Institute for Computational Mechanics in Propulsion. Lewis Research ... volume formulation makes use of a universal limiter for transient interpolation modeling of the .... upwind ULTIMATE scheme gives excellent practical results. Local higher ...... 119. Security Classif. (of this report). 120. Security Classif. (of this page). 121.
NASA Technical Memorandum 100916 ICOMP-88-11
Universal Limiter for Transient Interpolation Modeling of the Advective Transport Equations : The ULTIMATE Conservative Difference Scheme
B.P. Leonard Institute for Computational Mechanics in Propulsion Lewis Research Center Cleveland, Ohio _-
-
-
__
- -
( L A S A - T H - 1009 16) U N I V E R S A L L1B13EB PO8 IEALSIENI I I ' I E E E C L A T I C b HCDELIIG C P THB A L V E C T I V E T E A l S E 0 6 P BQUilTICYS: 3CE OLTXClASB CCNSERVATIVE D I E L E B B Y C E S C b E P E (IJaSA)
189-14744
117 p
017969U
September 1988
CSCL 1 2 A G3/6U
Unclas
UNIVERSAL LIMITER FOR TRANSIENT INTERPOLATION MODELING OF THE THE ULTIMATE
ADVECTIVE TRANSPORT EQUATIONS:
CONSERVATIVE DIFFERENCE SCHEME
B . P. Leonard* I n s t i t u t e f o r Computational Mechanics i n P r o p u l s i o n Lewis Research Center C l e v e l a n d , O h i o 44135 SUMMARY A f r e s h approach i s t a k e n t o t h e e m b a r r a s s i n g l y d i f f i c u l t problem of
adequately modeling simple pure advection.
An e x p l i c i t c o n s e r v a t i v e c o n t r o l -
volume f o r m u l a t i o n makes use of a u n i v e r s a l l i m i t e r f o r t r a n s i e n t i n t e r p o l a t i o n modeling o f t h e advective t r a n s p o r t equations.
T h i s ULTIMATE c o n s e r v a t i v e d i f -
f e r e n c e scheme i s a p p l i e d t o unsteady, one-dimensional
s c a l a r pure advection
a t constant velocity, using three c r i t i c a l t e s t p r o f i l e s : squared wave, a d i s c o n t i n u o u s s t e p , and a s e m i - e l l i p s e .
an i s o l a t e d s i n e The g o a l , o f course,
i s t o d e v i s e a s i n g l e r o b u s t scheme which achieves sharp monotonic r e s o l u t i o n
of t h e s t e p w i t h o u t c o r r u p t i n g t h e o t h e r p r o f i l e s .
The s e m i - e l l i p s e i s p a r t i c -
u l a r l y c h a l l e n g i n g because o f i t s c o m b i n a t i o n o f sudden and g r a d u a l changes i n gradient.
The ULTIMATE s t r a t e g y can be a p p l i e d t o e x p l i c i t c o n s e r v a t i v e
schemes o f any o r d e r o f accuracy.
Second-order schemes a r e u n s a t i s f a c t o r y ,
showing s t e e p e n i n g and c l i p p i n g t y p i c a l o f c u r r e n t l y p o p u l a r s o - c a l l e d " h i g h r e s o l u t i o n " s h o c k - c a p t u r i n g or TVD schemes. The ULTIMATE t h i r d - o r d e r upwind scheme i s h i g h l y s a t i s f a c t o r y for m o s t flows o f p r a c t i c a l importance. H i g h e r o r d e r methods g i v e p r e d i c t a b l y b e t t e r s t e p r e s o l u t i o n , a1 though even-order schemes g e n e r a t e a (monotonic) waviness i n t h e d i f f i c u l t s e m i - e l l i p s e sirnulation.
B u t l i t t l e i s t o be g a i n e d above ULTIMATE f i f t h - o r d e r u p w i n d i n g which
g i v e s r e s u l t s c l o s e t o t h e u l t i m a t e one m i g h t hope f o r . INTRODUCTION
I n a landmark s e r i e s o f papers i n t h e 1970's, Brain van Leer worked "Towards t h e U l t i m a t e C o n s e r v a t i v e D i f f e r e n c e Scheme" f o r c o m p u t a t i o n a l f l u i d dynamics ( r e f s .
1 t o 5).
T h i s work spawned a body o f l i t e r a t u r e i n t h e p r e s e n t
*Work funded under Space A c t Agreement C99066G; p r e s e n t l y a t Dept. of Mechanical E n g i n e e r i n g , The U n i v e r s i t y o f Akron, Akron, O h i o 44325.
decade involving advective modeling methods which are sometimes classified as "shock-capturing" schemes or , more recently, as "TVD" schemes, referring to the oscillation-suppression strategy of total-variation diminution (ref. 6). The ultimate conservative difference scheme for CFD has proven surprisingly elusive. There have been notable "successes" such as impressive demonstrations of nonoscillatory high resolution of steps and shock waves, for example; but progress has been unexpectedly slow. It seems that correction of one defect always introduces another, equally severe. Unphysical oscillations inherent in classical second-order methods were eliminated by switching to first-order upwinding; but this merely replaced unacceptable oscillations with (what was ultimately realized to be) unacceptable global artificial diffusion. By devising methods with locally varying artificial diffusion (small in smooth regions, larger in sharply varying regions), it is possible to achieve somewhat better resolution than global first-order upwinding without introducing spurious numerical oscillations. Some forms of shock-capturing (or TVD) schemes achieve their impressive results for step resolution by the use of locally varying positive artificial diffusion or viscosity (first-order upwinding) to suppress oscillations, combined with local negative viscosity (such as first-order downwinding) to artificially compress or steepen the front. Unfortunately, this inherent negative diffusion is responsible for artificial steepening of (what should be) gentle gradients, as well, as will be demonstrated. Because of the concomitant flattening of local extrema (due to the local positive artificial diffusion), this defect has become known as "clipping," although the problem is initiated by the artificial steepening introduced to give high resolution to simulated fronts. In some cases (for example, a step-function followed by a ramp), inherent oscillations, rather than being suppressed, are converted into a series of small monotonic steps, a phenomenon known as "stair-casing." will be clearly demonstrated here, currently popular forms of TVD schemes achieve monotonic (although not always sharp) resolution of steps at the expense of gross (albeit nonoscillatory) distortion of simple smooth profiles. Thus, A . R . Mitchell's characterization of advective modeling as computational dynamics' ultimate embarrassment (ref. 7) is still appropriate, given the current state of the art. The ultimate goal of the current research program is to inject some self-confidence (as opposed to self-satisfaction) into As
2
c o m p u t a t i o n a l f l u i d dynamics by d e v e l o p i n g a t r u l y r o b u s t ( i . e . ,
universally
a p p l i c a b l e ) framework on which f u r t h e r r e f i n e m e n t s can be c o n s t r u c t e d . The p r e s e n t paper i s t h e f i r s t i n a proposed s e r i e s p r e s e n t i n g a f r e s h approach t o s i m u l a t i n g t h e a d v e c t i v e t e r m , which, b e i n g an odd-order ( f i r s t - ) d e r i v a t i v e , i s t h e most s i g n i f i c a n t a s p e c t o f CFD ( r e f . 8). An e x t r e m e l y s i m p l e form o f " m o n o t o n i z i n g " u n i v e r s a l l i m i t e r i s d e s c r i b e d which can be a p p l i e d
t o e x p l i c i t c o n s e r v a t i v e d i f f e r e n c e schemes, w i t h no c o n s t r a i n t s on t h e o r d e r o f a c c u r a c y and r e s o l u t i o n .
The u n i v e r s a l l i m i t e r banishes u n p h y s i c a l o v e r -
shoots and nonmonotonic o s c i l l a t i o n s w i t h o u t c o r r u p t i n g t h e expected accuracy
o f t h e u n d e r l y i n g method.
I n p a r t i c u l a r , when used w i t h ( a r t i f i c i a l - v i s c o s i t y -
f r e e ) t h i r d or h i g h e r o r d e r base schemes, i t does n o t induce a r t i f i c i a l comp r e s s i o n ( s t e e p e n i n g ) or c l i p p i n g , t y p i c a l o f s o - c a l l e d " h i g h r e s o l u t i o n " ( a c t u a l l y , second-order)
s h o c k - c a p t u r i n g methods.
C o n s e r v a t i v e e x p l i c i t a d v e c t i o n schemes o f a r b i t r a r i l y h i g h o r d e r can be composed from " t r a n s i e n t i n t e r p o l a t i o n m o d e l i n g " ( T I M ) ; a t constant vector v e l o c i t y
t i o n of a scalar t i o n over t i m e
At
i.e.,
f o r p u r e advec-
v, t h e e x a c t t r a n s i e n t s o l u -
is Q(X,At)
(1 1
= Q(X-VAt,O)
where t h e accuracy ( i n b o t h space and t i m e ) o f t h e approximate n u m e r i c a l method i s determ ned sole y by ( m u l t i d i m e n s i o n a l ) s p a t i a l i n t e r p o l a t i o n a t t h e e a r l i e r time-level.
The ULTIMATE c o n s e r v a t i v e d i f f e r e n c e scheme t h e n c o n s i s t s o f u s i n g
t h e u n i v e r s a l l i m i t e r (UL) for t r a n s i e n t i n t e r p o l a t i o n modeling ( T I M ) o f t h e
advective t r a n s p o r t equations (ATE). S i n c e a c c u r a t e m o d e l i n g o f t h e a d v e c t i v e t e r m i s one o f t h e more c h a l l e n g i n g a s p e c t s o f CFD ( i n a d d i t i o n t o n o n l i n e a r i t i e s , m u l t i d i m e n s i o n a l i t y , e t c . ) , a t t e n t i o n w i l l be focused, i n t h i s f i r s t a r t i c l e , on t h e s u p e r f i c i a l l y s i m p l e b u t embarrassingly d i f f i c u l t
problem o f unsteady one-dimensional
o f scalar p r o f i l e s a t constant v e l o c i t y . sidered:
pure advection
Three c r i t i c a l t e s t p r o f i l e s a r e con-
an i s o l a t e d sine-squared wave, r e p r e s e n t i n g smooth f u n c t i o n s w i t h a
c o n t i n u o u s l y t u r n i n g g r a d i e n t ; a s t e p d i s c o n t i n u i t y i n v a l u e ; and a semie l l i p s e , combining d i s c o n t i n u o u s and c o n t i n u o u s changes i n g r a d i e n t .
At
f i r s t , s e v e r a l ( u n l i m i t e d ) e x p l i c i t schemes a r e surveyed, t o g e t h e r w i t h some p o p u l a r s h o c k - c a p t u r i n g methods which a r e e x p l a i n e d i n terms o f t h e N o r m a l i z e d
3
V a r i a b l e Diagram (NVD).
The NVD i s a l s o used as t h e b a s i s f o r development o f
t h e u n i v e r s a l l i m i t e r , which i s t h e n e x p l a i n e d i n terms o f u n n o r m a l i z e d v a r i a b l e s and a p p l i e d t o e x p l i c i t p o l y n o m i a l T I M methods o f second t h r o u g h e i g h t h For m o s t ( r e l a t i v e l y smooth) flows, t h e c o s t - e f f e c t i v e t h i r d - o r d e r
order.
upwind ULTIMATE scheme g i v e s e x c e l l e n t p r a c t i c a l r e s u l t s .
Local h i g h e r o r d e r
reso u t i o n can be a u t o m a t i c a l l y b u i l t i n w i t h a l m o s t n e g l i g i b l e a d d i t i o n a l cost SCALAR ADVECTION Unsteady velocity
u
one-dimensional, p u r e a d v e c t i o n o f a s c a l a r
0 a t constant
s d e s c r i b e d by
T h i s e q u a t i o n can be i n t e g r a t e d i n t i m e o v e r a t i m e - s t e p A t and n space o v e r a c o n t r o l volume (CV) a t s t a t i o n i from -Ax/2 t o +Ax/2, assum ng a u n i f o r m grid.
T h i s g i v e s t h e e x a c t c o n s e r v a t i v e d i f f e r e n c e scheme
where t h e b a r s r e p r e s e n t s p a t i a l averages, and t h e a s t e r i s k s t i m e averages, t h e superscripts designate time-levels,
c
i s t h e Courant number
and l e f t time-averaged f a c e v a l u e s a r e i n d i c a t e d .
u A t l A x , and r i g h t
The CV s p a t i a l averages can
be w r i t t e n i n t e r m s o f c e n t r a l node v a l u e s p l u s a d e v i a t i o n t e r m
Ti
=
oi
+ DEV
(4)
Now assume t h a t node v a l u e s a r e r e l a t e d by an e x a c t e q u a t i o n o f t h e form
0;+l
n - 0i =
-C(Or
(5)
- 0),
where t h e f a c e values now i n c l u d e t h e e f f e c t s o f t h e d e v i a t i o n t e r m s .
The
d e v i a t i o n t e r m s themselves w i l l t h e n s a t i s f y
DEV"'
- D E V ~=
-C(DEV,
- DEV~)
g i v e n s u i t a b l e d e f i n i t i o n s of t h e terms on t h e r i g h t .
(4), and ( 6 1 ,
4
(6)
Thus, from e q u a t i o n s (31,
which can be p u t i n t h e form of e q u a t i o n (5) p r o v i d e d t h e e f f e c t i v e average f a c e v a l u e a t any f a c e
f
i s d e f i n e d as
I n t h i s i n t e r p r e t a t i o n , e q u a t i o n (5) r e w r i t t e n as
can s t i l l be c o n s i d e r e d e x a c t f o r c o n s t a n t
o f t h e e f f e c t i v e face values.
c, g i v e n a p p r o p r i a t e d e f i n i t i o n s
I n p r a c t i c a l simulations, t h e f a c e values are
approximated and t h e form o f t h e e q u a t i o n i s extended t o v a r i a b l e a d v e c t i n g v e l o c i t y (including sign-reversals),
i.e.,
Transient I n t e r p o l a t i o n Modeling One method o f g e n e r a t i n g a d v e c t i v e a l g o r i t h m s i n t h e form o f e q u a t i o n ( 9 )
i s based on e q u a t i o n ( 1 ) . which i n one dimension can be w r i t t e n , f o r c o n s t a n t
u,
where
$‘((XI can be c o n s i d e r e d t o be a f u n c t i o n o f t h e n o r m a l i z e d l o c a l coor-
5
dinate,
=
(x
-
xi)/Ax,
and g i v e n by
The homogeneous s p a t i a l i n t e r p o l a t i o n f u n c t i o n
f
can i n t u r n be w r i t t e n as
f ( < >= ; compare e q u a t i o n s ( 2 3 > ,
( 2 7 1 , ( 3 0 > , and (34) w i t h (211, (241, (291, and ( 3 3 > , r e s p e c t i v e l y .
A l l the
methods c o n s i d e r e d h e r e g i v e e x a c t p o i n t - t o - p o i n t t r a n s f e r a t c :1 , as seen n n n n+ 1 from t h e f o r m u l a s f o r 4r; i . e . , = +i , 4% = 4i-l, and Oi = $i-l. For t h e h i g h e r o r d e r methods, p o i n t - t o - p o i n t
t r a n s f e r a l s o o c c u r s ( o v e r more t h a n
one mesh w i d t h ) for l a r g e r i n t e g e r v a l u e s o f
c; however, i n t h e absence o f
modeled p h y s i c a l d i f f u s i o n , t h e s e methods a r e n o t a l l s t a b l e o v e r a c o n t i n u o u s range, e x c e p t f o r 0
5 c 5 1. NORMALIZED VARIABLE DIAGRAM Normalized V a r i a b l e s
F i g u r e 13 shows a one-dimensional on one f a c e ( i n t h i s case, t h e l e f t ) .
control-volume w i t h a t t e n t i o n focussed I n d e t e r m i n i n g t h e e f f e c t i v e face v a l u e ,
t h e most i n f l u e n t i a l nodes a r e t h e two s t r a d d l i n g t h e f a c e and t h e n e x t +f, upstream node, t h e l a t t e r depending of course on t h e flow d i r e c t i o n a t t h e f a c e
i n question, i.e., (downstream), 0,
the sign of
uf.
(upstream), and
These t h r e e node v a l u e s can be l a b e l e d
$c
( c e n t r a l ) , as shown.
ence i n d e f i n i t i o n o f these nodes, depending on
SGN(uf).
