giles stability of ode solvers
TRANSCRIPT
-
8/3/2019 Giles Stability of ODE Solvers
1/26
R e p o r t n o . 9 5 / 0 4
S t a b i l i t y A n a l y s i s o f G a l e r k i n / R u n g e - K u t t a
N a v i e r - S t o k e s D i s c r e t i s a t i o n s o n U n s t r u c t u r e d G r i d s
M . B . G i l e s
O x f o r d U n i v e r s i t y C o m p u t i n g L a b o r a t o r y
N u m e r i c a l A n a l y s i s G r o u p
T h i s p a p e r p r e s e n t s a t i m e s t e p s t a b i l i t y a n a l y s i s f o r a c l a s s o f
d i s c r e t i s a t i o n s a p p l i e d t o t h e l i n e a r i s e d f o r m o f t h e N a v i e r - S t o k e s
e q u a t i o n s o n a 3 D d o m a i n w i t h p e r i o d i c b o u n d a r y c o n d i t i o n s . U s i n g
a s u i t a b l e d e n i t i o n o f t h e ` p e r t u r b a t i o n e n e r g y ' i t i s s h o w n t h a t t h e
e n e r g y i s m o n o t o n i c a l l y d e c r e a s i n g f o r b o t h t h e o r i g i n a l p . d . e . a n d
t h e s e m i - d i s c r e t e s y s t e m o f o . d . e . ' s a r i s i n g f r o m a G a l e r k i n d i s c r e t i s a -
t i o n o n a t e t r a h e d r a l g r i d . U s i n g r e c e n t t h e o r e t i c a l r e s u l t s c o n c e r n i n g
a l g e b r a i c a n d g e n e r a l i s e d s t a b i l i t y , s u c i e n t s t a b i l i t y l i m i t s a r e o b -
t a i n e d f o r b o t h g l o b a l a n d l o c a l t i m e s t e p s f o r f u l l y d i s c r e t e a l g o r i t h m s
u s i n g R u n g e - K u t t a t i m e i n t e g r a t i o n .
S u b j e c t c l a s s i c a t i o n s : A M S ( M O S ) : 6 5 M 1 0 , 6 5 M 2 0 , 6 5 M 6 0 , 7 6 - 0 8 , 7 6 N 1 0
K e y w o r d s a n d p h r a s e s : N a v i e r - S t o k e s , m e t h o d o f l i n e s , s t a b i l i t y a n a l y s i s
T h i s w o r k w a s s u p p o r t e d b y R o l l s - R o y c e p l c , D T I a n d E P S R C .
O x f o r d U n i v e r s i t y C o m p u t i n g L a b o r a t o r y
N u m e r i c a l A n a l y s i s G r o u p
W o l f s o n B u i l d i n g
P a r k s R o a d
O x f o r d , E n g l a n d O X 1 3 Q D
E - m a i l : g i l e s @ c o m l a b . o x f o r d . a c . u k A p r i l , 1 9 9 7
-
8/3/2019 Giles Stability of ODE Solvers
2/26
2
1 I n t r o d u c t i o n
O n e m o t i v a t i o n f o r t h e a n a l y s i s i n t h i s p a p e r w a s t h e o b s e r v a t i o n b y W i g t o n o f
i n s t a b i l i t i e s i n N a v i e r - S t o k e s c a l c u l a t i o n s o n s t r u c t u r e d g r i d s 1 ] . I t a p p e a r e d
t h a t t h e i n s t a b i l i t i e s m i g h t b e c o n n e c t e d t o l a r g e v a r i a t i o n s i n t h e l e v e l o f t u r -
b u l e n t v i s c o s i t y a r i s i n g q u i t e p r o p e r l y i n c e r t a i n p h y s i c a l s i t u a t i o n s . A p o s s i b l e
c a u s e o f t h e i n s t a b i l i t y w a s t h o u g h t t o b e t h e t i m e s t e p d e n i t i o n w h i c h w a s
b a s e d o n F o u r i e r s t a b i l i t y t h e o r y a s s u m i n g c o n s t a n t c o e c i e n t s . T h e r e f o r e , a n
o b j e c t i v e o f t h i s a n a l y s i s w a s t o d e t e r m i n e s u c i e n t c o n d i t i o n s f o r t h e s t a b i l i t y
o f d i s c r e t i s a t i o n s o f t h e N a v i e r - S t o k e s e q u a t i o n s w i t h n o n u n i f o r m v i s c o s i t y .
T h e s e c o n d m o t i v a t i o n w a s t h e r e q u i r e m e n t f o r t i m e s t e p s t a b i l i t y l i m i t s f o r
v i s c o u s c a l c u l a t i o n s o n u n s t r u c t u r e d g r i d s . I n v i s c i d c a l c u l a t i o n s a r e n o w b e i n g
p e r f o r m e d a l m o s t r o u t i n e l y o n u n s t r u c t u r e d g r i d s f o r c o m p l e t e a i r c r a f t g e o m e -
t r i e s ( e . g . 2 , 3 , 4 , 5 ] ) . U s i n g e n e r g y a n a l y s i s m e t h o d s , G i l e s d e v e l o p e d s u c i e n t
g l o b a l a n d l o c a l t i m e s t e p s t a b i l i t y l i m i t s f o r a G a l e r k i n d i s c r e t i s a t i o n o f t h e E u l e r
e q u a t i o n s o n a t e t r a h e d r a l g r i d w i t h t w o p a r t i c u l a r R u n g e - K u t t a t i m e i n t e g r a t i o n
s c h e m e s 6 ] ; t h i s h a s b e e n u s e d o n a n a d h o c b a s i s f o r c a l c u l a t i o n s u s i n g o t h e r
a l g o r i t h m s i n c l u d i n g v a r i o u s u p w i n d i n g a n d n u m e r i c a l s m o o t h i n g f o r m u l a t i o n s
3 , 5 ] . T h r o u g h p a r a l l e l c o m p u t i n g a n d e c i e n t m u l t i g r i d a l g o r i t h m s f o r u n s t r u c -
t u r e d g r i d s 5 ] , t h e r e i s n o w t h e c o m p u t a t i o n a l p o w e r t o p e r f o r m e x t r e m e l y l a r g e
N a v i e r - S t o k e s c a l c u l a t i o n s o n u n s t r u c t u r e d g r i d s , a n d s o t h e r e i s a n e e d f o r t h e
s u p p o r t i n g n u m e r i c a l a n a l y s i s t o g i v e a c c u r a t e g l o b a l a n d l o c a l t i m e s t e p s t a b i l i t y
l i m i t s .
F o u r i e r s t a b i l i t y a n a l y s i s c a n o n l y b e a p p l i e d t o l i n e a r n i t e d i e r e n c e e q u a -
t i o n s w i t h c o n s t a n t c o e c i e n t s o n s t r u c t u r e d g r i d s , a n d s o i t i s n o t a p p r o p r i -
a t e f o r t h i s a p p l i c a t i o n . T h e r e a r e t w o o t h e r w e l l - d o c u m e n t e d s t a b i l i t y a n a l y s i s
m e t h o d s w h i c h c a n b e u s e d w i t h l i n e a r d i s c r e t i s a t i o n s w i t h v a r i a b l e c o e c i e n t s
o n u n s t r u c t u r e d g r i d s . O n e i s t h e e n e r g y m e t h o d 7 ] w h i c h r e l i e s o n t h e c a r e f u l
c o n s t r u c t i o n o f a s u i t a b l y d e n e d ` e n e r g y ' w h i c h c a n b e p r o v e n t o m o n o t o n i c a l l y
d e c r e a s e . T h e d i c u l t y i s u s u a l l y i n c o n s t r u c t i n g a n a p p r o p r i a t e d e n i t i o n f o r
t h e e n e r g y , b u t w h e n t h i s m e t h o d c a n b e a p p l i e d i t i s v e r y p o w e r f u l i n g i v i n g a
v e r y s t r o n g f o r m o f s t a b i l i t y . I t i s u s e d i n t h i s p a p e r t o p r o v e t h e s t a b i l i t y o f t h e
o r i g i n a l l i n e a r i s e d f o r m o f t h e N a v i e r - S t o k e s p a r t i a l d i e r e n t i a l e q u a t i o n s , a n d
t h e s e m i - d i s c r e t i s e d s y s t e m o f c o u p l e d o . d . e . ' s t h a t i s p r o d u c e d b y t h e G a l e r k i n
s p a t i a l d i s c r e t i s a t i o n .
T h e o t h e r s t a b i l i t y a n a l y s i s t e c h n i q u e i n v o l v e s c o n s i d e r a t i o n o f t h e e i g e n -
v a l u e s o f t h e m a t r i x r e p r e s e n t i n g t h e d i s c r e t i s a t i o n o f t h e s p a t i a l d i e r e n t i a l
o p e r a t o r . T h i s l e a d s t o s u c i e n t c o n d i t i o n s f o r a s y m p t o t i c s t a b i l i t y , a s t ! 1
f o r u n s t e a d y c a l c u l a t i o n s o r a s n
! 1f o r c a l c u l a t i o n s u s i n g l o c a l t i m e s t e p s .
U n f o r t u n a t e l y , t h e r e a r e w e l l - d o c u m e n t e d e x a m p l e s s u c h a s t h e r s t o r d e r u p -
w i n d i n g o f t h e c o n v e c t i o n e q u a t i o n o n a n i t e 1 D d o m a i n ( e . g . 8 , 9 , 1 0 ] ) f o r
w h i c h t h i s i s n o t a p r a c t i c a l s t a b i l i t y c r i t e r i o n b e c a u s e i t a l l o w s a n u n a c c e p t a b l y
l a r g e t r a n s i e n t g r o w t h b e f o r e t h e e v e n t u a l e x p o n e n t i a l d e c a y . T h e n e x t s e c t i o n
-
8/3/2019 Giles Stability of ODE Solvers
3/26
-
8/3/2019 Giles Stability of ODE Solvers
4/26
4
u s i n g a n e x p l i c i t R u n g e - K u t t a m e t h o d w i t h t i m e s t e p k y i e l d s a d i e r e n c e e q u a -
t i o n o f t h e f o r m
u
( n + 1 )
= L ( k ) u
( n )
( 2 . 2 )
w h e r e L ( z ) i s a p o l y n o m i a l f u n c t i o n o f d e g r e e p
L ( z ) =
p
X
m = 0
a
m
z
m
; ( 2 . 3 )
w i t h a
0
= a
1
= 1 ; a
p
6= 0 . D i s c r e t e s o l u t i o n s o f t h i s d i e r e n c e e q u a t i o n o n a
n i t e t i m e i n t e r v a l 0
t
t
0
w i l l c o n v e r g e t o t h e a n a l y t i c s o l u t i o n a s k
!0 . I n
a d d i t i o n , t h e d i s c r e t i s a t i o n i s s a i d t o b e a b s o l u t e l y s t a b l e f o r a p a r t i c u l a r v a l u e o f
k i f i t d o e s n o t a l l o w e x p o n e n t i a l l y g r o w i n g s o l u t i o n s a s t ! 1 ; t h i s i s s a t i s e d
p r o v i d e d k l i e s w i t h i n t h e s t a b i l i t y r e g i o n S i n t h e c o m p l e x p l a n e d e n e d b y
S = f z : j L ( z ) j 1 g : ( 2 . 4 )
E x a m p l e s o f s t a b i l i t y r e g i o n s f o r d i e r e n t p o l y n o m i a l s a r e g i v e n i n A p p e n d i x A .
