The lip-Stability and Error Estimates for a Relaxation Scheme
The lip+-Stability and Error Estimates for a Relaxation Scheme Hailiang Liu
y
Jinghua Wang
z
Gerald Warnecke
x
Abstract
We show the discrete lip+ -stability for a relaxation scheme proposed by Jin and Xin (1995, Comm. Pure. Appl. Math. 48, 235-277) to approximate convex conservation laws. Equipped with the lip+ -stability we obtain global error estimates in the spaces W s;p for ?1 s 1=p, 1 p 1 and pointwise error estimates for the approximate solution obtained by the relaxation scheme. The proof uses the framework introduced by Nessyahu and Tadmor (1992, SIAM J. Numer Anal. 29, 1505-1519 ). We also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state.
Keywords: Relaxation scheme, convex conservation laws, lip+-stability, maximum principle. AMS Subject Classi cation: 35L65, 65M10, 65M15
1 Introduction Relaxation schemes are a class of nonoscillatory numerical schemes for systems of conservation laws proposed by Jin and Xin [3]. They are motivated by relaxation models for ows which are not in thermodynamic equilibrium, i.e. they constitute Manuscript version: July 5, 1999 IAN, Otto-von-Guericke-Universitat Magdeburg, PSF 4120, D-39016 Magdeburg, Germany. Email: [email protected] z Institute of Systems Science, Academia Sinica, Beijing 100080, P.R.China. Email: [email protected] x IAN, Otto-von-Guericke-Universit at Magdeburg, PSF 4120, D-39016 Magdeburg, Germany. Email: [email protected] y
1
more general and more accurate models of certain physical phenomena. The relaxation schemes provide a new way of perturbing, even regularizing, systems of conservation laws and approximating their solutions. In this sense they are to be seen as an interesting tool of analysis. The computational results that are available, see e.g. [3] as well as Aregba-Driollet and Natalini [2], indicate that the relaxed schemes obtained in the limit ! 0 provide a quite promising class of new schemes. We point out that that the main assets of these schemes are that they neither require the computation of the Jacobians of uxes for the conservation laws nor the use of Riemann-solvers. This is needed for ows in which a real gas law has to be used in place of the frequently used assumption of an ideal gas, e.g. the two phase ow for kryogenic gases. In such cases it may be too expensive or even impossible to calculate Riemann solutions or even ux Jacobians. This important property is schared by other schemes such as for instance the high resolution central schemes introduced by Nessyahu and Tadmor [15], see also Kurganov and Tadmor [5] for references on recent developments. To make things more precise we want to consider a scalar conservation law. We take a convex ux function f 2 C 3( ) and initial data u0 2 L1( ) and consider the Cauchy problem u + f (u) = 0 (1.1) with initial data u(x; 0) = u0(x): (1.2) For this problem we want to approximate the global weak entropy solution by a relaxation scheme. We choose a time step t, a spatial mesh size x, a parameter a which will be related to the characteristic speeds of the conservation law and a small relaxation parameter > 0. From these we de ne the mesh ratio = , the CFL parameter = pa 2]0; 1[ and the scale parameter k = . The mesh is given by the points (j x; nt) for j 2 and n 2 0. The approximate solution takes the discrete values u at the mesh points. Further, relaxation schemes involve the discrete relaxed uxes v . We want to consider the following semi-implicit relaxation scheme IR
IR
t
x
t x
t
Z Z
IN
n j
n j
u +1 ? u + 2 (v +1 ? v ?1) ? 2 (u +1 ? 2u + u ?1) = 0; n j
v
n j
n j
n j
n j
n j
+1 ? v n + a (un ? un ) ? (v n ? 2v n + v n ) j j j ?1 2 j+1 j?1 2 j+1
n j
2
j2 ; n2
n j
Z Z
IN
(1.3)
= ?k[v +1 ? f (u +1 )]: n j
n j
The discrete initial data are given by averaging the initial data u0 over mesh cells 1 I = (j ? 2 )x; (j + 21 )x , i.e. taking Z 1 0 u = x u0(x)dx; and setting v0 = f (u0): (1.4) j
j
j
Ij
j
The di erence v ? f (u) measures a deviation from an equilibrium in relaxation models. If we have v = f (u), we say that our variables are in an equilibrium state. We are interested in the relaxation limit where k is large, i.e. is small and the conservation law is being approximated. In this case the source term becomes sti . Moreover, we speci cally will have to require in our analysis that for some positive constant c the scale condition 0 0 for u 2 :
(3.1)
IR
Further we assume that the initial data satisfy the uniform bound ju0(x)j b < 1 for x 2 . Therefore, by (1.4) the discrete initial data inherit this bound, i.e. IR
ju0j b for j 2 :
(3.2)
Z Z
j
We choose a > 0 satisfying the subcharacteristic condition
p
sup jf 0(u)j < a:
(3.3)
jujb
It follows from Theorem 2.1 that the discrete solution (u ) 2 (1.3) satis es the same L1 -bound as initial data, i.e. n j j
2
Z Z;n
IN
given by the scheme
ju j b for j 2 ; n 2 : n j
Z Z
(3.4)
IN
Since f 2 C 3 and convex, there exist positive constants , 1, 2, K such that
p
sup jf 0(u)j = < a;
(3.5)
1 f 00(u) 2; for ? b u b; sup jf 000(u)j = K:
(3.6) (3.7)
jujb
jujb
Theorem 3.1. (lip+-Stability) Assume that ku0k
Lip
+ (IR)
=: L < 1;
(3.8)
the parameter a > 0 is suitably large, x is suitably small and the scale parameter k satis es the scale condition (1.5). Then the approximate solution (unj )j2 ;n2 given by the relaxation scheme (1.3) with initial data (1.4) satis es the lip+ -stability. More precisely, the following estimate holds Z Z
u ? u ?1 2Lx n j
IN
for j 2 ; n 2 :
n j
Z Z
8
IN
(3.9)
Proof. We de ne the di erence
R := R ? R ?1 : n
n j
j
(3.10)
n j
By the Mean Value Theorem we nd for any j 2 , n 2 1 and u ?+12 such that Z Z
IN
1 a value between u + 2 n j
n
j
n
j
1
1
1
1
u + 2 ? u ?+12 f 0( ) = f (u + 2 ) ? f (u ?+12 ): n
n
j
We set for any u 2
n
n j
j
IR
!
M10 (u) M0(u) := M 0 2 (u)
n
j
j
f 0 (u) 1 2 (1 ? 0pa ) f (u) 1 2 (1 + pa ):
=
!
(3.11)
(3.12)
The explicit form of the scheme (2.7) and using (3.11) together with (2.6) then gives 1 1 (3.13) R +1 = 1 +1 k R + 2 + 1 +k k M0( ) E R + 2 ; We choose a discrete vector function A as n
n
j
j
n
n j
j
n j
A := AA1 2
n ;j
n j
!
n ;j
=
!
M10 (u ?1) : M20 (u ) n j
n j
(3.14)
The heart of the matter is to prove the inequality
R A Lx; j 2 ; n 2 : n j
n j
Z Z
(3.15)
IN
This estimate combined with (3.5) and (3.14) yields (3.9) as follows f 0(u ?1 ) ? f 0(u ) pa u ? u ?1 = E R 1 ? Lx 2 1 + p a Lx 2Lx: n j
n j
n
n j
n j
j
It remains to prove (3.15). For this purpose we de ne
P := R ? A Lx: n j
n
(3.16)
n j
j
Then (3.15) becomes
P 0; i.e. P 0; j 2 ; n 2 ; i = 1; 2: n j
n i;j
Z Z
9
IN
(3.17)
We shall prove (3.17) by induction. First let us consider the case n = 0. We have by inserting (3.10) together with (2.1) and (3.14) into (3.16) and using the de nition of the discrete initial data (1.4) as well as the Mean Value Theorem 0 v0 1 0 v0?1 1 f 0(u0?1) 1 0 P1 = 2 u ? pa ? 2 u ?1 ? pa ? 2 1 ? pa Lx h f (u0) ?pf (u0?1) i 1 f 0(pu0?1) = 12 (u0 ? u0?1) ? ? 2 1 ? a Lx a h 0 0 i f 0(u0?1) 1 = 2 (u ? u ?1) ? Lx 1 ? pa ? 2p1 a f (u0) ? f (u0?1) ? f 0(u0?1)(u0 ? u0?1) h 0 0 i f 0(u0?1) 1 00 0 0 0 2 1 = 2 (u ? u ?1) ? Lx 1 ? pa ? 4pa f ( )(u ? u ?1) and 0 v0 1 0 v0?1 1 f 0(u0) 1 0 P2 = 2 u + pa ? 2 u ?1 + pa ? 2 1 + pa Lx h i f 0(u0) = 12 (u0 ? u0?1) ? Lx 1 + pa ? 2p1 a f (u0?1) ? f 0(u0) + f 0(u0)(u0 ? u0?1) h i f 0(u0) = 21 (u0 ? u0?1) ? Lx 1 + pa ? 4p1 a f 00(~0 )(u0 ? u0?1)2; where 0 and ~0 are intermediate values between u0?1 and u0. Thus (3.17) with n = 0 follows from the assumption (3.8), the convexity of the ux function, i.e. f 00 > 0, and the subcharacteristic condition (3.3). We now assume that (3.17) is true for n. It remains to prove (3.17) for n + 1, i.e. j
j
j
;j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
;j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
P
+1 0
n i;j
j
j
j
for i = 1; 2; j 2 :
(3.18)
Z Z
To this end we insert the relation (3.16) into (3.13) and get
P +1 = 1 +1 k n j
!
