un ? n Pn - Semantic Scholar
MATHEMATICS OF COMPUTATION Volume 66, Number 219, July 1997, Pages 1027{1053 S 0025-5718(97)00859-4
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION FOR CONSERVATION LAWS GUI-QIANG CHEN AND JIAN-GUO LIU Abstract. We are concerned with the convergence of Lax-Wendro type
schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendro scheme proposed by Lax and Wendro in 1960 and its two step versions{the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth ux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed L estimates of the approximate solutions, H ?1 compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws. p
1. Introduction
We are interested in the convergence of nite di erence schemes with high resolution for conservation laws (1.1) @t u + @x f(u) = 0 ; u(x; 0) = u0(x) 2 L2 (R) \ L1 (R) : One of the most fundamental and important second-order nite di erence schemes is the Lax-Wendro scheme [11]. It is de ned by ? un + u n unj +1 =unj ? 2n 0 f(unj ) + 2n ? f 0 j 2 j +1 + f(unj ) (1.2) ? + n ? jn+1=2 j+ f 0 (unj )j+ unj ; P where unj = u(xj ; tn), tn = nk=1 k , xj = jh, n = n =h, + uj = uj +1 ? uj , 0uj = uj +1 ? uj ?1, ? uj = uj ? uj ?1, and jn+1=2 ; 0 0 jn+1=2 1 < +1, is a smooth function of unj and unj+1 (e.g. jn+1=2 = const. in [11]). Moreover, this scheme has to satisfy the Courant-Friedrichs-Lewy condition ? max jf 0(un)j 1 "0 = max (1.3) j n n j 2
for stabilization in general. This scheme is designed to have the following desirable computational features: conservation form, three-point dependence, second-order accuracy on the smooth
Received by the editor April 1, 1996. 1991 Mathematics Subject Classi cation. Primary 65M12; Secondary 35L65. Key words and phrases. Conservation laws, convergence, entropy solution, Lax-Wendro scheme. 1027
c 1997 American Mathematical Society
1028
GUI-QIANG CHEN AND JIAN-GUO LIU
regions of solutions, and stabilization for the convex case for 0 > 0 and "0 1. When jn+1=2 = 0, the Lax-Wendro scheme serves as not only a simple mode for the physical phenomenon of the dissipation-dispersion couplings, but also an example of dispersive schemes that do not converge in the sense of strong topology (cf. [10]). Indeed, as observed by Harten-Hyman-Lax [7] and Majda-Osher [14], [15], the second-order numerical viscosity jn+1=2 0 > 0 in this scheme is essential to guarantee that the numerical solutions are nonlinearly stable and converge to the physical solutions. The main role of the factor j+ f 0 (unj )j in the third term of the scheme (1.3) is to reduce e ectively viscosity in the smooth regions of solutions to produce sharp discontinuities in numerical computations. Such a scheme does not have a TVD property. The main objective of this paper is to prove the convergence of the Lax-Wendro approximate solutions to the entropy solutions and to provide an analytical approach for such a convergence analysis for high-order nite di erence schemes, which do not preserve BV (and even L1 bound). This is motivated by the fact that any high-resolution di erence scheme cannot, in general, preserve BV for the hyperbolic systems of conservation laws, especially for the nonstrictly hyperbolic case and the multidimensional case. Since our analysis is qualitative, we will always assume that 0 is suitably large and 0 is suitably small to make our analysis more convenient without loss of our purpose. One can follow our analysis to get the optimal constants for 0 and 0 from this approach for some concrete equations. Our analysis is based on careful Lp estimates of the approximate solutions, H ?1 compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. In Section 2 we describe some compensated compactness theorems for conservation laws, which guarantee the convergence of the nite di erence schemes provided that the corresponding approximate solutions satisfy these frameworks. Since these compactness frameworks do not need the BV estimates, this enables us to carry through our analysis by using the Lp estimates and the H ?1 compactness estimates, which are weaker than the BV estimates in general. Section 3 is devoted to the estimates of the Lax-Wendro approximate solutions to the convex scalar conservation laws. We obtain the uniform Lp estimates of the approximate solutions and the H ?1 compactness estimates of the corresponding entropy dissipation measures by analyzing carefully the properties of this scheme and by developing some useful estimate techniques. A global entropy error estimate is also obtained to ensure the consistency of this scheme with the scalar conservation laws. In our analysis we need a technical assumption of the algebraic growth of ux functions. This is because we do not require the L1 bound or the BV bound for the Lax-Wendro approximate solutions to achieve our goal. One diculty in the analysis is the fact that the dissipation is third-order in the Lax-Wendro scheme, comparing with the second-order dissipation in the rst-order schemes. All these terms need to be carefully combined into a third-order term or a term consisting of square products of second-order di erences with a favorable sign. Moreover, the factor in each combined third-order di erence needs to be bounded by the growth factor in the dissipative term in each cell. Some special treatment is also made since the Lax-Wendro scheme is a three-point scheme. A requirement of small CFL number is made here to use the grid ratio to control some growth factors.
