Existence of the Discrete Travelling Waves for a Relaxing Scheme

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E x i s t e n c e o f t h e D i s c r e t e T r a v e l l i n g W a v e s for a R e l a x i n g S c h e m e

HAILIANG LIU

Department of Mathematics, Henan Normal University Xinxiang 453002, P.R. China

JINGHUA WANG

Institute of Systems Science, Academia Sinica Beijing 100080, P.R. China

T O N G Y A N G D e p a r t m e n t of Mathematics City University of Hong Kong, Hong Kong (Received M a y 1996; accepted August 1996)

Communicated by M. Slemrod

A b s t r a c t - - T h e existence of a discrete travelling wave is proved for the relaxing scheme. The main idea is to change the original scheme such that the resulting scheme is monotonic to which Jennings' result can be applied. The equivalence of the resulting scheme and the original one is shown when 1/ -- 1/q. The u-component of the discrete travelling wave thus obtained is a discrete shock for a monotone conservative difference scheme, which approximates the corresponding conservation law.

K e y w o r d s - - R e l a x i n g scheme, Existence, Discrete travelling wave.

1 . I N T R O D U C T I O N W e s t u d y a r e l a x i n g s c h e m e o f t h e f o r m

~ + I n )~ V n n l 2 un n

-- Uj "]- -~ ( jq-1 -- Vj--1) -- -~ ( j + l -- 2U~ "q- U j _ I ) --~ O,

(I.i)

aA #

V~q-1 -- V~ "1- "~- (ujn.kl -- U~_ 1) - ~ (V3_t_ 1 -- 2V~-['- j - l ) ~ - - k ( v V n 7 _ f ( u r ~ ) ) ,

w h e r e A = ( A t / A x ) , I~ = v ~ A , k = ( A t ~ e ) , a n d ( u ~ , v ~ ) = ( u ( j A x , n A t ) , v ( j A x , n A t ) ) , a n d ( A x , A t ) a r e t h e n u m e r i c a l a p p r o x i m a t i o n s o l u t i o n a n d t h e g r i d sizes of t h e s p a c e - t i m e . E q u a - t i o n (1.1) is i n t r o d u c e d in [1] as t h e f i r s t - o r d e r a p p r o x i m a t i o n t o t h e s y s t e m

Ut + Vx = 0 , x E R 1,

(1.2) v, + a u x = - 1 - ( v - f ( u ) ) ,

e

w h i c h a p p r o x i m a t e s s c a l a r c o n s e r v a t i o n laws ut + f ( u ) x = 0 w h e n t h e r e l a x a t i o n r a t e e is s m a l l . T h e s t u d y o f t h e e x i s t e n c e a n d s t a b i l i t y o f d i s c r e t e s h o c k waves is i m p o r t a n t in u n d e r s t a n d i n g Research by the authors was supported in part by the RGC Competitive Earmarked Research Grant #9040150.

Research by the first two authors was also supported in part by the National Natural Science Foundation of China.

Typeset by ,AA,,~TEX 117

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the convergence behavior of numerical shock computations. Jennings [2] proved the existence of discrete shock waves for the general first-order monotone scheme, see also [3,4] for system case.

The stability of travelling waves of (1.2) and discrete travelling waves (DTW) of (1.1) was studied in [5,6]. Let (U, V ) ( x - st) be a travelling wave solution of (1.2) connecting (u+,v+) to ( u _ , v _ ) , its existence is ensured by the subcharacteristic condition (cf. [7])

- v / a < f ' ( u ) < v ~ , the Rankine-Hugoniot condition

for all u, (1.3)

- s ( u + - u _ ) + / ( u + ) - I ( u _ ) = 0, (1.4)

and the generalized entropy condition

< 0 , Q ( u ) = f ( u ) - I ( u ~ ) - s ( u - u ± ) > 0,

f o r u + < u < u _ ,

(1.5) f o r u _ < u < u + ,

here, v± = f ( u ± ) . From now on, we impose the CFL condition # < 1.

