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.
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