Appi. Math.-JCU
THE
9:B(1994),213-222
CHEBYSHEV S P E C T R A L METHOD WITH A
RESTRAINT
O P E R A T O R FOR BURGERS EQUATION
MA HEPING
GUO BENYU
A b s t r a c t In this p a p e r , a restraint o p e r a t o r is u s e d t o improve the stability of the C h e b y s h e v s p e c t r a l m e t h o d . T h e g e n e r a l i z e d s t a b i l i t y of this n e w m e t h o d is p r o v e d a n d the rate of convergence is a n a l y z e d . T h e n u m e r i c a l r e s u l t s show t h e a d v a n t a g e of the method.
1. I n t r o d u c t i o n W e consider t h e following p r o b l e m
OU
OU 02U vow- ;+u:-vox2
=f'
in
( - 1 , 1 ) × (0, T),
V ( - 1 ) = V(1) = 0,
in [0,T],
U(x,O) = V 0 ( x ) ,
in [0,1],
(1.1)
where v is a positive constant. T h e F o u r i e r s p e c t r a l m e t h o d is p o w e r f u l for periodic problem, see [2,15,11,10]. As f o r non-periodic p r o b l e m s , t h e Chebyshev s p e c t r a l m e t h o d is widely u s e d , see [8,14,1,13]. To improve t h e s t a b i l i t y o f t h e F o u r i e r pseudospectral m e t h o d , we have successfully used a r e s t r a i n t o p e r a t o r b a s e d on t h e B o c h n e r s u m m a t i o n , see [9,12,4,5]. This t e c h n i q u e is a kind o f filtering. In t h e case o f t h e C h e b y s h e v approximations o f n o n - p e r o d i c p r o b l e m s , t h e filtering seems t o be more needful.When t h e parabolic e q u a t i o n is solved by the explicit Chebyshev s p e c t r a l m e t h o d , in w h i c h t h e Chebyshev approximation is used in s p a c e a n d t h e forward difference approximation is used in t i m e , t h e time-step is r e s t r i c t e d for t h e stability. I n d e e d , if v is the m e s h size in time a n d N is t h e numb e r o f t h e bases in s p a c e , t h e n we require t h a t ~- = O(N 4) (see [3]). This is m u c h more severe t h a n t h e r e s t r i c t i o n for s t a b i l i t y in t h e F o u r i e r m e t h o d , w h i c h is o f t h e form r = O ( N 2 ) . Therefore some filtering o r s m o o t h i n g t e c h n i q u e should be used t o improve t h e s t a b i l i t y o f the C h e b y s h e v approximation. In this p a p e r , we develop a C h e b y s h e v s p e c t r a l m e t h o d with a r e s t r a i n t o p e r a t o r for solving Burgers e q u a t i o n . T h e generalized s t a b i l i t y a n d the r a t e o f convergence are R e c e i v e d Oct. 2 8 , 1 9 9 3 . 1991 MR Subject Classification: 65N30, 76D99. Key words: B u r g e r s e q u a t i o n , C h e b y s h e v approximation, restraint operator.
213
214
MA H E P I N G
GUO B E N Y U
a n a l y z e d . T h e numerical r e s u l t s with v a r i o u s parameters o f the r e s t r a i n t o p e r a t o r are g i v e n , w h i c h show t h e effect o f such o p e r a t o r . 2. T h e S c h e m e s Let I = ( - 1 , 1), a n d let L 2 ( I ) b e e q u i p p e d with t h e i n n e r p r o d u c t ( . , . ) a n d t h e 2 _1_ n o r m [[-[[ as u s u a l . Supposew(x) = ( 1 - x ) 2. Set
L ~ ( I ) = { u : I --* R : u is measurable and ( u , u ) ~ < c~}, where t h e i n n e r p r o d u c t a n d t h e n o r m are d e f i n e d by 1
I1,11,,,
(u, v)~ = J~l u(x)v(x)w(x)dx,
= (=,
F o r any p o s i t i v e i n t e g e r m, d e f i n e
H y ( I ) = {u • L ~ ( I ) :
l[5- -t ,.,/
=
<
l=O
Hol~(I) = {u • H i ( I ) : u ( - 1 ) = u(1) = 0}. F o r any positive i n t e g e r N , let SN be t h e s p a c e o f algebraic polynomials o f d e g r e e a t m o s t N a n d let
V N = SN N H I , , , ( I ) . Let P N : L 2 ( I ) ---+ VN b e t h e L2(I)-orthogonal p r o j e c t i o n o p e r a t o r with
(PNU, v)~ = (u, v)~,
Vv E VN.