A D T W connecting (u±, v+) is a special numerical solution of the difference scheme such t h a t 1) = (u

~, 3--rl ' 3 - - ~ 1 '

(1.6)

where ~ = sA. Since 7/ is not necessarily an integer, the minimal domain for (1.6) is '£,7 -- {mr] + n [ 7/= sA; m, n E Z}.

From (1.1) and (1.6), the one-parameter discrete travelling wave ( u s , v s ) , x E £ , , defined by (1.6) will satisfy

A #

u ~ _ , - u~ + ~ ( v s + l - v x - x ) - ~ ( u s + z - 2 u x + u ~ - l ) = 0,

(1.7)

aA #

V x - W -- Vx -~- y (Ux+I -- Uz--1) -- ~ (Vs+I -- 2Vz Jr- Vx--1) = - k (v s - f ( u s ) ) ,

and lims-.±oo(ux,vx) = ( u ± , v ± ) . First, we consider the case when ~? = l/q, q E Z\{0}, and prove the following theorem. Without loss of generality, we assume u+ < u_ hereafter.

THEOREM 1. Let f ( u ) satis/y (1.3)-(1.5). Suppose that 7/ = 1/q and the re/axation rate e is sufficiently small. Then, for each uo E ( u + , u _ ) , there is a unique [unction ( u s , v x ) which is continuous on £ , . ux takes the value uo at x = 0 and satisfies (1.7) with (u+oo, v±oo) = (u±, v±), where v+ = f ( u± ). Furthermore, the u-component of the D T W thus obtained is a discrete shock for a monotone conservative difference scheme which approximates ut + f ( u )s = O.

2. E X I S T E N C E O F D T W

In this section, we will prove Theorem 1, and thus, establish the existence of D T W to the system (1.1) for the case ~ = 1/q. The proof will be divided into the following five steps.

STEP 1. DETERMINE Ux. From the first equation of (1.7), we have

_ #

A2 (vs+l - vs-1) = us - u s - , + ~ (us+l - 2us + u s - l ) • (2.1) Rewrite the second equation of (1.7) into the following two forms:

aA u #

V s + l - t ' l - - V s + l " ~ - ( S + 2 - - U s ) - - ~ ( U S + 2 - - 2 V s + I Jr- V s ) - - ' . - - k ( U s + l - f (Us+l)) , (2.2) and

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Discrete Travelling Waves 119

a~

V x - I - n - - V z - I ' I - T (Uz--Ux ^- 2 ) - ~ ( V x - 2 V z - l - [ - v x - 2 ) = - k ( v x - l - f ( u x - 1 ) ) (2.3) Then, by substituting (2.1) into A/2{(2.2) -

(2.3)},

and eliminating the terms of v~s, we obtain

[ ]

( u x - 2 n - 2U=-n + u=) - . 2 (u=+l - 2u= + Ux_l) '1- k (Ux_ n - Ux) --}- -~ ( f (ux+l) - f (?~x-1)) : . [(2u x ^{- -} Ux+ 1 ^{- -} Ux_I) ^{- -} (2Ux_ n ^{- -} Ux.t_I_7 I ^{- -} Ux-l--n) ] n u ~ (Ux-]-I ^{- -} 2Ux Jr- U,x-1) , (2.4) which is a scheme for u~ only, where x E/:n"

STEP 2. CHANGE TO A MONOTONE SCHEME. The existence theorem in [2] can only be applied to the monotone and conservative scheme. In order to have a monotone scheme, we are going to change our scheme to a monotonic one by using an iteration based on the shift operator. To this end, we rewrite (2.4) as

a(x O) ~__ a(x O) (Ux_l_n, Ux_l, Ux_2n, Ux_rl, Ux, Ux+l_n, Ux+l) = O, (2.5)

where

a(~ °) = . ~ x - l - . + ~ U x _ l + ~ f / u x - 1 ) - . ( 1 - . ) ux_, - ~ - . . - / k + 2 . - 2) ~ x - .

