **Mathematics Subject Classification (2010)**

65L80 65M12 35Q30 ## References

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