(1 ? )P1 + P1 +1 (1 ? )P2 + P2 ?1 + 1 +k k M0( )E (1 ? )P1 + P1 +1 (1 ? )P2 + P2 ?1 n ;j
n ;j
n ;j
n ;j
n j
10
n ;j
n ;j
n ;j
n ;j
!
+ (Q1 + Q2 )Lx: n
n
(3.19)
The vectors Q1 and Q2 are given as follows n
n
!
!
? )A1 + A1 +1 Q1 = QQ1 1 := ?A +1 + (1 (1 ? )A2 + A2 ?1 1 0 211 0 +1 0 p f (u 1 ) ? (1 ? )f (u ?1) ? f (u ) A = @ 2 1 0 ?+1 (3.20) ? 2p f (u ) ? (1 ? )f 0(u ) ? f (u ?1 ) and, using the fact that by (3.12) we have M20 (u) = 1 ? M10 (u), ! 0 (1 ? )A1 + A1 +1 ! Q k 1 2 := 1 + k M ( )E ? Id Q2 = Q 22 0 1? (1 ?0( )A?12) + A2 ?1 0( ) 1 ! 0 ( ) M 0 ( ) 1 ? p + 2 1 ? p A ? M k 2 2 1 @ =1 + k : 1? 1 + 0p( ) + 1 + 0 (p ?1 ) M20 ( ) ?M10 ( ) 2 2 n ;
n j
n ;
n j
a
n ;
n ;j
n ;j
n ;j
n j
n j
a
n ;j
n j
n j
n j
n j
n ;
n j
n j
f
un j
n j
n j
f
a un j
n ;j
n ;j
n ;j
n ;j
f
un j a
f
a
un j
a
(3.21)
For the last term in (3.19) we have the following estimate Lemma 3.2. (Key Lemma) If the assumptions of Theorem 3.1 and the induction assumption (3.17) hold, then there exists a constant c() > 0 such that with 2 given as in (3.6) we have the following estimate (3.22) Q + Q ? c(p) 2 P + P + P + P E : 1
2
n
n
1;j
a
n
1;j +1
2;j ?1
n
2;j
We continue the argument and postpone the proof of this lemma to the end of this section. It follows from (3.19) and Lemma 3.2 that h i P1 +1 k +1 1 (1 ? )P1 + P1 +1 f 0( ) + k +k 1 21 1 ? pa (1 ? )(P1 + P2 ) + (P1 +1 + P2 ?1 ) ? c(p) 2 (P + P + P + P )Lx: n ;j
n ;j
n ;j
n j
a
n
n ;j
n
1;j
n
1;j +1
2;j ?1
n ;j
n ;j
n ;j
n
2;j
The induction assumption P 0 yields the estimate 1 +1 P1 k + 1 (1 ? )P1 + P1 +1 k 1 f 0( ) c ( ) L x 2 + k + 1 2 1 ? p minf; (1 ? )g ? p n i;j
n ;j
n ;j
n ;j
n j
(P1 + P1 n ;j
n ;j
a
+1 + P2;j ?1 + P2;j ): n
11
n
a
(3.23)
Using the scale condition (1.5), i.e. the assumption that k is bounded away from zero, we see that +1 is also bounded away from zero. Due to this and the subcharacteristic condition (3.3) the coecient of the second term on the right hand side of (3.23) is nonnegative when x is small enough. Thereby P1 +1 0 follows immediately. Analogously, we obtain P2 +1 0; j 2 : These estimates complete the proof of the theorem. k
k
n ;j
n ;j
Z Z
Remark. A slightly more general stability estimate of the form u ? u ?1 nt + 2(Lx)?1 n j
n j
for a positive constant 1 can be analogously obtained by just proving that P := R ? A nt + 1(Lx)?1 0 instead of (3.17) with P de ned as in (3.16). Proof of the Key Lemma 3.2: First we collect three identities deduced from the relaxation scheme (2.3) and (2.6) n
n j
n j
j
n j
u +1 ? u = E (R n j
n j
1 n+ 2
j
!
+1 ? R1 n 2 ?1 ? R2;j
? R ) = E RR1
n ;j
n j
n ;j
= (R1 +1 ? R2 ); n
1 1 u +1 ? u ?+11 = E (R + 2 ? R ?+12 ) = E R + R1 +1 ? R1 R2 ?1 ? R2 = (1 ? )(R1 + R2 ) + (R1 +1 + R2 ?1); u ? u ?1 = E (R ? R ?1) = R1 + R2 : n j
n j
n
n
n
j
j
j
n
n
;j
n j
n j
n j
n
;j
n j
n
n
;j n
;j n
;j
;j
;j
!!
;j
n
;j
(3.24) (3.25) (3.26)
n
;j
n
n
;j
n ;j
;j
These identities will be used repeatedly. Using the Mean Value Theorem we have for the rst component Q1 1 of Q1 in (3.20) 1 Q1 1 = 2p1 a f 00( ?+12 )(u ?+11 ? u ?1) ? f 00( ? 12 )(u ? u ?1 ) : n ;
n ;
n
j
n j
n j
12
n j
n j
n j
n
Taking (3.24), (3.26) and the relation R = P + A Lx one obtains n
n j
j
n j
Q1 1 = 2p1 a f 00( ?+12 )(R1 ? R2 ?1) ? f 00( ? 12 )(R1 + R2 ) h i 1 1 = ? 2 f 00( ?+12 )P2 ?1 + f 00( ? 12 )P2 ? f 00( ?+12 ) ? f 00( ? 12 ) P1 00 + 12 00 0 2 3 00 0 f ( ) f ( u ) + 2 f ( ) ? f ( ) f ( u ) 1 1 +1 ?1 +2 ?2 5 Lx: p ? 2 4f 00( ? 12 ) + 2 a n ;
n
1
n
n
;j
j
n
;j
n ;j
j
n j
n
;j
;j
n
n ;j
n j
n j
n
n j
n j
j
n
n j
n ;j
n j
j
n j
Recalling the induction hypothesis (3.17), i.e. P 0, and using the bounds for f 0 f 00 given in (3.5), (3.6) one obtains the estimate (3.27) Q1 1 ? 2 2 (P2 ?1 + P2 + P1 ) ? 2 1 ? 32 p2a Lx: Note that if f 00 = 1 then the last term in (3.27) may alternatively be estimated sharper as ? 2 1 1 ? p a Lx: n j
n ;j
n ;
n ;j
n ;j
p
It is negative under the subcharacteristic condition < a. Now we proceed to estimate Q1 2. A straightforward evaluation of Q12 in (3.21) yields f 0(u ) + f 0(u ?1) ? 2f 0( ) (2 ? 1)f 0( )(f 0(u ) ? f 0(u ?1)) pa : + Q1 2 = 4(1 k+ k) a (3.28) n ;
n j
n ;
n j
n j
n j
n j
n j
The estimate to be derived from this inequality, which will be (3.36) below, will be obtained in four steps. Three of these steps are needed to estimate the rst quotient. Our rst two steps are estimates of the di erences appearing in the rst quotient in (3.28). We make repeated use of the Mean Value Theorem in the following form
f (u) ? f (v) =
Z1 0
f 0(u + (1 ? )v)d (u ? v):
Using the de nition of f 0( ) in (3.11) and (2.6) we obtain n j
f 0(unj?1) ? f 0(jn ) = ?
