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 strangeness. The theoretical considerations are backed and illustrated by numerical examples.

Date

Oct 2, 2019 16:10 — 16:35

Event

Location

Eegmond an Zee, The Netherlands

We consider the time discretization of the semi-discrete incompressible Navier-Stokes equations (NSE)

$ \begin{aligned} \quad \quad \quad M \dot v(t) &= N(v(t)) + J^Tp(t) + f(t), \\\\ 0 &= Jv(t), \end{aligned} $

formulated in the velocity $v(t) \in \mathbb R^{n_v}$ and pressure $p(t) \in \mathbb R^{n_p}$, with $M\in \mathbb R^{n_v, n_v}$ being the mass matrix from the spatial discretization, $N\colon \mathbb R^{n_v} \to \mathbb R^{n_v}$ modelling the discretized convection and diffusion, and with $J\in \mathbb R^{n_p, n_v}$ and $J^T$ representing the discrete divergence and gradient operators.

It is commonly known that the semi-discrete incompressible Navier-Stokes equations can be classified as a differential-algebraic equation (DAE) of *differentiation index* $\nu=2$ so that a straight-forward time discretization, e.g. by the *implicit-Euler* method, will likely suffer from instabilities. To overcome these numerical difficulties, a large number of time stepping schemes, with *Chorin*’s projection or the *SIMPLE* scheme as notable examples, have been developed.

In this talk, we trace down and illustrate the instability that origins from the index-2 structure and show that commonly and succesfully applied time-stepping schemes correspond to a reformulation of the semi-discrete NSE as an equivalent system of index-1. Also we show how a finite-element discretization by *Taylor-Hood* or *Crouzeix-Raviart* schemes can be modified such that the resulting semi-discrete NSE already is of index-1 so that standard time-integration schemes lead to stable pressure approximations.

**References**

Altmann, R. & Heiland, J.: *Finite Element Decomposition and Minimal Extension for Flow Equations*, ESAIM: M2AN, **2015**, 49, 1489–1509

Altmann, R. & Heiland, J.: *Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations*, DAE-Forum, Issue: *Applications of Differential-Algebraic Equations: Examples and Benchmarks*, Springer, **2018**