Let pO : C(_f) --* VN be t h e i n t e r p o l a t i o n o p e r a t o r with jTr
xj = c o s ~ - ,
P % ( * d =
1_< j < N - 1.
W e now i n t r o d u c e the r e s t r a i n t o p e r a t o r . Let T,~(x) b e t h e C h e b y s h e v polynomials N
o f d e g r e e n . F o r u(x) = ~ anT,~(x), we d e f i n e t h e o p e r a t o r R - R ( a , 3 ) by n--0 N
a>l,
3~1
n~-0
and d e f i n e R0 - R 0 ( a , 3) by N-2
R o u ( x ) = ~ (1 - ] N [ ' ~ ) 3 a n T n ( x ) + b N - 1 T N - I ( x ) + bNTN(X), rt~O
where b N - l , b N are determined by R 0 u ( - 1 ) = u ( - 1 ) , Rou(1) = u(1).
CHEBYSHEV SPECTRAL METHOD FOR BURGERS EQUATION
215
Let r b e t h e m e s h size in time discretization a n d let u k ( x ) = u ( x , k'r) b e d e n o t e d by uk for simplicity. Let ~2k = 2X-(uk+l + u k - 1 ) , a n d d e f i n e u~
= uZ(~ + l - ~ ~ ) ,
~'ik = 1~ (u~+ 1 _ u k _ l )
Now, we give two s c h e m e s with e i t h e r P = P N , corresponding to t h e s p e c t r a l m e t h o d , o r P -- pO, corresponding t o t h e pseudospectral m e t h o d . (1 t A scheme o f f i r s t - o r d e r in time d i s c r e t i z a t i o n . It is t o find u k ~_ VN such t h a t
~
+
( n P ( u ~ ) ~) - ~,Pn--O-~(no~) = P R P I ~,
k >_ o,
(2.1)
u ° = PUo.
(2) A scheme o f second-order in time d i s c r e t i z a t i o n . It is to find u k ~_ V N such t h a t 02 ui +
( n P ( u k ) 2) - vP~-x2(fik ) -- p n p ] k,
P
k>_l,
(2.2)
u 1 = P [ U o + rO_U (0)],
/Jr u°
PUo,
where -~(o)°U = V oz2o2Uo(x) - U - ~ x ( x ) + f ( x , O ) . 3.
Numerical Results
T h e Burgers e q u a t i o n with f _-_ 0 has t h e s o l u t i o n
1[
U(x,t) = ~ 1 - t a n h
4v
(3.1)
J"
P u t v -- 0.005, T = 0.0005 and N -- 64. T h e s o l u t i o n is c o m p u t e d by t h e scheme (2.1) with P = pO. T h e errors involved are E2(t) =
N + 1
lu(xj't) - U(xj't)12
'
j=o Eoo(t) =
max
O<_j<_N
lu(x~,t)- U(x~,t)l.
To test the effect o f t h e r e s t r a i n t o p e r a t o r , v a r i o u s v a l u e s o f t h e parameters a a n d fl are used. If a = ~ , fl = 0, then (2.11 b e c o m e s t h e u s u a l C h e b y s h e v approximation. In this case, E2(0.25) = 1.302 and E ~ ( 0 . 2 5 ) = 3.907. T h e numerical r e s u l t s with o t h e r values of a a n d / 3 are p r e s e n t e d in Tables 1 a n d 2. Clearly t h e r e s t r a i n t o p e r a t o r improves t h e accuracy. T h e b e t t e r choices are w h e n a goes from 6 to 10 and fl from 2 t o 3. This is also t r u e for some o t h e r problems (see [7]). Table 1. T h e errors E2(0.25). I0
1 2 3
0.883 0.393 0.016
8
0.787 0.016 0.018
Table 2. T h e errors Eoo(0.25).