S u p p o s e n o w t h a t a r e a l s q u a r e m a t r i x C h a s a c o m p l e t e s e t o f e i g e n v e c t o r s
a n d c a n t h u s b e d i a g o n a l i s e d ,
C = T T
1
; ( 2 . 5 )
w i t h b e i n g t h e d i a g o n a l m a t r i x o f e i g e n v a l u e s o f C , a n d t h e c o l u m n s o f T
b e i n g t h e a s s o c i a t e d e i g e n v e c t o r s . T h e R u n g e - K u t t a d i s c r e t i s a t i o n o f t h e c o u p l e d
s y s t e m o f o . d . e . ' s ,
d U
d t
= C U ; ( 2 . 6 )
c a n b e w r i t t e n a s
U
( n + 1 )
= L ( k C ) U
( n )
= T L ( k ) T
1
U
( n )
; ( 2 . 7 )
s i n c e
C
m
=
T T
1
m
= T
m
T
1
: ( 2 . 8 )
H e n c e
U
( n )
= T ( L ( k ) )
n
T
1
U
( 0 )
: ( 2 . 9 )
T h e n e c e s s a r y a n d s u c i e n t c o n d i t i o n f o r a b s o l u t e s t a b i l i t y a s n ! 1 , r e -
q u i r i n g t h a t t h e r e a r e n o d i s c r e t e s o l u t i o n s w h i c h g r o w e x p o n e n t i a l l y w i t h n , i s
t h e r e f o r e t h a t j L ( k ) j 1 , o r e q u i v a l e n t l y k l i e s i n S , f o r a l l e i g e n v a l u e s o f
C . I f t h i s c o n d i t i o n i s s a t i s e d , t h e n u s i n g L
2
v e c t o r a n d m a t r i x n o r m s i t f o l l o w s
t h a t
k U
( n )
k k T k k L ( k ) k
n
k T
1
k k U
( 0 )
k ( T ) k U
( 0 )
k ; ( 2 . 1 0 )
w h e r e ( T ) i s t h e c o n d i t i o n n u m b e r o f t h e e i g e n v e c t o r m a t r i x T .
-
8/3/2019 Giles Stability of ODE Solvers
5/26
5
I f t h e m a t r i x C i s n o r m a l , m e a n i n g t h a t i t h a s a n o r t h o g o n a l s e t o f e i g e n -
v e c t o r s t h e n t h e e i g e n v e c t o r s c a n b e n o r m a l i s e d s o t h a t ( T ) = 1 . I n t h i s c a s e ,
k U
( n )
k i s a n o n - i n c r e a s i n g f u n c t i o n o f n a n d k U
( n )
k
2
r e p r e s e n t s a n o n - i n c r e a s i n g
` e n e r g y ' w h i c h c o u l d b e u s e d i n a n e n e r g y s t a b i l i t y a n a l y s i s .
I f C i s n o t n o r m a l , t h e n t h e g r o w t h i n k U
( n )
k i s b o u n d e d b y t h e c o n d i t i o n
n u m b e r o f t h e e i g e n v e c t o r m a t r i x , ( T ) . U n f o r t u n a t e l y , t h i s c a n b e v e r y l a r g e
i n d e e d , a l l o w i n g a v e r y l a r g e t r a n s i e n t g r o w t h i n t h e s o l u t i o n e v e n w h e n f o r
e a c h e i g e n v a l u e k l i e s s t r i c t l y i n s i d e t h e s t a b i l i t y r e g i o n S a n d s o k U
( n )
k m u s t
e v e n t u a l l y d e c a y e x p o n e n t i a l l y . T h i s p r o b l e m c a n b e p a r t i c u l a r l y a c u t e w h e n t h e
m a t r i x C c o m e s f r o m t h e s p a t i a l d i s c r e t i s a t i o n o f a p . d . e . i n w h i c h c a s e t h e r e i s
t h e n a f a m i l y o f d i s c r e t i s a t i o n s a r i s i n g f r o m a s e q u e n c e o f c o m p u t a t i o n a l g r i d s o f
d e c r e a s i n g m e s h s p a c i n g h . I t i s p o s s i b l e i n s u c h c i r c u m s t a n c e s f o r t h e s e q u e n c e
o f c o n d i t i o n n u m b e r s ( T ) t o g r o w e x p o n e n t i a l l y , w i t h a n e x p o n e n t i n v e r s e l y
p r o p o r t i o n a l t o t h e m e s h s p a c i n g 8 ] . T h e r e a r e t w o p r a c t i c a l c o n s e q u e n c e s o f t h i s
e x p o n e n t i a l g r o w t h . I n a p p l i c a t i o n s c o n c e r n e d w i t h t h e b e h a v i o u r o f t h e s o l u t i o n
a s t ! 1 , i t p r o d u c e s a n u n a c c e p t a b l y l a r g e a m p l i c a t i o n o f m a c h i n e r o u n d i n g
e r r o r s i n l i n e a r c o m p u t a t i o n s a n d c o m p l e t e f a i l u r e o f t h e d i s c r e t e c o m p u t a t i o n i n
n o n l i n e a r c a s e s . I n a p p l i c a t i o n s c o n c e r n e d w i t h a n i t e t i m e i n t e r v a l , 0 t t
0
, i t
p r e v e n t s c o n v e r g e n c e o f t h e d i s c r e t e s o l u t i o n t o t h e a n a l y t i c s o l u t i o n a s h ; k ! 0
e x c e p t i n c e r t a i n e x c e p t i o n a l s i t u a t i o n s u s i n g s p e c t r a l s p a t i a l d i s c r e t i s a t i o n s .
T h e s t a b i l i t y o f d i s c r e t i s a t i o n s o f s y s t e m s o f o . d . e . ' s w i t h n o n - n o r m a l m a t r i c e s
h a s b e e n a m a j o r r e s e a r c h t o p i c i n t h e n u m e r i c a l a n a l y s i s c o m m u n i t y i n r e c e n t
y e a r s 8 , 9 , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 ] ; A r e c e n t r e v i e w a r t i c l e b y v a n D o r s s e l a e r e t a l 1 0 ]
p r o v i d e s a n e x c e l l e n t o v e r v i e w o f t h e s e a n d m a n y o t h e r r e f e r e n c e s . T h e a p p l i c a -
t i o n i s o f t e n t o f a m i l i e s o f n o n - n o r m a l m a t r i c e s a r i s i n g f r o m s p a t i a l d i s c r e t i s a t i o n s
o f p . d . e . ' s . I d e a l l y , o n e w o u l d h o p e t o p r o v e s t r o n g s t a b i l i t y ,
k U
( n )
k k U
( 0 )
k ; ( 2 . 1 1 )
w i t h b e i n g a c o n s t a n t w h i c h i s n o t o n l y i n d e p e n d e n t o f n b u t i s a l s o a u n i f o r m
b o u n d a p p l y i n g t o a l l m a t r i c e s i n t h e f a m i l y o f s p a t i a l d i s c r e t i s a t i o n s f o r d i e r e n t
m e s h s p a c i n g s h b u t w i t h t h e t i m e s t e p k b e i n g a f u n c t i o n o f h . O n e r e a s o n w h y
s t r o n g s t a b i l i t y i s v e r y d e s i r a b l e i s t h a t t h e L a x E q u i v a l e n c e T h e o r e m p r o v e s t h a t
i t i s a n e c e s s a r y a n d s u c i e n t c o n d i t i o n f o r c o n v e r g e n c e o f d i s c r e t e s o l u t i o n s t o
t h e a n a l y t i c s o l u t i o n o n a n i t e t i m e i n t e r v a l f o r a l l p o s s i b l e i n i t i a l d a t a , p r o v i d e d
t h a t t h e d i s c r e t i s a t i o n o f t h e p . d . e . i s c o n s i s t e n t f o r s u c i e n t l y s m o o t h i n i t i a l d a t a
7 ] .
A t p r e s e n t , t h e c o n d i t i o n s u n d e r w h i c h s t r o n g s t a b i l i t y c a n b e p r o v e d a r e
t o o r e s t r i c t i v e t o b e u s e f u l i n p r a c t i c a l c o m p u t a t i o n s . I n s t e a d , a t t e n t i o n h a s
f o c u s s e d o n w e a k e r d e n i t i o n s o f s t a b i l i t y w h i c h a r e m o r e e a s i l y a c h i e v e d a n d
a r e s t i l l u s e f u l f o r p r a c t i c a l c o m p u t a t i o n s . O n e i s a l g e b r a i c s t a b i l i t y 8 , 1 1 , 1 2 ]
w h i c h a l l o w s a l i n e a r g r o w t h i n t h e t r a n s i e n t s o l u t i o n o f t h e f o r m
k U
( n )
k n k U
( 0 )
k ; ( 2 . 1 2 )
-
8/3/2019 Giles Stability of ODE Solvers
6/26
6
w h e r e i s a g a i n a u n i f o r m c o n s t a n t . A n o t h e r , d u e t o K r e i s s a n d W u 9 ] , i s
g e n e r a l i s e d s t a b i l i t y w h i c h i s b a s e d o n e x p o n e n t i a l l y w e i g h t e d i n t e g r a l s o v e r t i m e
f o r a i n h o m o g e n e o u s d i e r e n c e e q u a t i o n w i t h h o m o g e n e o u s i n i t i a l c o n d i t i o n s .
F o r b o t h o f t h e s e d e n i t i o n s , a s u c i e n t c o n d i t i o n f o r s t a b i l i t y i s t h a t
( k C ) S ; ( 2 . 1 3 )
w h e r e t h e n u m e r i c a l r a n g e ( k C ) i s a s u b s e t o f t h e c o m p l e x d o m a i n d e n e d b y
( k C ) =
k
W
C W
W
W
: W
6= 0
( 2 . 1 4 )
w h e r e W c a n b e a n y n o n - z e r o c o m p l e x v e c t o r o f t h e r e q u i r e d d i m e n s i o n a n d W
i s i t s H e r m i t i a n , t h e c o m p l e x c o n j u g a t e t r a n s p o s e . T h e p r o o f o f s u c i e n c y f o r
a l g e b r a i c s t a b i l i t y i s g i v e n b y L e n f e r i n k a n d S p i j k e r 1 2 ] . I t p r o c e e d s i n t w o p a r t s ,
r s t s h o w i n g t h a t a c e r t a i n r e s o l v e n t c o n d i t i o n i s s u c i e n t f o r a l g e b r a i c s t a b i l -
i t y , a n d t h e n s h o w i n g t h a t t h i s r e s o l v e n t c o n d i t i o n i s s a t i s e d i f t h e n u m e r i c a l
r a n g e l i e s i n s i d e S . R e d d y a n d T r e f e t h e n 8 ] p r o v e t h a t t h e r e s o l v e n t c o n d i t i o n
i s n e c e s s a r y a s w e l l a s s u c i e n t f o r a l g e b r a i c s t a b i l i t y , a n d t h e e q u i v a l e n c e t o
g e n e r a l i s e d s t a b i l i t y f o l l o w s a l m o s t i m m e d i a t e l y g i v e n t h e r e s o l v e n t c o n d i t i o n
r e q u i r e d b y K r e i s s a n d W u 9 ] .
