7/30/2019 Taylor Expansion

1/6

8 T h i r d o r d e r e x p a n s i o n s

8 . 1 G e n e r a l

A s s o m e t h i n g n e w w e w i l l i n t r o d u c e t h i r d o r d e r t e r m s i n t h e b o u n d a r y e l e -

m e n t m o d e l . S o f r o m h e r e l e t t h e p e r t u r b a t i o n e x p a n s i o n s o f a n d b e o f

t h e f o r m

= "

1

+ "

2

2

+ "

3

3

+ O "

4

8 . 1

= "

1

+ "

2

2

+ "

3

3

+ O "

4

8 . 2

T h e s e r e d e n i t i o n s o f t h e e x p a n s i o n s o f a n d d o n o t c o n t r a d i c t t h e f o r m e r

d e n i t i o n s 2 . 4 4 a n d 2 . 4 4 . T h u s t h e r e s u l t s o b t a i n e d i n t h e p r e v i o u s p a r t s

o f t h i s w o r k a r e s t i l l v a l i d b u t t h e y m a y b e i m p r o v e d b y t h e i n t r o d u c t i o n o f

t h i r d o r d e r t e r m s .

I t i s e a s i l y s h o w n t h a t t h e t h i r d o r d e r p o t e n t i a l

3

m u s t s a t i s f y t h e L a p l a c e -

e q u a t i o n a n d t h u s t h e b o u n d a r y i n t e g r a l e q u a t i o n 2 . 4 9 i s v a l i d f o r t h e t h i r d

o r d e r p o t e n t i a l . A n a l y t i c a l l y t h i s i s w r i t t e n a s

r

2

3

= 0 8 . 3

+

Z

,

0

3

@ G

@ n

0

, G

@

3

@ n

0

d s =

3

x 8 . 4

I n d i s c r e t e f o r m t h e b o u n d a r y i n t e g r a l e q u a t i o n s o b t a i n e d f o r

3

w i l l h a v e

t h e s a m e c o e c i e n t s a s t h e e q u a t i o n s f o r

1

a n d

2

a s d e s c r i b e d i n s e c t i o n

4 . 5 . T h i s m e a n s t h a t t h e c a l c u l a t i o n t i m e p e r t i m e s t e p w i l l b e i n c r e a s e d b y

r o u g h l y 5 0 b y i n t r o d u c i n g t h i r d o r d e r t e r m s . F o r v e r y w e a k l y n o n l i n e a r

w a v e s i t m a y n o t b e w o r t h t h e e x t r a c a l c u l a t i o n t i m e t o o b t a i n t h e t h i r d

o r d e r p o t e n t i a l b u t f o r w a v e s o f i n t e r m e d i a t e n o n l i n e a r i t y t h e t h i r d o r d e r

t e r m s m a y m a k e t h i s m o d e l f e a s i b l e a n d t h i s m o d e l w i l l s t i l l b e a l o t f a s t e r

t h a n a f u l l n o n l i n e a r B E M .

8 . 2 F r e e s u r f a c e a n d o p e n b o u n d a r y c o n d i t i o n s

8 . 2 . 1 G e n e r a l

T h e m o s t d e m a n d i n g t a s k b y i n t r o d u c i n g t h i r d o r d e r t e r m s i n t h e B E M i s

t h e e v a l u a t i o n o f n e w b o u n d a r y c o n d i t i o n s b a s e d o n t h e d y n a m i c 3 . 3 1 a n d

k i n e m a t i c 3 . 3 2 f r e e s u r f a c e b o u n d a r y c o n d i t i o n s m o d i e d b y d i s s i p a t i v e

t e r m s f o r a s p o n g e l a y e r f o r i n s t a n c e g i v e n b y 3 . 4 2 .

6 8

7/30/2019 Taylor Expansion

2/6

T a y l o r e x p a n d i n g t h e k i n e m a t i c b o u n d a r y c o n d i t i o n f r o m z = 0 w e g e t