Note t h e d i f f e r I n terms of original
v a r i a b l e s , t h e r e a r e c l e a r l y a v e r y l a r g e number o f cases t o c o n s i d e r : nations of p o s i t i v e o r negative
uf, p o s i t i v e or n e g a t i v e
4D
combi-
4 , and p o s i t i v e or
13
n e g a t i v e v a l u e s o f g r a d i e n t and c u r v a t u r e .
and s c a l e can be n o r m a l i z e d o u t by d e f i n i n g t h e n o r m a l i z e d v a r i a b l e ( a t any p o i n t ) as
i s a f u n c t i o n o f 4; , 4: , 4; , and Courant number, the normalized face value i s o n l y a f u n c t i o n o f i t s adjacent normalized upstream node v a l u e and c Now, i n a case when
I
4f
s i n c e t h e o t h e r n o r m a l i z e d node v a l u e s a r e c o n s t a n t : (45)
E q u a t i o n ( 4 4 ) i n c l u d e s f i r s t - o r d e r methods, second-order c e n t r a l and upwind schemes, and t h i r d - o r d e r upwinding, i n a d d i t i o n t o second-order (and t h i r d o r d e r ) shock-capturing algorithms. nodes b u t t h e n o r m a l i z e d
Gf
H i g h e r o r d e r methods i n v o l v e more d i s t a n t
w i l l s t i l l depend most s t r o n g l y on
5: .
Second-Order T i m e A v e r a g i n g I n g e n e r a l , f o r t r a n s i e n t - i n t e r p o l a t i o n - m o d e l i n g methods based on
I
I
V a r i a t i o n s i n s i g n , flow d i r e c t i o n ,
e q u a t i o n (111, t h e t i m e accuracy of t h e r e s u l t i n g c o n t r o l - v o l u m e a l g o r i t h m (eq. ( 1 8 ) ) ,
i s t h e same o r d e r as t h e degree of t h e s p a t i a l i n t e r p o l a t i o n used
i n equation (12).
Many a d v e c t i o n schemes i n common use ( i n c l u d i n g TVO schemes)
a r e based on second-order t i m e a v e r a g i n g , which can be made e x p l i c i t by straight-forward integration.
S p e c i f i c a l l y , f o r s p a t i a l l y second-order-
a c c u r a t e methods, t h e d e v i a t i o n t e r m i n e q u a t i o n ( 4 ) i s f o r m a l l y n e g l e c t e d ; t h u s , from e q u a t i o n (8>, t h e e f f e c t i v e face v a l u e s a r e j u s t t h e e s t i m a t e d t i m e averages. I f these t i m e averages a r e based on l o c a l l y advected f a c e - v a l u e l i n e a r b e h a v i o r s p a t i a l l y , t h e method i s second-order a c c u r a t e i n t i m e , as w e l l .
i s t h e i n s t a n t a n e o u s f a c e v a l u e , t h e time-averaged f a c e
For example i f value over
14
At
is
(46)
where
0;
i s the estimated face value a t time-level
n.
I n most cases, t h e
s p a t i a l g r a d i e n t i s e s t i m a t e d as
(47)
so t h a t
0f
-
= t$f
(48)
or, i n terms o f n o r m a l i z e d v a r i a b l e s ,
..
9,
= (1
-
-n + ct$c -n c)t$f
where, now, t h e l i n e a r Courant number w e i g h t i n g i s e x p l i c i t and o n l y on
(49)
6;
depends
5;
F i g u r e 14 shows t h e n o r m a l i z e d v a r i a b l e diagram (NVD), p l o t t i n g c h a r a c t e r i s t i c s o f t h e form o f e q u a t i o n (501, f o r ( 1 ) F i r s t - o r d e r u p w i n d i n g (1U):
-n
0,
=
-n
+c
(51)
( 2 ) The Lax-Wendroff method (2C): (52)
(3) Second-order u p w i n d i n g (2U):
a;
=
3 -n
2 0c
(53)
and
15
( 4 ) F r o m m ' s method (2F):
Note t h a t t h e l a t t e r t h r e e s p a t i a l l y second-order methods a l l pass t h r o u g h t h e -n -n p o i n t (0.5, 0.75); i n f a c t , any method whose 4)f(ec> c h a r a c t e r i s t i c passes t h r o u g h t h i s p o i n t w i t h a f i n i t e s l o p e i s ( a t l e a s t ) second-order a c c u r a t e i n space, s i n c e
$IF
can t h e n be w r i t t e n
-' (1 + 4 -):)
4)f=T where t h e c u r v a t u r e f a c t o r
(55)
-) : ;2
i s a c o n s t a n t or, i n t h e case o f n o n l i n e a r
CF
6:.
schemes, a f u n c t i o n o f
- CF(1
T h i s i s more a p p a r e n t i n terms o f unnormalized
v a r i ab1 es n
4)f
71
=
(4);
+
-
4);
cF(4);
-
24);
(56)
+ 4);
or 4);
=
1
7
(4);
(57)
+
t h u s , d e v i a t i n g from t h e second- r d e - a c c u r a t e l i n e a r i n t e r p o l a t i o n b y second
i s finite.
or h i g h e r o r d e r terms, p r o v i d e d CF
Note t h a t f i r s t - o r d e r
upwinding, e q u a t i o n ( 5 1 1, and
( 5 ) F i r s t - o r d e r downwinding ( 1 D ) : -n
4)f
=
(58)
1
cannot be w r i t t e n i n t h e form o f e q u a t i o n ( 5 5 ) f o r f i n i t e
CF.
The c h a r a c t e r i s t i c s shown i n f i g u r e 14 a r e i n t h e form o f e q u a t i o n ( S O ) , i.e.,
the estimated face value a t time-level
n
6:
as a f u n c t i o n o f
.
a l s o i m p o r t a n t t o p o r t r a y t h e time-averaged face v a l u e i n t h e same way, superscript), according to the l i n e a r weighting, equation (49). 17 show t h e
NVDs c o r r e s p o r , j i n g t o
$f($:
f o r f i v e d i f f e r e n t v a l u e s o f Courant number:
(no
second-
The c h a r a c t e r i s t i c s a r e shown
c + 0, c
=
1 / 4 , 1 / 2 , 3 / 4 , and 1 .
The zero-value i s d i s t i n g u i s h e d by a dashed l i n e , s i n c e t h i s can o n l y be
16
Gf
F i g u r e s 15 t o
, c) f o r t h e Lax-Wendroff,
o r d e r upwinding and Fromm methods, r e s p e c t i v e l y .
It i s
1
approached i n an e x p l i c i t c a l c u l a t i o n . Note t h a t a l l these methods r e v e r t t o -n $f = $c when c = 1 , as seen from e q u a t i o n ( 4 9 > , g i v i n g e x a c t p o i n t - t o - p o i n t w
t r a n s f e r as d i s c u s s e d p r e v i o u s l y . Nonl inear "Shock-Capturi ng" Schemes
For l a c k o f b e t t e r c a t e g o r i z i n g t e r m i n o l o g y , a number o f c u r r e n t l y p o p u l a r a l g o r i t h m s have become known as " s h o c k - c a p t u r i n g "
schemes.
When a p p l i e d t o t h e
pure a d v e c t i o n problems s t u d i e d h e r e , these second-order methods ( i n space and t i m e ) can be p o r t r a y e d e i t h e r i n t h e
$!
NVD
i n g t o e q u a t i o n s (50) or (491, r e s p e c t i v e l y .
o r i n the
NVD, correspond-
However, i n c o n t r a s t t o t h e
l i n e a r c h a r a c t e r i s t i c s o f f i g u r e 14 (or o f f i g s . 15 t o 1 7 ) . these methods a r e d i s t i n g u i s h e d by n o n l i n e a r c h a r a c t e r i s t i c s , a l t h o u g h t h e y remain l i n e a r i n according t o equatlon (49).
c
A l l schemes r e v e r t t o f i r s t - o r d e r u p w i n d i n g o u t -
f o r $: < 0 or 6; > 1. They a l l pass t h r o u g h t h e o r i g i n (0,O) and t h e p o i n t ( 1 , l ) i n e i t h e r NVD. For t h e NVD, t h e y a l l s i d e t h e monotonic range, i . e . ,
6;
pass t h r o u g h (0.5, 0.751, as r e q u i r e d f o r second-order methods. F i g u r e 18 shows t h e s o - c a l l e d minimum-modulus (Minmod) method ( r e f . 13) -n which i s seen t o f o l l o w second-order upwinding f o r 0 5 @c5 0.5 and Lax-
Gn
$
. Note t h a t b o t h o f these f i g u r e s i n v o l v e t h e time-averaged normalt o be t h e smal l e r o f t h e i z e d f a c e v a l u e , 5, . The Super-C scheme r e q u i r e s
+f
upper u n i v e r s a l l i m i t e r boundary (developed i n t h e n e x t s e c t i o n ) and LaxWendroff, i . e . ,
the smaller of
(62)
for
0 5
G:
5 112; w i t h l i m i t e d second-order upwinding, i . e . , t h e smal e r o f
-
$f = 1 I
6;
and
5f
=
1 $3
- c ) o-nC
5 1 , w i t h f i r s t - o r d e r u p w i n d i n g elsewhere. Hyper-C ( l i m t e d for 1/2 < f i r s t - o r d e r downwinding) s i m p l y r e q u i r e s t o be t h e s m a l l e r o f
(63)
5; C I
I
and
1
(64)
..
in the monotonic regime, with +f = 5; elsewhere, as usual. finally, note that the Courant-number-dependence of Super-C and Hyper-C is no longer linear. Results for the Nonlinear Schemes figures 24 through 30 show the results of simulating the three test problems with each of the nonlinear schemes just described; again, the two representative Courant numbers (0.05 and 0.5) are used. As seen in figure 24, although Minmod resolves the step monotonically, the structure is not at all sharp and the other profiles are rather diffusive, especially at smaller Courant numbers. In the Chakravarthy-Osher simulations, shown in figure 25, the leading-edge steepening effects of the first-order downwinding are seen, with concomitant clipping due to first-order upwinding; again, profiles are more diffusive at small c values. The two van Leer schemes, MUSCL and CLAM (figs. 26 and 27), give similar results to each other, as perhaps expected from the qualitative similarity of their NVDs (figs. 20 and 21). The MUSCL scheme in particular is one of the more successful of the well-known second-order nonlinear schemes, considering overall performance for all three test problems. Once again, both MUSCL and CLAM deteriorate at small Courant-number values, due primarily to the unnecessarily restrictive TVD limiter, equation (59). Superbee (fig. 28) gives significantly sharper results for the step simulation at all c values. The smooth-function (sine-squared) simulation i s slightly better than that of MUSCL; however, there is a degree of artificial steepening inherent in this method, as seen in the semi-ellipse computation. In the presence of rapid changes in gradient (large curvature) - near the leading and trailing edges of the profile - the scheme has a tendency to convert all gentle slopes into sharp steps followed by plateaus. This defect is purposely taken to extremes with Super-C (fig. 29) and Hyper-C (fig. 30). Clearly, in terms of step performance, Super-C supersedes Superbee and Hyper-C supersedes Super-C. Super-C also does an excellent job of simulating the sine-squared profile; however, the semi-ellipse results are rather bizarre, showing stair-casing for small c values and gross artificial steepening at
19
I
incredible step simulation of these artificial-compression methods should not be used as a standard for judging the overall performance of truly robust schemes. In order to show the effect of Courant number over a wider range, figure 31 gives a log-log plot of ABSERROR for the sine-squared profile (lower curves) and the semi-ellipse (upper curves) versus ABSERROR for the step simulation for Minmod, MUSCL, Superbee, and Super-C, with Courant number as a parameter ranging from 0.01 to 0.978, with points shown at values o f 0.1,0.5, and 0.9 on each curve. At very small c values, Minmod produces large errors for all test profiles. The other schemes' semi-ellipse errors are comparable, with step errors decreasing in the order: MUSCL, Superbee, Super-C. The s i n e squared error follows the same order. Superbee and, in particular, Super-C, show much less sensitivity to Courant number than the other schemes. Although all methods' trajectories in this plane approach the origin (exact point-topoint transfer) as c + 1 , the tendency is rather slow, and even at c = 0.9, the sine-squared error in particular is unacceptably large, compared with what can be achieved with higher order methods.
I
THE UNIVERSAL LIMITER
Gf
The normalized-variable plane, with 6: as abscissa and (no superscrip) as ordinate, can be used to construct a very simple diagram representing constraints on the effective time-averaged normalized face value so as to guarantee maintenance of monotonic profiles, thereby suppressing extraneous overshoots or nonmonotonic oscillations, but allowing arbitrarily high resolution depending on the formal order of accuracy (in both space and time) of the base method. A s seen in the previous section, nonlinear second-order schemes (linplane for any ear in c) can be represented by a single curve in the (6: , fixed value of the Courant number. This is true for the nonlinear third-order scheme as well (to be described), but the Courant-number dependence is then a1 so nonl inear, as wi 1 1 be seen. A1 1 the nonlinear second-order schemes
a,>
I
I
20
(except Super-C) satisfy Sweby's sufficient TVD criteria (ref. 10) (translated into present notation)
- = 5;
+f
i
=
-n +c
-n +c
for
0
and
-n 2 +c
1
(65)
and (66)
1 I
In particular, all nonlinear characteristics pass through (0,O) and (1,l). These criteria are in terms of 5; ; they are thus limited t o temporally second-order schemes linear i n c, according t o equation ( 4 9 ) . T o allow higher order accuracy, it is extremely important t o work directly with the effective time-averaged normalized face value 5f, rather than 5; . By imposing simple monotonicity-maintenance criteria, much less restrictive constraints - the universal limiter - are placed on the allowable face values. The computational strategy is then, in principle, as follows: ( 1 ) formulate +f by some desired high-order method; ( 2 ) compute the actual normalized $ value and the intended normalized value, ( 3 ) i f falls within the allowable universal limiter range, simply proceed; ( 4 ) if Gf lies outside this range, reset (limit) its value t o that of the nearest constraint boundary at the given 5; value; (5) reconstruct the new Qf from 6, ; (6) repeat the procedure for all other faces; (7) update according t o equation (10). In actual practice, it is less expensive computationally t o carry out this strategy directly in terms of unnormalized variables, as will be described in detai 1 .
Gf
Gf
Monotonicity-Maintenance Criteria Figure 32 shows normalized node values with $ in the monotonic range, -n 0 5 +c 1 . A s suggested by the cross-hatching, monotonic behavior requires necessary conditions on Gf :
5;
L
Gf 5
(67)
1
and on the corresponding face value of the adjacent upstream CV face,
Gu
:
21
5;
Consider e q u a t i o n (9), w r i t t e n f o r -n+l +C
=
i n normalized variables
5;
-
-
c(6f
6C
I n o r d e r t o m a i n t a i n m o n o t o n i c i t y , t h e new -n+l
6u)
(69)
v a l u e must be c o n s t r a i n e d by
-n+l
(70)
L 0,
i 0,
For p u r e a d v e c t i o n a t c o n s t a n t v e l o c i t y , t h e r i g h t - h a n d i n e q u a l i t y i s l e s s r e s t r i c t i v e than
Gf
2
5;
, but the left-hand inequality r e s u l t s i n
And s i n c e
i s given
GU i s by gU =
-
-n+ 1
1
(71 1
-n+ 1 n o n p o s i t i v e , t h e worst-case c o n d i t i o n n o n n e g a t i v e and I$" -n+ 1 0 and 4" =0, i . e . ,
T h i s , combined w i t h inequa i t y (67) (73) constitutes the universal
:6 .
i m i t e r i n t h e monotonic range o f
For
5:
< 0
or > 1, n u m e r i c a l e x p e r i m e n t a t i o n has shown t h a t t h e s i m p l e c o n d i t i o n (74) g i v e s t h e most s a t i s f a c t o r y o v e r a l l performance; t h i s , o f c o u r s e , i s e q u i v a l e n t
t o f i r s t - o r d e r upwinding as used b y o t h e r n o n l i n e a r (second-order) TVD schemes. I t does n o t erode t h e accuracy o f t h e o v e r a l l scheme, which i s d e t e r m i n e d s o l e l y by b e h a v i o r i n t h e smooth r e g i o n , near
5;
+ 0.5.
The u n i v e r s a l l i m i t e r , e q u a t i o n (74) t o g e t h e r w i t h i n e q u a l i t i e s ( 7 2 ) and (731, i s shown i n diagrammatic form i n f i g u r e 33; t h e Courant-numberdependent boundary,
Gf
=
$/c,
i s shown dashed t o s t r e s s t h e f a c t t h a t i t s
s l o p e changes w i t h d i f f e r e n t v a l u e s o f approaches t h e v e r t i c a l a x i s ; w h i l e f o r
22
c.
c
Note t h a t for =
c
0, t h i s boundary -n 1 , i t degenerates i n t o Gf Z +
(everywhere), c o r r e s p o n d i n g t o e x a c t p o i n t - t o - p o i n t
t
For
t r a n s f e r as u s u a l .
r e f e r e n c e t o p r e v i o u s work, t h e c o r r e s p o n d i n g c r i t e r i a i n terms of Sweby's variables, r
and
'9, a r e g i v e n i n t h e appendix.