Z 1Z 1 0
0
h
i
f100(; 1) (u +1 ? u ?1) + (1 ? )(u ?+11 ? u ?1) d1d; n j
n j
n j
n j
(3.29) 13
where
f100(; 1) := f 00(1u ?1 + (1 ? 1)(u +1 + (1 ? )u ?+11 )): n j
n j
n j
By (3.25) and (3.24) we have J := (u +1 ? u ?1 ) + (1 ? )(u ?+11 ? u ?1) = (u +1 ? u ?+11 ) + (u ?+11 ? u ?1) ? = (R1 +1 + R2 ?1) + (1 ? )(R1 + R2 ) + (R1 ? R2 ?1): Inserting the relation (3.16) gives h J = (P1 +1 + P2 ?1) + (1 ? )(P1 + P2 ) ?f 0(u ?1) ? f 0(u ) (1 ? )?f 0(u ) ? f 0(u ?1) i p p + + + 1 Lx 2 a 2 a f 0 (u ) + (P1 ? P2 ?1) ? pa?1 Lx: n j
n j
n j
n
n j
n
;j
n
;j
n ;j
n j
n
;j
n ;j
n j
n j
n
;j
n ;j
n ;j
n j
n j
n
;j
;j
n ;j
n j
n j
n j
n j
n ;j
Having obtained this expression for J we use (3.3), (3.5), (3.6) and the induction hypothesis (3.17) to obtain from (3.29) the estimate h i f 0(u ?1) ? f 0( ) ? 2 2 (P1 +1 + P2 ?1) + 1 ?2 (P1 + P2 ) + P1 Z 1Z 1 ( j 2 ? 1 j + )
p + Lx ? f 00(; )d dLx: n j
n j
n ;j
n ;j
2
n ;j
a
0
n ;j
0
n ;j
1
1
1
(3.30) Now we get to the second step of estimating the second di erence in the rst quotient in (3.28). De ning analogously as above f200(; 1) := f 00(1u + (1 ? 1)(u +1 + (1 ? )u ?+11 )) one has Z 1Z 1 h i 0 0 f (u ) ? f ( ) = ? f200(; 1) (u +1 ? u ) + (1 ? )(u ?+11 ? u ) d1d n j
n j
n j
= ?
Z 1Z 1
n j
n j
0 0
n j
n j
n j
h +1 ? R2 ) 0 0 i +(1 ? ) ? (R1 +1 + R2 ?1) ? (1 ? )(R1 + R2 ) d1d " Z 1Z 1 0 f200(; 1) (R1
n
n
;j
;j
n
n
;j
= ?
n j
0
0
f200(; 1) (P1
n ;j
n
;j
n
;j
;j
f (u ) +1 ? P2 ) ? pa Lx n j
n ;j
+(1 ? ) ? (P1 +1 + P2 ?1) ? (1 ? )(P1 + P2 ) n ;j
n ;j
n ;j
n ;j
!# f 0 (u ?1) ? f 0(u ) f 0(u ) ? f 0(u ?1) p p ? + (1 ? ) + 1 Lx d d: n j
2 a
n j
n j
14
2 a
n j
1
This gives us the following inequality
(j1 ? 2j + )
pa Lx +1 + 2 Z 1Z 1 + f200(; 1)(1 ? )d1dLx: 0 0 the estimate for Qn1;2 we have to take
f 0(u ) ? f 0( ) ? 2P1 n j
n j
n ;j
(3.31)
care of the As the third step in deriving integrals involving second derivatives f 00 in (3.30) and (3.31). For this purpose we de ne Z 1Z 1 I := f200(; 1)(1 ? ) ? f100(; 1) d1d 0
0
which becomes zero for the case f 00 =const. For general convex ux functions this term has to be treated carefully since the integral in (3.31) is positive. It is not obviously dominated by other negative terms. We estimate I as follows. Using the Mean Value Theorem again we get
Z 1Z 1 h ? +1 +1 00 I= f 1u + (1 ? 1) (1 ? )u + u ?1 0 0 i ? +1 +1 00 ? f 1u ?1 + (1 ? 1) u + (1 ? )u ?1 d1 d Z 1 Z 1 Z 1 000 n j
n j
n j
=
where
0
0
0
n j
n j
n j
f 21 + (1 ? 2)2 (1 ? 2)d2d1d;
(3.32)
1 ? 2 := 1(u ? u ?1) + (1 ? 1)(1 ? 2)(u +1 ? u ?+11 ): Using (3.26) and R = P + A Lx we have n j
n
n j
j
n j
n j
n j
h
n j
1 ? 2 = 1(R1 + R2 ) + (1 ? 1)(1 ? 2) (1 ? )(R1 + R2 ) + (R1 +1 + R2 ?1) f 0(u ?1) ? f 0(u ) pa = 1(P1 + P2 ) + 1 1 ? Lx 2 h i + (1 ? 1)(1 ? 2) (P1 +1 + P2 ?1) + (1 ? )(P1 + P2 ) h (2 ? 1)(f 0 (u ) ? f 0(u ?1)) i p + (1 ? 1)(1 ? 2) 1 ? Lx: (3.33) 2 a Substituting K for supj j jf 000 (u)j and (3.33) into (3.32) gives the desired estimate for I h i I K ? 41 (P1 + P2 ) ? 121 (P1 +1 + P2 ?1) + (1 ? )(P1 + P2 ) h 1 1 j2 ? 1j i + 4 1 + p + 12 1 + p Lx : (3.34) a a n
;j
n ;j
n
n
;j
;j
n ;j
;j
n ;j
n ;j
n j
u
n
n
;j
;j
n j
n j
n ;j
n ;j
n
n ;j
n j
b
n ;j
n ;j
15
n ;j
n ;j
n ;j
i
As the fourth step in estimating Q1 2 we have to consider the second quotient in (3.28). Using (3.26) and similar arguments as above we get (2 ? 1)f 0( ) 00 (2 ? 1)f 0( ) 0 0 f (u ) ? f (u ?1) = f ( ? 12 )(u ? u ?1) a a (2 ? 1)f 0 ( )f 00( ? 21 ) = a h f 0(u ?1) ? f 0(u ) i p P2 + P1 + 1 ? Lx 2 a (3.35) ? 2 j1 ?a 2j (P2 + P1 ) + 2 2j1 ?a2j Lx: n ;
n j
n j
n j
n j
n j
n ;j
n j
n j
n j
n j
n j
n ;j
n ;j
n j
n ;j
Now we get back to (3.28). We insert the estimates (3.30) and (3.31) together with (3.34) and also (3.35) to give Q ? c()( 2p+ KLx) (P + P + P + P ) n ;
n ;j
12
n ;j
n ;j
n ;j
1 1 +1 2 ?1 2 a 2 2 pa x ; (3.36) + c()a 2 Lx + c()KL p where c() is a generic constant depending on . Using = a one obtains from
(3.27) and (3.36) that
Q1 1 + Q1 2 ? c() 2p+aKLx (P1 + P1 +1 + P2 ?1 + P2 ) ? 2 1 ? 32 p2a ? c(p) a2 ? c()KLx Lx: Choosing a suitably large in order to make the last brackets non-positive and x n ;
n ;
n ;j
n ;j
n ;j
n ;j
suitably small one arrives at the estimate ) 2 (P + P + P + P ): Q1 1 + Q1 2 ? c(p 1 +1 2 ?1 2 a 1 n ;
n ;
n ;j
n ;j
n ;j
n ;j
(3.37)
Analogously, we obtain such an estimate for Q2 1 + Q2 2. Thus the proof of the Key Lemma is complete. 2 n ;
n ;
Remark. In the proof of the Key Lemma we can see that there exists a positive
constant c(; 1; 2; ) such that a > c(; 1; 2; ) is sucient for all arguments in the proof of the theorem related to the choice of a. The smallness assumption for x depends on the quantities ; 1; 2; ; L and K . 16
4 Error Estimates In this section we will consider error estimates for the discrete solution given by the relaxation scheme (1.3) with the initial data (1.4) as an approximation to the solution u of the Cauchy problem for the conservation law (1.1) and the initial condition (1.2). We are following the Lip0 theory developed by Nessyahu and Tadmor [14], [16]. First we extend our discrete solution (u ; v ) 2 2 given at the grid points to a piecewise bilinear function by setting n j
n j j
Z Z;n
IN
?u (x; t); v (x; t) := X (u ; v ;
;
j
2
2IN
n; j
n; j
) (x; t); n j
Z Z;n
where (x; t) := (x) (t) with (x) = 1x min(x ? x ?1; x +1 ? x)+; (t) = 1t min(t ? t ?1; t +1 ? t)+: n j
n
j
j
j
n
j
j
By Theorem 3.1 we have
ku k x;
j
+ (IR)
Lip
2L:
In order to use the results in [14, Theorem 2.1], we still have to discuss the Lip0consistency.