B y c o n s i d e r i n g W t o b e a n e i g e n v e c t o r o f C , i t c a n b e s e e n t h a t k 2 ( k C )
f o r e a c h e i g e n v a l u e o f C a n d s o t h e r e q u i r e m e n t t h a t ( k C ) S i s a t i g h t e r
r e s t r i c t i o n o n t h e m a x i m u m a l l o w a b l e t i m e s t e p t h a n a s y m p t o t i c s t a b i l i t y . I n
c o m p a r i s o n t o s t r o n g s t a b i l i t y , a l g e b r a i c a n d g e n e r a l i s e d s t a b i l i t y a l l o w g r e a t e r
g r o w t h i n t r a n s i e n t s w h e n c o n s i d e r i n g t h e s o l u t i o n b e h a v i o u r a s t ! 1 . O n
t h e n i t e t i m e i n t e r v a l , i t c a n b e s h o w n t h a t u n d e r s o m e v e r y m i l d t e c h n i c a l
c o n d i t i o n s t h e y a r e s u c i e n t f o r c o n v e r g e n c e o f d i s c r e t e s o l u t i o n s t o t h e a n a l y t i c
s o l u t i o n a s h ; k ! 0 p r o v i d e d t h e i n i t i a l d a t a i s s m o o t h a n d t h e d i s c r e t i s a t i o n
i s c o n s i s t e n t . I t t h u s a p p e a r s t h a t t h e s e s t a b i l i t y d e n i t i o n s a r e u s e f u l t o o l s i n
a n a l y s i n g n u m e r i c a l d i s c r e t i s a t i o n s , b u t a d d i t i o n a l r e s e a r c h i s s t i l l r e q u i r e d .
I n t h e N a v i e r - S t o k e s a p p l i c a t i o n i n t h i s p a p e r w e w i l l n e e d t o c o n s i d e r a s l i g h t
g e n e r a l i s a t i o n t o a s y s t e m o f o . d . e . ' s o f t h e f o r m
M
d U
d t
= C U ; ( 2 . 1 5 )
i n w h i c h M i s a r e a l s y m m e t r i c p o s i t i v e - d e n i t e m a t r i x . T h e ` e n e r g y ' i s d e n e d
a s U
M U w h i c h s u g g e s t s t h e d e n i t i o n o f n e w v a r i a b l e s ,
V = M
1 = 2
U ; ( 2 . 1 6 )
s o t h a t k V k
2
= U
M U . I f M i s d i a g o n a l t h e n M
1 = 2
i s t h e d i a g o n a l m a t r i x w h o s e
e l e m e n t s a r e t h e p o s i t i v e s q u a r e r o o t o f t h e c o r r e s p o n d i n g e l e m e n t s o f M . I f M
i s n o t d i a g o n a l t h e n M
1 = 2
i s e q u a l t o T
1
1 = 2
T w h e r e i s t h e d i a g o n a l m a t r i x o f
-
8/3/2019 Giles Stability of ODE Solvers
7/26
7
e i g e n v a l u e s o f M a n d T i s t h e c o r r e s p o n d i n g m a t r i x o f o r t h o n o r m a l e i g e n v e c t o r s .
T
1
= T
a n d h e n c e b o t h M
1 = 2
a n d M
1 = 2
a r e s y m m e t r i c a n d p o s i t i v e d e n i t e .
U n d e r t h e c h a n g e o f v a r i a b l e s , t h e s y s t e m o f o . d . e . ' s b e c o m e s
d V
d t
= M
1 = 2
C M
1 = 2
V ; ( 2 . 1 7 )
w h i c h i s a l g e b r a i c a l l y s t a b l e p r o v i d e d ( k M
1 = 2
C M
1 = 2
) S . I f C i s e i t h e r s y m -
m e t r i c o r a n t i - s y m m e t r i c t h e n s o t o o i s M
1 = 2
C M
1 = 2
b e c a u s e o f t h e s y m m e t r y
o f M
1 = 2
. T h e r e f o r e , a s d i s c u s s e d e a r l i e r t h e c o n d i t i o n t h a t t h e n u m e r i c a l r a n g e
l i e s i n s i d e S a l s o e n s u r e s t h a t t h e e n e r g y , k V k
2
= U
M U , w i l l b e n o n - i n c r e a s i n g .
3 A n a l y t i c e q u a t i o n s
T h e s t a r t i n g p o i n t f o r t h e a n a l y s i s i s t h e n o n l i n e a r N a v i e r - S t o k e s e q u a t i o n s ,
@ U
@ t
+
@ F
x
@ x
+
@ F
y
@ y
+
@ F
z
@ z
= 0 : ( 3 . 1 )
U i s t h e v e c t o r o f c o n s e r v a t i o n v a r i a b l e s ( ; u ; v ; w ; E )
T
a n d t h e u x t e r m s
a r e a l l d e n e d i n A p p e n d i x B t o g e t h e r w i t h t h e e q u a t i o n o f s t a t e f o r a n i d e a l
g a s a n d t h e d e n i t i o n s o f t h e s t r e s s t e n s o r a n d t h e v i s c o u s h e a t u x v e c t o r . T h e
e q u a t i o n s a r e t o b e s o l v e d o n a u n i t c u b i c d o m a i n w i t h p e r i o d i c b o u n d a r y
c o n d i t i o n s . T h e c h o i c e o f p e r i o d i c b . c . ' s a v o i d s t h e c o m p l i c a t i o n o f a n a l y s i n g t h e
i n u e n c e o f d i e r e n t a n a l y t i c a n d d i s c r e t e b o u n d a r y c o n d i t i o n s .
T h e r s t s t e p i s t o l i n e a r i s e t h e N a v i e r - S t o k e s e q u a t i o n s b y c o n s i d e r i n g p e r -
t u r b a t i o n s t o a s t e a d y o w w h i c h i s u n i f o r m a p a r t f r o m s p a t i a l v a r i a t i o n s i n t h e
v i s c o s i t y p a r a m e t e r s ; ; k . P e r t u r b a t i o n s t o t h e c o n s e r v e d v a r i a b l e s a r e r e l a t e d
t o t h e v e c t o r o f p r i m i t i v e p e r t u r b a t i o n s , V = ( ~ ; ~u ; ~v ; ~w ; ~p )
T
, b y t h e e q u a t i o n
e
U = R V : ( 3 . 2 )
T h e u n i f o r m t r a n s f o r m a t i o n m a t r i x R i s g i v e n i n A p p e n d i x B . T o g e t h e r , t h e
l i n e a r i s a t i o n a n d t h e c h a n g e o f v a r i a b l e s y i e l d s
@ V
@ t
+ A
0
x
@ V
@ x
+ A
0
y
@ V
@ y
+ A
0
z
@ V
@ y
=
@
@ x
D
0
x x
@ V
@ x
+ D
0
x y
@ V
@ y
+ D
0
x z
@ V
@ z
!
+
@
@ y
D
0
y x
@ V
@ x
+ D
0
y y
@ V
@ y
+ D
0
y z
@ V
@ z
!
( 3 . 3 )
+
@
@ z
D
0
z x
@ V
@ x
+ D
0
z y
@ V
@ y
+ D
0
z z
@ V
@ z
!
:
A l l m a t r i c e s i n t h i s e q u a t i o n a r e l i s t e d i n A p p e n d i x B . T h e s e c o n d s t e p i s t o
d e n e a f u r t h e r t r a n s f o r m a t i o n o f v a r i a b l e s ,
V = S W : ( 3 . 4 )
-
8/3/2019 Giles Stability of ODE Solvers
8/26
8
T h e t r a n s f o r m a t i o n m a t r i x S , a l s o g i v e n i n A p p e n d i x B , i s d u e t o A b a r b a n e l a n d
G o t t l i e b 1 6 ] . I t h a s t h e p r o p e r t y t h a t t h e c o r r e s p o n d i n g t r a n s f o r m e d e q u a t i o n s ,
@ W
@ t
+ A
x
@ W
@ x
+ A
y
@ W
@ y
+ A
z
@ W
@ y
=
@
@ x
D
x x
@ W
@ x
+ D
x y
@ W
@ y
+ D
x z
@ W
@ z
!
+
@
@ y
D
y x
@ W
@ x
+ D
y y
@ W
@ y
+ D
y z
@ W
@ z
!
+
@
@ z
D
z x
@ W
@ x
+ D
z y
@ W
@ y
+ D
z z
@ W
@ z
!
;
( 3 . 5 )
a r e s u c h t h a t t h e m a t r i c e s A
x
; A
y
; A
z
a n d t h e c o m b i n e d d i s s i p a t i o n m a t r i x
0
B
B
B
@
D
x x
D
x y
D
x z
D
y x
D
y y
D
y z
D
z x
D
z y
D
z z
1
C
C
C
A
a r e a l l s y m m e t r i c . T h e m a t r i c e s a r e l i s t e d i n d e t a i l i n A p p e n d i x B a n d i t i s a l s o
p r o v e d t h a t t h e c o m b i n e d d i s s i p a t i o n m a t r i x i s p o s i t i v e s e m i { d e n i t e p r o v i d e d
t h a t 0 , 2 + 3 0 a n d k 0 . T h e s e t h r e e c o n d i t i o n s a r e s a t i s e d b y t h e
l a m i n a r v i s c o s i t y c o e c i e n t s ; i t w i l l b e a s s u m e d t h a t t h e y a r e a l s o s a t i s e d b y
t h e c o e c i e n t s d e n e d b y t h e t u r b u l e n c e m o d e l l i n g .
T h e p e r t u r b a t i o n ` e n e r g y ' i s d e n e d a s
E =
Z
1
2
W
W d V ; ( 3 . 6 )
w h e r e W
a g a i n d e n o t e s t h e H e r m i t i a n o f W , a n d i t s r a t e o f c h a n g e i s
d E
d t
=
Z
1
2
W
@ W
@ t
+
@ W
@ t
W
!
d V =
Z
1
2
W
@ W
@ t
+
W
@ W
@ t
!