For c l a r i t y , t h e p r e c i s e steps i n a p p l y i n g t h e u n i v e r s a l l i m i t e r t o t r a n s i e n t i n t e r p o l a t i o n modeling o f the advective t r a n s p o r t equations are g i v e n , as follows, u s i n g n o r m a l i z e d v a r i a b l e s . I
ULTIMATE s t r a t e q y ( n o r m a l i z e d v a r i a b l e s ) : ( 1 ) Designate upstream ( U ) , downstream ( D ) ,
basis of
and c e n t r a l (C) nodes on t h e
SGN(uf) f o r each f a c e i n t u r n .
+n +"U '-
( 2 ) Compute DEL = D proceed t o t h e n e x t f a c e .
5;
( 3 ) Otherwise, compute
+f
g r e a t e r than 1, again s e t
gf
(4) I f n o t , compute
=
=
=
if
[DELI
1 , r e s e t ( t h e a b s o l u t e upper l i m i t )
0f -- 1 .
(6) R e c o n s t r u c t update a g o r i thm).
+f
+f u
=
DEL +
+un
( t h i s i s t h e f a c e v a l u e used i n t h e
(7) A f t e r f i n d i n g a 1 f a c e v a l u e s i n t h i s way, e x p l i c i t l y update a c c o r d i n g t o equat on (10). I n o r d e r t o a v o i d d i v i s i o n s and m u l t i p l i c a t i o n s i n v o l v e d i n c o n s t r u c t i n g n o r m a l i z e d v a r i a b l e s and r e c o n s t r u c t i n g t h e unnormalized v a r i a b l e s , i t i s b e t t e r t o work d i r e c t l y w i t h t h e unnormalized v a r i a b l e s .
23
ULTIMATE s t r a t e g y ( u n n o r m a l i z e d v a r i a b l e s ) : ( 1 ) Designate upstream (U),
basis of
and c e n t r a l (C) nodes on t h e
downstream ( D ) ,
SGN(uf) f o r each f a c e i n t u r n , and compute
ADEL = IDEL1
DEL =
+;
-
and
for each f a c e .
( 2 ) Compute
ACURV
(nonmonotonic), s e t
0,
= =
I+: +:
+
- 29:
+:I
ACURV 2 ADEL
f o r each f a c e ; i f
and proceed t o t h e n e x t f a c e .
+REF
(3) Compute t h e r e f e r e n c e face v a l u e
=
+I + (4:
-
for each
$i)/c
face. ( 4 ) Set up some d e s i r e d h i g h o r d e r f a c e v a l u e
$1
(5) I f above. (6) I f
DEL > 0, l i m i t
+f
DEL < 0, l i m i t 0,
+f .
by
+:
below and t h e s m a l l e r o f
by
+:
above and t h e l a r g e r o f
+,,,
and
+REF
and
0;
below. ( 7 ) Update a c c o r d i n g t o e q u a t i o n (10). I n o r d e r t o g e t some f e e l i n g f o r t h e ULTIMATE s t r a t e g y , f i g u r e s 34 t o 37 show n o r m a l i z e d v a r i a b l e diagrams, a p p l i e d t o Lax-Wendroff,
6f
f($
,c)
for the universal l i m i t e r
second-order upwinding, Fromm's method, and t h e t h i r d -
o r d e r QUICKEST scheme, r e s p e c t i v e l y .
I
=
Note s i m i l a r i t i e s (and d i f f e r e n c e s )
between f i g u r e s 35 and 1 9 ( b > , and between f i g u r e s 36 and 2 0 ( b > .
Also n o t e
q u a l i t a t i v e s i m i l a r i t i e s between t h e l i m i t e d v e r s i o n s o f Fromm's method and QUICKEST ( f i g s . 36 and 37) r e s p e c t i v e l y ; t h e y a r e i d e n t i c a l a t
w e l l as a t
c = 0.5 (as
c = 1, o f course). R e s u l t s for t h e ULTIMATE Schemes
F i g u r e s 38 t o 46 show t h e r e s u l t s for t h e t h r e e s t a n d a r d t e s t problems I
o b t a i n e d by a p p l y i n g t h e u n i v e r s a l l i m i t e r t o t h e Lax-Wendroff scheme, secondo r d e r upwinding, Fromm's method, QUICKEST, f o u r t h - o r d e r
~
central, fifth-order
upwinding, s i x t h - o r d e r c e n t r a l , seventh-order upwinding, and e i g h t h - o r d e r
24
c e n t r a l , r e s p e c t i v e l y , each f o r t h e two Courant numbers I
c = 0.05 and 0 . 5 .
Note t h e inadequacies o f t h e l i m i t e d Lax-Wendroff scheme ( f i g . 38) and l i m i t e d second-order u p w i n d i n g ( f i g . 39) s i m i l a r i n some r e s p e c t s t o t h e n o n l i n e a r s h o c k - c a p t u r i n g schemes.
c
E s p e c i a l l y n o t e t h e poor performance a t t h e s m a l l e r
v a l u e ; t h e r e v e r s e d asymmetry o f t h e p r o f i l e s corresponds t o t h e r e s p e c t i v e
asymmetry o f t h e s e two methods' NVDs ( f i g s . 34 and 35). The l i m i t e d Fromm method ( f i g . 40) i s c l e a r l y an improvement o v e r t h e I
s i m p l e second-order schemes, and has q u a l i t a t i v e l y s i m i l a r performance t o l i m i t e d QUICKEST, as expected from t h e s i m i l a r i t y o f t h e i r NVDs ( f i g s . 36 and 37).
The l i m i t e d QUICKEST scheme ( f i g . 41) g i v e s r e s u l t s which a r e
p r o b a b l y e n t i r e l y adequate f o r most p r a c t i c a l s i t u a t i o n s . e r r o r , i n p a r t i c u l a r , i s now w i t h i n t o l e r a b l e bounds.
The s i n e - f u n c t i o n
Although the l i m i t e d
f o u r t h - o r d e r method ( f i g . 42) g i v e s lower ABSERROR f o r each p r o f i l e , t h e r e i s a c l e a r l y d i s c e r n i b l e i n c r e a s e i n WAVINESS i n t h e s e m i - e l l i p s e s i m u l a t i o n , e s p e c i a l l y a t small Courant number v a l u e s ; t h i s i s a t y p i c a l shortcoming of even-order methods.
O v e r a l l , t h e limited-QUICKEST scheme seems t o be t h e b e s t
o f t h e schemes u s i n g t h e f i v e - p o i n t s t e n c i l o f f i g u r e 1 ; i t i s c e r t a i n l y f a r s u p e r i o r t o any o f t h e second-order s h o c k - c a p t u r i n g schemes ( w h i c h i n v o l v e t h e same s t e n c i l ) .
The a r t i f i c i a l waviness o f t h e l i m i t e d f o u r t h - o r d e r method
( w h i c h a l s o uses t h i s s t e n c i l ) d e t r a c t s from an o t h e r w i s e e x c e l l e n t performance.
I f b e t t e r step r e s o l u t i o n i s r e q u i r e d , the l i m i t e d f i f t h - o r d e r upwinding scheme ( f i g . 43) g i v e s h i g h l y s a t i s f a c t o r y r e s u l t s , a l t h o u g h o f course t h i s r e q u i r e s a seven-point s t e n c i l i n g e n e r a l ( a l l o w i n g f o r v e l o c i t y r e v e r s a l s ) . The l i m i t e d s i x t h - o r d e r c e n t r a l method ( w h i c h uses t h e same seven-point s t e n c i l ) , f i g u r e 44, g i v e s s l i g h t l y b e t t e r s t e p r e s o l u t i o n , b u t worse performance
f o r t h e s e m i - e l l i p s e ( i n t e r m s o f b o t h ABSERROR and WAVINESS) and a s l i g h t l y h i g h e r ABSERROR i n t h e sine-squared s i m u l a t i o n a t s m a l l e r Courant numbers (due
t o s l i g h t asymmetric c l i p p i n g , t y p i c a l o f even-order methods). The h i g h e r o r d e r schemes follow a p r e d i c t a b l e p a t t e r n , w i t h b e t t e r s t e p r e s o l u t i o n , and e s s e n t i a l l y e x a c t smooth-function s i m u l a t i o n , b u t w i t h annoyi n g waviness i n t h e c h a l l e n g i n g s e m i - e l l i p s e problem
-
now even n o t i c e a b l e i n
t h e upwind schemes, b u t s t i l l much worse w i t h t h e even-order c e n t r a l methods,
25
e s p e c i a l l y a t small
c
values.
C l e a r l y , t h e ULTIMATE s t r a t e g y c o u l d be con-
t i n u e d t o a r b i t r a r i l y h i g h o r d e r , e i t h e r w i t h p o l y n o m i a l T I M schemes o f t h e t y p e c o n s i d e r e d h e r e or w i t h a l t e r n a t e forms o f i n t e r p o l a t i o n such as s p l i n e s , I ~
f o r example.
The o n l y s t i p i l a t i o n i s t h a t t h e base method must be e x p l i c i t i n
determining the intended
+f
- s t e p ( 4 ) o f t h e ULTIMATE s t r a t e g y .
Once a g a i n , t o see t h e e f f e c t of Courant number o v e r t h e complete s t a b l e range, f i g u r e 47 shows ABSERROR of t h e sine-squared and s e m i - e l l i p s e p r o f i l e s p l o t t e d on a l o g - l o g s c a l e a g a i n s t ABSERROR o f t h e s t e p f o r t h e ULTIMATE Fromm, QUICKEST, f i f t h - and seventh-order upwind schemes, w i t h Courant number as a parameter r a n g i n g from 0.01 t o 0.978, w i t h p o i n t s shown a t 0.9 on each c u r v e . omitted.
c = 0.1, 0.5, and
For c l a r i t y , t h e ULTIMATE even-order schemes have been
The l i m i t e d v e r s i o n o f Fromm's method i s perhaps o f academic i n t e r e s t
( b e i n g s l i g h t l y b e t t e r t h a n MUSCL i n t h e r e g i o n near
c
=
0.5); b u t since the
l i m i t e d QUICKEST t h i r d - o r d e r upwind scheme r e q u i r e s t h e same s t e n c i l and e s s e n t i a l l y t h e same number and t y p e o f computations, i t i s c l e a r l y a more a t t r a c t i v e method. F i g u r e 47 s h o u l d be compared w i t h f i g u r e 31 g i v i n g t h e c o r r e s p o n d i n g r e s u l t s for second-order s h o c k - c a p t u r i n g methods.
The o b v i o u s
g l o b a l c h a r a c t e r i s t i c for t h e h i g h e r - o r d e r ULTIMATE schemes i s t h e i r much I
lower e r r o r f o r t h e s m o o t h - f u n c t i o n s i m u l a t i o n , due t o t h e i r l a c k o f a r t i f i c i a l s t e e p e n i n g and c o n c o m i t a n t c l i p p i n g .
A s expected, s t e p - s i m u l a t i o n error
decreases m o n o t o n i c a l l y w i t h t h e o r d e r o f t h e u n d e r l y i n g base method.
The
s e m i - e l l i p s e and s t e p errors o f t h e h i g h e r o r d e r ULTIMATE schenes a r e s t r o n g l y I
c o r r e l a t e d , whereas for t h e second-order a r t i f i c i a l compression methods o f f i g u r e 31, t h e s e m i - e l l i p s e e r r o r i s r o u g h l y t h e same f o r each scheme, a g a i n r e f l e c t i n g t h e a r t i f i c i a l s t e e p e n i n g and c l i p p i n g o f these methods. S i m p l i f i e d ULTIMATE QUICKEST S t r a t e g y R e f e r r i n g t o f i g u r e 37, i t i s seen t h a t i n t h e range
5:
0.2
i 0.8
(75)
t h e ULTIMATE QUICKEST scheme i s i n f a c t i d e n t i c a l t o t h e u n c o n s t r a i n e d QUICKEST scheme.
Thus, c o n s i d e r a b l e s i m p l i f i c a t i o n can be made i n t h e a l g o r i t h m w i t h o u t
any a p p r o x i m a t i o n or e f f e c t on r e s u l t s .
; ;1 26
-
I n e q u a l i t y ( 7 5 ) can be r e w r i t t e n as
0.51 5 0 . 3
(76)
o r , m u l t i p l y i n g b y 2, I
11
-
-n 24cI
5 0.6
R e w r i t i n g i n terms o f unnormalized v a r i a b l e s r e s u l t s i n
where
CURV
i s t h e upwind-biased s e c o n d - d i f f e r e n c e n CURV = $,
and
DEL
- 24cn + 0,n
i s the normalization difference DEL
=
n
OD - 0;
(80)
Thus, i f i n e q u a l i t y (78) i s s a t i s f i e d , t h e u n c o n s t r a i n e d QUICKEST scheme can be used d i r e c t l y , w i t h no need f o r t e s t i n g o f u n i v e r s a l - l i m i t e r c o n s t r a i n t s . I n any p r a c t i c a l flow, t h i s c r i t e r i o n w i l l be s a t i s f i e d i n t h e overwhelming b u l k o f t h e flow domain, b e i n g v i o l a t e d ( i f a t a l l ) o n l y a t a small f r a c t i o n
o f g r i d p o i n t s near where sharp changes i n g r a d i e n t o c c u r . O f course, i f
a l l ULTIMATE schemes ( o f any o r d e r ) w i l l use ( i n terms o f u n n o r m a l i z e d
variables)
4f
=
n
4,
(82)
I n e q u a l i t i e s (81) are equivalent t o (83)
or, e q u i v a l e n t l y (84)
27
Thus, t h e s i m p l i f i e d ULTIMATE QUICKEST s t r a t e g y i s as follows: ( 1 ) Designate I'D,"
"C,"
and "U" nodes based on
SGN(uf)
i n t h e usua
way,
f o r each f a c e . ( 2 ) Compute
~
[DELI
and
ICURVl.
(3) Ifi n e q u a l i t y (78) i s s a t i s f i e d , use t h e u n c o n s t r a i n e d and unnormali z e d QUICKEST f a c e v a l u e ,
+f
=
$ (+; + +:) - Icl (4:
- )4:
-
( 4 ) Otherwise, i f i n e q u a l i t y ( 8 4 ) i s s a t i s f i e d ,
(1
-
c2)CURV
(85)
use e q u a t i o n ( 8 2 ) .
(5) Otherwise, compute t h e l i m i t e d QUICKEST f a c e v a l u e a c c o r d i n g t o t h e (unnormalized) ULTIMATE s t r a t e g y . I
(6) Proceed t o t h e n e x t CV
face.
(7) Update i n t h e u s u a l way. Note t h a t s t e p ( 5 ) o c c u r s o n l y i n t h e small ranges
C o s t - E f f e c t i v e H i g h - R e s o l u t i o n H y b r i d Scheme I
The ULTIMATE QUICKEST scheme i s a s i m p l e , r o b u s t a l g o r i t h m u s i n g t h e same s t e n c i l as second-order s h o c k - c a p t u r i n g schemes b u t w i t h much b e t t e r g l o b a l accuracy.
I f h i g h e r r e s o l u t i o n of n e a r - d i s c o n t i n u i t i e s i s deemed necessary, i t
i s c l e a r l y p o s s i b l e t o use h i g h e r o r d e r (upwind) schemes g l o b a l l y .
However,
s i n c e t h e need f o r h i g h e r r e s o l u t i o n o c c u r s o n l y i n small l o c a l i z e d r e g i o n s , a c o s t - e f f e c t i v e s t r a t e g y i s t o use t h e e f f i c i e n t s i m p l i f i e d ULTIMATE QUICKEST scheme as a base method, a u t o m a t i c a l l y s w i t c h i n g t o a h i g h e r r e s o l u t i o n ,
ULTIMATE scheme o n l y where needed.
I
28
Numerical e x p e r i m e n t a t i o n has shown t h a t t h e need f o r a h i g h e r r e s o l u t i o n
1
scheme can be d e t e r m i n e d by m o n i t o r i n g t h e a b s o l u t e v a l u e o f t h e u n n o r m a l i z e d 2 average " c u r v a t u r e , " 9 , , d e f i n e d i n e q u a t i o n (25). I f t h i s i s l e s s t h a n a s p e c i f i e d c o n s t a n t , t h e l i m i t e d QUICKEST scheme i s used; i f t h e c r i t i c a l v a l u e
i s exceeded l o c a l l y , t h e a l g o r i t h m branches t o a h i g h e r o r d e r scheme ( t y p i c a l l y 2 f i f t h - or seventh-order u p w i n d i n g ) , r e t u r n i n g t o ULTIMATE QUICKEST wherever gr drops below t h e t h r e s h o l d .
By j u d i c i o u s c h o i c e o f t h e t h r e s h o l d c o n s t a n t , t h e
a l g o r i t h m w i l l use t h e ULTIMATE ( p r o b a b l y u n l i m i t e d ) QUICKEST a l g o r i t h m a l m o s t everywhere ( i . e . ,
i n smooth r e g i o n s ) and s w i t c h t o t h e h i g h e r o r d e r scheme a t
j u s t the r i g h t g r i d points to give the desired high resolution.
