Fluids: Liquids and Gases - Computer Animation: Algorithms and Techniques - page 260

Kass and Miller [ 10 ] find that a first-order implicit method (Eqs. 8.20 and 8.21 ) is sufficient for this
application. Using these equations to solve for h ( n ) and substituting Equation 8.19 for the second deriv-
ative ( h ( n )) results in Equation 8.22 .

hðnÞhðn 1 Þ

_ ðnÞ¼

(8.20)

Dt

_ ðnÞ _ ðn 1 Þ
Dt

hðnÞ¼

(8.21)

h i ðnÞ¼

2 h i ðn

Þh i ðn Þ

1 2

2 d i 1 þ d i
2 Dx

gðDtÞ ðh i ðnÞh i 1 ðnÞÞ

2

(8.22)

2 d i 1 þ d i
2

þgðDtÞ ðh i þ 1 ðnÞh i ðnÞÞ

Dx

2

Assuming d is constant during the iteration, the next value of h can be calculated from previous
values with the symmetric tridiagonal linear system represented by Equations 8.23-8.25 .

Ah i ðnÞ¼ 2 h i ðn 1 Þþh i ðn 2 Þ (8.23)

2
4

3
5

e 0 f 0 000 0 0

f 0 e 0 f i 00 0 0

0 f i e 2 ... 0 0 0

A ¼ 00 ... ... ... 0 0 (8.24)

000 ... e n 3 f n 3 0

0000 f n 3 e n 2 f n 2

00000 f n 2 e n 1

!

d 0 þ d 1
2

2

e 0 ¼ 1 þ gðDtÞ

2

ðDxÞ

!
ð

d i 1 þ 2 d 1 þ d i þ 1

2

e i ¼ 1 þ gðDtÞ 0 < i < n 1 Þ

2

ðDxÞ

2

!

d n 2 þ d n 1
2

2

e n 1 ¼ 1 þ gðDtÞ

2

ðDxÞ

! !

d i þ d i þ 1
2

2

f i ¼ gðDtÞ (8.25)

2

ðDxÞ

To simulate a viscous fluid, Equation 8.23 can be modified to incorporate a parameter that controls
the viscosity, thus producing Equation 8.26 . The parameter t ranges between 0 and 1. When t¼ 0,

Equation 8.26 reduces to Equation 8.3 .

Ah i ðnÞ¼h i ðn 1 Þþð 1 tÞðh i ðn 1 Þh i ðn 2 ÞÞ (8.26)