6
ct
0.023 0.018 0.021
1 2 3
I0
3.409 2.596 0.065
8
3.283 0.065 0.071
6
0.102 0.073 0.082
216
MA HEPING
GUO BENYU
4. S o m e L e m m a s In this section, we list some lemmas w h i c h will paly a n important role in t h e errors analysis. W e d e n o t e by C a p o s i t i v e c o n s t a n t i n d e p e n d e n t of N , r a n d any f u n c t i o n , w h i c h may b e different in different cases. L e m m a l(see [13]). If u • HI,,~(I), then
Ou,O ox
) 1 (~u)
(4.1)
__ ~li~il~,~
L e m m a 2 ( s e e [13]). If u • L ~ ( I ) a n d r • H],~(I), then (4.2)
(u, ~ ( ~ v ) ) < 2'1~fi~l'lx.~ L e m m a 3 ( s e e [13]). I [ u • SN, then
(4.3) Let Pc : C ( ] ) -~ SN b e t h e i n t e r p o l a t i o n o p e r a t o r with
Fc~(x~) = ~ ( ~ ) ,
jr
x~ = cos ~ - ,
o < j < N
a n d let P1,N : H ] , ~ ( I ) --* VN be d e f i n e d by
V v e yN.
L e m m a 4 ( s e e [14]). I f u 6 H m ( I ) , then [ [ P N U --
u[[,~ _<
CN-m[[u[[
(4.4)
....
][Pcu - uHj,,~ < C N 2 J - m l l u l i m , ~ ,
0 <_ j <_ m ,
1
m > -. 2
(4.5)
m>l.
(4.6)
I f u • H~,~ (1) ('l H 2 (I), then liP1.Nu - uib.~ _< CN~-'fl~,lim.~,
j = 0,1,
W e now i n t r o d u c e t h e following d i s c r e t e i n n e r p r o d u c t a n d n o r m N 1
(u,v)N,~ = ~ _ . w : , ( x j ) v ( x j ) ,
IlullN,., = (~,,u)~,~,
j=O
where ~o = ~ N =
~ and ~ j = ~-(1 < j < N - 1).
C H E B Y S H E V S P E C T R A L M E T H O D FOR B U R G E R S E Q U A T I O N
217
L e m m a 5 ( s e e [1]). W e have N VU E S 2 N - 1 .
(4.7)
Ilull~ _< tt~1t~,.~ --- '/2-11~11,~, I(~,~)N,~ - ( u , v ) ~ l < CN-'llull,-,,.,,,llvll~.