!
d V : ( 3 . 7 )
U s i n g t h e f a c t t h a t A
x
i s r e a l a n d s y m m e t r i c , a n d t h e n i n t e g r a t i n g b y p a r t s
u s i n g t h e p e r i o d i c b o u n d a r y c o n d i t i o n s ,
Z
W
A
x
@ W
@ x
!
d V =
Z
@ W
@ x
A
x
W d V =
Z
W
A
x
@ W
@ x
d V
= )
Z
W
A
x
@ W
@ x
+
W
A
x
@ W
@ x
!
d V = 0 : ( 3 . 8 )
S i m i l a r l y ,
Z
W
A
y
@ W
@ y
+
W
A
y
@ W
@ y
!
d V = 0 ;
Z
W
A
z
@ W
@ z
+
W
A
z
@ W
@ z
!
d V = 0 : ( 3 . 9 )
-
8/3/2019 Giles Stability of ODE Solvers
9/26
9
I n t e g r a t i n g t h e d i u s i o n t e r m s b y p a r t s a n d n o t i n g t h a t
2
6
6
4
0
B
B
@
@ W
@ x
@ W
@ y
@ W
@ z
1
C
C
A
0
B
B
@
D
x x
D
x y
D
x z
D
y x
D
y y
D
y z
D
z x
D
z y
D
z z
1
C
C
A
0
B
B
@
@ W
@ x
@ W
@ y
@ W
@ z
1
C
C
A
3
7
7
5
=
0
B
B
@
@ W
@ x
@ W
@ y
@ W
@ z
1
C
C
A
0
B
B
@
D
x x
D
x y
D
x z
D
y x
D
y y
D
y z
D
z x
D
z y
D
z z
1
C
C
A
0
B
B
@
@ W
@ x
@ W
@ y
@ W
@ z
1
C
C
A
( 3 . 1 0 )
s i n c e t h e c o m b i n e d d i s s i p a t i o n m a t r i x i s r e a l a n d s y m m e t r i c , y i e l d s t h e n a l
r e s u l t ,
d E
d t
=
Z
0
B
B
B
@
@ W
@ x
@ W
@ y
@ W
@ z
1
C
C
C
A
0
B
B
B
@
D
x x
D
x y
D
x z
D
y x
D
y y
D
y z
D
z x
D
z y
D
z z
1
C
C
C
A
0
B
B
B
@
@ W
@ x
@ W
@ y
@ W
@ z
1
C
C
C
A
d V : ( 3 . 1 1 )
S i n c e t h e c o m b i n e d d i s s i p a t i o n m a t r i x i s p o s i t i v e s e m i { d e n i t e , t h e p e r t u r b a t i o n
` e n e r g y ' i s n o n - i n c r e a s i n g t h e r e b y p r o v i n g s t a b i l i t y i n t h e e n e r g y n o r m .
4 S e m i { d i s c r e t e e q u a t i o n s
U s i n g a n u n s t r u c t u r e d g r i d o f t e t r a h e d r a l c e l l s w i t h W d e n e d b y l i n e a r i n t e r p o -
l a t i o n b e t w e e n n o d a l v a l u e s , t h e s t a n d a r d G a l e r k i n s p a t i a l d i s c r e t i s a t i o n o f t h e
t r a n s f o r m e d p . d . e . i s
M
G
d W
d t
+ A W = D W ; ( 4 . 1 )
w h e r e
m
G
i j
=
Z
N
i
N
j
I d V
a
i j
=
Z
N
i
A
x
@ N
j
@ x
+ A
y
@ N
j
@ y
+ A
z
@ N
j
@ x
!
d V
d
i j
=
Z
D
x x
@ N
i
@ x
@ N
j
@ x
+ D
x y
@ N
i
@ x
@ N
j
@ y
+ D
x z
@ N
i
@ x
@ N
j
@ z
( 4 . 2 )
+ D
y x
@ N
i
@ y
@ N
j
@ x
+ D
y y
@ N
i
@ y
@ N
j
@ y
+ D
y z
@ N
i
@ y
@ N
j
@ z
+ D
z x
@ N
i
@ z
@ N
j
@ x
+ D
z y
@ N
i
@ z
@ N
j
@ y
+ D
z z
@ N
i
@ z
@ N
j
@ z
!
d V :
T h e v e c t o r W o f d i s c r e t e n o d a l v a r i a b l e s h a s 5 - c o m p o n e n t s u b v e c t o r s w
i
a t e a c h
n o d e i . F o r a p a r t i c u l a r p a i r o f n o d e s i ; j , m
G
i j
, a
i j
a n d d
i j
d e n o t e t h e c o r r e -
s p o n d i n g 5 5 s u b m a t r i c e s o f t h e m a t r i c e s M
G
, A a n d D , r e s p e c t i v e l y . N
i
i s t h e
p i e c e w i s e l i n e a r f u n c t i o n w h i c h i s e q u a l t o u n i t y a t n o d e i a n d z e r o a t a l l o t h e r
n o d e s , a n d t h e v i s c o s i t y p a r a m e t e r s , a n d k w i t h i n t h e d i s s i p a t i o n m a t r i c e s
a r e d e n e d t o b e c o n s t a n t o n e a c h t e t r a h e d r o n .
-
8/3/2019 Giles Stability of ODE Solvers
10/26
1 0
A s t a n d a r d m o d i c a t i o n i s t o ` m a s s - l u m p ' t h e m a t r i x M
G
, t u r n i n g i t i n t o a
d i a g o n a l m a t r i x M w i t h
m
i i
=
X
j
m
G
i j
=
Z
N
i
I d V = V
i
I ; ( 4 . 3 )
w h e r e V
i
i s t h e v o l u m e a s s o c i a t e d w i t h n o d e i , d e n e d a s o n e q u a r t e r o f t h e s u m
o f t h e v o l u m e s o f t h e s u r r o u n d i n g t e t r a h e d r a .
A n o t h e r s t a n d a r d m o d i c a t i o n w h e n i n t e r e s t e d i n a c c e l e r a t i n g c o n v e r g e n c e
t o a s t e a d y - s t a t e s o l u t i o n , i s t o p r e c o n d i t i o n t h e ` m a s s - l u m p e d ' m a t r i x s o t h a t
m
i i
=
V
i
t
i
I : ( 4 . 4 )
T h e o b j e c t i v e o f t h i s p r e c o n d i t i o n i n g i s t o u s e l o c a l t i m e s t e p s , t
i
, w h i c h a r e
l a r g e r i n l a r g e c o m p u t a t i o n a l c e l l s t h a n i n s m a l l o n e s , s o t h a t f e w e r i t e r a t i o n s
o f t h e f u l l y - d i s c r e t e e q u a t i o n s w i l l b e n e e d e d t o c o n v e r g e t o t h e s t e a d y - s t a t e
s o l u t i o n t o w i t h i n s o m e s p e c i e d t o l e r a n c e .
T h e m a t r i x A i s a n t i s y m m e t r i c s i n c e , i n t e g r a t i n g b y p a r t s ,
a
i j
=
Z
A
x
@ N
i
@ x
N
j
+ A
y
@ N
i
@ y
N
j
+ A
z
@ N
i
@ z
N
j
d V
=
Z
N
j
( A
T
x
@ N
i
@ x
+ A
T
y
@ N
i
@ y
+ A
T
z
@ N
i
@ z
) d V
= ( a
j i
)
T
: ( 4 . 5 )
T h e m a t r i x D i s c l e a r l y s y m m e t r i c . F u r t h e r m o r e , f o r a n y v e c t o r W ,
W
D W =
Z
0
B
B
@
@ W
@ x
@ W
@ y
@ W
@ z
1
C
C
A
0
B
B
@
D
x x
D
x y
D
x z
D
y x
D
y y
D
y z
D
z x
D
z y
D
z z
1
C
C
A
0
B
B
@
@ W
@ x
@ W
@ y
@ W
@ z
1
C
C
A
d V ; ( 4 . 6 )
w h e r e
@ W
@ x
=
X
i
@ N
i
@ x
w
i
@ W
@ y
=
X
i
@ N
i
@ y
w
i
( 4 . 7 )
@ W
@ z
=
X
i
@ N
i
@ z
w
i
:
S i n c e t h e c o m b i n e d d i s s i p a t i o n m a t r i x i s p o s i t i v e s e m i - d e n i t e , i t f o l l o w s t h e r e -
f o r e t h a t D i s a l s o p o s i t i v e s e m i - d e n i t e .
-
8/3/2019 Giles Stability of ODE Solvers
11/26
1 1
D e n i n g t h e ` e n e r g y ' f o r a r b i t r a r y c o m p l e x W a s e i t h e r E =
1
2
W
M
G
W o r
E =
1
2
W
M W , d e p e n d i n g w h e t h e r o r n o t m a s s - l u m p i n g i s u s e d ,
d E
d t
=
1
2
( W
( A + D ) W + W
( A + D )
W )
=
1
2
( W
( A + D ) W + W
( A + D ) W )
= W
D W 0 ( 4 . 8 )
a n d s o t h e e n e r g y i s n o n - i n c r e a s i n g . S i n c e b o t h M
G
a n d M a r e s y m m e t r i c a n d
p o s i t i v e d e n i t e t h i s i n t u r n i m p l i e s s t a b i l i t y f o r t h e s e m i - d i s c r e t e e q u a t i o n s .
N o t e t h a t o t h e r d i s c r e t i s a t i o n s o f t h e N a v i e r - S t o k e s e q u a t i o n s w i l l r e s u l t i n
e q u a t i o n s o f t h e f o r m ,
M
d U
d t
= C U ; ( 4 . 9 )
w h e r e M i s a s y m m e t r i c p o s i t i v e d e n i t e ` m a s s ' m a t r i x a n d C c a n b e d e c o m p o s e d
i n t o i t s s y m m e t r i c a n d a n t i { s y m m e t r i c c o m p o n e n t s ,
C = ( A + D ) ; A =
1
2
( C C
T
) ; D =
1
2
( C + C
T
) : ( 4 . 1 0 )
I n g e n e r a l A w i l l n o w c o n t a i n s o m e t e r m s d u e t o t h e v i s c o u s d i s c r e t i s a t i o n ,
a n d D w i l l c o n t a i n s o m e t e r m s d u e t o t h e n u m e r i c a l s m o o t h i n g a s s o c i a t e d w i t h
t h e c o n v e c t i v e d i s c r e t i s a t i o n . D m u s t s t i l l b e p o s i t i v e s e m i { d e n i t e t o e n s u r e
s t a b i l i t y .
5 F u l l y d i s c r e t e e q u a t i o n s
U s i n g R u n g e - K u t t a t i m e i n t e g r a t i o n t h e f u l l y d i s c r e t e e q u a t i o n s u s i n g o n e o f t h e
t w o d i a g o n a l m a s s m a t r i c e s a r e
W
( n + 1 )
= L ( k M
1
C ) W
( n )
( 5 . 1 )
w h e r e L ( z ) i s t h e R u n g e - K u t t a p o l y n o m i a l w i t h s t a b i l i t y r e g i o n S a s d e n e d i n
S e c t i o n 2 a n d C = ( A + D ) . A s e x p l a i n e d i n S e c t i o n 2 , s u c i e n t c o n d i t i o n s f o r
a l g e b r a i c a n d g e n e r a l i s e d s t a b i l i t y a r e t h a t
( k M
1 = 2
C M
1 = 2
) S ( 5 . 2 )
w h e r e
( k M
1 = 2
C M
1 = 2
) =
(
k
W
M
1 = 2
C M
1 = 2
W
W
W
: W 6= 0
)
: ( 5 . 3 )
F o r u n s t e a d y c a l c u l a t i o n s w i t h t h e d i a g o n a l m a s s - l u m p e d m a t r i x , t h e a i m
i s s i m p l y t o n d t h e l a r g e s t k s u c h t h a t t h e c o n s t r a i n t , E q . ( 5 . 2 ) , i s s a t i s e d .