F i g u r e s 48
and 49 show r e s u l t s f o r t h e usual t e s t - p r o b l e m s f o r a h y b r i d - 3 / 5 and h y b r i d -
317 s t r a t e g y , r e s p e c t i v e l y , u s i n g a v a l u e o f 0.015 f o r t h e t h r e s h o l d c o n s t a n t . Node v a l u e s i n v o l v e d i n t h e h i g h e r o r d e r component on e i t h e r l e f t or r i g h t
CV
f a c e s (or b o t h ) a r e shown by b l a c k d o t s ; open c i r c l e s d e s i g n a t e t h a t t h e s i m p l e r l i m i t e d t h i r d - o r d e r scheme i s t o be used on b o t h f a c e s o f t h e respect i v e c o n t r o l volume. SOME ASPECTS OF GENERALIZATION
C l e a r l y i n t h i s i n i t i a l paper o u t l i n i n g t h e p r i c i p l e s o f t h e u n i v e r s a l l i m i t e r , a t t e n t i o n has been n a r r o w l y f o c u s s e d on t h e i d e a l i z e d academic ( y e t
s t i l l v e r y c h a l l e n g i n g ) problem o f one-dimensional v e l o c i t y on a u n i f o r m g r i d .
pure advection a t constant
T h i s was done p u r p o s e l y , o f c o u r s e , t o i d e n t i f y
t h e b a s i c problems a s s o c i a t e d w i t h t h e a d v e c t i o n t e r m , t h e m o d e l i n g o f which i s by f a r t h e most d i f f i c u l t n u m e r i c a l a s p e c t and major p a c i n g i t e m i n t h e development o f c o m p u t a t i o n a l f l u i d dynamics.
[ I t makes a b s o l u t e l y no sense, l o g i c a l l y or p r a c t i c a l l y , t o s i m u l a t e a flow u s i n g h i g h l y s o p h i s t i c a t e d m u l t i -
e q u a t i o n t u r b u l e n c e models, f o r example, w i t h an a d v e c t i o n method which essent i a l y r e p l a c e s modeled p h y s i c a l v i s c o s i t y w i t h i n h e r e n t ( o r e x p l i c i t ) a r t f i c i a l v i s c o s i t y t h r o u g h o u t t h e b u l k o f t h e flow domain - b u t t h i s i s s t i l l the "state-of-the-art" cia
i n a d i s t u r b i n g l y l a r g e (and g r o w i n g ) number o f commer-
and r e s e a r c h codes, e s p e c i a l l y i n h e a t t r a n s f e r and r e l a t e d i n d u s t r i e s . ]
But, o b v i o u s l y , i n o r d e r f o r t h e u n i v e r s a l l i m i t e r t o be o f p r a c t i c a l v a l u e a general-purpose code, s e v e r a l g e n e r a l i z a t i o n s w i l l need t o be made.
n
Space i n
a s i n g l e ( a l r e a d y l e n g t h y ) a r t i c l e does n o t p e r m i t d e t a i l e d e x p o s i t i o n s o f such g e n e r a l i z a t i o n s or v e r i f i c a t i o n u s i n g c l a s s i c a l t e s t - p r o b l e m s . be t a k e n up i n f u t u r e papers
-
This w i 1
b o t h by t h e a u t h o r and presumably b y o t h e r
29
r e s e a r c h e r s who may w i s h t o e x t e n d and a p p l y t h e t h e o r y i n v a r i o u s ways.
How-
e v e r , some g e n e r a l i z a t i o n s a r e f a i r l y s t r a i g h t - f o r w a r d and w i l l be sketched i n t h i s s e c t i o n , w i t h o u t showing s p e c i f i c r e s u l t s .
Other g e n e r a l i z a t i o n s o f a
more obvious n a t u r e ( d i f f u s i o n , n o n u n i f o r m g r i d s ) or a more c o n t r o v e r s i a l one (systems o f n o n l i n e a r e q u a t i o n s ) w i l l be b r i e f l y addressed i n t h e c l o s i n g ~
sect ion. Variable Advecting V e l o c i t y The ULTIMATE s t r a t e g y i s e a s i l y extended t o unsteady one-dimensional a d v e c t i o n , where t h e a d v e c t i n g v e l o c i t y i s a f u n c t i o n o f space and t i m e .
For
s i m p l i c i t y , assume t h a t t h e r i g h t - f a c e Courant number i s p o s i t i v e
> o and t h a t i n a l o c a l r e g i o n , 6
(87)
i s increasing monotonically (88)
The update a l g o r i t h m i s based on e q u a t i o n ( l o ) , r e p e a t e d h e r e f o r convenience
where t h e f a c e Courant numbers a r e c o n s i d e r e d t o be known q u a n t i t i e s a t t i m e level method.
n.
A s u s u a l , $,
i s f i r s t e s t i m a t e d u s i n g some d e s i r e d h i g h o r d e r
T h i s i s l i m i t e d b y t h e a d j a c e n t node v a l u e s (90)
Now r e q u i r e , c o n s e r v a t i v e l y , f o r m o n o t o n i c i t y maintenance
T h i s can be r e w r i t t e n , u s i n g e q u a t i o n ( 8 9 > , as (92)
i
and u s i n g a worst-case e s t i m a t e o f
30
$Q, t h i s becomes
for cQ > 0. (If cQ < 0, it may not be appropriate to require persistence of monotonicity.) Of course, for constant velocity, inequality ( 9 3 ) is equivalent to inequality ( 7 2 ) . One further condition is necessary in the variablevelocity case, guaranteeing
which results in a condition on $ Q , for control-volume (i),
or, in terms of +,, viewed as the left face o f CV(i+l), estimate for the "far-right" face value,
given by
using a worst-case
assuming crr > 0. In the constant-velocity case, this is superseded by the right-hand inequality of ( 9 0 ) . The equivalent restrictions for monotonic decreasing regions should be clear. Local extrema are treated in the usual way. Nonlinear Advection Consider one-dimensional nonlinear advection
*+af(4) ax = o
(97)
at
where f is monotonic increasing. This can be rewritten in control-volume form as
$;+I where X that
=
AtlAx, fQ
=
f ( g Q ) , and
=
$7 + X(fQ fr
=
f(+r).
- fr)
(98)
Again, for definiteness, assume
(99)
First, estimate by some desired high order method. As usual, this will be limited by interpolative monotonicity
31
(100)
which is equivalent to (101)
Then, to assure inequality (91),
which is the generalization of equation (93). The condit on analogous to inequality (96) is superseded by the right-hand member of equation (101). Clearly if (103) f($) = u+ for constant u, the limiting conditions revert back to inequality (72). Multidimensional Algorithm Because of the control-volume formulation of the one-dimensional algorithm, it is a straight-forward procedure to extend the ULTIMATE strategy to two and three dimensions. For two dimensions, the explicit update step analogous to equation (10) i s
where "bottom" and "top" CV faces have been introduced, in addition to "left" and "right." Because of strict conservation, QQ(i+l,j)
(105)
Qt(i ,j) = Qb(i ,j+l)
(106)
$r(i,j)
=
and
(and similarly with the Courant numbers). Cell-averaged source-terms can be added to the right-hand side of equation (104), if appropriate. Focusing attention on one CV face (say the left), the first step is to compute some explicit high (third o r higher) order multidimensional estimate for 4%. This is then limited in a manner similar to figure 3 3 , where :G is
32
based on t h r e e node v a l u e s i n a d i r e c t i o n normal t o t h e f a c e :
t h e two s t r a d -
I
d l i n g t h e f a c e t o g e t h e r w i t h t h e n e x t upstream-biased node i n t h e normal d i r e c t i o n , determined b y
SGN(un), where
the face i n question.
un
i s t h e normal component o f v e l o c i t y a t
Thus, i n g e n e r a l t e r m s , t h e l i m i t i n g s t e p i s " l o c a l l y
one-dimensional" i n t h e normal d i r e c t i o n for each f a c e , even though t h e h i g h o r d e r base method i s m u l t i d i m e n s i o n a l .
For example, f i g u r e 50 shows t h e nodes i n v o l v e d f o r t h e l e f t f a c e ( f o r and
uQ as shown) u s i n g t h e a u t h o r ' s u n i f o r m l y t h i r d - o r d e r p o l y n o m i a l i n t e r p o -
vQ l a t i o n a l g o r i t h m (UTOPIA) as t h e base method, w i t h t h e l i m i t e r - n o d e s shown as
b l a c k d o t s and t h e r e m a i n i n g nodes as open c i r c l e s .
Allowing for a l l v e l o c i t y
d i r e c t i o n s on a l l f o u r f a c e s r e s u l t s i n a 1 3 - p o i n t s t e n c i l ( t h e same as t h a t used f o r t h e second-order d i s c r e t e b i h a r m o n i c o p e r a t o r ) . can be designed i n a s i m i l a r way.
H i g h e r o r d e r schemes
Making use o f a d d i t i o n a l nodes i n t h e normal
d i r e c t i o n f o r each f a c e appears t o be more e f f e c t i v e t h a n i n v o l v i n g o t h e r nodes i n the transverse d i r e c t i o n . s i m i l a r procedure.
E x t e n s i o n t o t h r e e dimensions follows an e n t i r e l y
D e t a i l s and r e s u l t s o f r i g o r o u s t e s t problems w i l l be
p r e s e n t e d i n a subsequent paper.
DISCUSSION AND FORECAST The u n i v e r s a l l i m i t e r , p o r t r a y e d i n f i g u r e 33, r e p r e s e n t s an e x t r e m e l y s i m p l e way t o produce e x p l i c i t "monotonic" a d v e c t i o n schemes o f a r b i t r a r i l y h i g h o r d e r . P o t e n t i a l n u m e r i c a l o s c i l l a t i o n s a r e suppressed w i t h o u t c o r r u p t i n g t h e expected r e s o l u t i o n of t h e u n d e r l y i n g scheme. I t has been seen t h a t second-order methods ( i n c l u d i n g well-known s h o c k - c a p t u r i n g or TVD schemes) a r e s i g n i f i c a n t l y i n f e r i o r t o t h e t h i r d - o r d e r ULTIMATE QUICKEST scheme, which uses t h e same f i v e - p o i n t s t e n c i l .
B e t t e r r e s o l u t i o n o f sharp changes i n g r a d i e n t
can be a c h i e v e d by u s i n g a h i g h e r o r d e r base method, and c o n c o m i t a n t l y w i d e r computational s t e n c i l .
However, h i g h e r (even) o r d e r c e n t r a l methods a r e sus-
c e p t i b l e t o a degree of waviness under some circumstances, as seen w i t h t h e semi-ellipse simulation.
I n terms o f o v e r a l l performance, t h e r e seems l i t t l e
t o be gained beyond t h e ULTIMATE f i f t h - o r d e r upwind scheme, or t h e h i g h l y c o s t e f f e c t i v e h y b r i d scheme which uses ULTIMATE QUICKEST i n smooth r e g i o n s and a u t o m a t i c a l l y s w i t c h e s t o a h i g h e r o r d e r ULTIMATE upwind scheme i n i s o l a t e d regions of high curvature.
More p r e c i s e s i m u l a t i o n o f narrow l o c a l extrema can
33
be b u i l t i n t o t h e a l g o r i t h m b y u s i n g a w e i g h t i n g s t r a t e g y which r e l i e s more h e a v i l y o n an u n l i m i t e d h i g h e r o r d e r scheme i n such r e g i o n s , p r o v i d e d a s u i t a b l e m o n i t o r can be d e v i s e d t o d i s c r i m i n a t e between p h y s i c a l and n u m e r i c a l l y produced extrema.
T h i s now appears t o be p o s s i b l e , u s i n g simple p a t t e r n -
r e c o g n i t i o n techniques. Some i n s i g h t i n t o t h e s t e p - r e s o l u t i o n o f v a r i o u s schemes can be g a i n e d by
(6;
c o r r e l a t i n g portions o f a step simulation w i t h regions o f the Gf) p l a n e . F i g u r e 51(a> shows s c h e m a t i c a l l y a s i m u l a t e d s t e p p r o f i l e , i n v e r t e d for more d i r e c t r e f e r e n c e t o t h e NVD.
I n t h e leading-edge (LE) r e g i o n ,
G:
9
i s slightly
l e s s t h a n 1 ; i n t h e t r a i l i n g - e d g e ( T E ) r e g i o n , i t i s s l i g h t l y g r e a t e r t h a n 0. This i s r e f l e c t e d i n f i g u r e 51(b>. corresponds t o l a r g e r both regions.
Gf
A s seen i n f i g u r e 5 1 ( c > , sharper r e s o l u t i o n
values r e l a t i v e t o t h e corresponding
values, i n
T h i s c o r r e l a t i o n can be q u i t e c l e a r l y seen, f o r example, w i t h
t h e Chakravarthy-Osher scheme ( f i g s . 19 and 25) or ULTIMATE second-order u p w i n d i n g ( f i g s . 35 and 39) where t h e NVD c h a r a c t e r i s t i c i s r e l a t i v e l y low i n t h e TE r e g i o n c o r r e s p o n d i n g t o a " b l u n t " t r a i l i n g edge s t e p s i m u l a t i o n , and r e l a t i v e l y h i g h i n t h e LE r e g i o n , g i v i n g a sharp l e a d i n g edge.
The r e v e r s e i s
t r u e f o r t h e ULTIMATE Lax-Wendroff scheme, r e s u l t i n g i n a b l u n t l e a d i n g edge and a sharp t r a i l i n g edge.
The Minmod scheme s e l e c t s t h e lower c h a r a c t e r i s t i c
i n each r e g i o n , l e a d i n g t o r a t h e r d i f f u s e performance; whereas Superbee (or more e f f e c t i v e l y , Super-C> r e l i e s on t h e h i g h e r c h a r a c t e r i s t i c i n each r e g i o n
o f t h e NVD, and t h i s r e s u l t s i n t h e c o m b i n a t i o n o f sharp l e a d i n g and t r a i l i n g edges w i t h c o n c o m i t a n t narrow r e s o l u t i o n of t h e s t e p .
The a r t i f i c i a l l y h i g h v a l u e s o f
Gf
f o r t h e l a t t e r schemes i s e q u i v a l e n t
t o a r t i f i c i a l compression, or ( t o t h e e x t e n t t h a t t h e NVD c h a r a c t e r i s t i c l i e s above a well-behaved scheme such as ULTIMATE QUICKEST) t o n e g a t i v e a r t i f i c i a l diffusion.
For example, t h e l i m i t e d Lax-Wendroff scheme e x h i b i t s a r t i f i c i a l
negative d i f f u s i o n for
5;
< 0.5; whereas f o r 1 i m i t e d second-order u p w i n d i n g
and t h e Chakravarthy-Osher scheme, t h i s appears i n t h e l e a d i n g edge r e g i o n , where t h e NVD c h a r a c t e r i s t i c i s based on f i r s t - o r d e r downwinding.
This
i n t e r p r e t a t i o n c o r r e l a t e s w i t h t h e observed t r a i l i n g o s c i l l a t i o n s o f t h e u n l i m i t e d Lax-Wendroff scheme ( f i g . 4) and t h e l e a d i n g o s c i l l a t i o n s o f u n l i m i t e d second-order u p w i n d i n g ( f i g . 5 ) .
34
The g r e a t advantage of t h e ULTIMATE s t r a t e g y i s t h a t i t can be used w i t h I
e x p l i c i t schemes o f a r b i t r a r i l y h i g h o r d e r , g i v i n g c o n c o m i t a n t r e s o l u t i o n o f sudden changes i n g r a d i e n t w i t h o u t a r t i f i c i a l s t e e p e n i n g and c l i p p i n g .
The
I
l i m i t e d p o l y n o m i a l schemes r e v i e w e d here a l r e a d y g i v e e x c e l l e n t r e s u l t s (above second o r d e r ) , b u t c l e a r l y t h e ULTIMATE p h i l o s o p h y can be used w i t h o t h e r more s o p h i s t i c a t e d and more a p p r o p r i a t e forms o f s p a t i a l i n t e r p o l a t i o n .
One such
method c u r r e n t l y under i n v e s t i g a t i o n i n v o l v e s e x p o n e n t i a l ( " t e n s i o n " ) s p l i n e s a p p l i e d l o c a l l y ; t h i s technique offers
p o s s i b i l i t i e s o f very high r e s o l u t i o n
on a compact s t e n c i l . E x t e n s i o n t o m u l t i d i m e n s i o n a l flow i s s t r a i g h t f o r w a r d and, as mentioned, t h i s w i l l form t h e s u b j e c t o f f u t u r e papers.
A l t h o u g h n o t addressed i n t h i s
paper ( s i n c e a t t e n t i o n has been focused on t h e c r i t i c a l problem o f p u r e a d v e c t i o n ) i t s h o u l d be c l e a r t h a t t h e i n t r o d u c t i o n o f modeled p h y s i c a l d i f f u s i o n t o an o r d e r c o n s i s t e n t w i t h t h e a d v e c t i o n scheme p r e s e n t s a b s o l u t e l y no problems.