(4.8)
j=O
L e m m a 6 ( s e e [13]). I f u , v E S N , then
(4.9)
L e m m a 7. Ht~,13 >_ 1, O< a <_ ~ and u E S N , then (4.10)
I I R u - ~ 1 1 ~ ___ C ~ N - ~ l J u l l ~ ° ~ •
N
Proof. Let u ( x ) = ~_, a n T n ( x ) . T h e n n=0
N
IIR u
N
7r
- -
n
uil 2 < ~ E /321--~
2aa 2
~o
. < C~2N-2~ E
n~l
<-C/~2N-2a
E
n
2a 2
al. I _< C / 3 2 N - 2 a
n= 0
~ a,,Tn(cos(0))
n
2~ a n2
n,=l
~
--lr
~S,,id"°)
n=-N
dO <_ C Z ~ N - ~
~
71=0
L e m m a 8 . I f u E SN, then
Cg2v/-A(~,~)!lull~,
JRull,~ <
r(2¢~ + 1)r(3c~ -1) A(~, fl) =
ar(25
(4.11)
+ 1 + 3a-1)'
and i f u E VN, then
IRou[t.~ <_ Cog2 v/ Ao(a,~)llu[l~,
r(2Z + 1 ) r ( ~ -1) Ao(a,¢~) = c~r(2¢~ + 1 + c~-l)"
(4.12)
N
Proof. Let u ( x ) = Y~ a ~ T n ( x ) . W e have rim0
0
N-1
n=0
2 bn -- - Cn
N
E
m(1-
m=n+l m+~
odd
m~e
)
am,
218
MA H E P I N G
GUO BENYU
where Co = 2, c , = l(n > 1). Therefore 71"
N-1
N
IRul2'' = 7 Z cnbZn <- C N ( Z n=0 N
n2( 1 -
m
r e ( l - I ~ 1 )~am) 2
n=l
I 1°) r~
N
N
S, a2n < CN3
n=l
n:l
Z
n
N n
1~-12( 1 -
~
n=l
n=l
z2(1 -
z'~)2adz = r(2/~ + 1 ) r ( 3 a -~)
a r ( 2 / ~ + 1 -4- 3 o ~ - 1 )
f0
T h e second result (4.12) can b e p r o v e d in t h e same way.
Lemma
[]
9. We have (RU, V)ta
= (U, R ' U ) w ,
( R u , v)w = ( u , R o ' v ) w ,
(4.13)
U,V • ~ N , u • SN_2,
(4.14)
v • ~N.
5. S t a b i l i t y and C o n v e r g e n c e To fix the problem, we suppose t h a t P = p o in s c h e m e s (2.1) a n d (2.2). W e f i r s t consider the generalized s t a b i l i t y o f (2.1). A s s u m e t h a t u k a n d f k have t h e errors fz~ E VN a n d f k respectively. T h e n from (2.1), (4.?) a n d (4.14) we have f o r all v E VN,
(~, ~k,V)N,~ - (RP°$"k, -~x (,,,Roy)) + ~'(~(Ro~ 0 k ), ~ ( ~ R o v ) ) = ( R Po~ f-k ,V)N,,.,, where f k = ½(2u k + ilk)ilk. By t a k i n g v = 2fik + a n d (4.8) t h a t
mTf~kt(m
> 1), we o b t a i n from (4.1)
+ 7-(m - 1)llut -k IIN,~ 2 2 (11~ [IN,~)t 2 + ~u l l R o u-k II1,~
<(RP°fk,2~k+ mrfi~)N,~ + (RP°$'~, O ( W R o ( 2 ~ k + m r ~ t k ) ) ) 0
k
.
Let e > 0. W e have from (4.8) t h a t
(
I ( R P ° ] k , 2 ~ k + m T - u-k, ) ~ , ~ l - - - I I ~ k l l ~ + r~lla~ll~ + 2 +
Cm2T~ ~
/ ii/k IIN,~" 2
W e get from (4.2) t h a t
I(Rpo~k,O ~ z ( ~ R ° ( 2 u-k +
m7-z2t~))) I_ 1<
,,~kl,~
(21Ro~2~1~,~ + m7-1RoUt- k I1,~) 2
// ~k 2 2 2 ~k 2 <~llRou ]11,~ + um 7- ]Rout I1,~ + d(p, IlukllL~(/))(ll~kl[~ + NIl~kll~).
C H E B Y S H E V S P E C T R A L M E T H O D FOR B U R G E R S E Q U A T I O N
219
Also, from (4.2),
m~-~,
(n0'sk), ~ ( ~ R o ' s t )
< 411aod'll~,,,, + ~m2~21Ro'5~l~,,,,.
By s u b s t i t u t i n g t h e a b o v e estimates into (5.1), we o b t a i n
2 - k I1,,, ~. + 8 IIRouk - I1~,,,, ~ - 2'~m2~-21R°'5~l~'' (llu- k IIN,,,,h + r ( m - 1 - c)llu~ (5.2) _ < d ( v , IlukllL~(~))(ll'skl[~ + N I I ~ I I 4 ) +
2 +
4~
/ I l l [IN,~.
