Discrete Exterior Calculus (DEC) for the Surface Navier-Stokes Equation

Abstract

We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on staggered grids in flat space for which we demonstrate second order convergence. We compare computational results with a vorticity-stream function approach for surfaces with genus $g(\mathcal{S}) = 0$ and demonstrate the interplay between topology, geometry and flow properties. Our discretization also allows to handle harmonic vector fields, which we demonstrate on a torus.

Appendices

Appendix 1: Notation for DEC

We often use the strict order relation $\succ$ and $\prec$ on simplices, where $\succ$ is proverbial the “contains” relation, i.e. $e \succ v$ means: the edge $e$ contains the vertex $v$. Correspondingly $\prec$ is the “part of” relation, i.e. $v \prec T$ means: the vertex $v$ is part of the face $T$. Hence, we can use this notation also for sums, like $\sum _{f\succ e}$, i.e. the sum over all faces $T$ containing the edge $e$, or $\sum _{v\prec e}$, i.e. the sum over all vertices $v$ being part of the edge $e$. Sometimes we need to determine this relation for edges more precisely with respect to the orientation. Therefore, sign functions are introduced,

$$ \displaystyle\begin{array}{rcl} s_{T,e}&:=& \left \{\begin{array}{@{}l@{\quad }l@{}} +1\quad &\text{if }e \prec T\mbox{ and $T$ is on the left side of $e$}\\ -1\quad &\text{if } e \prec T\mbox{ and $T$ is on the right side of $e$}\,\text{,} \end{array} \right. {}\\ s_{e,\tilde{e}}&:=& \left \{\begin{array}{@{}l@{\quad }l@{}} +1\quad &\text{if }\measuredangle (\mathbf{e},\tilde{\mathbf{e}}) <\pi \\ -1\quad &\text{if } \measuredangle (\mathbf{e},\tilde{ \mathbf{e} })>\pi \end{array} \right. {}\\ s_{v,e}&:=& \left \{\begin{array}{@{}l@{\quad }l@{}} +1\quad &\text{if }v \prec e\mbox{ and $e$ points to $v$}\\ -1\quad &\text{if } v \prec e\mbox{ and $e$ points away from $v$}\,\text{,} \end{array} \right.{}\\ \end{array} $$

to describe such relations between faces and edges, edges and edges or vertices and edges, respectively. Figure 7.9 gives a schematic illustration.

Fig. 7.9

These formations always yield positive signs $+1$ for $s_{T,e}$ (top left), $s_{v,e}$ (bottom left) and $s_{e,\tilde{e}_{i}}$ (right) for $i \in \left \{1,2,3,4\right \}$, respectively. Every odd-numbered change in edge orientations results in a change of the sign $s_{\cdot,\cdot }$

The property of a primal mesh to be well-centered ensures the existence of a Voronoi mesh (dual mesh), which is also an orientable manifold-like simplicial complex, but not well-centered. The basis of the Voronoi mesh are not simplices, but chains of them. To identify these basic chains, we apply the (geometrical) star operator $\star$ on the primal simplices, i.e. $\star v$ is the Voronoi cell corresponding to the vertex $v$ and inherits its orientation from the orientation of the polyhedron $\left \vert \mathcal{K}\right \vert$. From a geometric point of view, $\star v$ is the convex hull of circumcenters $c(T)$ of all triangles $T \succ v$. The Voronoi edge $\star e$ of an edge $e$ is a connection of the right face $T_{2} \succ e$ with the left face $T_{1} \succ e$ over the midpoint $c(e)$. The Voronoi vertex $\star T$ of a face $T$ is simply its circumcenter $c(T)$, cf. Fig. 7.1. For a more detailed mathematical discussion see e.g. [20, 39].

The boundary operator $\partial$ maps simplices (or chains of them) to the chain of simplices that describes its boundary with respect to its orientation (see [20]), e.g. $\partial (\star v) = -\sum _{e\succ v}s_{v,e}(\star e)$ (formal sum for chains) and $\partial e =\sum _{v\prec e}s_{v,e}v$.

The expression $\left \vert \cdot \right \vert$ measures the volume of a simplex, i.e. $\left \vert T\right \vert$ the area of the face $T$, $\left \vert e\right \vert$ the length of the edge $e$ and the 0-dimensional volume $\left \vert v\right \vert$ is set to be 1. Therefore, the volume is also defined for chains and the dual mesh, since the integral is a linear functional.