The i n c l u s i o n o f p h y s i c a l d i f f u s i o n i n f a c t p u t s l e s s demands on t h e
a d v e c t i v e a l g o r i t h m ; i n most cases o f p r a c t i c a l i n t e r e s t , t h e ULTIMATE QUICKEST scheme g i v e s r e s u l t s which a r e g r a p h i c a l l y i n d i s t i n g u i s h a b l e from t h e e x a c t s o l u t i o n (when known). p r e s e n t s no problems.
I n a s i m i l a r way, e x t e n s i o n t o s p a t i a l l y v a r y i n g g r i d s The h i g h e r o r d e r accuracy o f t h e base method can be
f o r m a l l y maintained ( r e f s .
9 and 16); o r , p r o v i d e d t h e expansion r a t i o does n o t
exceed about 125 p e r c e n t , t h e s i m p l e r u n i f o r m - g r i d f o r m u l a s f o r t h i r d - and h i g h e r o r d e r methods can be used d i r e c t l y w i t h o u t r e d u c i n g t h e p r a c t i c a l o r d e r
o f accuracy
-
a l t h o u g h t h e f o r m a l o r d e r i s , o f course, reduced ( r e f . 17).
One o f t h e most i m p o r t a n t q u e s t i o n s , o f course, concerns how w e l l t h e ULTIMATE s t r a t e g y g e n e r a l i z e s t o h a n d l e s y s t e m s o f n o n l i n e a r e q u a t i o n s such as t h e E u l e r (or Navier-Stokes) e q u a t i o n s o f gasdynamics, and t h e " s h a l l o w - w a t e r " and r e l a t e d e q u a t i o n s o f oceanography and m e t e o r o l o g y .
T h i s i s by no means a
t r i v i a l g e n e r a l i z a t i o n and w i l l no doubt form t h e s u b j e c t o f f u t u r e p a p e r s ; however, i t i s f a i r t o p r e d i c t t h a t ULTIMATE s i m u l a t i o n s w i l l be a t l e a s t as s u c c e s s f u l as t h e b e t t e r s h o c k - c a p t u r i n g schemes, s i n c e t h e u n d e r l y i n g p h i l o s o phy i s very s i m i l a r .
I n p a r t i c u l a r , because o f n a t u r a l p h y s i c a l compression,
shock-wave or h y d r a u l i c jump ( o r atmospheric f r o n t ) r e s o l u t i o n can be expected
t o be narrower t h a n t h e s c a l a r s t e p r e s o l u t i o n s t u d i e d h e r e .
And because of
35
t h e a b i l i t y t o use a r b i t r a r i l y h i g h o r d e r r e s o l u t i o n , unsteady gasdynamic simul a t i o n s can be expected t o g i v e e x t r e m e l y r e l i a b l e r e s u l t s , even a t v e r y h i g h Mach numbers and n o n i d e a l c o n d i t i o n s .
Because of t h e c o n s e r v a t i v e c o n t r o l -
volume f o r m u l a t i o n , m u l t i d i m e n s i o n a l gasdynamic, and g e o p h y s i c a l a l g o r i t h m s a r e The g e n e r a l p h i l o s o p h y h e r e i s t o model t h e
expected t o be s i m i l a r l y r o b u s t .
c o n t r o l - v o l u m e a d v e c t i v e modes p r e c i s e l y and l e t t h e p h y s i c s (as r e f l e c t e d i n t h e g o v e r n i n g e q u a t i o n s ) t a k e c a r e o f t h e " a c o u s t i c " or wave modes.
This idea
seems n o t t o work w e l l w i t h c u r r e n t l y p o p u l a r second-order f l u x - l i m i t e d
schemes
because o f these schemes' a d d i c t i v e r e l i a n c e on l o c a l l y v a r y i n g ( p o s i t i v e or n e g a t i v e ) a r t i f i c i a l v i s c o s i t y , r e q u i r i n g p r e c i s e knowledge o f c h a r a c t e r i s t i c speeds and d i r e c t i o n s .
But t h e p r i n c i p l e has g r e a t p o t e n t i a l when used w i t h
h i g h e r o r d e r ULTIMATE m u l t i d i m e n s i o n a l a l g o r i t h m s , which, by d e s i g n , r e q u i r e o n l y advective v e l o c i t i e s a t the
CV
c e l l f a c e s , as p r e s c r i b e d by t h e govern-
i n g equations. F i n a l l y , t h e ULTIMATE s t r a t e g y can be a p p l i e d t o s t e a d y - f l o w multidimensional s i m u l a t i o n s (even though t h e acronym m i g h t be somewhat o u t o f place).
Q u i t e s i m p l y , t h e s t e a d y - s t a t e l i m i t e r c o n s i s t s o f ULTIMATE w i t h t h e
Courant number s e t t o z e r o .
T h i s , of course, r e s u l t s i n t h e e x t r e m e l y s i m p l e
c o n d i t i o n i n t h e monotonic regime
with
+f-
=
iC, or s i m i l a r
t h e nonmonotonic range.
c h a r a c t e r i s t i c p a s s i n g t h r o u g h (0,O) and (l,l), in I n o t h e r words, f o r l o c a l l y monotonic b e h a v i o r i n t h e
d i r e c t i o n normal t o a g i v e n estimate o f
4f
CV
face, a f t e r a high-order multidimensional
i s made, t h e f a c e v a l u e a c t u a l l y used i s c o n s t r a i n e d s i m p l y t o
l i e between t h e a d j a c e n t upstream and downstream node v a l u e s : respectively.
4u and +D,
The a u t h o r ' s m u l t i d i m e n s i o n a l SHARP a l g o r i t h m ( r e f . 151, s i m p l e
h i g h accuracy r e s o l u t i o n program, i s a t h i r d - o r d e r n o n l i n e a r scheme c o n f o r m i n g t o these r e q u i r e m e n t s ; i t i s based on an e x p o n e n t i a l u p w i n d i n g or l i n e a r e x t r a p o l a t i o n r e f i n e m e n t o f t h e m u l t i d i m e n s i o n a l QUICK scheme ( r e f . 1 6 ) .
But, o f
course, t h e p r i n c i p l e can be extended t o a r b i t r a r i l y h i g h o r d e r u s i n g t h e u n i v e r s a l l i m i t e r f o r t i g h t r e s o l u t i o n and accuracy, t h u s g i v i n g ULTRA-SHARP simul a t i o n o f steady m u l t i d i m e n s i o n a l flows c o n t a i n i n g n e a r - d i s c o n t i n u i t i e s .
In a
p r a c t i c a l a l g o r i t h m , i t i s i m p o r t a n t t o c o n s t r u c t a s i n g l e - v a l u e d upper boundary
36
I
for the allowable NVD region near
5C + O+.
This is most conveniently achieved
by requiring
in addition t o inequality (107), with suitably large values of the slope constant (-10 o r 100). The piecewise linear NVD characteristic of the steady twodimensional SMART algorithm of Gaskell and Lau (ref. 18) is also of this form. Steady-state methods based o n second- and third-order schemes (including TVD limiters) can be solved with straight-forward AD1 scalar penta-diagonal matrix algorithms using Gaskell-and-Lau's modified (nonlinear) curvature-factor technique, similar t o the curvature factor, CF(G:), appearing i n equation (55). Alternatively, arbitrarily high order limited schemes can be solved by popular AD1 tridiagonal methods using the author's downwind weighting factor technique (ref. 191, thus rendering the ULTRA-SHARP very-high-order multidimensional nonoscillatory steady-flow schemes immediately compatible with many of the general-purpose commercial and research steady-state elliptic solvers currently in use, which are typically based on low-accuracy blended combinations of first- and second-order advection methods. ACKNOWLEDGMENTS AND PERSPECTIVE Much of the programming and computer graphics manipulation for the large number of test cases shown (and countless others not included) was ably handled by Hassan S. Niknafs. As i s often the case in scientific investigations, the research reported here was developed independently of other similar earller work, i n this case during the latter part o f 1987 while the author was an ICOMP Visiting Research Engineer. The idea of the universal limiter stems from the author's development in 1974 of exponential upwinding (refs. 20 and 21), a third-order nonlinear nonoscillatory advection scheme for steady and unsteady flows which predated (ref. 22) QUICK and QUICKEST (ref. 9 > , and from extensive experience over the past decade o r more with the normalized variable diagram and associated monotonicity criteria (ref. 20). The author is indebted t o Dr. S.H. Chang, another ICOMP researcher, for pointing out some of the more intellig ble shock-capturing and TVD literature - particularly several articles by Helen Yee (see ref. 23) and the 1984 survey paper by P.K. Sweby (ref. 10). As descr bed here (in the appendix), Sweby works with a much more restrictive
37
"TVD r e g i o n , " c o n f o r m i n g t o c u r r e n t p r a c t i c e i n o p e r a t i o n a l schemes, a l t h o u g h
t h e l e s s r e s t r i c t i v e c o n d i t i o n s would have been known t o him, s i n c e t h e y had appeared i n a GAMM proceedings paper b y Roe and Baines ( r e f . 24) i n 1982 and i n o t h e r proceedings papers b y Roe ( r e f s . 6 and 25) i n 1983. a t t e n t i o n was d i r e c t e d toward t h e l a t t e r papers by S . T . 1987 and b y P . L . Roe i n May 1988.
The a u t h o r ' s
Zalesak i n November
I n t e r e s t i n g l y enough, Roe's 1986 r e v i e w
a r t i c l e ( r e f . 26) makes no mention o f t h e l e s s r e s t r i c t i v e l i m i t e r b o u n d a r i e s , b u t r a t h e r d e s c r i b e s i n s t e a d o n l y some o f t h e more r e s t r i c t i v e l i m i t e r s ( s u c h as Minmod, "CLAM," and Superbee), even though t h e e a r l i e r proceedings papers a r e referenced. Z a l e s a k ' s IMACS paper ( r e f .
12) d e s c r i b e s work b y h i s c o l l e a g u e , John
Lyon, a t t h e Naval Research L a b o r a t o r y , who appears t o be t h e o n l y o t h e r r e s e a r c h e r t o have used a r b i t r a r i l y h i g h (up t o e i g h t h ) o r d e r a d v e c t i o n schemes c o n s t r a i n e d by an e q u i v a l e n t of t h e u n i v e r s a l l i m i t e r . and A . K . C .
F i n a l l y , P.H. G a s k e l l
Lau, a t t h e U n i v e r s i t y o f Leeds, have ( a g a i n i n d e p e n d e n t l y , follow-
i n g from t h e i r s t e a d y - f l o w work ( r e f . 1 8 ) ) developed an e q u i v a l e n t l i m i t i n g s t r a t e g y f o r unsteady flow a p p l i e d t o a q u a s i - t h i r d - o r d e r
a d v e c t i o n scheme.
They r e p o r t h i g h l y a c c u r a t e s i m u l a t i o n s o f s t a n d a r d shock-tube t e s t problems w i t h o u t r e s o r t i n g t o f l u x v e c t o r s p l i t t i n g , Riemann s o l v e r s , or any of t h e o t h e r c o m p l e x i t i e s a s s o c i a t e d w i t h c u r r e n t l y p o p u l a r forms o f f i r s t / s e c o n d o r d e r s h o c k - c a p t u r i n g schemes.
T h i s c o n f i r m s t h e a u t h o r ' s b e l i e f t h a t charac-
t e r i s t i c decomposition i s n o t a n e c e s s i t y , b u t r a t h e r a symptom o f p o o r l y designed a d v e c t i o n schemes, based on v a r i a b l e a r t i f i c i a l v i s c o s i t y , which r e q u i r e d e t a i l e d e x p l i c i t knowledge o f a l l c h a r a c t e r i s t i c v e l o c i t i e s and a m p l i tudes.
I f t h e a d v e c t i o n (macroflux) terms a p p e a r i n g i n t h e c o n s e r v a t i o n equa-
t i o n s for mass momentum and energy a r e c a r e f u l l y modeled w i t h s u i t a b l y l i m i t e d h i g h e r o r d e r methods and m i c r o f l u x terms ( p r e s s u r e t e n s o r , e t c . ) t r e a t e d approp r i a t e l y , t h e g o v e r n i n g e q u a t i o n s themselves w i l l a u t o m a t i c a l l y g e n e r a t e t h e c o r r e c t a d d i t i o n a l wave modes.
The f a r - r e a c h i n g
i m p l i c a t i o n s o f t h i s , espe-
c i a l l y for m u l t i d i m e n s i o n a l flows, s h o u l d be c l e a r .
38
I
APPENDIX
I
I I
I
I
In a well-known paper (ref. lo), Sweby introduced sufficient conditions for schemes to have total-variation-diminishing properties, and portrayed these conditions in what has become known as the "Sweby diagram," a plot of the fluxlimiter factor, Q , against the local gradient ratio, r. For constant Courant number, the flux-limiter factor is defined as
where 9, is the limited face value, 9;
is the adjacent upstream node value,
and 9;' is the face value according to second-order central differencing (Lax-Wendroff) 0kW --
;(9;
+
9;)
-
y
(9;
- 9;)
(A.2)
where 9,n is the adjacent downstream value, as usual. All of the schemes considered by Sweby are linear in Courant number in the form of equation ( 4 9 ) , rewritten for c 0, as
9,
=
(1 - Icl)9;
+
Id+;
(A. 3)
The Lax-Wendroff face value can also be written in this form
4y
=
(1 -
Icl)+f CEN +
IC/+;
(A. 4)
where +f Substituting ( A . 3 ) and is the linear interpolation 1/2(9: + 4);. ( A . 4 ) into ( A . 1 ) results in an equivalent expression for the flux-limiter factor
which can be rearranged as
interpreted as first-order upwinding modified by adding a term proportional to the difference between linear interpolation and first-order upwinding. 39
~-
~
I n terms o f t h e n o r m a l i z e d v a r i a b l e s as d e f i n e d i n t h e p r e s e n t paper, i t i s n o t d i f f i c u l t t o show t h a t
'9=
(a; y2 1
-
a;) a;)
-
(af - )a; $1
-
Jcp(1
(A.7)
- )a;
S i m i l a r l y , t h e g r a d i e n t r a t i o used by Sweby i s
(A.8)
The s o - c a l l e d TVD r e g i o n proposed by Sweby consists o f f , r s t - o r d e r ing
((P
= 0)
for negative
upwind-
r, f o l l o w e d by for
O(cp(2r
O
~
r
i
l
and t h e n l i m i t e d by f i r s t - o r d e r downwinding a t t i m e - l e v e l
(A.9) -n n, 9,
=
1 i n equa-
t i o n (A.7), g i v i n g 0 5 ~ 5 2 for
r -> l
(A. 10)
Note t h a t e q u a t i o n (A.9) can be i n t e r p r e t e d as (A.11) Figure A . l
compares Sweby's TVD r e g i o n (shaded) i n t e r p r e t e d i n t h e
p l a n e and i n t h e ( r ,
(P)
plane.
(6; , $)
For r e f e r e n c e , Roe's Superbee scheme i s shown,
p a s s i n g t h r o u g h t h e "second-order" p o i n t , (0.5, 0.75) i n f i g u r e A . l ( a > , and (1,l)
i n figure A.l(b).
By c o n t r a s t , f i g u r e A.2 g i v e s t h e "extended Sweby d i a -
gram" c o r r e s p o n d i n g t o t h e ULTIMATE s t r a t e g y o f f i g u r e 33; r e c a l l t h a t t h a t f i g u r e involved s l o p i n g boundary
Gf OB
r a t h e r than
$:
, as used i n f i g . A . l ( a ) .
Note t h a t t h e
i n t h e extended Sweby diagram has a s l o p e o f
2/lcl
as
Also n o t e t h a t t h e upper boundary o f t h e extended diagram i n c r e a s e s i n h e i g h t as
compared w i t h t h e s l o p e o f 2 f o r
OC
i n t h e o r i g i n a l Sweby diagram.
I c l + 1; i n f a c t ( A . 12)
I
40
I
whereas, in the original diagram figure A.l(b), dent of c:
the upper boundary is indepen-
As the Courant number varies, the point B in figure A.2 "slides" along the line 'pB = 2(1 + r B ) ,where rB = Icl/(l - Icl). For reference, the Super-C scheme is shown in figure A.2. Hyper-C simply follows the upper boundary: AOBA. As i s immediately obvious, Sweby's TVD region is grossly overrestrictive, resulting in predicted (normalized) face values which are too small, especially for 6; values slightly larger than 0 o r slightly smaller than 1. This is one reason for the poor performance o f currently popular second-order TVD schemes.
41
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1. van Leer, B.:
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11.