Now, let e b e s u i t a b l y small and Co > 0. W e take m -- 1 + e q- 2Co. T h e n it follows from (4.12) t h a t
-k II~ ~ T(m -- 1 -- ~)llut
2 > Co~.ll'sgll 2, 2~m ~. r 2 [Rou- tkI1,,,,
provided t h a t
1
TN 4 <
-
(5.3)
2vC02A0(a,/~)(1 + e + 2C0) 2"
Let n-1
p , = c ( 1 1 ' 5 o 1 1 2 + ~ . ~ Ill-k IIN,,,,), 2 k----0 n-1
E " = 11'5"11~ + r ~ ( C o ~ l l ' a ~ l l ~ + 8 1 1 R o U' - [ 1 12, , , , ) ,
IllulllL°° =
k----O
max
kr<_T
IlukllL~(i).
Then (5.2) implies rt--1
E n < p'~ + v d ( v , IllulllLoo)
~
[E k + N ( E k ) 2 ] -
k=0
Therefore we get t h e following s t a b i l i t y results by the n o n l i n e a r i n e q u a l i t y in L e m m a 4.16 of [6]. T h e o r e m 1. L e t r b e s u i t a b l y s m a l l and (5.3) h o l d . Then there exist p o s i t i v e cons t a n t s d and ~ d e p e n d i n g only on [[[u[[[L~ a n d u such t h a t i f pn <_ ~ / N f o r all n T ~ T, then E n <_ p n e d n r .
W e next consider t h e generalized s t a b i l i t y o f (2.2). In this case we have ( f i ~ , v ) g , ~ -- ( R P ° F k ,
( w v ) ) + V(~xx'5 ,
(wv)) =
V)y,~,
Vv e VN,
where ~k = ½(2u k + 'sk)12k" we take v = u k a n d o b t a i n l(H'sk 2
~ ^k
( R p O [ k ^k 0 .k c., , '5 ),,, + ( R P ° F'k, -Z-ox (w'5 )).
[IN,,~)i + 711'5 II21,,., -,<
220
MA H E P I N G
GUO B E N Y U
Summing this inequality for all 1 < k _< (n - 1), we get from (4.8) .--1
Ilu IIT~ +
II2 + ~'~ ~ II~kl121,~ k=l
(5.4)
.-1
_<2(1t~°11~ + II~Xll~) + 4~- E I ( R P 2 ]k ,u.k )w + (RP°c if"k, ~xx(WUk))[. o k=l
We have from (4.2) and (4.3) that 0 ^k v 32 -k 2 i4( R I ' ° p k , -~x(co~ ))l < 811P°Fkll~luklL~ < ~'llukll~,~ + -/,/- I I F IIx,,,, 1,1
^k
2
<~11 ~2 Ill,~ + d(v,
IlukllL~(t))(ll~kll~ + NIl~kll~).
Moreover. ] 4 ( R p ° f k , u k ) ~ [ < []~2k+111~ + [1~k-1[t2 + 2 [ I f-~ [[Y,~" L e t T < 1/2. It follows 2 from (5.4) and the above estimates that n--i
I1~"11~ + ~2 r
n--1
2 -< C(ll~°ll~ + I1~11~ + 7- ~ II]~ II.,v,~) ~ II~kll~,~ 2 k=l
k=l
(5.5)
n-1
+ "rd(v, IIlulllL~) ~--~. (ll~kll~ + NII~II~). k=l
Let n--1
n-1
P" = c(11~°112 + 11~1112 + ~ ~ II]kllN,~),2
~k
E " = I1~112 + 2
k=l
2
k=l
Then (5.5) reads n--I
E " ~ p~ + Td(~, ItlulIIL~) ~ [Ek + N(Ek)2]. k=l
By applying the nonlinear inequality in [6], we get the stability result expressed as follows. T h e o r e m 2. Let ¢~ be suitably s m a l l Then there exist positive constants d and 5 depending only on [[[u]]lL~ and v such that i f pn <_ 5 / N for a11 n r < T, then E n < pnednr. We c a n derive the convergence of (2.1) and (2.2) similarly. For simplicity, we only analyse the convergence of (2.2) with P = Pc°.