F o r s t e a d y - s t a t e c a l c u l a t i o n s u s i n g t h e p r e - c o n d i t i o n e d m a s s m a t r i x , o n e u s e s
-
8/3/2019 Giles Stability of ODE Solvers
12/26
1 2
a p s e u d o - t i m e s t e p k = 1 a n d t h e n t h e o b j e c t i v e i s t o d e n e t h e l o c a l t i m e s t e p s
t
i
t o b e a s l a r g e a s p o s s i b l e , a g a i n s u b j e c t t o t h e s u c i e n t s t a b i l i t y c o n s t r a i n t ,
E q . ( 5 . 2 ) .
T h e d i c u l t y i s t h a t d i r e c t e v a l u a t i o n o f ( k M
1 = 2
C M
1 = 2
) i s n o t p o s s i b l e .
I n s t e a d , a b o u n d i n g s e t i s c o n s t r u c t e d t o e n c l o s e t h e n u m e r i c a l r a n g e a n d s u f -
c i e n t c o n d i t i o n s a r e d e t e r m i n e d f o r t h i s b o u n d i n g s e t t o l i e i n s i d e S . T h e r e
a r e t w o c h o i c e s o f b o u n d i n g s e t w h i c h a r e r e l a t i v e l y e a s i l y c o n s t r u c t e d , a h a l f -
d i s k a n d a r e c t a n g l e . T h e c o n s t r u c t i o n o f t h e b o u n d i n g h a l f - d i s k s t a r t s w i t h t h e
o b s e r v a t i o n t h a t
W
M
1 = 2
C M
1 = 2
W
W
W
k M
1 = 2
C M
1 = 2
k : ( 5 . 4 )
L e t t h e v a r i a b l e r b e d e n e d b y
r = m a x
i
8
>
:
V
i
; m a s s - l u m p e d m a t r i x
V
i
t
i
; p r e c o n d i t i o n e d m a s s - l u m p e d m a t r i x
( 5 . 6 )
C o n s i d e r i n g a n a r b i t r a r y v e c t o r V , w i t h s u b v e c t o r v
i
a t e a c h n o d e i ,
kM
1 = 2
C M
1 = 2
V
k
2
=
X
i
m
1
i
X
j
c
i j
( m
1 = 2
j
v
j
)
2
X
i ; j ; k
m
1
i
k c
i j
k m
1 = 2
j
k v
j
k k c
i k
k m
1 = 2
k
k v
k
k
X
i ; j ; k
m
1
i
m
1
j
k v
j
k
2
k c
i j
k k c
i k
k
r
X
i ; j
m
1
j
kv
j
k
2
kc
i j
k
r
2
k V k
2
;
= ) k M
1 = 2
C M
1 = 2
k r : ( 5 . 7 )
T h e t h i r d l i n e i n t h e a b o v e d e r i v a t i o n u s e s t h e i n e q u a l i t y
m
1 = 2
j
k v
j
k m
1 = 2
k
k v
k
k
1
2
m
1
j
k v
j
k
2
+ m
1
k
k v
k
k
2
; ( 5 . 8 )
f o l l o w e d b y a n i n t e r c h a n g e o f s u b s c r i p t s t o r e p l a c e m
1
k
k v
k
k
2
b y m
1
j
k v
j
k
2
g i v e n
t h a t k c
i j
k k c
i k
k i s s y m m e t r i c i n j a n d k .
A l s o , f o r a n a r b i t r a r y v e c t o r W ,
W
C W + ( W
C W )
= W
( C + C
) W = 2 W
D W 0 ( 5 . 9 )
-
8/3/2019 Giles Stability of ODE Solvers
13/26
1 3
a n d s o t h e r e a l c o m p o n e n t o f W
C W m u s t b e z e r o o r n e g a t i v e . C o m b i n e d w i t h
t h e p r e v i o u s b o u n d , t h i s m e a n s t h a t ( k M
1 = 2
C M
1 = 2
) m u s t t h e r e f o r e l i e i n t h e
h a l f - d i s k
fz = x + i y : x
0 ;
jz
j k r
g:
F o r u n s t e a d y c a l c u l a t i o n s , t h e n e c e s s a r y a n d s u c i e n t c o n d i t i o n f o r t h e h a l f -
d i s k t o l i e i n s i d e S , a n d t h u s a s u c i e n t c o n d i t i o n f o r a l g e b r a i c a n d g e n e r a l i s e d
s t a b i l i t y i s
k r r
c
; ( 5 . 1 0 )
w h e r e r
c
i s t h e r a d i u s o f t h e h a l f - d i s k i n s c r i b i n g S , a s d e n e d a n d i l l u s t r a t e d i n
A p p e n d i x A .
F o r p r e c o n d i t i o n e d s t e a d y - s t a t e c a l c u l a t i o n s w i t h l o c a l t i m e s t e p s , k = 1 a n d
s o t h e l a r g e s t v a l u e f o r r f o r w h i c h t h e h a l f - d i s k l i e s i n s i d e S i s r
c
. F o r e a c h n o d e
i , t
i
i s t h e n m a x i m i s e d s u b j e c t t o t h e d e n i t i o n o f r b y
t
i
=
r
c
V
i
m a x
8
-
8/3/2019 Giles Stability of ODE Solvers
14/26
1 4
F o r u n s t e a d y c a l c u l a t i o n s , a s u c i e n t s t a b i l i t y l i m i t i s o b t a i n e d b y r e q u i r i n g
t h a t R S . I f t h e b o u n d a r y o f S c a n b e r e p r e s e n t e d b y z = r e x p ( i ) w i t h r ( )
b e i n g a s i n g l e - v a l u e d f u n c t i o n f o r
2
3
2
t h e n t h i s c a n w r i t t e n a s
k
q
x
2
d
+ y
2
a
r ( ) ; t a n ( ) =
y
a
x
d
: ( 5 . 1 5 )
F o r p r e c o n d i t i o n e d s t e a d y - s t a t e c a l c u l a t i o n s , w e a g a i n l e t k = 1 a n d c a n t h e n
c h o o s e a n y r e c t a n g l e R w h i c h i n s c r i b e s S . A p p e n d i x A s h o w s t h e p a r t i c u l a r
e x a m p l e o f a h a l f - s q u a r e f o r w h i c h x
d
= y
a
. T h e m a x i m u m l o c a l t i m e s t e p t
i
s u b j e c t t o t h e d e n i t i o n s o f b o t h x
D
a n d y
A
i s t h e n
t
i
= m i n
8
>
>
>
>
>
:
x
d
V
i
m a x f
X
j
k d
i j
k ;
X
j
k d
j i
k g
;
y
a
V
i
X
j
k a
i j
k
9
>
>
>
=
>
>
>
;
: ( 5 . 1 6 )
T h e n a l f o r m o f t h e s t a b i l i t y l i m i t i s a g a i n o b t a i n e d b y u s i n g t h e r e s u l t s o f
A p p e n d i x C t o e v a l u a t e k a
i j
k a n d p l a c e a n u p p e r b o u n d o n k d
i j
k a n d k d
j i
k .
I t i s d i c u l t t o p r e d i c t a p r i o r i w h i c h b o u n d i n g s e t w i l l g i v e t h e l e a s t r e s t r i c -
t i v e s u c i e n t s t a b i l i t y c o n d i t i o n s . I t d e p e n d s i n p a r t o n t h e p a r t i c u l a r R u n g e -
K u t t a m e t h o d w h i c h i s u s e d . A p p e n d i x A s h o w s t h a t f o r s o m e m e t h o d s t h e
i n s c r i b i n g h a l f - d i s k a l m o s t c o n t a i n s t h e i n s c r i b i n g h a l f - s q u a r e a n d o t h e r r e c t a n -
g l e s l y i n g i n s i d e S ; i n t h i s c a s e t h e h a l f - d i s k s u c i e n t s t a b i l i t y c o n d i t i o n s w i l l
p r o b a b l y b e l e s s r e s t r i c t i v e . W i t h o t h e r m e t h o d s , t h e h a l f - s q u a r e a l m o s t c o n -
t a i n s t h e i n s c r i b i n g h a l f - d i s k a n d f o r t h e s e t h e h a l f - s q u a r e s t a b i l i t y c o n d i t i o n s
w i l l p r o b a b l y b e l e s s r e s t r i c t i v e .
I n e i t h e r c a s e , t h e t i m e s t e p l i m i t s a r e s u c i e n t c o n d i t i o n s f o r a l g e b r a i c a n d
g e n e r a l i s e d s t a b i l i t y , b u t w i l l a l m o s t c e r t a i n l y n o t b e n e c e s s a r y . T h i s p o i n t i s
w e l l i l l u s t r a t e d b y c o n s i d e r i n g t h e s t a b i l i t y l i m i t s i n t h e h y p e r b o l i c a n d p a r a b o l i c
e x t r e m e s . I n t h e h y p e r b o l i c c a s e i n w h i c h D = 0 , c o r r e s p o n d i n g t o a d i s c r e t i s a t i o n
o f t h e i n v i s c i d E u l e r e q u a t i o n s , t h e b e s t s t a b i l i t y c o n d i t i o n o b t a i n e d f r o m t h e
a n a l y s i s i n t h i s p a p e r c o m e s f r o m a n e x t r e m e l i m i t o f t h e r e c t a n g u l a r b o u n d i n g
s e t . S e t t i n g x
d
= 0 a n d y
a
= r
a
, w h e r e r
a
i s d e n e d i n A p p e n d i x A t o b e t h e
l e n g t h o f t h e p o s i t i v e i m a g i n a r y a x i s l y i n g i n s i d e t h e s t a b i l i t y r e g i o n S , g i v e s t h e
l o c a l t i m e s t e p s t a b i l i t y l i m i t
t
i
r
a
V
i
X
j
k a
i j
k
: ( 5 . 1 7 )
A s e x p l a i n e d i n S e c t i o n 2 , b e c a u s e A i s a n t i - s y m m e t r i c , t h i s w i l l a l s o e n s u r e t h a t
t h e ` e n e r g y ' W
M W w i l l b e n o n - i n c r e a s i n g . T h i s r e p r e s e n t s a g e n e r a l i s a t i o n t o
a r b i t r a r y R u n g e - K u t t a m e t h o d s o f t h e e a r l i e r e n e r g y a n a l y s i s b y G i l e s f o r t w o
s p e c i c R u n g e - K u t t a m e t h o d s 6 ] . I n t h a t e a r l i e r w o r k , t h e s u c i e n t s t a b i l i t y
-
8/3/2019 Giles Stability of ODE Solvers
15/26
1 5
l i m i t d e r i v e d b y e n e r g y a n a l y s i s w a s c o m p a r e d t o t h e n e c e s s a r y a n d s u c i e n t
F o u r i e r s t a b i l i t y l i m i t f o r a u n i f o r m m e s h . A t w o r s t , w h e n t h e M a c h n u m b e r i s
z e r o a n d t h e g r i d s p a c i n g i s t h e s a m e i n e a c h d i r e c t i o n , t h e t i m e s t e p l i m i t f r o m
t h e e n e r g y a n a l y s i s i s 4 0 % l e s s t h a n t h a t f r o m t h e F o u r i e r a n a l y s i s . A t b e s t ,
a t h i g h M a c h n u m b e r s o r o n s t r e t c h e d g r i d s , t h e t w o t i m e s t e p l i m i t s a r e a l m o s t
e q u a l .