+ ( k - k , - 1 + 2 , - 2 , 2 ) u= + . u x - l - l - n -t-

?Ux-t-1-

^{? f} (Ux+I) - - , ( 1 -- , ) Ux-I-1.

For k very large, t h a t is, e being sufficiently small for given grid sizes, G (°) is monotone increasing function of each of its arguments except for u x - 2 n and u=_n; therefore, (2.5) cannot be written directly as a monotone scheme. To overcome this difficulty, we consider r / = 1 / q as the grid size in the space direction, and transform G (°) = 0 into 0 x = 0 such t h a t G= depends on every argument explicitly defined on every grid point. And G= has only one t e r m with a negative coefficient. To this end, we multiply ,~(0) by a positive constant 00 and add it to G (°) i.e.,

L ~ X _ n z ,

a(2)

-:-

a(~ ° ) - 0 a (°)

"I- 0 x-n,

(2.6/

t h e n the coefficients of ux-3n, ttx-2n, and u x - n are -00, - O o k + 2(1 - #)0o - 1 and [00(1 - #) - 1]

k + 2(1 - , ) ( 1 + 0 o , ) - 0o, respectively. We choose 00 > (1/1 - #) such t h a t there are still two terms with negative coefficients (here the coefficient means the derivative of G (1) with respect to its corresponding argument). Repeating the same process yields the following induction formulae:

= 0 re(m) (2.7)

--x~(m+l) G(m) + m ~ = - n , m = 1, 2, .. . q - 4, f2(rn+l).

where Om is chosen so t h a t there are always two negative terms in ,~x , precisely, we have

OG(m ) m - 1

= - - H Oi,

OUz-(m+2)n i=o

OG(rn) m-1 ~ rn-1

= - ( ~ 0 , ) k - O m + 2 ( 1 - # ) l - I 0 , ,

O U x - ( m + l ) n i=O

Ou--~:-:-_~, =

~ - . ) ( I I o, -e,.,,

k - o ( ~ ) > o ,

i=0

O G (~)

> 0 , f o r l = - ( q + m + l ) , . . . , - q and Oux+ln

and

q - m - 1 , . . . , q ,

- - r e + l , . . . , 0

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here, O(1) depends only on 0i and # and O m =

Y]o_<h<i2<...<~,,,_,_<m-x

^0~,0i2... Oi,,_~. W e n o w change the negative term uz-1+n in G (q-s) to be positive and introduce a n e w positive term ux+n, by setting

(~x ----: (~(q-3) .a- ~(2'_(q -3)

- ~ + , • ( 2 . 8 )

Then,

Ouz_(q_ 1)n

o0

OUx--(q--2)n

0G

~x--(q--3)v/

a¢

Ouz+ln

(?

--t~ - / z ( 1 - #) + f ' - H 0 , > 0 , i----0

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i----O

[ Cg) ]

- ( 1 - # ) Oi - O q - 3 k - 0 O~ k - O ( 1 ) > 0 , k i = 0

> 0 , f o r l = - 2 q + 2 , . . . , - q and - q + 4 , . . . , q + l ,

provided e is suitably small and 0 < 0 < [(1 -- #) YL=0 0i -- q-a](1-L=0 8,) q-4 O q-4 . --1 . Furthermore, we multiply Gz = 0 by 1 / k first, and then add c~ux_l+~v on both sides of G x / k = 0, where

(1 + ~) q-4

= I-I~=0 (1 + 0~). Then, by multiplying 1 / ~ on both sides of the resulting scheme, we obtain

U x - l T 2 n = G (Ux_2T2r D U x _ 2 + 3 ~ , . . . , U x @ l . t . ~ l ) , (2.9)

which is a strict monotone scheme.

S T E P 3. PASSAGE TO A CONSERVATIVE SCHEME. Next we will prove the following lemma.