@

@ t

,

@

@ z

+

@

@ x

@

@ x

+ x

+

@

@ z

@

@ t

,

@

@ z

+

@

@ x

@

@ x

+ x

+

2

1

2

@

2

@ z

2

@

@ t

,

@

@ z

+

@

@ x

@

@ x

+ x

+

3

1

6

@

3

@ z

3

@

@ t

,

@

@ z

+

@

@ x

@

@ x

+ x

= O

4

a t z = 0 8 . 5

+

@

@ t

,

@

@ z

+

@

@ x

@

@ x

+ x

,

@

2

@ z

2

+

@

2

@ x @ z

@

@ x

,

1

2

2

@

3

@ z

3

+

1

2

2

@

3

@ x @ z

2

@

@ x

+

3

1

6

@

3

@ z

3

@

@ z

,

@

@ t

,

@

@ x

@

@ x

= O

4

a t z = 0 8 . 6

S i m i l a r l y f r o m t h e d y n a m i c b o u n d a r y c o n d i t i o n w e g e t

@

@ t

+ g +

1

2

r

2

+ x

+

@

@ z

@

@ t

+ g +

1

2

r

2

+ x

+

2

1

2

@

2

@ z

2

@

@ t

+ g +

1

2

r

2

+ x

+

3

1

6

@

3

@ z

3

@

@ t

+ g +

1

2

r

2

+ x

= O

4

a t z = 0 8 . 7

+

@

@ t

+ g +

1

2

r

2

+ x

+

@

2

@ t @ z

+

1

2

@

@ z

,

r

2

+ x

@

@ z

+

1

2

2

@

3

@ t @ z

2

+

1

4

2

@

2

@ z

2

,

r

2

+

1

2

x

2

@

2

@ z

2

+

3

1

6

@

3

@ z

3

@

@ t

+ g +

1

2

r

2

+ x

= O

4

a t z = 0 8 . 8

6 9

7/30/2019 Taylor Expansion

3/6

P e r t u r b a t i o n e x p a n d i n g a n d i n 8 . 6 a n d k e e p i n g o n l y t e r m s o f t h i r d

o r d e r i n " w e g e t

@

3

@ t

,

@

3

@ z

+ x

3

= ,

@

1

@ x

@

2

@ x

,

@

2

@ x

@

1

@ x

+

1

@

2

2

@ z

2

+

2

@

2

1

@ z

2

,

1

@

2

1

@ x @ z

@

1

@ x

+

1

2

2

1

@

3

1

@ z

3

8 . 9

+

@

3

@ t

,

@

3

@ n

0

+ x

3

= ,

@

1

@ x

@

2

@ x

,

@

2

@ x

@

1

@ x

,

1

@

2

2

@ x

2

,

2

@

2

1

@ x

2

,

1

@

1

@ x

@

@ x

@

1

@ n

0

,

1

2

2

1

@

2

@ x

2

@

1

@ n

0

8 . 1 0

S i m i l a r l y p e r t u r b a t i o n e x p a n d i n g 8 . 8 a n d k e e p i n g o n l y t e r m s o f t h i r d o r d e r

w e g e t

@

3

@ t

+ g

3

+ x

3

= ,

@

1

@ x

@

2

@ x

,

@

1

@ z

@

2

@ z

,

1

@

2

2

@ t @ z

,

2

@

2

1

@ t @ z

,

1

2

1

@

@ z

,

r

1

2

, x

1

@

2

@ z

, x

2

@

1

@ z

,

1

2

2

1

@

3

1

@ t @ z

2

,

1

2

x

2

1

@

2

1

@ z

2

8 . 1 1

+

@

3

@ t

+ g

3

+ x

3

= ,

@

1

@ x

@

2

@ x

,

@

1

@ n

0

@

2

@ n

0

,

1

@

@ t

@

2

@ n

0

,

2

@

@ t

@

1

@ n

0

,

1

@

1

@ x

@

@ x

@

1

@ n

0

+

1

@

1

@ n

0

@

2

1

@ x

2

, x

1

@

2

@ n

0

, x

2

@

1

@ n

0

+

1

2

2

1

@

@ t

@

2

1

@ x

2

+

1

2

x

2

1

@

2

1

@ x

2

8 . 1 2

T h e e q u a t i o n s 8 . 9 a n d 8 . 1 1 s h o w t h e t e r m s a s t h e y w i l l l o o k d i r e c t l y a f t e r

t h e e x p a n s i o n w h i l e t h e e q u a t i o n s 8 . 1 0 a n d 8 . 1 2 a r e r e w r i t t e n t o f o r m s

r e a d y f o r i m p l e m e n t a t i o n i n a c o m p u t e r p r o g r a m .