Lemma 4.1. (Lip0-Consistency) The approximation generated by the relaxation scheme
(1.3) with the initial data (1.4) on a time interval [0; T ] satis es the following truncation error estimate for u0 2 BV ( ) \ L1( ) IR
ku + f (u ) t
;
;
x
IR
[0 ]) CT (x + );
0(
(4.1)
Lip IR; ;T
where CT is a positive constant depending on the nal time T . Proof. Let N denote the number of time steps on [0; T ], i.e., T = tN = N t. We set
Z = f (u ) ? v ; for (j; n) 2 f1; ; N g: n j
n j
n j
Z Z
Then it follows from the rst equation of (1.3) that (4.2) x(u +1 ? u ) = ? 2t f (u +1) ? f (u ?1) p + x2 a (u +1 ? u ) ? (u ? u ?1) + 2t [Z +1 ? Z ?1 ]: n j
n j
n j
n j
n j
n j
17
n j
n j
n j
n j
We consider the test function 2 C01( 2), set t = nt and de ne the piecewise P bilinear interpolant ^(x; t) = 2 2 0 (x ; t ) (x; t). We further set IR
j
Z Z;n
n
j
IN
n j
n
4 X x H := Tk4 k =1
with T1 ; ; T4 as de ned by Nessyahu and Tadmor [14, (3.5)]. Here we additionally need XX (x ; t ) 2t (Z +1 ? Z ?1 ): H := x
x
N
j
=0
2
j
n j
n
n j
n
Z Z
Then we have, as in [14], the relation
?@ u (x; t) + @ f ?u (x; t); ;
t
= H + H :
;
x
x
x;t
(4.3)
For the relation between (4.2) and (4.3) see the Appendix of [14]. The following estimate is shown in [14, (3.7)]
H Const xkuk x
L
1 ([0;T ];BV
) k kLip(IR[0;T ]):
x
(4.4)
Now we estimate H , which comes from the relaxation term. Using summation by parts
X X t Z +1 [ (x ; t ) ? (x +2; t )] jH j = 2 2 =0 XX
tx (; t) ( [0 ]) jZ +1 j: N
n j
j
Z Z
j
n
j
n
n
N
Lip IR; ;T
j
P
2
Z Z
=0
n j
n
Recall that jf (u ) ? v jx C was shown in [23, Lemma 6]. This combined with the above estimate leads to n j
j
n j
jH j CT (; t)
( [0;T ]):
(4.5)
Lip IR;
Equipped with the above estimates (4.4), (4.5) we have
? @ u (x; t) + @xf (u ); C (x + )k k t
;
;
x;t
which implies (4.1). 18
T
( [0 ])
Lip IR; ;T
Furthermore, we show that the approximate solutions u are also Lip0-consistent with the initial data. We rst note that the u are clearly conservative, for by (4.2) and our choice of the discrete initial data, Z X X 0 0 Z x x u (x; t )dx = 2 (u + u +1) = 2 (u + u +1) = u0(x)dx: 2 2 ;
;
;
n j
n
IR
j
n j
j
j
Z Z
j
IR
Z Z
Moreover, these initial conditions are Lip0-consistent. In fact we have ?u (x; 0) ? u (x); (x) = ?u (x; 0) ? u ); (x) ? (x 1 ) 0 0 +2 ;
;
xk k
( [0;T ])
XZ
Lip IR;
C (x)2 u0(x) k k j
BV
which yields
u (x; 0) ? u (x)
0 x;
Lip0 (IR)
j
xj
+1
xj
u (x; 0) ? u (x) dx 0 ;
( [0;T ])
Lip IR;
C u0 (x)2:
(4.6)
BV
Now we can use the result in [14, Theorem 2.1] and get the error estimate h
u (; T ) ? u(; T )
u (; T ) ? u0(x) 0( ) 0( ) C ;
x;
T
+ u + f (u )
Lip IR
t
;
;
x
i
Lip IR
0(
[0 ])
Lip IR; ;T
C (x + ) = O(x + ):
(4.7) The Lip0 error estimate (4.7) may now be interpolated into the W -error estimates along the lines of Nessyahu and Tadmor [14, Corollary 2.2, 2.4]. The resulting error estimates are summarized as follows. Theorem 4.2. Consider the convex scalar conservation law (1.1) with Lip+-bounded initial data u0 and v0 = f (u0 ). Then the relaxation scheme with discrete initial data (u0; f (u0)) 2 converges. The piecewise linear interpolants u satisfy the convergence rate estimates ku (; T ) ? u(; T )k C (x + ) 1?2 ; for ? 1 s 1 ; 1 p 1; T
s;p
j
j
j
;
Z Z
sp
;
W s;p
p
p
T
(4.8)
as well as
ju (x; T ) ? u(x; T )j Const (x + ) 31 ; ;
with
Const
x;T
(4.9)
x;T
1 + u (; T ) 1 ( ?( + )1 3 x
L
19
x
x
=
+(x+)1 3 ) :
;x
=
Remarks:
1. When (s; p) = (?1; 1) the error estimate (4.8) turns into the Lip0 error estimate
ku (; t) ? u(; t)k x
0(
) O( + x):
Lip IR
2. p When (s; p) = (0; 1) the error estimate (4.8) yields an L1-convergence rate of order O( x + ) which is consistent with the result obtained in [10] for conservation laws with possibly nonconvex ux functions.
Acknowledgement: The authors are grateful to Tao Tang for helpful discussions
on this work. H. Liu was supported partially by the German-Israeli Foundation for Research and Development (GIF) and partially by the Deutsche Forschungsgemeinschaft (DFG) grant Wa 633/11-1. J. Wang was supported by the National Natural Science Foundation of China, visitor funds of the German Priority Research Program on Conservation Laws (ANumE), DFG-grant Wa 633/9-1, and by the U.S.- China Joint Research proposal No. INT-9601376.
References [1] D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996), 163-190. [2] D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional conservation laws, preprint (1997). [3] S. Jin, and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure. Appl. Math. 48 (1995), 235-277. [4] A. Kurganov and E. Tadmor, Sti systems of hyperbolic conservation laws. Convergence and error estimates, SIAM J. Math. Anal., 28 (1997), 1446-1456. [5] A. Kurganov and E. Tadmor, New High-Resolution central schemes for nonlinear conservation laws and convection-di usion equations, preprint (1999). [6] M. Katsoulakis and A. Tzavaras, Contractive relaxation systems and scalar multidimensional conservation laws, Commun. P.D.E., 22 (1997), 195233. 20
[7] M. Katsoulakis, K. Kossioris and Ch. Makridakis, Convergence and error estimates of relaxation schemes for multidimensional conservation laws, Commun. P. D. E., 24 (1999), 395{424. [8] T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), 153{175. [9] H. L. Liu and R. Natalini, Asymptotic behavior for a hyperbolic relaxation system, in preparation. [10] H. L. Liu and G. Warnecke, Convergence rates for relaxation schemes approximating conservation laws, submitted to SIAM J. Numer Anal. [11] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), 795-823. [12] R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Di erential Equations 148 (1998), 292{317 [13] R. Natalini, Recent mathematical results on hyperbolic relaxation problems, Quaderno IAC 7 (1998). [14] H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal. 29 (1992), 15051519. [15] H. Nessyahu and E. Tadmor, Non-oscillatory central di erencing for hyperbolic conservation laws, J. Comp. Phys. 87 (1990), 408-463. [16] E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891-906. [17] E. Tadmor and T. Tang , Pointwise error estimates for relaxation approximations to conservation laws, preprint (1998). [18] E. Tadmor and T. Tang , The optimal convergence rate of nite di erence solutions for nonlinear conservation laws, Proceedings of Seventh International Conference on Hyperbolic Conservation Laws, Vol 2 (1998), 925-934. 21
[19] Z.-H. Teng, First-order L1?convergence for relaxation approximations to conservations, Comm. Pure Appl. Math. 51 (1998), 857-895. [20] A. Tveito and R. Winther, On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term, SIAM J. Math. Anal. 28 (1997), 136-161. [21] H.Z. Tang and H.M. Wu, On a cell entropy inequality for relaxing schemes of conservation laws, to appear in J. Comput. Math. [22] G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [23] J. Wang and G. Warnecke, Convergence of relaxing schemes for conservations laws, Advances in Nonlinear Partial Di erential Equations and Related Areas, G.-Q. Chen, Y. Li, X. Zhu, and D. Cao (eds.), pp. 300-325, World Scienti c, Singapore, 1998 [24] W. A. Yong, Numerical analysis of relaxation schemes for scalar conservation laws, Technical Report 95-30 (SFB 359), IWR, University of Heidelberg, 1995.