I n t h e p a r a b o l i c c a s e i n w h i c h A = 0 , w h i c h w o u l d c o r r e s p o n d t o a s i m p l e
d i u s i o n p r o b l e m , o r t h e i n c o m p r e s s i b l e N a v i e r - S t o k e s e q u a t i o n s a t a v e r y l o w
R e y n o l d s n u m b e r , t h e c o r r e s p o n d i n g s t a b i l i t y l i m i t c o m e s f r o m s e t t i n g y
a
= 0
a n d x
d
= r
d
, w h e r e r
d
i s d e n e d i n A p p e n d i x A t o b e t h e l e n g t h o f t h e n e g a t i v e
r e a l a x i s l y i n g i n s i d e t h e s t a b i l i t y r e g i o n S . T h e s u c i e n t t i m e s t e p s t a b i l i t y l i m i t
i s t h e n
t
i
r
d
V
i
m a x f
X
j
k d
i j
k ;
X
j
k d
j i
k g
: ( 5 . 1 8 )
A n a d h o c t i m e s t e p l i m i t w h i c h c o u l d p e r h a p s b e u s e d c o m e s f r o m c o m b i n i n g
t h e s e l a s t t w o l i m i t s t o g i v e
1
t
2
i
=
1
t
2
a i
+
1
t
2
d i
; ( 5 . 1 9 )
w h e r e t
a i
a n d t
d i
a r e t h e h y p e r b o l i c a n d p a r a b o l i c t i m e s t e p l i m i t s g i v e n b y
E q . ( 5 . 1 7 ) a n d E q . ( 5 . 1 8 ) . I t i s p o s s i b l e t o r i g o r o u s l y j u s t i f y t h i s c o m b i n e d l i m i t
i f t h e g r i d i s u n i f o r m , t h e v i s c o u s c o e c i e n t s a r e u n i f o r m , a n d t h e s t a b i l i t y r e g i o n
S c o n t a i n s t h e h a l f - e l l i p s e p a s s i n g t h r o u g h t h e p o i n t s i r
a
; r
d
; i r
a
. H o w e v e r , i n
g e n e r a l t h i s t i m e s t e p f o r m u l a t i o n c a n n o t b e j u s t i e d a n d s o s h o u l d o n l y b e u s e d
w i t h c a r e . I t s a d v a n t a g e o v e r t h e r i g o r o u s s t a b i l i t y l i m i t s u s i n g t h e h a l f - d i s k a n d
t h e r e c t a n g l e i s t h a t i t w i l l g i v e a l a r g e r t i m e s t e p w h i c h i s h o p e f u l l y s t i l l s t a b l e .
6 C o n c l u s i o n s
T h i s p a p e r h a s a n a l y s e d t h e s t a b i l i t y o f o n e c l a s s o f d i s c r e t i s a t i o n s o f t h e N a v i e r -
S t o k e s e q u a t i o n s o n a t e t r a h e d r a l g r i d . T h e s u c i e n t s t a b i l i t y l i m i t s f o r b o t h
g l o b a l a n d l o c a l t i m e s t e p s a r e b a s e d o n r e c e n t a d v a n c e s i n n u m e r i c a l a n a l y s i s .
A d d i t i o n a l r e s e a r c h i s n e e d e d t o v a l i d a t e t h e u s e f u l n e s s o f t h e s e l i m i t s , w h e t h e r
t h e y a r e c l o s e e n o u g h t o t h e n e c e s s a r y s t a b i l i t y l i m i t s t o b e a v a l u a b l e p r a c t i c a l
c r i t e r i o n o n w h i c h t o b a s e t h e t i m e s t e p i n a c t u a l c o m p u t a t i o n s .
A n o t h e r d i r e c t i o n f o r f u t u r e r e s e a r c h i s t h e e x t e n s i o n o f t h e a n a l y s i s t o o t h e r
d i s c r e t i s a t i o n s . U p w i n d a p p r o x i m a t i o n s o f t h e i n v i s c i d u x e s w o u l d b e a p a r t i c u -
l a r l y i n t e r e s t i n g t o p i c f o r s t u d y . A s i n d i c a t e d a t t h e e n d o f S e c t i o n 4 , t h i s w o u l d
c h a n g e t h e d e n i t i o n o f t h e d i s s i p a t i o n m a t r i x D , b u t t h e o v e r a l l a p p r o a c h t o
t h e s t a b i l i t y a n a l y s i s w o u l d r e m a i n v a l i d . I t m a y a l s o b e p o s s i b l e t o i n v e s t i g a t e
t h e s t a b i l i t y o f d i e r e n t N a v i e r - S t o k e s b o u n d a r y c o n d i t i o n i m p l e m e n t a t i o n s b y
i n c o r p o r a t i n g t h e s e w i t h i n t h e c o u p l e d s y s t e m o f o . d . e . ' s .
-
8/3/2019 Giles Stability of ODE Solvers
16/26
1 6
A c k n o w l e d g e m e n t s
I w i s h t o t h a n k L a r r y W i g t o n f o r s t i m u l a t i n g t h i s r e s e a r c h a n d E l i T u r k e l , E i t a n
T a d m o r , B i l l M o r t o n , E n d r e S u l i , N i c k T r e f e t h e n a n d S a t i s h R e d d y f o r t h e i r
h e l p w i t h t h e n u m e r i c a l a n a l y s i s l i t e r a t u r e o n t h e s t a b i l i t y o f s y s t e m s o f o . d . e . ' s
w i t h n o n - n o r m a l m a t r i c e s , a n d f o r t h e i r v a l u a b l e c o m m e n t s o n t h e p a p e r . T h e
n a n c i a l s u p p o r t o f R o l l s - R o y c e p l c , D T I a n d E P S R C i s g r a t e f u l l y a c k n o w l e d g e d .
R e f e r e n c e s
1 ] L . W i g t o n . P e r s o n a l c o m m u n i c a t i o n , 1 9 9 4 .
2 ] N . P . W e a t h e r i l l , O . H a s s a n , M . J . M a r c h a n t , a n d D . L . M a r c u m . A d a p t i v e
i n v i s c i d o w s o l u t i o n s f o r a e r o s p a c e g e o m e t r i e s o n e c i e n t l y g e n e r a t e d u n -
s t r u c t u r e d t e t r a h e d r a l m e s h e s . A I A A P a p e r 9 3 - 3 3 9 0 , 1 9 9 3 .
3 ] J . P e r a i r e , J . P e i r o , a n d K . M o r g a n . F i n i t e e l e m e n t m u l t i g r i d s o l u t i o n o f
E u l e r o w s p a s t i n s t a l l e d a e r o - e n g i n e s . C o m p u t . M e c h . , 1 1 : 4 3 3 { 4 5 1 , 1 9 9 3 .
4 ] R . D . R a u s c h , J . T . B a t i n a , a n d H . T . Y . Y a n g . T h r e e - d i m e n s i o n a l t i m e -
m a r c h i n g a e r o e l a s t i c a n a l y s e s u s i n g a n u n s t r u c t u r e d - g r i d E u l e r m e t h o d .
A I A A J . , 3 1 ( 9 ) : 1 6 2 6 { 1 6 3 3 , 1 9 9 3 .
5 ] P . C r u m p t o n a n d M . B . G i l e s . A i r c r a f t c o m p u t a t i o n s u s i n g m u l t i g r i d a n d a n
u n s t r u c t u r e d p a r a l l e l l i b r a r y . A I A A P a p e r 9 5 - 0 2 1 0 , 1 9 9 5 .
6 ] M . B . G i l e s . E n e r g y s t a b i l i t y a n a l y s i s o f m u l t i - s t e p m e t h o d s o n u n s t r u c t u r e d
m e s h e s . T e c h n i c a l R e p o r t T R - 8 7 - 1 , M I T D e p t . o f A e r o . a n d A s t r o . , 1 9 8 7 .
7 ] R . D . R i c h t m y e r a n d K . W . M o r t o n . D i e r e n c e M e t h o d s f o r I n i t i a l V a l u e
P r o b l e m s . W i l e y - I n t e r s c i e n c e , 2 n d e d i t i o n , 1 9 6 7 .
8 ] S . C . R e d d y a n d L . N . T r e f e t h e n . S t a b i l i t y o f t h e m e t h o d o f l i n e s . N u m e r .
M a t h . , 6 2 : 2 3 5 { 2 6 7 , 1 9 9 2 .
9 ] H . O . K r e i s s a n d L . W u . O n t h e s t a b i l i t y d e n i t i o n o f d i e r e n c e a p p r o x i m a -
t i o n s f o r t h e i n i t i a l b o u n d a r y v a l u e p r o b l e m . A p p l . N u m . M a t h . , 1 2 : 2 1 3 { 2 2 7 ,
1 9 9 3 .
1 0 ] J . L . M . v a n D o r s s e l a e r , J . F . B K r a a i j e v a n g e r , a n d M . N . S p i j k e r . L i n e a r s t a -
b i l i t y a n a l y s i s i n t h e n u m e r i c a l s o l u t i o n o f i n i t i a l v a l u e p r o b l e m s . A c t a
N u m e r i c a , p a g e s 1 9 9 { 2 3 7 , 1 9 9 3 .
1 1 ] J . F . B . M . K r a a i j e v a n g e r , H . W . J . L e n f e r i n k , a n d M . N . S p i j k e r . S t e p s i z e r e -
s t r i c t i o n s f o r s t a b i l i t y i n t h e n u m e r i c a l s o l u t i o n o f o r d i n a r y a n d p a r t i a l d i f -
f e r e n t i a l e q u a t i o n s . J . C o m p u t . A p p l . M a t h . , 2 0 : 6 7 { 8 1 , N o v 1 9 8 7 .
-
8/3/2019 Giles Stability of ODE Solvers
17/26
1 7
1 2 ] H . W . J . L e n f e r i n k a n d M . N . S p i j k e r . O n t h e u s e o f s t a b i l i t y r e g i o n s i n t h e
n u m e r i c a l a n a l y s i s o f i n i t i a l v a l u e p r o b l e m s . M a t h . C o m p . , 5 7 ( 1 9 5 ) : 2 2 1 { 2 3 7 ,
1 9 9 1 .
1 3 ] S . C . R e d d y a n d L . N . T r e f e t h e n . L a x - s t a b i l i t y o f f u l l y d i s c r e t e s p e c t r a l m e t h -
o d s v i a s t a b i l i t y r e g i o n s a n d p s e u d o - e i g e n v a l u e s . C o m p u t . M e t h o d s A p p l .
M e c h . E n g r g . , 8 0 : 1 4 7 { 1 6 4 , 1 9 9 0 .