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1029
In order to get the compactness of entropy dissipation measures, we estimate certain entropy inequalities with a higher growth rate. These techniques are generalized to study the convergence of the Lax-Wendro scheme for the nonconvex scalar case in Section 4 and for the hyperbolic systems of conservation laws in Section 5, and to prove the convergence of the Richtmyer scheme [22] and the MacCormack scheme [13]{two step versions of the Lax-Wendro schemes in Section 6, which are widely used in industry and engineering. In connection with earlier work on the Lax-Wendro type schemes, we recall that Majda and Osher [14], [15] showed the L2 -stability for general scalar conservation laws, the entropy consistency for the boundedly convergent approximate solutions for the semi-discrete cases as well as the complete-discrete scheme for the timeindependent cases of general systems endowed with a convex entropy, the ecient choices of arti cial viscosity such as the switching techniques, and the validity of the CFL number in their analysis. Many of their techniques have been melted into our analysis. We also refer to [25] for the stability of the local discrete shock pro le for the Lax-Wendro scheme. Regarding work on the convergence analysis of full discrete high-resolution nite di erence schemes, we refer to [4] and the references cited therein for the ux-limit schemes with slope modi cation or antidi usive ux approach, which preserve L1 bound. Since the Lax-Wendro type schemes do not have a TVD property, our analysis consists of two steps: One is to prove the convergence, which is an essential diculty here and is, however, automatically ensured by the Helly principle for the TVD or TVB schemes, and the other is to verify the entropy consistency. The convergence analysis of TVD or TVB schemes focus mainly upon the second step for the scalar case, that is, the consistency proof. The convergence of a class of semi-discrete generalized MUSCL schemes for the strictly convex case was obtained in [19]. For the semi-discrete MUSCL scheme for the convex scalar conservation laws, some consistency results were announced in [12], [31]. 2. Compactness frameworks
In this section we discuss some compactness frameworks ? for the approximate solutions for subsequent developments. A pair of functions (u); q(u) is called an entropy-entropy ux pair if they satisfy q0 (u) = 0 (u)f 0 (u) . For the scalar case, any function is an entropy function. In the following theorem and the analysis in Section 3 and Section 4, we will denote entropy 0 (u) = 21 u2 for the convex case and 0(u) = f(u) for the general case. Theorem 2.1. Consider the scalar conservation laws (1:1) satisfying measu : f 00 (u) = 0 = 0. Let uh (x; t) be numerical approximate solutions of (1:1) satisfying the following conditions: ? (1) uh is bounded in Lp for some p 2 and f(uh ); 0(uh ); q0(uh ) is bounded in L2loc ; (2) The dissipation measures @tuh +@x f(uh ) and @t 0(uh )+@x q0(uh ) are compact ?1; in Hloc ? (3) For any C 2 convex entropy pair (u); q(u) , @t (uh )+@x q(uh ) o(1) in D0, provided that j(u)j + jq(u)j M(1 + jujr ); r < p. Then there is a subsequence (still denoted as ) uh such that uh (x; t) ! u(x; t) a.e. as h ! 0 and u is the entropy solution of (1:1) satisfying @t(u)+@x q(u) 0 in D0 for any C 2 convex entropy pair (; q).
1030
GUI-QIANG CHEN AND JIAN-GUO LIU
We remark that, for general conservation laws, the conditions (1) and (2) imply that w- limf(uh ) = f(w- limuh ) in L2 and the condition (3) implies the following entropy condition for the Young measures x;t, determined by the Lax-Wendro approximate solutions uh (x; t), @t hx;t; i + @x hx;t; qi 0 in the sense of distributions. We refer the reader to Chen-Lu [3] for a detailed discussion. The proof of the L1 version of Theorem 2.1 can be found in [2], [3], [28] using the div-curl lemma of Tartar and Murat [28]. For a 2 2 system of conservation laws endowed with global Riemann invariants, we have the following similar framework. Theorem 2.2. Let uh (x; t) be numerical approximate solutions of the 2 2 genuinely nonlinear and strictly hyperbolic system of conservation laws (1:1) satisfying the following conditions: (1) uh is bounded in L1 ; ?1; (2) For any C 2 entropy pair (; q), @t(uh ) + @x q(uh ) is compact in Hloc (3) For any C 2 convex entropy pair (; q), @t(uh ) + @x q(uh ) o(1) in D0 . Then there is a subsequence (still denoted as ) uh such that uh (x; t) ! u(x; t) a.e. as h ! 0 and u is the entropy solution of (1:1) satisfying @t(u)+@x q(u) 0 in D0 for any C 2 convex entropy (; q).