22
y
Jinghua Wang
z
Gerald Warnecke
x
Abstract
We show the discrete lip+ -stability for a relaxation scheme proposed by Jin and Xin (1995, Comm. Pure. Appl. Math. 48, 235-277) to approximate convex conservation laws. Equipped with the lip+ -stability we obtain global error estimates in the spaces W s;p for ?1 s 1=p, 1 p 1 and pointwise error estimates for the approximate solution obtained by the relaxation scheme. The proof uses the framework introduced by Nessyahu and Tadmor (1992, SIAM J. Numer Anal. 29, 1505-1519 ). We also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state.
Keywords: Relaxation scheme, convex conservation laws, lip+-stability, maximum principle. AMS Subject Classi cation: 35L65, 65M10, 65M15
1 Introduction Relaxation schemes are a class of nonoscillatory numerical schemes for systems of conservation laws proposed by Jin and Xin [3]. They are motivated by relaxation models for ows which are not in thermodynamic equilibrium, i.e. they constitute Manuscript version: July 5, 1999 IAN, Otto-von-Guericke-Universitat Magdeburg, PSF 4120, D-39016 Magdeburg, Germany. Email: [email protected] z Institute of Systems Science, Academia Sinica, Beijing 100080, P.R.China. Email: [email protected] x IAN, Otto-von-Guericke-Universit at Magdeburg, PSF 4120, D-39016 Magdeburg, Germany. Email: [email protected] y
1
more general and more accurate models of certain physical phenomena. The relaxation schemes provide a new way of perturbing, even regularizing, systems of conservation laws and approximating their solutions. In this sense they are to be seen as an interesting tool of analysis. The computational results that are available, see e.g. [3] as well as Aregba-Driollet and Natalini [2], indicate that the relaxed schemes obtained in the limit ! 0 provide a quite promising class of new schemes. We point out that that the main assets of these schemes are that they neither require the computation of the Jacobians of uxes for the conservation laws nor the use of Riemann-solvers. This is needed for ows in which a real gas law has to be used in place of the frequently used assumption of an ideal gas, e.g. the two phase ow for kryogenic gases. In such cases it may be too expensive or even impossible to calculate Riemann solutions or even ux Jacobians. This important property is schared by other schemes such as for instance the high resolution central schemes introduced by Nessyahu and Tadmor [15], see also Kurganov and Tadmor [5] for references on recent developments. To make things more precise we want to consider a scalar conservation law. We take a convex ux function f 2 C 3( ) and initial data u0 2 L1( ) and consider the Cauchy problem u + f (u) = 0 (1.1) with initial data u(x; 0) = u0(x): (1.2) For this problem we want to approximate the global weak entropy solution by a relaxation scheme. We choose a time step t, a spatial mesh size x, a parameter a which will be related to the characteristic speeds of the conservation law and a small relaxation parameter > 0. From these we de ne the mesh ratio = , the CFL parameter = pa 2]0; 1[ and the scale parameter k = . The mesh is given by the points (j x; nt) for j 2 and n 2 0. The approximate solution takes the discrete values u at the mesh points. Further, relaxation schemes involve the discrete relaxed uxes v . We want to consider the following semi-implicit relaxation scheme IR
IR
t
x
t x
t
Z Z
IN
n j
n j
u +1 ? u + 2 (v +1 ? v ?1) ? 2 (u +1 ? 2u + u ?1) = 0; n j
v
n j
n j
n j
n j
n j
+1 ? v n + a (un ? un ) ? (v n ? 2v n + v n ) j j j ?1 2 j+1 j?1 2 j+1
n j
2
j2 ; n2
n j
Z Z
IN
(1.3)
= ?k[v +1 ? f (u +1 )]: n j
n j
The discrete initial data are given by averaging the initial data u0 over mesh cells 1 I = (j ? 2 )x; (j + 21 )x , i.e. taking Z 1 0 u = x u0(x)dx; and setting v0 = f (u0): (1.4) j
j
j
Ij
j
The di erence v ? f (u) measures a deviation from an equilibrium in relaxation models. If we have v = f (u), we say that our variables are in an equilibrium state. We are interested in the relaxation limit where k is large, i.e. is small and the conservation law is being approximated. In this case the source term becomes sti . Moreover, we speci cally will have to require in our analysis that for some positive constant c the scale condition 0 0 for u 2 :
(3.1)
IR
Further we assume that the initial data satisfy the uniform bound ju0(x)j b < 1 for x 2 . Therefore, by (1.4) the discrete initial data inherit this bound, i.e. IR
ju0j b for j 2 :
(3.2)
Z Z
j
We choose a > 0 satisfying the subcharacteristic condition
p
sup jf 0(u)j < a:
(3.3)
jujb
It follows from Theorem 2.1 that the discrete solution (u ) 2 (1.3) satis es the same L1 -bound as initial data, i.e. n j j
2
Z Z;n
IN
given by the scheme
ju j b for j 2 ; n 2 : n j
Z Z
(3.4)
IN
Since f 2 C 3 and convex, there exist positive constants , 1, 2, K such that
p
sup jf 0(u)j = < a;
(3.5)
1 f 00(u) 2; for ? b u b; sup jf 000(u)j = K:
(3.6) (3.7)
jujb
jujb
Theorem 3.1. (lip+-Stability) Assume that ku0k
Lip
+ (IR)
=: L < 1;
(3.8)
the parameter a > 0 is suitably large, x is suitably small and the scale parameter k satis es the scale condition (1.5). Then the approximate solution (unj )j2 ;n2 given by the relaxation scheme (1.3) with initial data (1.4) satis es the lip+ -stability. More precisely, the following estimate holds Z Z
u ? u ?1 2Lx n j
IN
for j 2 ; n 2 :
n j
Z Z
8
IN
(3.9)
Proof. We de ne the di erence
R := R ? R ?1 : n
n j
j
(3.10)
n j
By the Mean Value Theorem we nd for any j 2 , n 2 1 and u ?+12 such that Z Z
IN
1 a value between u + 2 n j
n
j
n
j
1
1
1
1
u + 2 ? u ?+12 f 0( ) = f (u + 2 ) ? f (u ?+12 ): n
n
j
We set for any u 2
n
n j
j
IR
!
M10 (u) M0(u) := M 0 2 (u)
n
j
j
f 0 (u) 1 2 (1 ? 0pa ) f (u) 1 2 (1 + pa ):
=
!
(3.11)
(3.12)
The explicit form of the scheme (2.7) and using (3.11) together with (2.6) then gives 1 1 (3.13) R +1 = 1 +1 k R + 2 + 1 +k k M0( ) E R + 2 ; We choose a discrete vector function A as n
n
j
j
n
n j
j
n j
A := AA1 2
n ;j
n j
!
n ;j
=
!
M10 (u ?1) : M20 (u ) n j
n j
(3.14)
The heart of the matter is to prove the inequality
R A Lx; j 2 ; n 2 : n j
n j
Z Z
(3.15)
IN
This estimate combined with (3.5) and (3.14) yields (3.9) as follows f 0(u ?1 ) ? f 0(u ) pa u ? u ?1 = E R 1 ? Lx 2 1 + p a Lx 2Lx: n j
n j
n
n j
n j
j
It remains to prove (3.15). For this purpose we de ne
P := R ? A Lx: n j
n
(3.16)
n j
j
Then (3.15) becomes
P 0; i.e. P 0; j 2 ; n 2 ; i = 1; 2: n j
n i;j
Z Z
9
IN
(3.17)
We shall prove (3.17) by induction. First let us consider the case n = 0. We have by inserting (3.10) together with (2.1) and (3.14) into (3.16) and using the de nition of the discrete initial data (1.4) as well as the Mean Value Theorem 0 v0 1 0 v0?1 1 f 0(u0?1) 1 0 P1 = 2 u ? pa ? 2 u ?1 ? pa ? 2 1 ? pa Lx h f (u0) ?pf (u0?1) i 1 f 0(pu0?1) = 12 (u0 ? u0?1) ? ? 2 1 ? a Lx a h 0 0 i f 0(u0?1) 1 = 2 (u ? u ?1) ? Lx 1 ? pa ? 2p1 a f (u0) ? f (u0?1) ? f 0(u0?1)(u0 ? u0?1) h 0 0 i f 0(u0?1) 1 00 0 0 0 2 1 = 2 (u ? u ?1) ? Lx 1 ? pa ? 4pa f ( )(u ? u ?1) and 0 v0 1 0 v0?1 1 f 0(u0) 1 0 P2 = 2 u + pa ? 2 u ?1 + pa ? 2 1 + pa Lx h i f 0(u0) = 12 (u0 ? u0?1) ? Lx 1 + pa ? 2p1 a f (u0?1) ? f 0(u0) + f 0(u0)(u0 ? u0?1) h i f 0(u0) = 21 (u0 ? u0?1) ? Lx 1 + pa ? 4p1 a f 00(~0 )(u0 ? u0?1)2; where 0 and ~0 are intermediate values between u0?1 and u0. Thus (3.17) with n = 0 follows from the assumption (3.8), the convexity of the ux function, i.e. f 00 > 0, and the subcharacteristic condition (3.3). We now assume that (3.17) is true for n. It remains to prove (3.17) for n + 1, i.e. j
j
j
;j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
;j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
P
+1 0
n i;j
j
j
j
for i = 1; 2; j 2 :
(3.18)
Z Z
To this end we insert the relation (3.16) into (3.13) and get
P +1 = 1 +1 k n j
!