Toward t h e U l t i m a t e C o n s e r v a t i v e D i f f e r e n c e Scheme. 111. Upstream-Centered F i n i t e - D i f f e r e n c e Schemes f o r I d e a l Compressible Flow. J. Comput. Phys., v o l . 23, no. 3, Mar. 1977, pp. 263-275.
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A S t a b l e and A c c u r a t e C o n v e c t i v e M o d e l l i n g Procedure Based on Q u a d r a t i c Upstream I n t e r p o l a t i o n . Computer Methods i n A p p l i e d Mechanics and E n g i n e e r i n g , v o l . 19, no. 1 , June 1979, pp. 59-98.
A Method f o r Reducing D i s p e r s i o n i n C o n v e c t i v e D i f f e r e n c e 10. Fromm, J.E.: Schemes. J. Comput. Phys., v o l . 3, no. 2, O c t . 1968, pp. 176-189.
High R e s o l u t i o n Schemes U s i n g F l u x L i m i t e r s f o r H y p e r b o l i c 11. Sweby, P . K . : C o n s e r v a t i o n Laws. S I A M J . Numer. A n a l . , v o l . 21, no. 5, O c t . 1984, pp. 995-101 1 . A P r e l i m i n a r y Comparison o f Modern Shock-Capturing 12. Zalesak, S.T.: Schemes: L i n e a r A d v e c t i o n . Advances i n Computer Methods f o r P a r t i a l D i f f e r e n t i a l Equations V I , R . V i c h n e v e t s k y and R . S . Stepelman, eds., IMACS, Rutgers U n i v e r s i t y , 1987.
42
1
1
13. Harten, A . : H i g h R e s o l u t i o n Schemes for H y p e r b o l i c C o n s e r v a t i o n Laws. Comput. Phys., v o l . 49, no. 3, Mar. 1983, pp. 357-393.
J.
14. Chakravarthy, S.R.; and Osher, S . : High Resolution A p p l i c a t i o n s o f the Osher Upwind Scheme for t h e E u l e r E q u a t i o n s . A I A A Paper 83-1943, July 1983. 15. Leonard, B.P.: SHARP S i m u l a t i o n o f D i s c o n t i n u i t i e s i n H i g h l y C o n v e c t i v e Steady Flow. NASA TM-100240, 1987. E l l i p t i c Systems: F i n i t e D i f f e r e n c e Method I V . Handbook 16. Leonard, B.P.: of Numerical Heat T r a n s f e r , W.J. Minkowicz, E.M. Sparrow, G.E. S c h e i d e r , and R.H. P l e t c h e r , eds., W i l e y , 1988, Chapter 9, pp. 347-378. S t u d i e s i n Numerical Computations o f 17. C a s t r o , I . P . ; and Jones, J.M.: R e c i r c u l a t i n g Flows. I n t . J. Numer. Methods F l u i d s , v o l . 7, no. 8, Aug. 1987, pp. 793-823. 18. G a s k e l l , P.H.; and Lau, A.K.C.: C u r v a t u r e Compensated C o n v e c t i v e T r a n s p o r t : SMART, a New Boundedness P r e s e r v i n g T r a n s p o r t A l g o r i t h m . J . Numer. Methods F l u i d s , v o l . 8, no. 6, June 1988, pp. 617-641. Sharp Numerical 19. Leonard, B.P.: C o n v e c t i v e F l o w s . I n BAIL V -Conference on Computational and I n t e r i o r Layers, Shanghai, June Boole Press, D u b l i n ( i n p r e s s ) .
Int.
S i m u l a t i o n o f T h i n Layers i n H i g h l y Proceedings o f t h e F i f t h I n t e r n a t i o n a l A s y m p t o t i c Methods f o r Boundary and 1988, J.J.H. M i l l e r and B.Y. Guo, eds.,
20. Leonard, B . P . : A d j u s t e d Q u a d r a t i c Upstream A l g o r i t h m s f o r T r a n s i e n t I n c o m p r e s s i b l e C o n v e c t i o n . Computational F l u i d Dynamics Conference, A I A A , 1979, pp. 226-233. A Survey o f F i n i t e D i f f e r e n c e s w i t h Upwinding f o r Numerical 21. Leonard, B . P . : M o d e l l i n g o f t h e I n c o m p r e s s i b l e C o n v e c t i v e D i f f u s i o n E q u a t i o n . Computat i o n a l Techniques i n T r a n s i e n t and T u r b u l e n t Flow, C. T a y l o r and K . Morgan, eds., P i n e r i d g e Press, Swansea, UK, 1981, pp. 1-23.
22. Leonard, B . P . ; Vachtsevanos, G.J.; and Abood, K . A . : Unsteady-State, Two Dimensional I n t r u s i o n Model f o r an E s t u a r y . A p p l i e d Numerical M o d e l l i n g , C.A. B r e b b i a , ed., Wiley, 1978, pp. 113-123. 23. Yee, H.C.: 1987.
Upwind and Symmetric Shock-Capturing Schemes.
NASA TM-89464,
24. Roe, P.L.; and Baines, M.J.: A l g o r i t h m s f o r A d v e c t i o n and Shock Problems. Proceedings o f t h e 4 t h GAMM Conference on Numerical Methods i n F l u i d Mechanics, H. V i v i a n d , ed., Vieweg, West Germany, 1982, pp. 281-290. 25. Roe, P.L.; and Baines, M.J.: A s y m p t o t i c Behaviour o f Some Non-Linear Schemes f o r L i n e a r A d v e c t i o n . Proceedings o f t h e 5 t h GAMM Conference on Numerical Methods i n F l u i d Mechanics, M. P a n d o l f i , and R. P i v a , eds., Vieweg, West Germany, 1983, pp. 283-290.
43
26. Roe, P . L . : C h a r a c t e r i s t i c - B a s e d Schemes for t h e E u l e r Equations. Annual Reviews o f F l u i d Mechanics, Vol. 18, M . Van Dyke, J . V . Wehausen, and J . L . Lumley, e d s . , Annual Reviews I n c . , 1986, p p . 337-365.
r-----
a
0
0
0
'
0
L----A
:
.
a
0
0
(d) FIGURE 1. - CONTROL-VOLUME STENCILS FOR ( a ) FIRST-ORDER UPWINDING AND SECOND-ORDER CENTRAL; (b) SECOND- AND THIRD-ORDER UPWINDING, FROMM'S METHOD AND FOURTH-ORDER CENTRAL: ( C ) FIFTH-ORDER UPWINDING AND SIXTH-ORDER CENTRAL: ( d ) SEVENTH- AND EIGHTH-ORDER SCHEMES.
44
SINE-SQUARED PROF I LE
I N I T I A L CONDITIONS
00
0'.25
0'.50
0.75
00
X
( a ) ISOLATED SINE-SQUARED WAVE.
TI
UNIT - S T E P PROF1 L E
0
I
I Y I T L A L CONDITIONS
I
?I
01
'0'. 00
0.25
0.50
0.75
I .DO
X
(b) UNIT STEP. Ln 0
-I
I PSE PROF I L E
SEMI -ELL
t 1
I N I T I A L CONDITIONS v, 0 0
00
t c ) SEMI-ELLIPSE.
FIGURE 2. - INITIAL CONDITIONS FOR THE THREE TEST PROFILES.
45
u 0 ,
-
I
FIRST-ORDER
-1
UPWINDING
F I R S T - O R D E R LJPWINDING
I
I
s:i d
1
( 0 00
l
,
0.25
0 50
0.75
X ln 0
-
F I R S T - ORDEf 0
-
Lo 0
-0
I
a
C O U R NUM = 0 . 0 5 ABSERROR = 7 . 7 6 WAVINESS = 2.00
0
0
m 0 0
'[
X
(a) c = 0 . 0 5 . FIGURE 3.
46
- RESULTS FOR FIRST-ORDER UPWINDING.
i 00
I
I 0 C
-
FIRST-ORDER UPWINDING 0
-
Lo 0
I -0 [L
COUR NUM = 0 . 5 0 ABSERROR = 4.52 WAVINESS = 1.46
0 D
0
v) 0
0
'[
0.25
IO
0'. 50
0'. 7 5
I '. 00
X
-1
Lo 0
F I R S T - O R D E R UPWINDING
COUR NUM
=
0.50
X Lo D
-
FIRST-ORDER UPWINDING 0 0
COUR NUM Ln 0
I
-0
0
=
0.50
ABSERROR = 5 . 2 3 WPVINESS = 1.74
0 0
0
Lo 0
0
10
0.50
0.25
0.75
00
X
(b) c FIGURE 3 .
-
=
0.5.
CONCLUDED.
47
LD 0
LAX-WENDROFF 0 0
COUR NUM = 0 . 0 5 LD 0
ABSERROR = 2.07 WAVINESS = 0 . 9 5
-0
I
a
0
0
Lo 0
0
'(
0.50
0.25
10
0.75
00
X LT c
,
L A X - WEN DROFF 0 0
C O U R NUM LD 0
=
0.05
ABSERROR = 5 . 2 0 WAVINESS = 4 . 5 4
I -0
a 0
1
0
VI 0
0
(I 10
0'. 2 5
0'. 50
. 00
0'. 75
X LD 0
I
LAX-WENDROFF
COUR NUM = 0 . 0 5
?b--1
I
0.00
0.25
0.50
0.75
X
(a) c
=
0.05.
FIGURE 4 . - RESULTS FOR THE LAX-WENDROFF METHOD (SECOND-ORDER CENTRAL).
48
I
.oo
In 0
-
L AX-WENDROFF 0
-
Y) 0
-6 I a
COUR NUN = 0 . 5 0 ABSERROR = 1 . 4 4 WAVINESS = 0.65
0 0
P
0
I 0 n 0
0'. 25
' I 00
0'. 50 X
0'. 75
.oo
Y) 0
L
LAX-WENDROFF 0 0
-
Y) 0
I -0
a
COUR NUM = 0 . 5 0 ABSERROR = 2.82 WAVINESS = 2.42
0 0
0 0
'(
0.25
IO
0.50
0.75
I .oo
0.75
1
X n 0
-1
LAX-WENDROFF
COUR NUM = 0 . 5 0
ABSERROR
=
2.62
31
n
0
'0.00
0.25
0.50
1. o o
X
(b) c = 0.5. FIGURE 4.
- CONCLUDED.
49
10 0
:I
SECOND-ORDER
COUR NUM
=
UPWINDING
0.05
,
!]
,
0
‘0.00
I
0.25
0.50
, 0.75
00
X I 0 n
SECOND-ORDER
UPWINDING
0
COUR NUM = 0.05 AESERROR = 3.56 WAVINESS = 2.32
m 0 -G I
[L
0
0
Ln 0
0
’(
IO
6. 25
0.50
0’.7s
1
X 0 10
SECOND-ORDER
UPWINDING
0 a
n 0
-6 I n
0 0
0
0 10 0
‘I 10
0.25
0.50
0.75
X
( a ) c = 0.05. FIGURE 5.
50
- RESULTS FOR SECOND-ORMR UPWINDING.
I .00
Lo 0
-
SECOND-ORDER
UPWINDING
0
COUR N U M = 0 . 5 0 ABSERROR = 1 . 4 4 WAVINESS = 0.65 0
0
h
SECOND-ORDER
UPWINDING
0
CO?IR NUM = 0 . 5 0 ABSERROR = 2 . 8 2 WAVINESS = 2 . 4 2
Z
lo.oo
L
0.25
0.50
0.75
I
1
.oo
X Lo 0
COUR N U M = 0 . 5 0
0.50
0.25
0.75
I .oo
X
(b) c = 0.5. FIGURE 5 .
- CONCLUDED.
51
0 Y)
FROMM'S METHOD
I
COUR N b M = 0 . 0 5
ASSERXCR = 0.P5
WAVINESS = 0.39 0
0
0.50
0.75
00
0.75
I .GO
X v) 5
FROMM'S METHOD
COUR N U M = 0 . 0 5
-
-0 ?L
WbV;NESS
+I-.0 . 0 0
= '.?5
:-
0.25
2.:
FROMM'S METHOD
COUR N U N = 0 . 0 5 ABSERROR = 1 . 7 0 WAVINESS = 1.51
\
r
d
10
0.25
0.50
0. 75
X
( a ) c = 0.05.
FIGURE 6. -
52
RESULTS FOR FRCM'S ETHOD.
00
-1
FROMM'S
METHOD
COUR NUM = 0 . 5 0 ABSERROR = 0 . 2 6 WAVINESS = 0.15 0 0
-
" 0 0.00
.
0.25
0.50
0.75 00
X 0
Y)
COUR NUM
'0'. 00
0'. ' 2
=
0.50
0'. 25
0.50
0.75
11.00
0.75
00
X Lo 0
FROMM'S
METHOD
0 0
-
Y) 0
-0
I
a
COUR NUM = 0 . 5 0 ABSERROR = 1.16 W A V I N E S S = 1.29
0 0
0
I0 n 0
'(
10
0.25
0.50
X
(b) c = 0.5.
FIGURE 6. - CONCLUDED.
53
VI 0
QUICKEST(TH1RD-ORDER UPWINDING)
COUR NUM = 0 . 0 5 ABSERROR = 0 . 4 1 WAVINESS = 0.22
m 0 -0 I
a 0 0
0
71
,
0
@O.OO
0.50
0.25
,
0.75
00
X 0 0
QUICKEST(TH1RD-ORDER 0
-
'
2
UPWINDING)
COUR NUM = 0 . 0 5 ABSERROR = 1 . 6 6 WAVINESS = ;.83
I
0
QUICKEST (THIRD-ORDER UPWINDING) 0
*
0
-6 I
a
COUR NUM = 0 . 0 5 ABSERROR = 1 . 4 1 WAVINESS = 1.40
D 0
,
0
a1 '0.00
,
,
0.25
,
0.50
0.75
X
(a) c
=
0.05.
FIGURE 7. - RESULTS FOR QUICKEST (THIRD-ORDER UPWINDING).
54
.oo
-1
v) 0
QUICKEST (THIRD-ORDER
UPWINDING)
0
COUR N U M ABSERROR ABSERROR WAVINESS
-!2
= 0.50 = 0.26 = 0.26 = = 0.15
,
0
lo.oo
0.25
, 0.50
I
0.75
00
X v) 0
OUICKEST(TH1RD-ORDER
UPWINDING)
COUR NUM = 0 . 5 0
0
'0.00
0.25
0.50
0.75
I
.oo
55
-1
Y) 0
FOURTH-ORDER CENTRAL
0 n
COUR N J M Lo 0
=
0.05
ABSERR3R = 0.21 WAVINESS = G . 2 1
-0-
I
Q
0
0
Y) 0
I
'
0.00
I
COUR R U M
=
ABSERRCP =
k'P\ ;hESS
I
Q
0.50
9'.25
0.75
I .oo
0.35 3.22
= L.03
I
'0'. 00
0.25
0.50
0.75
I .oo
0.75
00
X Y) 0
FOURTH-ORDER CENTRAL 0
-
Y) 0
-0
I
WAVINESS = 2.76
IL
0 0
Y) 0
0
'(
IO
0.25
0'. 50
X
( a ) c = 0.05. F I G U R E 8. - R E S U L T S FOR FOURTH-ORDER CENTRAL.
56
FOURTH-ORDER CENTRAL
COUR NUM = 0 . 5 0
00
FOURTH-ORDER
CENTRAL
0
-
i
C O U R N U M = 0.50 ABSERROR = 1 . 5 7 , d A \ / i & E S S = 2.09
-2i
0 '
s
~
:I
---/d '0.00
0.25
0.50
I .oo
0.75
X
-1
FOURTH-ORDER CENTRAL
51 lo.oo
I
0.25
, 0.50
__ 0.75
I 00
X
(b) c = 0.5 FIGURE 8. - CONCLUDED.
57
-1
FIFTH-ORDER UPWINDING
C O U R N U M = 0.05 ABSERSOR = 0 . 0 7 WAVINESS = 0 . 0 8
a! lo.oo
0.25
0.50
0.75
00
X m 0
-1
FIFTH-ORDER UPWINDING
0
-
0.50
0.25
0.75
.oo
0.75
00
X Lo 0
FIFTH-ORDER UPWINDING 0
Lo 0
I -0 [L
COUR NUM = 0.05 ABSERROR = 0 . 9 5 WAVINESS = 1 . 3 1
0
0
0 Lo 0
'(
IO
6. 25
0.50
X
(a) c
=
0.05.
FIGURE 9. - RESULTS FOR FIFTH-ORDER UPWINDING.
58
-1
LD 0
FIFTH-ORDER UPWINDING
COUR NUM = 0 . 5 0 ABSERROR = 0 . 0 5 WAVINESS = 0.06 0
0
a/
,
,
,
0
'0.00
0.50
0.25
I 00
0.75
X
-1 Lo 0
-0
I
a
FIFTH-ORDER UPWINDING
COL'R NL'M = 0 . 5 0 ABSERROK = 1 . 0 7 WAVINESS = 1.6E
'6.00
0'. 25
0.50
0.75
0.50
0.75
I
.oo
Y) 0
-
FIFTH-ORDER UPWINDING 0
In 0
-6
I
a
COUR NUM = 0 . 5 0 A B S E R R O R = 0.81 W A V I N E S S = 1.22
D 0'
L 0 n
0.