Appendix 2: Second Order Convergence

In this section we show that the discretization equation (7.14) of $\boldsymbol{\varDelta }^{\text{RR}}$, defined in Eq. (7.13) on a staggered grid, has a truncation error of order two. Without loss of generality, by a quarter turn of the difference scheme in Fig. 7.3 (left), we only elaborate on the discretization of $(\boldsymbol{\varDelta }^{\text{RR}}u)^{x}$ along the horizontal x-direction. The first three terms in Eq. (7.14) show the well-known second order central difference approximation in vertical direction of the first term in Eq. (7.13), i.e.

$$\displaystyle\begin{array}{rcl} \frac{1} {h^{2}}\left (u_{i,j+1}^{x} + u_{ i,j-1}^{x} - 2u_{ i,j}^{x}\right )& =& \left (\partial _{ y}^{2}u^{x}\right )_{ i,j}^{x} + \mathcal{O}(h^{2})\,. {}\\ \end{array}$$

For the remaining terms, we first carry out a Taylor expansion on central vertices $v_{i+k,j} \in \mathcal{V}$ for $k \in \left \{0,1\right \}$ in the vertical edge columns, i.e.

$$\displaystyle\begin{array}{rcl} u_{i+k,j-l}^{y}& =& \left (u^{y} + (-1)^{l}\frac{h} {2}\partial _{y}u^{y} + \frac{h^{2}} {8} \partial _{y}^{2}u^{y} + (-1)^{l}\frac{h^{3}} {48}\partial _{y}^{3}u^{y} + \frac{h^{4}} {384}\partial _{y}^{4}u^{y}\right )_{ i+k,j} {}\\ & & +\mathcal{O}(h^{5}) {}\\ \end{array}$$

for all $l \in \left \{0,1\right \}$. An additional horizontal expansion of sufficient order at the edge midpoint $c(e_{i,j}^{x})$ results in

$$\displaystyle\begin{array}{rcl} u_{i+k,j-l}^{y}& =& \Big(\!u^{y} +\! (-1)^{k+1}\frac{h} {2}\partial _{x}u^{y} + \frac{h^{2}} {8} \partial _{x}^{2}u^{y} +\! (-1)^{k+1}\frac{h^{3}} {48}\partial _{x}^{3}u^{y} + \frac{h^{4}} {384}\partial _{x}^{4}u^{y} {}\\ & & +\!(-1)^{l}\frac{h} {2}\partial _{y}u^{y} +\! (-1)^{l+k+1}\frac{h^{2}} {4} \partial _{x}\partial _{y}u^{y} +\! (-1)^{l}\frac{h^{3}} {16}\partial _{x}^{2}\partial _{ y}u^{y} {}\\ & & +\!(-1)^{l+k+1}\frac{h^{4}} {96}\partial _{x}^{3}\partial _{ y}u^{y} + \frac{h^{2}} {8} \partial _{y}^{2}u^{y} +\! (-1)^{k+1}\frac{h^{3}} {16}\partial _{x}\partial _{y}^{2}u^{y} {}\\ & & +\frac{h^{4}} {64}\partial _{x}^{2}\partial _{ y}^{2}u^{y} +\! (-1)^{l}\frac{h^{3}} {48}\partial _{y}^{3}u^{y} +\! (-1)^{l+k+1}\frac{h^{4}} {96}\partial _{x}\partial _{y}^{3}u^{y} + \frac{h^{4}} {384}\partial _{y}^{4}u^{y}\!\Big)_{ i,j}^{x} {}\\ & & +\mathcal{O}(h^{5}) {}\\ \end{array}$$

for all $l,k \in \left \{0,1\right \}$. Finally, we obtain

$$\displaystyle\begin{array}{rcl} & & \frac{1} {h^{2}}\left (u_{i,j}^{y} - u_{ i+1,j}^{y} + u_{ i+1,j-1}^{y} - u_{ i,j-1}^{y}\right ) {}\\ & & \quad = -\left (\partial _{x}\partial _{y}u^{y} + \frac{h^{2}} {96}\partial _{x}\partial _{y}\left (\partial _{x}^{2}u^{y} + \partial _{ y}^{2}u^{y}\right )\right )_{ i,j}^{x} + \mathcal{O}(h^{3}) {}\\ \end{array}$$

and thus a truncation error at most $\mathcal{O}(h^{2})$ regarding $(\boldsymbol{\varDelta }^{\text{RR}}u)_{i,j}^{x}$ generally.