1 4 ] S . C . R e d d y . P s e u d o s p e c t r a o f O p e r a t o r s a n d D i s c r e t i z a t i o n M a t r i c e s a n d a n
A p p l i c a t i o n t o S t a b i l i t y o f t h e M e t h o d o f L i n e s . P h D t h e s i s , M a s s a c h u s e t t s
I n s t i t u t e o f T e c h n o l o g y , C a m b r i d g e , M a s s a c h u s e t t s 0 2 1 3 9 , 1 9 9 1 . N u m e r i c a l
A n a l y s i s R e p o r t 9 1 - 4 .
1 5 ] C . L u b i c h a n d O . N e v a n l i n n a . O n r e s o l v e n t c o n d i t i o n s a n d s t a b i l i t y e s t i -
m a t e s . B I T , 3 1 : 2 9 3 { 3 1 3 , 1 9 9 1 .
1 6 ] S . A b a r b a n e l a n d D . G o t t l i e b . O p t i m a l t i m e s p l i t t i n g f o r t w o - a n d t h r e e -
d i m e n s i o n a l N a v i e r - S t o k e s e q u a t i o n s w i t h m i x e d d e r i v a t i v e s . J o u r n a l o f
C o m p u t a t i o n a l P h y s i c s , 3 5 : 1 { 3 3 , 1 9 8 1 .
-
8/3/2019 Giles Stability of ODE Solvers
18/26
1 8
A p p e n d i x A R u n g e - K u t t a s t a b i l i t y c u r v e s
A n e x a m p l e o f a R u n g e - K u t t a t y p e o f a p p r o x i m a t i o n o f t h e o . d . e .
d u
d t
= u ; ( A . 1 )
i s t h e f o l l o w i n g t w o - s t a g e p r e d i c t o r - c o r r e c t o r m e t h o d ,
u
( 1 )
= u
n
+ k u
n
u
n + 1
= u
n
+ k u
( 1 )
: ( A . 2 )
C o m b i n i n g t h e s e t w o e q u a t i o n s g i v e s
u
n + 1
= L ( k ) u
n
; ( A . 3 )
w h e r e t h e R u n g e - K u t t a p o l y n o m i a l f u n c t i o n i s L ( z ) = 1 + z + z
2
. F i g u r e 1 a )
s h o w s t h e s t a b i l i t y r e g i o n S w i t h i n w h i c h j L j 1 . I t a l s o s h o w s t h e l a r g e s t
h a l f - d i s k ,
f z = x + i y : x 0 ; j z j r
c
g ;
a n d t h e l a r g e s t h a l f - s q u a r e ,
(
z = x + i y :
r
s
p
2
x 0 ; j y j
r
s
p
2
)
;
w h i c h l i e i n s i d e S . I f t h e b o u n d a r y o f S i s d e n e d a s z = r e x p ( i ) t h e n r
c
a n d
r
s
c a n b e d e n e d a s
r
c
= m i n
2
3
2
r ( ) ; r
s
= r (
3
4
) : ( A . 4 )
T h e v a l u e s o f r
c
a n d r
s
a r e l i s t e d t o t h e r i g h t o f t h e g u r e a l o n g w i t h t h o s e o f
t w o o t h e r i m p o r t a n t p a r a m e t e r s , r
a
= r (
1
2
) , w h i c h i s t h e l e n g t h o f t h e p o s i t i v e
i m a g i n a r y a x i s s e g m e n t w i t h i n S , a n d r
d
= r ( ) , w h i c h i s t h e l e n g t h o f t h e n e g a -
t i v e r e a l a x i s s e g m e n t w i t h i n S . T h e i m p o r t a n c e o f a l l f o u r o f t h e s e p a r a m e t e r s
i s d i s c u s s e d i n t h e m a i n t e x t i n S e c t i o n 5 .
F i g u r e s 1 b ) a n d 1 c ) s h o w t h e c o r r e s p o n d i n g c u r v e s a n d d a t a f o r t w o o t h e r
p o p u l a r m u l t i s t a g e i n t e g r a t i o n s c h e m e s .
-
8/3/2019 Giles Stability of ODE Solvers
19/26
1 9
0 5- 0 . 5 - 1 . 5
- 1 . 5
- 0 . 5
0 5
1 5
a ) P r e d i c t o r - c o r r e c t o r
u
( 1 )
= u
n
+ t u
n
u
n + 1
= u
n
+ t u
( 1 )
r
c
= 1 : 0
r
s
= 1 : 4 1 4
r
a
= 1 : 0
r
d
= 1 : 0
1 0- 1 . 0 - 3 . 0
- 3 . 0
- 1 . 0
1 0
3 0
b ) T h r e e - s t a g e s c h e m e
u
( 1 )
= u
n
+
1
3
t u
n
u
( 2 )
= u
n
+
1
2
t u
( 1 )
u
n + 1
= u
n
+ t u
( 2 )
r
c
= 1 : 7 3 1
r
s
= 2 : 3 7 5
r
a
= 1 : 7 3 1
r
d
= 2 : 5 1 3
1 0- 1 . 0 - 3 . 0
- 3 . 0
- 1 . 0
1 0
3 0
c ) F o u r - s t a g e s c h e m e
u
( 1 )
= u
n
+
1
4
t u
n
u
( 2 )
= u
n
+
1
3
t u
( 1 )
u
( 3 )
= u
n
+
1
2
t u
( 2 )
u
n + 1
= u
n
+ t u
( 3 )
r
c
= 2 : 6 1 6
r
s
= 2 : 7 0 4
r
a
= 2 : 8 2 8
r
d
= 2 : 7 8 5
F i g u r e 1 : S t a b i l i t y b o u n d a r y a n d i n s c r i b i n g h a l f - d i s k a n d h a l f - s q u a r e f o r t h r e e
R u n g e - K u t t a m e t h o d s
-
8/3/2019 Giles Stability of ODE Solvers
20/26
2 0
A p p e n d i x B v e c t o r s , m a t r i c e s a n d p o s i t i v i t y
S t a r t i n g w i t h t h e c o n s e r v a t i v e f o r m o f t h e N a v i e r - S t o k e s e q u a t i o n s , t h e s t a t e
v e c t o r a n d u x v e c t o r s a r e
U =
0
B
B
B
B
B
B
@
u
v
w
E
1
C
C
C
C
C
C
A
;
F
x
=
0
B
B
B
B
B
B
@
u
u
2
+ p
x x
u v
y x
u w
z x
u ( E +
p
) u
x x
v
y w
w
z x
+ q
x
1
C
C
C
C
C
C
A
F
y
=
0
B
B
B
B
B
B
@
v
u v
x y
v
2
+ p
y y
v w
z y
v ( E +
p
) u
x y
v
y y
w
z y
+ q
y
1
C
C
C
C
C
C
A
F
z
=
0
B
B
B
B
B
B
@
w
u w
x z
v w
y z
w
2
+ p
z z
w ( E +
p
) u
x z
v
y z
w
z z
+ q
z
1
C
C
C
C
C
C
A
: ( B . 1 )
; u ; v ; w ; p ; E a r e t h e d e n s i t y , t h r e e C a r t e s i a n v e l o c i t y c o m p o n e n t s , p r e s s u r e a n d
t o t a l i n t e r n a l e n e r g y , r e s p e c t i v e l y . T o c o m p l e t e t h e s y s t e m o f e q u a t i o n s r e q u i r e s
a n e q u a t i o n o f s t a t e f o r a n i d e a l g a s ,
p = R T = ( 1 ) ( E
1
2
( u
2
+ v
2
+ w
2
) ) ; ( B . 2 )
i n w h i c h R ; T ; a r e t h e g a s c o n s t a n t , t e m p e r a t u r e a n d u n i f o r m s p e c i c h e a t
r a t i o , r e s p e c t i v e l y , a s w e l l a s e q u a t i o n s d e n i n g t h e h e a t u x e s ,
q
x
= k
@ T
@ x
; q
y
= k
@ T
@ y
; q
z
= k
@ T
@ z
; ( B . 3 )
-
8/3/2019 Giles Stability of ODE Solvers
21/26
2 1
a n d t h e v i s c o u s s t r e s s t e r m s ,
x x
= 2
@ u
@ x
+
@ u
@ x
+
@ v
@ y
+
@ w
@ z
!
;
x y
=
y x
=
@ u
@ y
+
@ v
@ x
!
;
y y
= 2
@ v
@ y
+
@ u
@ x
+
@ v
@ y
+
@ w
@ z
!
;
x z
=
z x
=
@ u
@ z
+
@ w
@ x
!
;
z z
= 2
@ w
@ z
+
@ u
@ x
+
@ v
@ y
+
@ w
@ z
!
;
y z
=
z y
=
@ v
@ z
+
@ w
@ y
!
: ( B . 4 )
T h e t r a n s f o r m a t i o n f r o m c o n s e r v a t i v e t o p r i m i t i v e v a r i a b l e s , ( u v w p )
T
, i s
a c c o m p l i s h e d b y t h e m a t r i x
R =
0
B
B
B
B
B
B
@
1 0 0 0 0
u 0 0 0
v 0 0 0
w 0 0 0
u
2
+ v
2
+ w
2
2
u v w
1
1
1
C
C
C
C
C
C
A
: ( B . 5 )
T h e l i n e a r i s e d , t r a n s f o r m e d e q u a t i o n s a r e
@ V
@ t
+ A
0
x
@ V
@ x
+ A
0
y
@ V
@ y
+ A
0
z
@ V
@ z
=
@
@ x
D
0
x x
@ V
@ x
+ D
0
x y
@ V
@ y
+ D
0
x z
@ V
@ z
!
+
@
@ y
D
0
y x
@ V
@ x
+ D
0
y y
@ V
@ y
+ D
0
y z
@ V
@ z
!
+
@
@ z
D
0
z x
@ U
@ x
+ D
0
z y
@ V
@ y
+ D
0
z z
@ V
@ z
!
