Page 345 - Computational Fluid Dynamics for Engineers

11.4 Artificial Compressibility Method: INS2D 335

11.4.4 System of Discretized Equation

n
With Eqs. (11.4.14) and (11.4.20), the residual vector i? -term in Eq. (11.4.5)
can be written as

E
Rn _ E i+l/2,j "" i-l/2J F zJ+l/2 ~ F i,j-l/2
hj
~ Ax Ay
E
E
F
F
_ ( v)i+l/2j ~ ( v)i-l/2,j _ ( v)i,j+l/2 ~ ( v)i,j-l/2 (n 4 2 1 ,
Ax Ay ^ ' ' }
The calculation of the exact Jacobian matrix of the residual vector ( § § ) n
can be very expensive, particularly when higher order upwind methods are used.
Therefore, it is more economical to approximate the exact Jacobian from the
residual R n resulting from the first order upwind method. Applying the first
order upwind method to the convective terms (Eqs. (11.4.16)-(11.4.19)) and
maintaining the central differences for the viscous terms (Eq. 11.4.20) yields
n
the following residual R :

+
F• • i - F• i • AE+ / 0 - - AE~ /Q • - AE ! /Q . + AE~ , /Q .
r>n ^ ^i+l,j ^ i - l j ^+l/2j ^+l/2j z-l/2j ^-l/2,3
^ ~ 2AX 2AX
+
F'• • L1 - F 1 ^ F+ -^ /o ~ AF7.. /Q - AF . -, / 9 + AF~. „ /Q
r r
i,3 + l i,3~l i,3 + l/2 i,3 + l/2 i,3-l/2 zj-1/2
'zny 'lay
(En) A i i in A — (ED)- I /o „• ( i ^ j L „• i 1 /o — (i*?> L „• i /o
(11.4.22)

The exact Jacobian matrix of the residual given by Eq. (11.4.22) forms a
banded matrix B and can be written in the form

dm, dm, dm, dm, dm,
(£)- % i3 c\ r\ L ->3 l i3 l iJ r\ r\ L j3
U
U
, U , . . . , U , 1 r\T^ > 0 7 ^ 5 5 - - * ? 5
(11.4.23)
As discussed in [9], the discrete form of Eq. (11.2.5) can be written as

B[V, 0,..., 0, X, y, Z, 0,..., 0, W]AD = -R (11.4.24)

where V, X, Y, Z and W denote vectors of 3 by 3 blocks which lie on the diagonals
of the banded matrix, with the Y vector lying on the main diagonal. These
vectors can be approximated as