LEMMA 1. S u p p o s e e is s u i t a b l y small, (2.9) can be written as the [ollowing conservative scheme:

% _ . = u u - 0 { g u - g u - n } , y e £ n ,

(2.10)

where 0 ---- ( A t ~ w A x ) , gy ---- g(uy+2-2n, u y + 2 - 3 n , . . . , u y - 1 ) and g(u, u, . . . , u) = f ( u ) .

PROOF. If we define a shift operator S u as Su(gz) = gz+u, then the process in Step 2 is equivalent to applying the operator P to G ( ° ) / k = O, where

P = Ot - 1 (I .-b OSn) (I + Oq_4S_n) (I .-]- Oq_sS_n)... (I + 00S_,7) ,

(2.11)

here, I is an identity operator and c~ is defined above. It is easy to see that, for any u independent of x, P ( u ) = u.

Multiplying G (°) by 1 / k and rearranging the terms yields

O= -ux-" +ux- A ( l(u~+l--) + ~-~

+(u_?_u2)(u~+l_ux)_/~(u~+l_ _ux_,)] I(u~-1)2 + (u - ? - .2) (u~ - u~-l) - u (ux-~ - ux_l_~)] } .

i[

kA (ux-~ - u~-2~)

(2.12) T o have a conservative scheme, we rewrite (2.12) as

o +u=

o/¢°> ^{(o) T}

- - t ~ - - g x - n j '

(2.13)

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Discrete Travelling Waves 121 where

g(O) = ~l f (ux_bl ,'1 + f (Ux+l-~/) + " " + f ( U x - l + , ) -F --~ (ux -- U x - , ) 2

~ ( U x + l - ~ - - Ux--~? + U x + I - 2 ~ - - U x - 2 ~ ? + " "" + U x - - U x - 1 ) •

kA

T h e n we apply the operator P to (2.13),

o : - .

{. (,:,)

^{- .}

Since (2.9) is a monotone scheme and u x - : + 2 , = Sq_-~3ux-,, (2.14) can be written as

(2.14)

Set y = x - 1 + 377 and P i g (°)) - 1 / 8 ( P - Sq_-v3)u~ = gy, (2.10) follows immediately. On the other hand,

g y ( U , U , . . . _{, ~ ) :}

~ ~.., (u,~, ((0 ,~1,

where we have used P ( f ( u ) ) = f ( u ) and ~ = 1/q. |

Now Jennings' result [2] can be applied to our case.

LEMMA 2. Under the assumptions of Theorem 1, for each Uo E ( u + , u _ ) , there is a unique function, continuous on f-',7, which takes on the value Uo at x = O, satisfies Gx = O, a n d has the limits u±oo = u+. T h e solution ux is a m o n o t o n e function of x E £.~ and d e p e n d s continuously at each value o f x on uo.

STEP 4. EQUIVALENCE BETWEEN TWO SCHEMES. W e next prove the solution uz constructed above is also a solution of (2.5), precisely, we will prove the following lemma.

LEMMA 3. L e t u~ be a unique solution of G~ = 0 determined in L e m m a 2, then u~ satisfy G(~ °) = 0, Vx ~ £ , .

PROOF. Since ux is a solution of (2.9), we have

o . = G ~ -~) + ~G~+-: ) = 0, Vx E £~, (2.15)

where G (q-3) is defined by the induction formulae (2.7). Thus,

q - - 3

G~q-s) = ~ , , . ~ ( o ) _,_ G(O)

~ X . - - i ~ I ~ X '

i=1

(2.16)

(2.1~)

Combining (2.16)

c~,-3) = ~ a(:+-:) I = e I~-( ~-3)_.., = .= °-~lim I~1 ° --.-,~-('-3)', _- 0,

where c4 is a positive constant depending only on (8,} q-3. By (2.5), we have G(°)(u±, u + , . . . , u ± )

= 0, t h a t is

lim G(f ) = 0,

~---*±OO

here the arguments of G (°) are substituted by the solution ux of Gx = 0.

and (2.17), we obtain limx-~+c~ G(x q-s) = 0.