8 . 2 . 2 U p d a t i n g - A p p l y i n g a n i m p l i c i t A d a m s m e t h o d

A s w e h a v e s e e n i n t h e p r e v i o u s s e c t i o n 8 . 2 . 1 t h e t h i r d o r d e r e x p r e s s i o n s

f o r t h e d y n a m i c a n d k i n e m a t i c f r e e s u r f a c e b o u n d a r y c o n d i t i o n s a r e q u i t e

7 0

7/30/2019 Taylor Expansion

4/6

c o m p l i c a t e d a n d t h u s t o a v o i d c o m p l i c a t i n g t h i n g s e v e n m o r e w e w i l l n o t

T a y l o r e x p a n d t h e s e e q u a t i o n s i n t i m e . I n s t e a d w e w i l l u s e a n e x p l i c i t A d a m s -

B a s h f o r t h m e t h o d f o r t h e u p d a t i n g o f

k

a n d t h e n u s e t h e k i n e m a t i c b o u n d -

a r y c o n d i t i o n f o r n d i n g n e w v a l u e s o f

@

k

@ t

a s d e s c r i b e d i n s e c t i o n 4 . 3 . 3 .

W e h a v e c h o s e n a n e x p l i c i t A d a m s - B a s h f o r t h m e t h o d o f h i g h e r o r d e r t h a n

b e f o r e k

a

= 3 a c c o r d i n g t o 4 . 2 5 w i t h t h e c o e c i e n t s A

0

= 5 5 = 2 4

A

1

= , 5 9 = 2 4 A

2

= 3 7 = 2 4 a n d A

3

= , 9 = 2 4 . S o s t i l l l e t t i n g i n d i c e s m ; j "

r e f e r t o t i m e t = t

m

a n d s p a c e x = x

j

t h e n e w u p d a t i n g o f w i l l b e

k m + 1 j

=

k m j

+ t

5 5

2 4

@

k

@ t

m j

,

5 9

2 4

@

k

@ t

m , 1 j

+

3 7

2 4

@

k

@ t

m , 2 j

,

9

2 4

@

k

@ t

m , 3 j

; k 2 f 1 ; 2 ; 3 g 8 . 1 3

A f t e r s o m e c o n s t r u c t i v e c o n v e r s a t i o n s w i t h J e s p e r S k o u r u p w e c h o s e t o

u p d a t e o n t h e f r e e s u r f a c e b y a n i m p l i c i t A d a m s m e t h o d . T h i s p r o c e -

d u r e i s o f t e n a t t r i b u t e d t o A d a m s - M o u l t o n a n d t h i s w i l l a l s o b e d o n e h e r e

e v e n t h o u g h t h e f o r m u l a i s d u e t o A d a m s . S i m i l a r l y t o t h e e x p l i c i t A d a m s -

B a s h f o r t h m e t h o d s a n i m p l i c i t A d a m s - M o u l t o n m e t h o d i s o f t h e f o r m

y

m + 1

= y

m

+ t B

0

f

m + 1

+ + B

k

a

f

m + 1 , k

a

8 . 1 4

C h o o s i n g k

a

= 3 w e h a v e t h e c o e c i e n t s B

0

= 9 = 2 4 B

1

= 1 9 = 2 4 B

2

=

, 5 = 2 4 a n d B

3

= 1 = 2 4 . T h e m e t h o d i s o f t e n u s e d a s t h e c o r r e c t o r s t e p i n a

p r e d i c t o r - c o r r e c t o r m e t h o d b u t w e u s e t h e m e t h o d f o r p r e d i c t i o n o f

k

o n l y .

T h e d y n a m i c f r e e s u r f a c e b o u n d a r y c o n d i t i o n w i t h F

k

a s r i g h t h a n d s i d e

c a n f o r k 2 f 1 ; 2 ; 3 g b e w r i t t e n a s

@

k

@ t

m + 1 j

+ g

k m + 1 j

+ x

k m + 1 j

= F

k m + 1 j

8 . 1 5

I n s e r t i n g

@

k

@ t

m + 1 j

i n t h e e q u a t i o n f o r t h e i m p l i c i t A d a m s - M o u l t o n m e t h o d

w e g e t

k m + 1 j

=

1

1 + t B

0

x

"

k m j

+ t

B

0

,

, g

k m + 1 j

+ F

k m + 1 j

+ B

1

@

k

@ t

m j

+ B

2

@

k

@ t

m , 1 j

+ B

3

@

k

@ t

m , 2 j

8 . 1 6

7 1

7/30/2019 Taylor Expansion

5/6

W h e n

k m + 1 j

a n d

k m + 1 j

a r e k n o w n t h e d y n a m i c s u r f a c e b o u n d a r y

c o n d i t i o n s w i l l p r o v i d e

@

k

@ t

m + 1 j

t o b e u s e d i n t h e f o l l o w i n g t i m e s t e p s .

