Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds - Fluid-Structure Interaction and Biomedical Applications - page 377

h K G k .K/ Z ;' h / D 1 X

K 2 T ht k C 1

LJ h . w h ; w k C 1

K r w k C 1

r' h dx (5.147)

h h

and

J h . w h ; w k C 1

;' h / (5.148)

h

2 G k .K .L/ / C G k .K .R/ Z

D 2 X
2 F

1

Πw k C 1
h

Œ' h dS

I
ht

with parameters 1 ; 2 D O.1/.
It is important that the indicator G k .K/ vanishes in regions where the solution
is regular and the artificial viscosity is applied only locally in the vicinity of
discontinuities or steep gradients. Therefore, the numerical solution does not lose
the accuracy in regions, where the exact solution is regular and, moreover, it does
not contain any nonphysical entropy production in these regions.
The complete resulting scheme has the following form:

.a/ w k C 1
h

2 S ht k C 1 ; (5.149)

.b/ w k C 1

!

w h
k

h

;' h

C b h . w h ; w k C 1 ;' h / Ca h . w h ; w k C 1

;' h /

h h

;' h / C d h w k C 1 ;' h

CJ h . w k C 1
h

h

C LJ h . w h ; w k C 1 ;' h / C J h . w h ; w k C 1 ;' h / D `. w B ;'/

h h

8' h 2 S ht k C 1 ;kD 0;1;::::

The above scheme is of the first-order accuracy in time. The linearized second-
order scheme can be written in the form

3 w k C 1
h

;' h

4 w h C w k 1
2t

C b h . w k C 1
h

; w k C 1
h

;' h / Ca h . w k C 1 ; w k C 1

h

;' h /

h

LJ h . w h ; w k C 1

C J h . w k C 1
h

;' h / C d h . w k C 1 ;' h / C ;' h /

h h

C J h . w h ; w k C 1 ;' h / D `. w k C 1 ;' h /; 8' h 2 S ht k C 1 ; (5.150)

h B

where

D 2 w h w k 1 D 2 w B w k 1 w k C 1

h

; w k C 1
B

: (5.151)

h B