C H E B Y S H E V S P E C T R A L M E T H O D FOR B U R G E R S E Q U A T I O N
221
3. A s s u m e that U C H I ( O , T ; H ~ , ~ ( I ) N H y ( I ) ) N H 3 ( O , T ; L 2 ( I ) ) , f C C(O, T, H,, (I)) and rn > 1. Then there e x i s t positive constants d and 6 depending only on U such that when T is s u i t a b l y s m a l l and T2 + N - m <_ 6 N - 1 / 2 , we have for a11 Theorem
nv <
T, [lu" - U'*[I,~ <_ d(~-2 + N - m ) .
Proof. P u t t i n g Uk. = P1,N Uk, we get from (1.1) that (~_~0 0k,v)~, - ( ( U k )2 + T2(Uk'2-2 ' ,tt, 0--~(wv)) + u ( 0 - ~ 0 k , ~----~(wv))= ( f k , v ) w .
(5.6)
Furthermore, let ek = u k - U.~. Then by (2.2) and (4.7), we have from (5.6) that
• O ~k,O(~)) (e~,v)N,~ - ( R P ° Fk , ~x(WV)) + U(~x k 2 - (Uk) 2 _ -~-, ~.2(Uk~2 = ( ~0U ^k ,v)~ - (Uk.i,V)N,~ + ~1 ( R P ~o (U.) ,d, ~---~(wv)) + ( R P ° c ]k,v)N,~, - (]k, v)~,
Vv • VN,
(5.7)
where ~k = ~',a(2r/k~. + ek)e k and ( U k ) ~ = 7~-1( ( u k + l ) 2 -- ( u k ) 2 ÷ ( V k - 1 ) 2 ) . According to (4.8) and (4.5),
I( ~ O k , v ) ~ -- (Uk,~,V)N,~
+ [(U/k, v)~o - (U~,v)N,~] + ,(U~ - U f f i , V)N,~l < Bk + C[lv,[~
0 ~ k - U ik I I2, o + C N - 2 m IIU~k I1~+ 2 [IUk - P1,NUk[[2. Thus by Taylor's formula with B k = [I-g7 and (4.6), we have r~--I k=l
II(
-
)t~lb
I[
Ot
[]L2(O,T;L~(1))
Ot
< d(IIUIIw',~(O,T;L~(n))(
L2(O,T;Hy(I))
L=(OT.L2(I))II-[-I)" ,
,
w
Moreover, k 2 - ( U k ) ~ l b < IIRP¢[(U.k) ~ IIRP3o (U.)
(Vk)~]lb + IIR(P~ - I)(U~)211,,,
+ II(n - I)(U~)211,,, _< d(llUkllm,,,,)N - " , Ile°ll,., _< II(P~ - I)Volk,+ll(Px,N - I ) U o l b _< CN-~IIUolI~,~,, ou
_< d(llUollm,~,,
r2
o z u II
(0) ,,,,,1[ ° t ~ L~(O,r;Lg(z))
I ( R P ° ] k , V)N,~ -- (]k, v)l~ _< C N
--2m
2
Ilfllc(o,w;.2(,)) + C l b l l ~ -
Finally we a p p l y Theorem 2 to (5.6) and use the triangle inequality with (4.6) to complete the proof. []
222
MA H E P I N G
R e m a r k 1. result for the R e m a r k 2.
If the
GUO BENYU
c o n d i t i o n s o f T h e o r e m 3 a n d ( 5 . 4 ) h o l d , t h e n we h a v e a s i m i l a r
c o n v e r g e n c e o f s c h e m e ( 2 . 1 ) w i t h P = p O , as in T h e o r e m 3. It is easy t o see t h a t w e c a n get t h e s a m e s t a b i l i t y a n d c o n v e r g e n c e r e s u l t s
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