'0
0
I
0.25
I . 00
X (b) c = 0 . 5 . FIGURE 9.
- CONCLUDED.
59
0
Y)
-1
SIXTH-ORDER
CENTRAL
0 Q
COUR NUM = 0.05 ABSERROR = 0.10 WAVINESS = 0 . 1 2 0 0
0
51
,
-1
SIXTH-ORDER
COUR NUM
,
0.25
'0.00
=
0.50
0.75
I
.oo
CENTRAL
0.05
ABSERROR = 2 . 4 9
WAViNESS = 3 . 8 s
I 0 0
-1
7
SI X T H - O R D E R C E N T R A L
COUR N U M = 0 . 0 5
L------,-i (0.00
0.25
0.50
0.75
X
( a ) c = 0.05.
FIGURE 10. - RESULTS FOR SIXTH-ORDER CENTRAL.
60
1.00
SIXTH-ORDER
C O U R NUM ABSERROR WAVINESS
=
CENTRAL
0.50
= 0.05 =
0.00
0.06
0.25
0.75
1 00
0.50
0.75
I .oo
0.50
0.75
I .oo
0.50
X Ln 0
SIXTH-ORDER
CENTRAL
0 0
-
2 ' 0 -0 I
a
C O U R N L i M = 0.50 ABSERROR = 1 . 1 7 WAVINESS = 1.93
k
;I 0
'0: 00
0.25
SIXTH-ORDER
COUR N U M
=
CENTRAL
0.50
0.25
X
(b) c FIGURE 10.
=
0.5.
- CONCLUDED.
61
Lo 0
SEVENTH-ORDER
UPWIND1 NG
I
ZI
1'
COUQ NUM = 0 . 0 5 ABSERROR = 0 . 0 4 WAVINESS = 0.06
0 0
? L : -1
'0.00
0.50
0.25
0.75
I
.oo
X
m a
SEVENTH-ORDER U P W I N D I N G
0
-
I
gi
.;
-c L
COUR NUM
=
0.05
ABSERROR = ' . 0 5
~
.
W4\/ilUESS =
I .
75
0
-1
SEVENTH-ORDER
UPWINDING
0
-
".
-0
0.05
C O U R NUM
=
ABSERROR WAVINESS
= 0.80 =
1.28
0.25
0.50
0.75
X
( a ) c = 0.05.
FIGURE 11. - RESULTS FOR SEVENTH-ORDER UPWINDING.
62
I
.oo
-
SEVENTH-ORDER UPWINDING 0
-
COUR N U M
-8-
=
0.50
ABSERROR = 0 . 0 3 WAVINESS = 0.04
2
0
0
, lo. 00
,
~
0.50
0.25
0.75
00
X
SEVENTH-ORDER UPWINDING 0
-
I
COUR N U M = 0 . 5 0 ABSERROR = 0 . 9 3 WAVINESS = 1 .64
x!
_
o+--
lo. 00
-1
_
~
I ,
0.25
0.50
0.75
I
.oo
SEVENTH-ORDER UPWINDING
0 01
COUR N U N = 0 . 5 0
-:.
2
ABSERROR = 0 . 7 0 WAVINESS = 1.16
'0.00
0.25
0.50
x
(b) c
=
0.5,
FIGURE 11.
-
CONCLUDED.
0.75
00
63
u-i 0
-1
EIGHTH-ORDER
CENTRAL
0
COUR NUM = 0 . 0 5 ABSERROR = 0 . 0 6 WAVINESS = 0.08
u-i 0
-0-
I [L
0 0
,
0
?I
,
l o . 00
,
0.25
,
0 50
0 . 75
00
X u-i 0
I
EIGHTH-ORDER CENTRAL
1
0 0
o /
"? -0
1I
2
COUR NUM
=
0.05
ABSERROR
=
2.23
WAVINESS = 3.65
I
I0 n 0
'0.00
0.25
0.50
0.75
X In D
COUR NUM
=
0.05
(a) c
=
0.05.
FIGURE 12. - RESULTS FOR EIGHTH-ORDER CENTRAL.
64
1
.oo
Lo 0
EIGHTH-ORDER
CENTRAL
0
COUR NUM = 0 . 5 0 ABSERROR = 0 . 0 3 WAVINESS = 0.04
Lo 0
-0
I LL
0 0 0
Lo 0
0
IO
'(
1
1
0.25
0.50
0.75
00
0.75
00
0.75
I .oo
X Lo 0
EIGHTH-ORDER
CENTRAL
0
-
,
2
C O l i R NUM = 0 . 5 0 ABSERROR = 1 . 0 2 WAVINESS = ' . 8 3
"1 l0.00
0.25
0.50
X Lo 0
EIGHTH-ORDER
CENTRAL
0
Lo 0
I -0 L
COUR NUM = 0 . 5 0 ABSERROR = 0 . 7 6 WAVINESS = 1 . 2 7
0
0
Lo 0
0
IC
10
0'. 50
0.25
X
(b) c
FIGURE 12.
=
0.5.
-
CONCLUDED.
65
@fi] Ifl I
I
I
I
(a) uf - 0 . FIGURE 13. - DEFINITION OF UPSTREM (U). (C) NODES. DEPENDING ON SGNtuf).
66
(b) uf
IO
0.25
0.50
0'.75
00
X
COUR-NUM = 0 . 5 0
%I lo.oo
,
I
0.25
,
0.50
0.75
00
X (b) c = 0.5.
FIGURE 28.
-
CONCLUDED.
81
Y) 0
-
SUPER
C
-
l COUR-NUM = 0 . 0 5 ABSERROR = 0 . 2 8 WAVINESS = 0.32
-"?
zol
so
0.75
so
0.75
0'. 50
0.75
0.
0.25
X
I
.oo
Lo 0
0 0
I
-'? 101 a
1
COUR-NUM
=
0.O S
ABSEeRUR = 0 . 6 0
WAVINESS = 1.18
a(
,
lo.oo
, 0.
0.25
x
1
, 1
.oo
SUPER - C
COUR-NUM = 0 . 0 5 PRSERROR = 1 . 2 3 WAVINESS = 1.37
IO
0.2s
X
(a) c FIGURE 29.
82
=
0.05.
- RESULTS FOR THE SUPER-C SCHEK.
00
Y) 0
-
SUPER - C
0
-
Lo 0 I.
I O LL
COUR-NUM = 0 . 5 0 ABSERROR = 0 . 2 7 WAVINESS = 0.32
0 0
Y) 0
0
'C
0.25
0
0.50
0.75
I
.oo
X LD 0
ro
I COUR-NUM = 0 . 5 0 ABSERR0R = 0 . 6 0 k'A\'IhJESS = 1 . 2 0
o
*O.OO
b
,
0.50
0.25
0.75
I .oo
X Lo 0
7 -? CL
COUR-NUM = 0 . 5 0 ABSERROR = 1 . 3 8 W A V I N E S S = 1 .3S
:I lo.oo
,
,
0.25
,
0.50
0.75
1 I .OD
X
(b) c
=
0.5.
FIGURE 29. - CONCLUDED.
83
ln 0
-
HYPER - C
0 D
COUR-NUM = 0 . 0 5
ABSERROR = 2 . 6 9 WAVINESS = 2.38 0 0
0 01
lo.oo
0.25
0.50
0.75
00
X
In 0
-
HYPER
C
-
"
0
0
0
In 0
0 .
00
HYPER
C
-
COUR-NUM = 0 . 0 5 ABSERROR = 1 . 3 4 WAVINESS = 2.12
0.50
0.25
10
0.75
X
(a) c
FIGURE 30.
84
-
=
0.05.
RESULTS FOR HYPER-C (LIMITED FIRST-ORDER DOWNWINDING).
I
.oo
u1 0
-
HYPER - C
COUR-NUM
0.50
=
Y) 0 0
1
-
HYPER - C
COUR-NUM = 0 . 5 0 ABSERROR = 0 . 0 0 WAVINESS = 0 . 0 0
0
0.25
00
0.50
0.75
X
1 .oo
Y) 0
-
HYPER
-
C
0 0
-
0
Y)
I .
I O
a
COUR-NUM = 0 . 5 0 ABSERROR = 3 . 1 0 WAVINESS = 2 . 8 9
0 0
U 0 J
0.
IC
0
0.50
0.25
(b) c
=
0.75
I .oo
0.5.
FIGURE 30. - CONCLUDED.
85
, 3
10.0
ABSERROR (STEP) FIGURE 31. - ABSERROR FOR THE SEMI-ELLIPSE PROFILE (UPPER CURVES) AND SINESQUARED PROFILE (LOWER CURVES) PLOTTED ON A LOG-LOG SCALE AGAINST ABSERROR FOR THE STEP. WITH COURANT NUMBER AS A P A R M T E R RANGING FROM 0.01 TO 0.978. WITH VALUES SHOWN AT 0.1, 0.5, AND 0.9. CURVES SHOW RESULTS FOR: (1) RINMOD, (2) RUSCL, ( 3 ) SUPERBEE, AND (4) SUPER-C. ARROWS SHOW D I RECTION OF INCREASING COURANT NUMBER.
86
CONTROL
I
VOLUME
i
7
"f
FIGURE 32. - NORMALIZED NODE VALUES FOR LOCALLY MONOTONIC BEHAVIOR. HATCHING SHOWS NECESSARY CONDITIONS ON THE FACE VALUE OF INTEREST, AND ON THAT OF THE CORRESPONDING UPSTREM FACE VALUE.
sf.
5,.
/ FIGURE 33. - NORMALIZED VARIABLE DIAGRAM SHOWING THE ULTIMATE STRATEGY. THE DASHED BOUNDARY HAS A COURANT-NUMBER-DEPENDENT SLOPE OF l/C. THE CASE SHOWN I S FOR c = 0.2.
87
*f
1
*
I
1
-n
a;
OC
FIGURE 34. - NVD SHOWING Sf AS A FUNCTION OF FOR THE ULTIMATE LAX-WENDROFF ETHOD.
$
AND c
FIGURE 35. - NVD SHOWING AS A FUNCTION OF FOR ULTIMATE SECOND-ORDER UPWINDING.
5;
AND c
I
'I
-n
/
OC
I
FIGURE 36. - NVD SHOWING Of AS A FUNCTION OF FOR THE ULTIMATE F R O M SCHEPZ
.
8," AND
c
FIGURE 37. - NVD SHOWING &f AS A FUNCTION FOR THE ULTIMATE QUICKEST METHOD.
OF
8:
AND
c
89
PREXEDING PAGE BLANK NOT FILMED
Lo 0
hl-1
U L T I M A T E LAX-WENDROFF
COUR NUM
=
0.05
I
I
L ‘0.00
-I
1
0.50
0.25
0.75
X
L
U L T I M A T E LAX-WENDROFF
xj
I
,
‘0.00
I
0 25
0.56
0.75
X Ln 0
U L T I M A T E LAX-WENDROFF
COUR NUN = 0 . 0 5
X
( a ) c = 0.05. FIGURE 38. - RESULTS FOR THE ULTINATE LM-WENDROFF RTHOD.
90
I .00
1
.oo
U L T I M A T E LAX-WENDROFF
COUR NUM = 0 . 5 0 ABSERROR = 0.88 WAVINESS = 0.60
10
0'. 25
0'. 50
0'. 75
1 00
0. 50
0.75
1
X
U L T I M A T E LAX-WENDROFF 0
-
'0.00
0.25
.oo
X
U L T I M A T E LAX-WENDROFF
COUR N U M = 0 . 5 0 ABSERROR = 1 . 6 3 WAVINESS = 1.47
00
0.25
0.50
0.75
I .oo
X
(b) c
=
0.5.
FIGURE 38. - CONCLUDED.
91
In 0
COUR N U M = 0 . 0 5
41 0
0.25
‘0.00
0.50
U L T I M A T E SECDND-ORDER
0.75
1.00
UPWINDING
0 e
COUR NUM
= I
=
0.05
ABSERROR = 2 . 0 9 WAVINESS = 1 . 7 3 1
0.50
0.25
0.75
I .oo
01.75
00
X 0
n
U L T I M A T E SECOND-ORDER
UPWINDING
0 0
0
n
-6 I
a
COUR N U N = 0 . 0 5 ABSERROR = 2 . 1 7 U A V I N E S S = 1.59
0 0
n 0 0
‘I
IO
6.25
01.50
X
(a) c FIGURE 39.
92
=
0.05.
- RESULTS FOR ULTIMATE SECOND-ORDER UPWINDING.
Y) 0
U L T I M A T E SECOND-ORDER
COUR
NUM
=
UPWINDING
0.50
0
'0.00
0.25
0.50
0.75
00
X Lo 0
-
U L T I M A T E SECOND-ORDER
UPWINDING
COUR N U V = 0 . 5 0 ABSERROR = 1 . 5 7 I" 1 WAVINESS = 1 . 6 4
"?
,
I
'0'. 00
'
I -1
0.50
0.25
U L T I M A T E SECOND-ORDER
0.25
0.75
I
.oo
UPWINDING
0.50
0.75
00
X
(b) c = 0.5.
FIGURE 39. - CONCLUDED.
93
C O U R NUM = 0 . 0 5 ABSERROR = 0 . 5 3 WAVINESS = 0.36 0 D
0
\
e
7 i -
0-25
0.00
0-50
0 -7 5
00
X 0
-1
ULTIMATE
FROIqK
ULT I NATE
FROMM
I
i
c
:I "I 2 1
-0
C O U R NUM = 0 . 0 5 ABSEPROR = I .42
WAV!NESS
=
1 .34
0 0 L
0
0
,ON 00
0 25
0.50
0.75
00
X
(a) c FIGURE 40.
=
0.05.
- RESULTS FOR THE ULTIMATE FROMM SCHEME.
ORIGLNRL PAGE IS OF: POOR QUALITY
94
Y) 0
U L T I M A T E FROMM 0 (3
-
C O U R NUM Y) 0
-6 I a
=
0.50
ABSERROR = 0 . 1 3 WAVINESS = 0 . 0 9
0 0
T
In
0.00
D
0.50
0.25
0.75 00
X In 0
-
U L T I MATE FRDMM 0 0
In 0
-0 I
CL
C O U R NUM = 0 . 5 0 ABSERROR = 1.12 W A V I N E S S = 1.50
0
0
0 Lc 0
’(
00
0.25
so
0.75
0.50
0.75
0.
I
.oo
X I 0 n
U L T I M A T E FROMM
COUR NUM = 0.50
0.25
00
X
(b) c FIGURE 40.
=
0.5
- CONCLUDED.
95
In 0
-
U L T I M A T E OUICKEST 0
-
Lo 0
-6I a
COUR NUM = 0 . 0 5 ABSERROR = 0 . 2 3 WAVINESS = 0 . 1 4
0 0
0
:
' 0.00
0.50
0.25
0.75
00
X
U L T I M A T E OUICKEST
COUR N U M = 0 . 0 5
:j
,
, lo.oo
,
0.50
0.25
0.75
. 00
X v 0 )
-1
ULTIMATE OUICKEST
COUR N U M = 0 . 0 5 ABSERRUR = 1 . 2 3 WAVINESS = 1.24 0 0
0
0
'0.00
0.50
0.25
0.75
X
(a) c FIGURE 41.
96
=
0.05.
- RESULTS FOR ULTIMATE QUICKEST
00
U L T I M A T E OUICKEST
COUR N U M = 0 . 5 0 ABSERROR = 0 . 1 3 WAVINESS = 0.09
IO
0'. 25
0'.50
I .oo
0'. 75
X u, 0
U L T I MATE O U I C K E S l
COUR
d lo.oo
-1
NUN = 0 . 5 0
l
0.25
-
r
0.50
P
0.75
1
.oo
U L T I M A T E OUICKEST
COUR N U N
51 0
'0.00
=
0.50
, 0.25
I
0.50
, 0.75
I . 00
97
Lo 0
U L T I M A T E F O U R T H - O R D E R CENTRA1 0 0
-
Ln 0
-0 I
a
COUR NUM = 0 . 0 5 AESERROR = 0 . 2 0 WAVINESS = 0.20
0 0
0 Ln. G
'C '0
0.25
0.50
0.75
00
X ui c
U L T I M A T E FOURTH-ORDER
:i
0.00
I
CENTRAL
0.25
0.50
0 75
1
.oo
X m 0
U L T I M A T E FOURTH-Oi?DER
CENTRAL
0 0
-
Lo 0
I -0
a
COUR N U M = 0 . 0 5 AESERROR = 1 . 0 4 WAVINESS = 1.29
0 0
0
Lo 0
0
0.25
10
0'. 5 0
0. 7 5
X
(a) c FIGURE 42.