( B . 6 )
w h e r e
A
0
x
=
0
B
B
B
B
B
B
@
u 0 0 0
0 u 0 0
1
0 0 u 0 0
0 0 0 u 0
0 p 0 0 u
1
C
C
C
C
C
C
A
; A
0
y
=
0
B
B
B
B
B
B
@
v 0 0 0
0 v 0 0 0
0 0 v 0
1
0 0 0 v 0
0 0 p 0 v
1
C
C
C
C
C
C
A
A
0
z
=
0
B
B
B
B
B
B
@
w 0 0 0
0 w 0 0 0
0 0 w 0 0
0 0 0 w
1
0 0 0 p w
1
C
C
C
C
C
C
A
( B . 7 )
-
8/3/2019 Giles Stability of ODE Solvers
22/26
2 2
a n d
D
0
x x
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0
2 +
0 0 0
0 0
0 0
0 0 0
0
p
P r
2
0 0 0
P r
1
C
C
C
C
C
C
C
C
A
; D
0
x y
= D
0 T
y x
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0 0
0 0
0
0 0 0
0 0 0 0 0
0 0 0 0 0
1
C
C
C
C
C
C
C
C
A
D
0
y y
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0
0 0 0
0 0
2 +
0 0
0 0 0
0
p
P r
2
0 0 0
P r
1
C
C
C
C
C
C
C
C
A
; D
0
x z
= D
0 T
z x
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0 0 0
0
0 0 0 0 0
0
0 0 0
0 0 0 0 0
1
C
C
C
C
C
C
C
C
A
D
0
z z
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0
0 0 0
0 0
0 0
0 0 0
2 +
0
p
P r
2
0 0 0
P r
1
C
C
C
C
C
C
C
C
A
; D
0
y z
= D
0 T
z y
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0 0 0 0 0
0 0 0
0
0 0
0 0
0 0 0 0 0
1
C
C
C
C
C
C
C
C
A
( B . 8 )
T h e P r a n d t l n u m b e r i s d e n e d a s
P r =
c
p
k
=
R
( 1 ) k
; ( B . 9 )
b u t i s n o t a s s u m e d t o b e u n i f o r m s i n c e a n d k i n g e n e r a l r e p r e s e n t c o m b i n a t i o n s
o f l a m i n a r a n d t u r b u l e n t v i s c o s i t i e s , e a c h w i t h t h e i r o w n P r a n d t l n u m b e r .
T h e s e c o n d t r a n s f o r m a t i o n m a t r i x i s
S =
0
B
B
B
B
B
B
B
B
@
p
c
0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
1
p
c 0 0 0
q
1
c
1
C
C
C
C
C
C
C
C
A
( B . 1 0 )
a n d t h e t r a n s f o r m e d m a t r i c e s a r e
-
8/3/2019 Giles Stability of ODE Solvers
23/26
2 3
A
x
= S
1
A
0
x
S =
0
B
B
B
B
B
B
B
B
B
@
u
1
p
c 0 0 0
1
p
c u 0 0
q
1
c
0 0 u 0 0
0 0 0 u 0
0
q
1
c 0 0 u
1
C
C
C
C
C
C
C
C
C
A
;
A
y
= S
1
A
0
y
S =
0
B
B
B
B
B
B
B
B
B
@
v 0
1
p
c 0 0
0 v 0 0 0
1
p
c 0 v 0
q
1
c
0 0 0 v 0
0 0
q
1
c 0 v
1
C
C
C
C
C
C
C
C
C
A
;
A
z
= S
1
A
0
z
S =
0
B
B
B
B
B
B
B
B
B
@
w 0 0
1
p
c 0
0 w 0 0 0
0 0 w 0 0
1
p
c 0 0 w
q
1
c
0 0 0
q
1
c w
1
C
C
C
C
C
C
C
C
C
A
; ( B . 1 1 )
a n d
D
x x
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0
2 +
0 0 0
0 0
0 0
0 0 0
0
0 0 0 0
P r
1
C
C
C
C
C
C
C
C
A
; D
x y
= D
T
y x
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0 0
0 0
0
0 0 0
0 0 0 0 0
0 0 0 0 0
1
C
C
C
C
C
C
C
C
A
;
D
y y
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0
0 0 0
0 0
2 +
0 0
0 0 0
0
0 0 0 0
P r
1
C
C
C
C
C
C
C
C
A
; D
x z
= D
T
z x
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0 0 0
0
0 0 0 0 0
0
0 0 0
0 0 0 0 0
1
C
C
C
C
C
C
C
C
A
;
-
8/3/2019 Giles Stability of ODE Solvers
24/26
2 4
D
z z
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0
0 0 0
0 0
0 0
0 0 0
2 +
0
0 0 0 0
P r
1
C
C
C
C
C
C
C
C
A
; D
y z
= D
T
z y
=
0
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0 0 0 0 0
0 0 0
0
0 0
0 0
0 0 0 0 0
1
C
C
C
C
C
C
C
C
A
:
( B . 1 2 )
A n i m p o r t a n t f e a t u r e o f t h e t r a n s f o r m e d e q u a t i o n s i s t h a t t h e c o m b i n e d d i s -
s i p a t i o n m a t r i x ,
0
B
B
B
@
D
x x
D
x y
D
x z
D
y x
D
y y
D
y z
D
z x
D
z y
D
z z
1
C
C
C
A
i s b o t h s y m m e t r i c a n d p o s i t i v e s e m i - d e n i t e . T h e s y m m e t r y i s c l e a r f r o m t h e
a b o v e d e n i t i o n s o f t h e c o m p o n e n t m a t r i c e s , a n d t h e p o s i t i v i t y c o m e s f r o m n o t -
i n g t h a t
x
T
0
B
@
D
x x
D
x y
D
x z
D
y x
D
y y
D
y z
D
z x
D
z y
D
z z
1
C
A
x =
( x
3
+ x
7
)
2
+
( x
4
+ x
1 2
)
2
+
( x
9
+ x
1 3
)
2
+
1
0
B
@
x
2
x
8
x
1 4
1
C
A
T
0
B
@
2 +
2 +
2 +
1
C
A
0
B
@
x
2
x
8
x
1 4
1
C
A
+
P r
( x
2
5
+ x
2
1 0
+ x
2
1 5
) : ( B . 1 3 )
T h e e i g e n v a l u e s o f
0
B
@
2 +
2 +
2 +
1
C
A
a r e 2 ; 2 ; 2 + 3 a n d h e n c e t h e c o m b i n e d d i s s i p a t i o n m a t r i x i s p o s i t i v e s e m i -
d e n i t e p r o v i d e d 0 , 2 + 3 0 a n d k 0 .
-
8/3/2019 Giles Stability of ODE Solvers
25/26
2 5
A p p e n d i x C L
2
n o r m s o f c o m p o n e n t m a t r i c e s
D e n i n g
Z
N
i
r N
j
d V = S ~ n ; ( C . 1 )
t h e n
a
i j
= S ( n
x
A
x
+ n
y
A
y
+ n
z
A
z
)
= S
0
B
B
B
B
B
B
B
B
B
@
~ u : ~ n
1
p
c n
x
1
p
c n
y
1
p
c n
z
0
1
p
c n
x
~ u : ~ n 0 0
q
1
c n
x
1
p
c n
y
0 ~ u : ~ n 0
q
1
c n
y
1
p
c n
z
0 0 ~ u : ~ n
q
1
c n
z
0
q
1
c n
x
q
1
c n
y
q
1
c n
z
~ u : ~ n
1
C
C
C
C
C
C
C
C
C
A
: ( C . 2 )
T h r e e o f t h e e i g e n v a l u e s o f S
1
a
i j
a r e e q u a l t o ~ u : ~ n a n d t h e o t h e r t w o a r e ~ u : ~ n c .
H e n c e ,
k a
i j
k = S ( j ~ u : ~ n j + c ) ( C . 3 )
u s i n g t h e f a c t t h a t f o r s y m m e t r i c m a t r i c e s t h e L
2
n o r m i s t h e m a g n i t u d e o f t h e
l a r g e s t e i g e n v a l u e .
T h e q u a n t i t y S ~n c a n b e i n t e r p r e t e d g e o m e t r i c a l l y . F i r s t n o t e t h a t r N
j
i s
n o n - z e r o o n l y o n t e t r a h e d r a s u r r o u n d i n g n o d e j , a n d t h a t o n s u c h a t e t r a h e d r o n ,
l a b e l l e d ,
r N
j
=
1
3 V
~
S
j
( C . 4 )
w h e r e
~
S
j
i s t h e i n w a r d - p o i n t i n g a r e a v e c t o r o f t h e f a c e o f o p p o s i t e n o d e j ,
a n d V
i s t h e v o l u m e o f t h e t e t r a h e d r o n . S u m m i n g o v e r a l l t e t r a h e d r a f o r w h i c h
b o t h i a n d j a r e c o r n e r n o d e s , g i v e s
S ~n =
1
1 2
X
~
S
j
( C . 5 )
D e n e d
i j
t o b e t h e c o n t r i b u t i o n t o d
i j
f r o m t h e i n t e g r a t i o n o v e r t e t r a h e d r o n
. T h e r e f o r e ,
d
i j
=
X
d
i j
= ) k d
i j
k
2
X
k d
i j
k
2
( C . 6 )
w h e r e a g a i n t h e s u m m a t i o n i s o v e r t e t r a h e d r a c o m m o n t o b o t h i a n d j . O n
-
8/3/2019 Giles Stability of ODE Solvers
26/26
2 6
t e t r a h e d r o n ; r N
i
a n d r N
j
a r e b o t h u n i f o r m a n d s o
d
i j
= V
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
0 0 0 0 0
0
+
@ N
i
@ x
@ N
j
@ x
+
@ N
i
@ x
@ N
j
@ y
+
@ N
i
@ x
@ N
j
@ z
0
+
rN
i
rN
j
0
+
@ N
i
@ y
@ N
j
@ x
+
@ N
i
@ y
@ N
j
@ y
+
@ N
i
@ y
@ N
j
@ z
0
+
r N
i
r N
j
0
+
@ N
i
@ z
@ N
j
@ x
+
@ N
i
@ z
@ N
j
@ y
+
@ N
i
@ z
@ N
j
@ z
0
+
r N
i
r N
j
0 0 0 0
P r
r N
i
r N
j
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
( C . 7 )
H e n c e ,
k d
i j
k V
m a x
(
j r N
i
r N
j
j +
+
j r N
i
j j r N
j
j ;
P r
j r N
i
r N
j
j
)
( C . 8 )
w h i c h c a n b e r e - e x p r e s s e d u s i n g t h e v a l u e s f o r r N
i
a n d r N
j
a s
k d
i j
k
1
9 V
m a x
(
j
~
S
i
~
S
j
j +
+
j
~
S
i
j j
~
S
j
j ;
P r
j
~
S
i
~
S
j
j
)
; ( C . 9 )
w h e r e
~
S
i
a n d
~
S
j
a r e a s d e n e d p r e v i o u s l y . N o t e t h a t t h e u p p e r b o u n d o n t h e
r i g h t - h a n d - s i d e o f E q . ( C . 9 ) i s u n c h a n g e d i f i a n d j a r e i n t e r c h a n g e d , a n d s o i t
i s a l s o a n u p p e r b o u n d f o r k d
j i
k . H e n c e ,
m a x f k d
i j
k ; k d
j i
k g
X
1
9 V
m a x
(
j
~
S
i
~
S
j
j +
+
j
~
S
i
j j
~
S
j
j ;
P r
j
~
S
i
~
S
j
j
)
:
( C . 1 0 )
T h e e x a c t v a l u e f o r
ka
i j
ka n d t h e u p p e r b o u n d s f o r
kd
i j
k;
kd
j i
kc a n t h e n b e
c o m b i n e d b y t h e t r i a n g l e i n e q u a l i t y ,
k c
i j
k = k a
i j
+ d
i j
k k a
i j
k + k d
i j
k ; ( C . 1 1 )
t o g e t u p p e r b o u n d s f o r k c
i j
k a n d k c
j i
k f o r u s e i n t h e s u c i e n t s t a b i l i t y l i m i t s
d e r i v e d i n S e c t i o n 5 i n t h e m a i n t e x t .