(1 ? )P1 + P1 +1 (1 ? )P2 + P2 ?1 + 1 +k k M0( )E (1 ? )P1 + P1 +1 (1 ? )P2 + P2 ?1 n ;j
n ;j
n ;j
n ;j
n j
10
n ;j
n ;j
n ;j
n ;j
!
+ (Q1 + Q2 )Lx: n
n
(3.19)
The vectors Q1 and Q2 are given as follows n
n
!
!
? )A1 + A1 +1 Q1 = QQ1 1 := ?A +1 + (1 (1 ? )A2 + A2 ?1 1 0 211 0 +1 0 p f (u 1 ) ? (1 ? )f (u ?1) ? f (u ) A = @ 2 1 0 ?+1 (3.20) ? 2p f (u ) ? (1 ? )f 0(u ) ? f (u ?1 ) and, using the fact that by (3.12) we have M20 (u) = 1 ? M10 (u), ! 0 (1 ? )A1 + A1 +1 ! Q k 1 2 := 1 + k M ( )E ? Id Q2 = Q 22 0 1? (1 ?0( )A?12) + A2 ?1 0( ) 1 ! 0 ( ) M 0 ( ) 1 ? p + 2 1 ? p A ? M k 2 2 1 @ =1 + k : 1? 1 + 0p( ) + 1 + 0 (p ?1 ) M20 ( ) ?M10 ( ) 2 2 n ;
n j
n ;
n j
a
n ;
n ;j
n ;j
n ;j
n j
n j
a
n ;j
n j
n j
n j
n j
n ;
n j
n j
f
un j
n j
n j
f
a un j
n ;j
n ;j
n ;j
n ;j
f
un j a
f
a
un j
a
(3.21)
For the last term in (3.19) we have the following estimate Lemma 3.2. (Key Lemma) If the assumptions of Theorem 3.1 and the induction assumption (3.17) hold, then there exists a constant c() > 0 such that with 2 given as in (3.6) we have the following estimate (3.22) Q + Q ? c(p) 2 P + P + P + P E : 1
2
n
n
1;j
a
n
1;j +1
2;j ?1
n
2;j
We continue the argument and postpone the proof of this lemma to the end of this section. It follows from (3.19) and Lemma 3.2 that h i P1 +1 k +1 1 (1 ? )P1 + P1 +1 f 0( ) + k +k 1 21 1 ? pa (1 ? )(P1 + P2 ) + (P1 +1 + P2 ?1 ) ? c(p) 2 (P + P + P + P )Lx: n ;j
n ;j
n ;j
n j
a
n
n ;j
n
1;j
n
1;j +1
2;j ?1
n ;j
n ;j
n ;j
n
2;j
The induction assumption P 0 yields the estimate 1 +1 P1 k + 1 (1 ? )P1 + P1 +1 k 1 f 0( ) c ( ) L x 2 + k + 1 2 1 ? p minf; (1 ? )g ? p n i;j
n ;j
n ;j
n ;j
n j
(P1 + P1 n ;j
n ;j
a
+1 + P2;j ?1 + P2;j ): n
11
n
a
(3.23)
Using the scale condition (1.5), i.e. the assumption that k is bounded away from zero, we see that +1 is also bounded away from zero. Due to this and the subcharacteristic condition (3.3) the coecient of the second term on the right hand side of (3.23) is nonnegative when x is small enough. Thereby P1 +1 0 follows immediately. Analogously, we obtain P2 +1 0; j 2 : These estimates complete the proof of the theorem. k
k
n ;j
n ;j
Z Z
Remark. A slightly more general stability estimate of the form u ? u ?1 nt + 2(Lx)?1 n j
n j
for a positive constant 1 can be analogously obtained by just proving that P := R ? A nt + 1(Lx)?1 0 instead of (3.17) with P de ned as in (3.16). Proof of the Key Lemma 3.2: First we collect three identities deduced from the relaxation scheme (2.3) and (2.6) n
n j
n j
j
n j
u +1 ? u = E (R n j
n j
1 n+ 2
j
!
+1 ? R1 n 2 ?1 ? R2;j
? R ) = E RR1
n ;j
n j
n ;j
= (R1 +1 ? R2 ); n
1 1 u +1 ? u ?+11 = E (R + 2 ? R ?+12 ) = E R + R1 +1 ? R1 R2 ?1 ? R2 = (1 ? )(R1 + R2 ) + (R1 +1 + R2 ?1); u ? u ?1 = E (R ? R ?1) = R1 + R2 : n j
n j
n
n
n
j
j
j
n
n
;j
n j
n j
n j
n
;j
n j
n
n
;j n
;j n
;j
;j
;j
!!
;j
n
;j
(3.24) (3.25) (3.26)
n
;j
n
n
;j
n ;j
;j
These identities will be used repeatedly. Using the Mean Value Theorem we have for the rst component Q1 1 of Q1 in (3.20) 1 Q1 1 = 2p1 a f 00( ?+12 )(u ?+11 ? u ?1) ? f 00( ? 12 )(u ? u ?1 ) : n ;
n ;
n
j
n j
n j
12
n j
n j
n j
n
Taking (3.24), (3.26) and the relation R = P + A Lx one obtains n
n j
j
n j
Q1 1 = 2p1 a f 00( ?+12 )(R1 ? R2 ?1) ? f 00( ? 12 )(R1 + R2 ) h i 1 1 = ? 2 f 00( ?+12 )P2 ?1 + f 00( ? 12 )P2 ? f 00( ?+12 ) ? f 00( ? 12 ) P1 00 + 12 00 0 2 3 00 0 f ( ) f ( u ) + 2 f ( ) ? f ( ) f ( u ) 1 1 +1 ?1 +2 ?2 5 Lx: p ? 2 4f 00( ? 12 ) + 2 a n ;
n
1
n
n
;j
j
n
;j
n ;j
j
n j
n
;j
;j
n
n ;j
n j
n j
n
n j
n j
j
n
n j
n ;j
n j
j
n j
Recalling the induction hypothesis (3.17), i.e. P 0, and using the bounds for f 0 f 00 given in (3.5), (3.6) one obtains the estimate (3.27) Q1 1 ? 2 2 (P2 ?1 + P2 + P1 ) ? 2 1 ? 32 p2a Lx: Note that if f 00 = 1 then the last term in (3.27) may alternatively be estimated sharper as ? 2 1 1 ? p a Lx: n j
n ;j
n ;
n ;j
n ;j
p
It is negative under the subcharacteristic condition < a. Now we proceed to estimate Q1 2. A straightforward evaluation of Q12 in (3.21) yields f 0(u ) + f 0(u ?1) ? 2f 0( ) (2 ? 1)f 0( )(f 0(u ) ? f 0(u ?1)) pa : + Q1 2 = 4(1 k+ k) a (3.28) n ;
n j
n ;
n j
n j
n j
n j
n j
The estimate to be derived from this inequality, which will be (3.36) below, will be obtained in four steps. Three of these steps are needed to estimate the rst quotient. Our rst two steps are estimates of the di erences appearing in the rst quotient in (3.28). We make repeated use of the Mean Value Theorem in the following form
f (u) ? f (v) =
Z1 0
f 0(u + (1 ? )v)d (u ? v):
Using the de nition of f 0( ) in (3.11) and (2.6) we obtain n j
f 0(unj?1) ? f 0(jn ) = ?