From (2.15), by virtue of ~ < 1,

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which holds for a n y x E / : ~ . So Us is also a solution ,,,,f G (q-3)x --- 0. R e p e a t i n g t h e s a m e process, one c a n prove t h a t us satisfies ,~xP-(m) = 0 ( m = 0, 1,. . . , q - 2) for a n y x E £ v .

N o w we o b t a i n a unique solution ux of (2.5) which is a m o n o t o n e function of x. A n d Ux, for

a n y fixed x, is a c o n t i n u o u s function of t h e value u0. |

STEP 5. DETERMINE Vs. Finally, we d e t e r m i n e t h e v - c o m p o n e n t of (1.7) based on t h e existence o f u - c o m p o n e n t ux. To this end, by s u m m i n g t h e first e q u a t i o n o f (1.7) over x from y - 2 N + 1 t o y - 1 w i t h step size 2, t h e n we have

2 y - 1 y - - 1

s = y - 2 N + l s = y - 2 N + l

[ ( u x - x-1) - - u . ) ] .

X'~Y° "U ~ U

C o n s i d e r z.~x=-oo( s - x o - ux), for a n y fixed x0 > 0, since u x - s o - ux > 0 a n d ~-~x=-oo( x - s o -

~--~yo "U

ux) < u _ - u + , t h e n L , x = - ~ ( x-xo - u x ) converges. Hence, v v _ 2 g converges as N -~ oo for a n y y E £:~, which implies t h a t t h e r e exists ~ such t h a t limx-~-c~ vx = ~. L e t x --, - o o in (1.7), we yield ~ = f ( u _ ) = v _ . Similarly, we have limx-.+oo vs = v+ = f ( u + ) . Now, vx c a n be expressed

a s

2 0

m = - c ¢ (2.18)

0

+ v ~ ~ (u~+2~ - 2ux+2m-1 + u~+2~-2).

m = - - ~:)

We c a n verify t h a t (ux, vx) given by L e m m a 2 and (2.18) is a solution of (1.7).

REMARK. W h e n ~ is a general rational number, we can also c o n s t r u c t a D T W which approxi- m a t e s t h e travelling wave of (1.2). T h a t is, u n d e r t h e s a m e conditions o f T h e o r e m 1, if ~? = p / q for a given grid size ( A x , At), we consider t h e s y s t e m (1.1) for A = ( 1 A t ~ p A x ) a n d k = ( A t / p e ) , d e n o t e d b y S 0). B y T h e o r e m 1, we know t h a t there exists a D T W which is u n i q u e l y defined on t h e grid points ( j A x , ( n / p ) A t ) . Hence, we define t h e D T W o n t h e grid points ( j A x , n A t ) as t h e one for a p p r o x i m a t i n g (1.2). In fact, t h e D T W t h u s o b t a i n e d is t h e one for t h e scheme S (p) which is t h e scheme b y iterating S (1) p times. T h e u - c o m p o n e n t of this D T W is also a m o n o t o n e discrete shock profile for scalar conservation law.

R E F E R E N C E S

1. S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure AppL Math. 48, 555-563 (1995).

2. G. Jennings, Discrete shocks, Comm. Pure Appl. Math. 27, 25-37 (1974).

3. B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp.

36, 321-351 (1981).

4. A. Majda and J. Ralston, Discrete shock profiles for systems of conservation laws, Comm. Pure Appl. Math.

32, 445-483 (1979).

5. H.L. Liu, J. Wang and T. Yang, Stability for a relaxation model with a nonconvex flux (preprint).

6. H.L. Liu, J. Wang and T. Yang, Nonlinear stability and existence of stationary discrete travelling waves for the relaxing schemes (preprint).

7. T.P. Liu, Hyperbolic conservation laws with relaxation, Commun. Math. Phys. 108, 153-175 (1987).

Existence of the Discrete Travelling Waves for a Relaxing Scheme

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