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.

Appendices

Appendix 1: Strangeness Index of Eq. (3.1)

We analyse in detail the strangeness index of the DAE (3.1); see [39, Def. 4.4] for the precise definition. Note that we do not ask for any assumptions on the nonlinearity K and that we allow the matrix B to be rank-deficient. This then also implies that the differentiation index of (3.1) equals 2 if it is well-defined, i.e., if B is of full rank.

1.1 Linear Case

Considering any linearisation of the Navier-Stokes equations, i.e., K(u) = Ku in (3.1), we deal with the matrix pair

Following [39, Th. 3.11], we can construct a to (E, A) (globally) equivalent pair $(\tilde E, \tilde A )$ of the form

with $A_{13} \in \mathbb {R}^{b,m}$ being of full rank. Thus, the original system (3.1) is equivalent to a system of the form

$$\displaystyle \begin{aligned} \dot x_1 = A_{12}x_2 + A_{13}x_3 + f_1, \qquad \dot x_2 = A_{23}x_3 + f_2, \qquad 0 = x_1 + f_3, \qquad 0 = f_4 \end{aligned}$$

with dimensions $x_1(t)\in \mathbb {R}^{b}$, $x_2(t)\in \mathbb {R}^{n-b}$, $x_3(t)\in \mathbb {R}^{m}$. Since we have a differential and an algebraic equation for x ₁ (this causes the ‘strangeness’), we use the derivative of 0 = x ₁ + f ₃ in order to eliminate $\dot x_1$ in the first equation. Hence, we consider the pair (E _mod, A _mod) with

Since A ₁₃ is of full rank, one can show that system (E _mod, A _mod) is strangeness-free, cf. the calculation in [39, Th. 3.7]. Since we have obtained a strangeness-free system with only one differentiation, system (3.1) has strangeness index one.

1.2 Nonlinear Case

The general form of a nonlinear DAE is given by

$$\displaystyle \begin{aligned} F(t, x, \dot x) = 0. \end{aligned}$$

In regard of system (3.1) we set x := [q ^T, p ^T]^T and define

with

In the sequel we show that (3.1) has strangeness index 1 also in the nonlinear case. For this, we assume that B has full rank such that there are no vanishing equations and the pressure variable is uniquely defined. In the case $\operatorname {\mathrm {rank}} B = b < m$, we consider the following transformation.

Let $C_0\in \mathbb {R}^{m, m-b}$ be the matrix of full rank satisfying B ^T C ₀ = 0. Furthermore, $C'\in \mathbb {R}^{m, b}$ defines any matrix such that $C = [C_0\ C'] \in \mathbb {R}^{m, m}$ is invertible. With this, we obtain the relation

with $\tilde B \in \mathbb {R}^{b, n}$ having full rank. With the matrix C in hand, we first introduce the new pressure variable $\tilde p := C^{-1}p$. Thus, we consider the pair $z := [q^T, \tilde p^T]^T$. As a second step, we multiply equation (3.1) by the block-diagonal matrix diag(I _n, C ^T) from the left. In total, this yields the equivalent DAE

Note that the constraint contains (m − b) consistency equations of the form 0 = g ₁. Assuming that system (3.1) is solvable, we suppose that these are in fact vanishing equations. Thus, they have no influence on the index of the system. Furthermore, the first (m − b) components of the transformed pressure $\tilde p$ do not influence the system. These components are underdetermined and may be omitted, again without changing the index. Leaving out the underdetermined parts as well as the vanishing equations, we obtain a system of the form (3.1) with a full rank matrix B.