=
0.05.
- RESULTS FOR THE ULTIMATE FOURTH-ORDER CENTRAL SCHEME.
00
Lo 0
U L T I M A T E FOURTH-ORDER
CENTRAL
0
COUR NUM ABSERROR WAVINESS
m 0 -0 I
a
=
= =
0.50 0.12 0.13
0 0 0
m 0 0
0.25
10
'(
0.50
0'. 75
00
X u, 0
U L T I M A T E FOURTH-ORDER
CENTRAL
0
-
j -&-
NUM ABSERROR
2
W P V I N E S S = 1 . 39
COUR
=
=
0.50 0.92
0.25
0. 50
0.75
X Lo 0
U L T I M A T E F O U R T H - O R D E R CENTRA1 0 0
m 0
I
-0
a
C O U R NUM ABSERROR WAVINESS
=
0.50
=
0.90 1.18
=
0 0
Lo 0 0
'c
10
0.50
0.25
1
0.75
00
X
(b) c FIGURE 42.
=
0.5.
- CONCLUDED.
99
Lo 0
-
ULTIMATE FIFTH-ORDER UPWINDING
COUR NUN = 0 . 0 5
0.50
0.25
I .oo
0.75
X
Ln 0
-
TE FIFTH-ORDER UPWINDING 0
' CGUR N U N O /
"4'
=
4
9.05
ABSERSGR = 9 . 9 1
WAV!WESS
=
1
1
.38
0
-1
1
ULTIMATE FIFTH-ORDER UPWINDING
, lo.oo
0.75
X
(a) c
=
0.05.
FIGURE 43. - RESULTS FOR ULTIMATE FIFTH-ORDER UPWINDING.
100
1
,
, 0.50
0.25
I
.oo
Y) 0
-1
ULTIMATE FIFTH-ORDER
UPWINDING
e a
1
COUR NUM = 0 . 5 0 A B S E R R O R = 0.09 W A V I N E S S = 0.09
-: 0 0
0
0
'0.00
0.25
0.75
0.50
X
I .OD
I 0 n
-
ULTIMATE FIFTH-ORDER UPWINDING
-2
1
COUR NUM = 0 . 5 0 ABSERROR = 0 . 8 3 WAVINESS = 1.32
0 0
,
0
lo.oo
0.5c
0.25
0.75
00
X
\
COUR N U N = 0 . 5 0
(b) c = 0.5.
FIGURE 43.
- CONCLUDED.
101
-1
Lo 0
U L T I M A T E S I XTH-ORDER
COUR NUM
=
01
0 Lo_
-0
E
0.05
1
‘0.00
-1
CENTRAL
0.25
0.50
ULTIMATE SIXTH-ORDER
COUR NUN ABSERROR
=
0.05
=
0.82
0.75
1
.oo
CENTRAL
WAVINESS = 1.32
0 1
O l
, lo. 00
0.25
0.50
0.75
X
. 00
lo 0
U L T I M A T E SIXTH-ORDER
CENTRAL
-J COUR NUM = 0 . 0 5 y _ A B S E R R O R = 0.83 0 -0
E
W A V I N E S S = 1.18
0 0 0
00
(a) c FIGURE 44.
102
=
0.05.
- RESULTS FOR THE ULTIMATE SIXTH-ORDER CENTRAL SCHEME.
Lo 0
U L T I M A T E SIXTH-ORDER CENTRAL 0
-I
i
COUR NUM = 0 . 5 0 ABSERROR = 0 . 0 8 WAVINESS = 0.09
zo
"/
0.50
0.25
'0.00
1
,
,
0
0.75
X
I .oo
Lo 0
-1
U L T I M A T E SIXTH-ORDER CENTRAL
I
CClUR NUP
'?I
=
0.50
ABSERRCIR = 0 . 7 7 WAVINESS = 1 . 2 7
;L lo.oo
-1
0. 2 5
0.50
1
0.75
00
X
m 0
-1
ULTIMATE SIXTH-ORDER
CENTRAL
COUR NUM = 0 . 5 0 AESERROR = 0 . 7 2 WAVINESS = 1.10 0 0
? ? _ _ i 0.00
0.50
0.25
0.75
1
.oo
X
( b ) c = 0.5. FIGURE 44.
-
CONCLUDED.
103
U L T I M A T E SEVENTH-ORDER
I
-2 ,"
UPWIhDING
fi COUR NUM = 0 . G 5 ABSERROR = 0.12 WAVINESS = 0.13
0
0
?I
,
0
lo. 00
,
0.25
0.50
, 0.75
1 00
0.75
00
X In 0
-
U L T I M A T E SEVENTH-ORDER
UPWINDING
0
COUR NUM YI 0
I -0
a
=
0.05
ABSERROR = 0.78
WAVINESS = 1 .28
0 0
I n 0 0
I
10
1
0.25
0.50
X Y) 0
U L T I M A T E SEVENTH-ORDER U P W I N D I N G 0 8
Y) 0
-6 I
n
COUR N U N = 0 . 0 5 ABSERROR = 0.72 U A V I N E S S = 1.09
0 0
0
v) 0
0
'(
to
0'.25
0'.50
0: 75
X
(a) c FIGURE 45.
104
- RESULTS FOR
= 0.05.
ULTIMATE SEVENTH-ORDER UPWINDING.
00
.I
U L T I M A T E SEVENTH-ORDER
U P W I N D NG
e
COUR NUM
=
01 ‘0.00
0.50
0.25
0.75
0.50
X
-1
I .oo
U L T I M A T E SEVENTH-ORDER UPWINDING
COUR NUN
=
0.50
0
‘0.00
0.25
0.50
0.75
I .oo
X ul 0
-1
U L T I M A T E SEVENTH-ORDER
COUR NUM
?! 0
‘0.00
UPWINDING
= 0.50
, 0.25
, 0.50
, 0.75
I 00
X (b) c = 0 . 5
FIGURE 45. - CONCLUDED.
105
-1
Lo 0
G L T I M A T E EIGHTH-ORDER CENTRAl
C O U R NUM = 0 . 0 5 ABSERROR = 0 . 1 5 WAVINESS = 0.19 I 0 0
r~
1
l0.00
0.25
0.50
00
0.75
X in (3
U L T I M A T E EIGHTH-ORDER
CENTRAL
0 D
-
4
A
I
.
.oo
0.00
U L T I M A E EIGHTH-ORDER
CENTRA1
COUR NUM = 0 . 0 5 ABSERROR = 0 . 7 5 W A V I N E S S = 1.16
0
1
0.25
0.50
0.75
X
(a) c
=
0.05.
FIGURE 46. - RESULTS FOR THE ULTIMATE EIGHT-ORDER CENTRAL SCHEME.
106
I.
00
Lo 0
ULTIMATE EIGHTH-ORDER
CENTRAL
COUR N U M = 0 . 5 0
I'.OO
X 10 0
U L T I M A T E EIGHTH-ORDER CENTRAL
COUR NUM
=
0.50
00
X ln 0
U L T I M A T E EIGHTH-ORDER
CENTRAL
e 0
COUR NUM LD 0 I 0
L
=
0.50
ABSERROR = 0 . 6 7 W A V I N E S S = 1.10
0
0
LD 0
0
'(
10
0.25
0. 50
0.75
00
X (b) c
=
0.5.
FIGURE 46. - CONCLUDED.
107
2 oL1.-
-01
.1
1 ABSERROR (STEP)
0
FIGURE 47. - ABSERROR FOR THE SEMI-ELLIPSE PROFILE (UPPER CURVES) AND SINESQUARED PROFILE (LOWER CURES) PLOTTED ON A LOG-LOG SCALE AGAINST ABSERROR FOR THE STEP. WITH COURANT NUMBER AS A PARAMETER RANGING FROM 0.01 TO 0.098. WITH VALUES SHOWN AT 0.1, 0.5. AND 0.9. CURVES SHOW: (1) ULTIMATE FROMM. (2) ULTIMATE QUICKEST, ( 3 ) ULTIMATE FIFTH-ORDER UPWINDING, AND (4) ULTIMATE SEENTH-ORDER UPWINDING. ARROWS SHOW DIRECTION OF INCREASING COURANT NUMBER.
PRBCEDING PAGE BT,ANR NOT FTLMED
109
Y 0 )
HYBRID 3RD/5TH 0
COUR NUM = 0 . 0 5 ABSEREOR = 0 . 1 4 WAVINESS = 0.13
Y) 0
-.
L O
0 3
,
0
y1 0
0. IC
10
0.50
0.25
0.75
00
1
X In 0
HYBRID 3RD/5TH
COUR N U M = 0 . 0 5 ABSERROR = 0.97 WAVINESS = 1.42
10
0'. 2 5
0'. 50
0'. 75
l'.OO
X
HYBRID 3RD/5TH
COUR NUM = 0 . 0 5 ABSERROR = 0 . 9 0 WAVINESS = 1.12
1
0
1
0
,
, '0.00
0.25
0.75
X
(a) c
=
0.05.
FIGURE 118. - HYBRID ULTIMATE THIRD-/FIFTH-ORDER UPWINDING RESULTS. SOLID CIRCLES SHOW NODES FOR WHICH EITHER THE LEFT OR RIGHT (OR BOTH) FACE VALUES FOR THE NEXT T I M STEP ARE TO BE COMPUTED USING THE HIGHER-ORDER SCHEME.
110
1
,
0.50
I
.oo
-1
HYBRID 3R0/5TH
COUR NUM
=
0.50
W A V I N E S S = 0.11
ro.oo
-1
0.25
I
0.50
0.75
0X. 5 0
0.75
1
.oo
HYBkID 3RD/5TH
COUR NUM = 0 . 5 0 ABSERROR = 0 . 9 0 W A V I N E S S = 1.38 0
0
; Lo
0.00
1
0.25
00
HYBRID 3RD/5TH
COUR N U M = 0 . 5 0 AESERROR = 0 . 7 2 WAVINESS = 1.05
D
1
lo.oo
0.25
0.50
0.75
00
X
(b) c FIGURE 48.
=
0.5.
- CONCLUDED,
111
~
~~
HYBRID 3RD/7TH
COUR N U M = 0 . 0 5 WAVINESS = 0.15
0.25
0.50
0.75
1 00
01.75
00
X n
HYBRID 3RD/7TH
COUR N U M = 0 . 0 5
WAVINESS = 1.35
0
m 0
HYBRID 3RD/7TH 0
I 0 n I.
Y O
COUR N U M = 0 . 0 5 ABSERROR = 0 . 7 9 W A V I N E S S = 1.07
0 0
0 In 0
'(
10
ol. 25
0'. 50
X
(a) c
=
0.05.
FIGURE 49. - HYBRID U L T I M T E THIRD-/SEVENTH-ORDER UPWINDING RESULTS. SOLID CIRCLES SHOW NODES FOR WHICH EITHER THE LEFT OR RIGHT (OR BOTH) FACE VALUES FOR THE NEXT T I E STEP ARE TO BE C W U T E D USING THE HIGHER-ORDER SCHEME.
112
I 0 n
COUR
NUM
= 0.50
W A V I N E S S = 0.14
1
00
X
HYBRID 3RD/7TH
COUR NUM = 0 . 5 0 ABSERROR = 0.82 W A V I N E S S = 1.32
1
00
0.50
0.25
0.75
1
X
,oo
-
-1
In
HYBRID 3RD/7TH
COUR NUM = 0 . 5 0 WAVINESS = 0.96
2
'0.00
0.50
0.25
0.75
2
00
X
(b) c
=
0.5.
FIGURE 49. - CONCLUDED.
113
(a) ul
>
0. vt = 0.
-
FIGURE 50. TWO-DIENSIONAL CONTROL-VOLUME SHOWING NODE STENCIL INVOLVED I N ESTIMATING THE LEFFFACE VALUE USING ULTIMATE UTOPIA. SOLID CIRCLES SHOW NODES USED FOR THE UNIVERSAL LIRITER.
4 I
I
L
I I
A
(a) DIFFUSE SOLUTION.
(b) NW REGIONS.
(C)
SHARP SIMULATION.
FIGURE 51. - S C H W T I C (INVERTED) STEP SIRULATION SHOWING LEADING-EDGE (LE) AND TRAILINGEDGE (TE) BEHAVIOR.
I
114
4
(a) THE
@, T
~ PLANE. )
(b) THE
( r,q)
PLANE.
FIGURE A.1. - SWEBY'S TVD REGION (SHADED) SHOWN. FOR REFERENCE. ROE'S SUPERBEE SCHEE IS SHOWN BY THE HEAVY PIECEWICE LINEAR CHARACTERISTIC.
0
1
2
r
FIGURE A.2. - "EXTENDED SWEBY DIAGRM" CORRESPONDING TO THE U L T I M T E STRATEGY OF FIGURE 33. FOR REFERENCE. THE SUPER-C SCHEME IS SHOWN BY THE HEAVY PIECEWISE LINEAR CHARACTERISTIC.
115
Report Documentation Page 1. Report No.
NASA TM- 10091 6 ICOMP-88-11
'
2. Government Accession No.
3. Recipient's Catalog No.
4. Title and Subtitle
Ii
5. Report Date
Universal Limiter for Transient I n t e r p o l a t i o n Modeling o f the Advective Transport Equations: ULTIMATE C o n s e r v a t i v e D i f f e r e n c e Scheme
September 1988 The
6. Performing Organization Code
I
7. Author@)
B.P.
8. Performing Organization Report No.
Leonard 10. Work Unit No.
I
1
9. Performing Organization Name and Address
11. Contract or Grant No.
N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n Lewis Research Center C l e v e l a n d , O h i o 44135-3191
13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address
T e c h n i c a l Memorandum N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n Washington, D.C. 20546-0001
14. Sponsoring Agency Code
16. Abstract
A f r e s h approach i s t a k e n t o t h e e m b a r r a s s i n g l y d i f f i c u l t problem o f a d e q u a t e l y m o d e l i n g s i m p l e p u r e a d v e c t i o n . An e x p l i c i t c o n s e r v a t i v e c o n t r o l - v o l u m e f o r m u l a t i o n makes use o f a u n i v e r s a l l i m i t e r f o r t r a n s i e n t i n t e r p o l a t i o n modeling o f t h e a d v e c t i v e t r a n s p o r t e q u a t i o n s . T h i s ULTIMATE c o n s e r v a t i v e d i f f e r e n c e scheme i s a p D l i e d t o u n s t e a d y , one-dimensional s c a l a r p u r e a d v e c t i o n a t c o n s t a n t v e l o c i t y , u s i n g t h r e e c r i t i c a l t e s t p r o f i l e s : an i s o l a t e d sine-squared wave, a d i s c o n t i n uous s t e p , and a s e m i - e l l i p s e . The g o a l , o f course, i s t o d e v i s e a s i n g l e r o b u s t scheme which a c h i e v e s sharp monotonic r e s o l u t i o n o f t h e s t e p w i t h o u t c o r r u p t i n g t h e o t h e r p r o f i l e s . The s e m i - e l l i p s e i s p a r t i c u l a r l y c h a l l e n g i n g because o f i t s c o m b i n a t i o n o f sudden and g r a d u a l changes i n g r a d i e n t . The ULTIMATE s t r a t e g y can be a p p l i e d t o e x p l i c i t c o n s e r v a t i v e schemes o f any o r d e r o f accuracy. Secondo r d e r schemes a r e u n s a t i s f a c t o r y , showing s t e e p e n i n g and c l i p p i n g t y p i c a l of c u r r e n t l y p o p u l a r s o - c a l l e d " h i g h r e s o l u t i o n " s h o c k - c a p t u r i n g or TVD schemes. The ULTIMATE t h i r d - o r d e r upwind scheme i s h i g h l y s a t i s f a c t o r y f o r most flows o f p r a c t i c a l i m p o r t a n c e . H i g h e r o r d e r methods g i v e p r e d i c t a b l y b e t t e r s t e p r e s o l u t i o n , a l t h o u g h even-order schemes g e n e r a t e a (monotonic) waviness i n t h e d i f f i c u l t s e m i - e l l i p s e s i m u l a t i o n . B u t l i t t l e i s t o be g a i n e d above ULTIMATE f i f t h - o r d e r u p w i n d i n g which g i v e s r e s u l t s c l o s e t o t h e u l t i m a t e one m i g h t hope f o r .
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ULTIMATE; Monotonic a d v e c t i o n ; U n i v e r s a l l i m i t e r ; Nonoscillatory simulation; C o n s e r v a t i v e d i f f e r e n c e ; W i g g l e - f r e e CFD
119. Security Classif. (of this report)
Unclassified
uisrriDurion sraremenr
Unclassified - Unlimited S u b j e c t Category 64
120. Security Classif. (of this page)
Unclassified
121.NO of pages
116
122. Price'
A06