Z 1Z 1 0
0
h
i
f100(; 1) (u +1 ? u ?1) + (1 ? )(u ?+11 ? u ?1) d1d; n j
n j
n j
n j
(3.29) 13
where
f100(; 1) := f 00(1u ?1 + (1 ? 1)(u +1 + (1 ? )u ?+11 )): n j
n j
n j
By (3.25) and (3.24) we have J := (u +1 ? u ?1 ) + (1 ? )(u ?+11 ? u ?1) = (u +1 ? u ?+11 ) + (u ?+11 ? u ?1) ? = (R1 +1 + R2 ?1) + (1 ? )(R1 + R2 ) + (R1 ? R2 ?1): Inserting the relation (3.16) gives h J = (P1 +1 + P2 ?1) + (1 ? )(P1 + P2 ) ?f 0(u ?1) ? f 0(u ) (1 ? )?f 0(u ) ? f 0(u ?1) i p p + + + 1 Lx 2 a 2 a f 0 (u ) + (P1 ? P2 ?1) ? pa?1 Lx: n j
n j
n j
n
n j
n
;j
n
;j
n ;j
n j
n
;j
n ;j
n j
n j
n
;j
n ;j
n ;j
n j
n j
n
;j
;j
n ;j
n j
n j
n j
n j
n ;j
Having obtained this expression for J we use (3.3), (3.5), (3.6) and the induction hypothesis (3.17) to obtain from (3.29) the estimate h i f 0(u ?1) ? f 0( ) ? 2 2 (P1 +1 + P2 ?1) + 1 ?2 (P1 + P2 ) + P1 Z 1Z 1 ( j 2 ? 1 j + )
p + Lx ? f 00(; )d dLx: n j
n j
n ;j
n ;j
2
n ;j
a
0
n ;j
0
n ;j
1
1
1
(3.30) Now we get to the second step of estimating the second di erence in the rst quotient in (3.28). De ning analogously as above f200(; 1) := f 00(1u + (1 ? 1)(u +1 + (1 ? )u ?+11 )) one has Z 1Z 1 h i 0 0 f (u ) ? f ( ) = ? f200(; 1) (u +1 ? u ) + (1 ? )(u ?+11 ? u ) d1d n j
n j
n j
= ?
Z 1Z 1
n j
n j
0 0
n j
n j
n j
h +1 ? R2 ) 0 0 i +(1 ? ) ? (R1 +1 + R2 ?1) ? (1 ? )(R1 + R2 ) d1d " Z 1Z 1 0 f200(; 1) (R1
n
n
;j
;j
n
n
;j
= ?
n j
0
0
f200(; 1) (P1
n ;j
n
;j
n
;j
;j
f (u ) +1 ? P2 ) ? pa Lx n j
n ;j
+(1 ? ) ? (P1 +1 + P2 ?1) ? (1 ? )(P1 + P2 ) n ;j
n ;j
n ;j
n ;j
!# f 0 (u ?1) ? f 0(u ) f 0(u ) ? f 0(u ?1) p p ? + (1 ? ) + 1 Lx d d: n j
2 a
n j
n j
14
2 a
n j
1
This gives us the following inequality
(j1 ? 2j + )
pa Lx +1 + 2 Z 1Z 1 + f200(; 1)(1 ? )d1dLx: 0 0 the estimate for Qn1;2 we have to take
f 0(u ) ? f 0( ) ? 2P1 n j
n j
n ;j
(3.31)
care of the As the third step in deriving integrals involving second derivatives f 00 in (3.30) and (3.31). For this purpose we de ne Z 1Z 1 I := f200(; 1)(1 ? ) ? f100(; 1) d1d 0
0
which becomes zero for the case f 00 =const. For general convex ux functions this term has to be treated carefully since the integral in (3.31) is positive. It is not obviously dominated by other negative terms. We estimate I as follows. Using the Mean Value Theorem again we get
Z 1Z 1 h ? +1 +1 00 I= f 1u + (1 ? 1) (1 ? )u + u ?1 0 0 i ? +1 +1 00 ? f 1u ?1 + (1 ? 1) u + (1 ? )u ?1 d1 d Z 1 Z 1 Z 1 000 n j
n j
n j
=
where
0
0
0
n j
n j
n j
f 21 + (1 ? 2)2 (1 ? 2)d2d1d;
(3.32)
1 ? 2 := 1(u ? u ?1) + (1 ? 1)(1 ? 2)(u +1 ? u ?+11 ): Using (3.26) and R = P + A Lx we have n j
n
n j
j
n j
n j
n j
h
n j
1 ? 2 = 1(R1 + R2 ) + (1 ? 1)(1 ? 2) (1 ? )(R1 + R2 ) + (R1 +1 + R2 ?1) f 0(u ?1) ? f 0(u ) pa = 1(P1 + P2 ) + 1 1 ? Lx 2 h i + (1 ? 1)(1 ? 2) (P1 +1 + P2 ?1) + (1 ? )(P1 + P2 ) h (2 ? 1)(f 0 (u ) ? f 0(u ?1)) i p + (1 ? 1)(1 ? 2) 1 ? Lx: (3.33) 2 a Substituting K for supj j jf 000 (u)j and (3.33) into (3.32) gives the desired estimate for I h i I K ? 41 (P1 + P2 ) ? 121 (P1 +1 + P2 ?1) + (1 ? )(P1 + P2 ) h 1 1 j2 ? 1j i + 4 1 + p + 12 1 + p Lx : (3.34) a a n
;j
n ;j
n
n
;j
;j
n ;j
;j
n ;j
n ;j
n j
u
n
n
;j
;j
n j
n j
n ;j
n ;j
n
n ;j
n j
b
n ;j
n ;j
15
n ;j
n ;j
n ;j
i
As the fourth step in estimating Q1 2 we have to consider the second quotient in (3.28). Using (3.26) and similar arguments as above we get (2 ? 1)f 0( ) 00 (2 ? 1)f 0( ) 0 0 f (u ) ? f (u ?1) = f ( ? 12 )(u ? u ?1) a a (2 ? 1)f 0 ( )f 00( ? 21 ) = a h f 0(u ?1) ? f 0(u ) i p P2 + P1 + 1 ? Lx 2 a (3.35) ? 2 j1 ?a 2j (P2 + P1 ) + 2 2j1 ?a2j Lx: n ;
n j
n j
n j
n j
n j
n ;j
n j
n j
n j
n j
n j
n ;j
n ;j
n j
n ;j
Now we get back to (3.28). We insert the estimates (3.30) and (3.31) together with (3.34) and also (3.35) to give Q ? c()( 2p+ KLx) (P + P + P + P ) n ;
n ;j
12
n ;j
n ;j
n ;j
1 1 +1 2 ?1 2 a 2 2 pa x ; (3.36) + c()a 2 Lx + c()KL p where c() is a generic constant depending on . Using = a one obtains from
(3.27) and (3.36) that
Q1 1 + Q1 2 ? c() 2p+aKLx (P1 + P1 +1 + P2 ?1 + P2 ) ? 2 1 ? 32 p2a ? c(p) a2 ? c()KLx Lx: Choosing a suitably large in order to make the last brackets non-positive and x n ;
n ;
n ;j
n ;j
n ;j
n ;j
suitably small one arrives at the estimate ) 2 (P + P + P + P ): Q1 1 + Q1 2 ? c(p 1 +1 2 ?1 2 a 1 n ;
n ;
n ;j
n ;j
n ;j
n ;j
(3.37)
Analogously, we obtain such an estimate for Q2 1 + Q2 2. Thus the proof of the Key Lemma is complete. 2 n ;
n ;
Remark. In the proof of the Key Lemma we can see that there exists a positive
constant c(; 1; 2; ) such that a > c(; 1; 2; ) is sucient for all arguments in the proof of the theorem related to the choice of a. The smallness assumption for x depends on the quantities ; 1; 2; ; L and K . 16
4 Error Estimates In this section we will consider error estimates for the discrete solution given by the relaxation scheme (1.3) with the initial data (1.4) as an approximation to the solution u of the Cauchy problem for the conservation law (1.1) and the initial condition (1.2). We are following the Lip0 theory developed by Nessyahu and Tadmor [14], [16]. First we extend our discrete solution (u ; v ) 2 2 given at the grid points to a piecewise bilinear function by setting n j
n j j
Z Z;n
IN
?u (x; t); v (x; t) := X (u ; v ;
;
j
2
2IN
n; j
n; j
) (x; t); n j
Z Z;n
where (x; t) := (x) (t) with (x) = 1x min(x ? x ?1; x +1 ? x)+; (t) = 1t min(t ? t ?1; t +1 ? t)+: n j
n
j
j
j
n
j
j
By Theorem 3.1 we have
ku k x;
j
+ (IR)
Lip
2L:
In order to use the results in [14, Theorem 2.1], we still have to discuss the Lip0consistency.