In the sequel, we assume that $\operatorname {\mathrm {rank}} B = m$ and show that [39, Hyp. 4.2] is satisfied for μ = 1. Note that this hypothesis is not satisfied for μ = 0, i.e., the system is not strangeness-free. We define the matrices

We now pass through the list of points of the hypothesis in [39, Hyp. 4.2]:

• 1. First, we note that the rank of M ₁ equals 2n and we set a := 2(n + m) − 2n = 2m. Thus, the system contains 2m algebraic variables (the pressure and the part of q, which is not divergence-free). Furthermore, we define $Z_2\in \mathbb {R}^{2(n+m),2m}$ by $Z_2^TM_1 = 0$, i.e.,

  

• 2. As a second step we define $\hat A_2:= Z_2^T N_1 [I_{n+m},\ 0]^T$, which yields

  

  This matrix has rank 2m, since the full-rank property of B implies that BM ⁻¹ B ^T is invertible. We define d := n − m as the number of differential variables and $T_2\in \mathbb {R}^{n+m,n-m}$ by $\hat A_2T_2 = 0$. Let $C\in \mathbb {R}^{n,n-m}$ be a matrix of full rank with BC = 0 and $C_2 := -(BM^{-1}B^T)^{-1}BM^{-1}K_uC \in \mathbb {R}^{m,n-m}$. Then, we set

  

• 3. Finally, we compute the rank of ET ₂. Since C has full rank, this equals $\operatorname {\mathrm {rank}} MC = n-m = d$. The matrix $Z_1^T := [C^T\ 0] \in \mathbb {R}^{n-m, n+m}$ satisfies

  $$\displaystyle \begin{aligned} \operatorname{\mathrm{rank}} Z_1^T E T_2 = \operatorname{\mathrm{rank}} C^TMC = n-m = d. \end{aligned}$$

Thus, the hypothesis in [39, Hyp. 4.2] is satisfied for μ = 1, which implies that the nonlinear DAE (3.1) has strangeness index one.

Appendix 2: Difference-Algebraic Equation Index of the Considered Systems

In this appendix, we derive the Kronecker index for the discrete schemes considered in Sect. 4.3.

1.1 Half-Explicit Euler

We start with the half-explicit Euler discretisation, that gives a scheme $\mathcal Ex^{k+1} = \mathcal A^k x^k+ h^k$ with the matrix pair

as in (4.4). For sufficiently small τ, due to the definiteness of M and the full-rank property of B, the matrix $\mathcal A$ is invertible and thus, the pair $(\mathcal E, \mathcal A)$ is regular. Let S denote the matrix BM ⁻¹ B ^T. If one applies

from the left and the right, one finds that $(\mathcal E, \mathcal A)$ is similar to

Since B is of full rank, there exists an orthogonal matrix Q and an invertible matrix R such that $BM^{-\frac 12}Q = \big [ 0\ \ R \big ]$ and, in particular,

Thus, the corresponding similarity transformation transforms $(\mathcal E, \mathcal A)$ into

where $\tilde a_{11}$ and $\tilde a_{21}$ stand for unspecified but possibly nonzero block matrix entries. With another few regular row and column transformations, one can eliminate the entry $\tilde a_{21}$ and read off the Kronecker index of ${\bigl ( \mathcal E, \mathcal A \bigr )}$ as the index of nilpotency of which is 2.

1.2 Projection Scheme

The matrix coefficient pair of the Projection scheme (4.9) reads

(6.1)

If we define S := BM ⁻¹ B ^T, if we move the second row and column to the left and bottom, respectively, and if we rescale certain rows and columns, we find that the pair is equivalent to

where the $\tilde e$’s and $\tilde a$’s stand for unspecified but possibly nonzero entries. Since, in particular, S is invertible, one can eliminate the entries $\tilde e_{24}$, $\tilde e_{34}$, $\tilde a_{41}$, and $\tilde a_{14}$ by regular row and column manipulations without affecting the invertibility of the left upper 3 × 3 block in the transformed $\mathcal E$ and read off the Kronecker index of (4.9) as the index of nilpotency of 0 which is 1.

1.3 SIMPLE

The matrix coefficient pair of the SIMPLE scheme (4.13) reads

If we define $S_A := B(\frac 1\tau M^{-1} + A)^{-1} B^T$, move the second row and column to the left and bottom, respectively, and rescale certain rows and columns, then we find that the pair is equivalent to

where, again, the $\tilde e$’s and $\tilde a$’s stand for unspecified but possibly nonzero entries. Since S _A is invertible for sufficiently small τ, we find that this matrix pair has the very same structure as the one of the projection scheme (see section “Projection Scheme” in Appendix) and, thus, is of index 1.