Applied Computational Fluid Dynamics Techniques - Wiley Episode 1 Part 9 doc

190 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES

ipoi1 ipoi2

unkno, rhspo

geoed, redge

rhspo

ipoi1 ipoi2

ipoi1 ipoi2

(a)

(b)

(c)

Figure 10.1. Edge-based Laplacian

10.2. First derivatives: first form

We now proceed to first derivatives. Typical examples are the Euler fluxes, the advective terms

for pollution transport simulations or the Maxwell equations that govern electromagnetic

wave propagation. The RHS is given by an expression of the form

ri = −

Ni

Nj

,k d Fk

j , (10.7)

where Fk

j denotes the flux in the kth dimension at node j . This integral is again separated into

shape functions that are not equal to Ni and those that are equal:

ri = −

j

=i



el

Ni

Nj

,k d 

Fk

j −

el

Ni

Ni

,k d Fk

i . (10.8)

As before, we use the conservation property (equation (10.4)) and get

ri = −

j

=i



el

Ni

Nj

,k d 

Fk

j +



el

Ni 

j

=i

Nj

,k d 

Fk

i . (10.9)

This may be restated as

ri = dij

k (Fk

i − Fk

j ), dij

k =

el

Ni

Nj

,k d , j

= i. (10.10)

One may observe that:

- for a change in indices ij versus ji we obtain

dji

k = −dij

k +



NjNi

nk d, (10.11)

this is expected due to the unsymmetric operator;

EDGE-BASED COMPRESSIBLE FLOW SOLVERS 191

- an extra boundary integral leads to a separate loop over boundary edges, adding

(unsymmetrically) only to node j .

The flow of information for this first form of the first derivatives is shown in Figure 10.2.

i j

+ Ŧ

j i

+ +

(a)

(b)

Figure 10.2. First derivatives: first form

Observe that we take a difference on the edge level, and then add contributions to both

endpoints. This implies that the conservation law given for the first derivatives is not reflected

at the edge level, although it is still maintained at the point level. This leads us to a second

form, which reflects the conservation property on the edge level.

10.3. First derivatives: second form

While (10.10) is valid for any finite element shape function, a more desirable form of the

RHS for the first-order fluxes is

ri = e

ij

k (Fk

j + Fk

i ), eij

k = −e

ji

k . (10.12)

In what follows, we will derive such an approximation for linear elements. As before, we start

by separating the Galerkin integral into shape functions that are not equal to Ni and those that

are equal:

ri = −

j

=i



el

Ni

Nj

,k d 

Fk

j −

el

Ni

Ni

,k d Fk

i . (10.13)

In the following, we assume that whenever a sum over indices i, j is to be performed, then

i

= j , and that whenever the index i appears repeatedly in an expression, no sum is to be

taken. Moreover, we use the abbreviation Fk

ij = Fk

i + Fk

j . Integration by parts for the first

integral yields

ri =

Ni

,kNj d Fk

j −



Ni

Njnk d Fk

j −

Ni

Ni

,k d Fk

i . (10.14)

After conversion of the last domain integral into a surface integral via

Ni

Ni

,k d Fk

i =



Ni

Ni

nk d Fk

i −

Ni

Ni

,k d Fk

i , (10.15)

equation (10.13) may be recast as

ri =

Ni

,kNj d Fk

ij −

Ni

,kNj d Fk

i −



Ni

Nj nk d Fk

j

− 1

2



Ni

Ni

nk d Fk

i . (10.16)