Lemma 4.1. (Lip0-Consistency) The approximation generated by the relaxation scheme
(1.3) with the initial data (1.4) on a time interval [0; T ] satis es the following truncation error estimate for u0 2 BV ( ) \ L1( ) IR
ku + f (u ) t
;
;
x
IR
[0 ]) CT (x + );
0(
(4.1)
Lip IR; ;T
where CT is a positive constant depending on the nal time T . Proof. Let N denote the number of time steps on [0; T ], i.e., T = tN = N t. We set
Z = f (u ) ? v ; for (j; n) 2 f1; ; N g: n j
n j
n j
Z Z
Then it follows from the rst equation of (1.3) that (4.2) x(u +1 ? u ) = ? 2t f (u +1) ? f (u ?1) p + x2 a (u +1 ? u ) ? (u ? u ?1) + 2t [Z +1 ? Z ?1 ]: n j
n j
n j
n j
n j
n j
17
n j
n j
n j
n j
We consider the test function 2 C01( 2), set t = nt and de ne the piecewise P bilinear interpolant ^(x; t) = 2 2 0 (x ; t ) (x; t). We further set IR
j
Z Z;n
n
j
IN
n j
n
4 X x H := Tk4 k =1
with T1 ; ; T4 as de ned by Nessyahu and Tadmor [14, (3.5)]. Here we additionally need XX (x ; t ) 2t (Z +1 ? Z ?1 ): H := x
x
N
j
=0
2
j
n j
n
n j
n
Z Z
Then we have, as in [14], the relation
?@ u (x; t) + @ f ?u (x; t); ;
t
= H + H :
;
x
x
x;t
(4.3)
For the relation between (4.2) and (4.3) see the Appendix of [14]. The following estimate is shown in [14, (3.7)]
H Const xkuk x
L
1 ([0;T ];BV
) k kLip(IR[0;T ]):
x
(4.4)
Now we estimate H , which comes from the relaxation term. Using summation by parts
X X t Z +1 [ (x ; t ) ? (x +2; t )] jH j = 2 2 =0 XX
tx (; t) ( [0 ]) jZ +1 j: N
n j
j
Z Z
j
n
j
n
n
N
Lip IR; ;T
j
P
2
Z Z
=0
n j
n
Recall that jf (u ) ? v jx C was shown in [23, Lemma 6]. This combined with the above estimate leads to n j
j
n j
jH j CT (; t)
( [0;T ]):
(4.5)
Lip IR;
Equipped with the above estimates (4.4), (4.5) we have
? @ u (x; t) + @xf (u ); C (x + )k k t
;
;
x;t
which implies (4.1). 18
T
( [0 ])
Lip IR; ;T
Furthermore, we show that the approximate solutions u are also Lip0-consistent with the initial data. We rst note that the u are clearly conservative, for by (4.2) and our choice of the discrete initial data, Z X X 0 0 Z x x u (x; t )dx = 2 (u + u +1) = 2 (u + u +1) = u0(x)dx: 2 2 ;
;
;
n j
n
IR
j
n j
j
j
Z Z
j
IR
Z Z
Moreover, these initial conditions are Lip0-consistent. In fact we have ?u (x; 0) ? u (x); (x) = ?u (x; 0) ? u ); (x) ? (x 1 ) 0 0 +2 ;
;
xk k
( [0;T ])
XZ
Lip IR;
C (x)2 u0(x) k k j
BV
which yields
u (x; 0) ? u (x)
0 x;
Lip0 (IR)
j
xj
+1
xj
u (x; 0) ? u (x) dx 0 ;
( [0;T ])
Lip IR;
C u0 (x)2:
(4.6)
BV
Now we can use the result in [14, Theorem 2.1] and get the error estimate h
u (; T ) ? u(; T )
u (; T ) ? u0(x) 0( ) 0( ) C ;
x;
T
+ u + f (u )
Lip IR
t
;
;
x
i
Lip IR
0(
[0 ])
Lip IR; ;T
C (x + ) = O(x + ):
(4.7) The Lip0 error estimate (4.7) may now be interpolated into the W -error estimates along the lines of Nessyahu and Tadmor [14, Corollary 2.2, 2.4]. The resulting error estimates are summarized as follows. Theorem 4.2. Consider the convex scalar conservation law (1.1) with Lip+-bounded initial data u0 and v0 = f (u0 ). Then the relaxation scheme with discrete initial data (u0; f (u0)) 2 converges. The piecewise linear interpolants u satisfy the convergence rate estimates ku (; T ) ? u(; T )k C (x + ) 1?2 ; for ? 1 s 1 ; 1 p 1; T
s;p
j
j
j
;
Z Z
sp
;
W s;p
p
p
T
(4.8)
as well as
ju (x; T ) ? u(x; T )j Const (x + ) 31 ; ;
with
Const
x;T
(4.9)
x;T
1 + u (; T ) 1 ( ?( + )1 3 x
L
19
x
x
=
+(x+)1 3 ) :
;x
=
Remarks:
1. When (s; p) = (?1; 1) the error estimate (4.8) turns into the Lip0 error estimate
ku (; t) ? u(; t)k x
0(
) O( + x):
Lip IR
2. p When (s; p) = (0; 1) the error estimate (4.8) yields an L1-convergence rate of order O( x + ) which is consistent with the result obtained in [10] for conservation laws with possibly nonconvex ux functions.
Acknowledgement: The authors are grateful to Tao Tang for helpful discussions
on this work. H. Liu was supported partially by the German-Israeli Foundation for Research and Development (GIF) and partially by the Deutsche Forschungsgemeinschaft (DFG) grant Wa 633/11-1. J. Wang was supported by the National Natural Science Foundation of China, visitor funds of the German Priority Research Program on Conservation Laws (ANumE), DFG-grant Wa 633/9-1, and by the U.S.- China Joint Research proposal No. INT-9601376.
References [1] D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996), 163-190. [2] D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional conservation laws, preprint (1997). [3] S. Jin, and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure. Appl. Math. 48 (1995), 235-277. [4] A. Kurganov and E. Tadmor, Sti systems of hyperbolic conservation laws. Convergence and error estimates, SIAM J. Math. Anal., 28 (1997), 1446-1456. [5] A. Kurganov and E. Tadmor, New High-Resolution central schemes for nonlinear conservation laws and convection-di usion equations, preprint (1999). [6] M. Katsoulakis and A. Tzavaras, Contractive relaxation systems and scalar multidimensional conservation laws, Commun. P.D.E., 22 (1997), 195233. 20
[7] M. Katsoulakis, K. Kossioris and Ch. Makridakis, Convergence and error estimates of relaxation schemes for multidimensional conservation laws, Commun. P. D. E., 24 (1999), 395{424. [8] T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), 153{175. [9] H. L. Liu and R. Natalini, Asymptotic behavior for a hyperbolic relaxation system, in preparation. [10] H. L. Liu and G. Warnecke, Convergence rates for relaxation schemes approximating conservation laws, submitted to SIAM J. Numer Anal. [11] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), 795-823. [12] R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Di erential Equations 148 (1998), 292{317 [13] R. Natalini, Recent mathematical results on hyperbolic relaxation problems, Quaderno IAC 7 (1998). [14] H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal. 29 (1992), 15051519. [15] H. Nessyahu and E. Tadmor, Non-oscillatory central di erencing for hyperbolic conservation laws, J. Comp. Phys. 87 (1990), 408-463. [16] E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891-906. [17] E. Tadmor and T. Tang , Pointwise error estimates for relaxation approximations to conservation laws, preprint (1998). [18] E. Tadmor and T. Tang , The optimal convergence rate of nite di erence solutions for nonlinear conservation laws, Proceedings of Seventh International Conference on Hyperbolic Conservation Laws, Vol 2 (1998), 925-934. 21
[19] Z.-H. Teng, First-order L1?convergence for relaxation approximations to conservations, Comm. Pure Appl. Math. 51 (1998), 857-895. [20] A. Tveito and R. Winther, On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term, SIAM J. Math. Anal. 28 (1997), 136-161. [21] H.Z. Tang and H.M. Wu, On a cell entropy inequality for relaxing schemes of conservation laws, to appear in J. Comput. Math. [22] G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [23] J. Wang and G. Warnecke, Convergence of relaxing schemes for conservations laws, Advances in Nonlinear Partial Di erential Equations and Related Areas, G.-Q. Chen, Y. Li, X. Zhu, and D. Cao (eds.), pp. 300-325, World Scienti c, Singapore, 1998 [24] W. A. Yong, Numerical analysis of relaxation schemes for scalar conservation laws, Technical Report 95-30 (SFB 359), IWR, University of Heidelberg, 1995.
22