A f t e r h a v i n g s o l v e d t h e b o u n d a r y i n t e g r a l e q u a t i o n s t h e k i n e m a t i c f r e e s u r -

f a c e b o u n d a r y c o n d i t i o n s s u p p l y

@

k

@ t

m + 1 j

a t ,

f 0

T h e m e t h o d d e s c r i b e d a b o v e h a s b e e n i m p l e m e n t e d a n d i s h e r e a f t e r u s e d

i n t h e t h i r d o r d e r m o d e l a l s o f o r t h e r s t a n d s e c o n d o r d e r t e r m s . T h e

m e t h o d s e e m s s t a b l e a n d r e l i a b l e w h e n t h e C o u r a n t n u m b e r i s b e l o w u n i t y

s a y C r = 0 8 . S i n c e t h e m o d e l m a y t u r n o u t t o b e u n s t a b l e w h e n t h e C o u r a n t

n u m b e r i s g r e a t e r t h a n u n i t y c a r e s h o u l d b e t a k e n t o e n s u r e t h a t t h i s d o e s

n o t h a p p e n .

I t s h o u l d b e n o t e d t h a t t h e A d a m s - B a s h f o r t h - M o u l t o n m e t h o d s a r e n o r -

m a l l y u s e d i n p r e d i c t o r - c o r r e c t o r m e t h o d s a n d s i n c e o u r m e t h o d r e p r e s e n t s

t h e p r e d i c t o r s t e p o n l y s t a b i l i t y c a n n o t b e g u a r a n t e e d . T o i m p r o v e s t a b i l i t y

a c o r r e c t o r s t e p c a n b e i n c l u d e d i n t h e m o d e l b u t s i n c e t h i s r o u g h l y w o u l d

d o u b l e t h e c a l c u l a t i o n t i m e a n d n o t i m p r o v e t h e a c c u r a c y b y m u c h w e h a v e

n o t i m p l e m e n t e d a c o r r e c t o r s t e p .

8 . 3 P i s t o n b o u n d a r y c o n d i t i o n

A p i s t o n t y p e w a v e - m a k e r s e e s e c t i o n s 3 . 3 a n d 3 . 4 c a n i n m a t h e m a t i c a l

t e r m s b e d e s c r i b e d b y a b o u n d a r y c o n d i t i o n o n t h e f o r m 3 . 1 3 a n d

@

@ z

= 0

i . e .

@

@ n

0

= ,

@

@ x

T a y l o r e x p a n d i n g 3 . 1 3 f r o m x = 0 w e g e t

,

@

@ x

+

@

@ x

,

@

@ x

+

1

2

2

@

2

@ x

2

,

@

@ x

+

1

6

3

@

3

@ x

3

,

@

@ x

+ O

4

= V

n

; a t ,

r 1 0

8 . 1 7

P e r t u r b a t i o n e x p a n d i n g a n d a n d k e e p i n g o n l y t e r m s o f t h i r d o r d e r w e

g e t

,

@

3

@ x

,

1

@

2

2

@ x

2

,

2

@

2

1

@ x

2

,

1

2

2

1

@

3

1

@ x

3

= V

n 3

8 . 1 8

7 2

7/30/2019 Taylor Expansion

6/6

U s i n g t h e L a p l a c e - e q u a t i o n

@

1

@ n

0

= V

n 1

a n d

@

@ z

= 0 w e g e t

@

3

1

@ x

3

=

@

@ x

,

@

2

1

@ z

2

= ,

@

2

@ z

2

@

1

@ x

=

@

2

@ z

2

@

1

@ n

0

=

@

2

@ z

2

V

n 1

=

@

2

@ z

2

,

@

1

@ t

= ,

@

2

@ z @ t

@

1

@ z

= 0

; a t ,

r 1 0

8 . 1 9

T h u s 8 . 1 8 c a n b e r e d u c e d t o

@

3

@ n

0

= V

n 3

+

1

@

2

2

@ x

2

+

2

@

2

1

@ x

2

= V

n 3

,

1

@

2

2

@ z

2

,

2

@

2

1

@ z

2

; a t ,

r 1 0

8 . 2 0

7 3