This theorem is proved by DiPerna [5] by estimating the entropy dissipation measures and by using the Lax entropy pairs and the compensated compactness method. An alternative extension proof can be found in [16], [23]. Finally we state the following two lemmas, which can be found in [17] and [2], respectively. ?1;q is completely Lemma 2.1. The embedding of the positive cone of W ?1;p in Wloc continuous for all q < p. Lemma 2.2. Let Rn be a bounded open set. Then ?compact set of W ?1;q ( ) \ ?bounded set of W ?1;r( ) loc loc ? ? 1 compact set of Hloc ( ) ; where q and r are constants, 1 < q 2 < r < 1. 3. Convex scalar conservation laws
In this section we are concerned with the Lax-Wendro approximate solutions of the scalar conservation laws (1.1) with ux function f(u) satisfying (3.1) f 00 (u) c0 > 0 ; f (k) (u) O(jujm?k ) ; for juj 1; k = 0; 1; 2 : For convenience, we write the Lax-Wendro scheme (1.2) into the form unj +1 = unj + Fjn + Hjn + Jjn ; (3.2) where ?+f(un )2 j n n 2 n 1 1 Fj = ? 2 n 0f(uj ) ; Hj = 2 n ? ; n + uj (3.3) ? Jjn = n? ~jn+1=2 j+a(unj )j+unj ;
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
with
? unj+unj a 2 ~jn+1=2 = jn+1=2 + n
+1
? f(un )= un + j + j = n
1031
0
j +1=2 + O(nja j):
j+ a(unj)j
From unj, we construct the Lax-Wendro approximate solutions on R2+: (3.4) uh (x; t) unj; for (x; t) 2 [xj ?1=2; xj +1=2) [tn; tn+1) : Then we have the following main convergence theorem of this section. Theorem 3.1. Let uh(x; t) be the Lax-Wendro approximate solutions (3:4) of the scalar conservation laws (1:1) and (3:1). Assume that the coecient 0 > 0 is suitably large and the CFL number "0 given by (1:3) is suitably small. Then there exists a subsequence strongly converging to the entropy solution of (1:1) satisfying the following entropy condition @t (u) + @x q(u) 0 in D0 for any C 2 convex entropy pair (; q) satisfying (k)(u) O(jujr?k ), for juj 1, 0 k r < 4m. The proof consists of the following three subsections in which we check the three conditions in Theorem 2.1, respectively. For simplicity of our proof, we drop the subscript n and use the following notations: ? un + u n fj = f(unj ) ; aj = a(unj) ; aj +1=2 = a j 2 j +1 ; j = (unj ) ; j0 = 0(unj ) ; j00 = 00 (unj) ; where is an entropy function of (1.1). We rst introduce the following three technical lemmas. Lemma 3.1. If g0 (u) c0 > 0 and g(k)(u) = ck um?k ?1+o(1) ; juj 1 ; for some constants ck ; k = 0; 1; 2, then there is u 1 and > 0 such that ( g(u) ? g(v) (max(juj; jvj)) ; with (s) = g0(s) ; s u ; u?v c0 ; s < u : The proof is straightforward and hence is omitted. Summing by parts, one has Lemma 3.2. If faj g and fbj g are two arbitrary sequences, then (3.5) + (aj 0bj ) = 0(aj + bj ) + ? (+ aj + bj ); and
(3.6)
2
X j
aj bj 0 bj =
X j
(+ bj )2 + aj ?
X j
bj20aj ;
provided that the above sums make sense. Proof. The formula (3.6) comes from a direct computation. Notice that
0 (aj bj ) = aj 0 bj + bj 0aj + ? (+ aj + bj ) : Multiplying both sides of the above equation by bj , taking sum for j, and summing by parts, one arrives at (3.7). Using the relation 0 f 0 = q0, expanding at uj in the intervals [uj ?1; uj ] and [uj ; uj +1], and using the integration by parts, one has
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GUI-QIANG CHEN AND JIAN-GUO LIU
Lemma 3.3. If (; q) is an entropy pair of (1:1), then
?
j0 0 fj = 0qj ? 12 j00(+ uj )2 + aj ? 12 j00 ? (+ uj )2 aj Z uj + 21 j00 (uj ? s)2 a0(s) ds
+1
uj?1
+ 21 000 (j )
Z uj
+1
uj
(uj ? s)2 a(s) ds + 12 000 (j ?1)
Z uj uj?1
(uj ? s)2 a(s) ds ;
where j ?1 and j are some values in [uj ?1; uj ] and [uj ; uj +1], respectively.
Proof. Using the relation 0 f 0 = q0 and expanding at uj in the intervals [uj ?1; uj ]
and [uj ; uj +1], one has j0 0fj = 0qj + j00
+ 12 000 (j )
Z uj
+1
Zuuj?j+11 uj
(uj ? s)a(s) ds (uj ? s)2 a(s) ds + 21 000 (j ?1)
Z uj uj?1
(uj ? s)2 a(s) ds;
for some j 2 [uj ; uj +1] and j ?1 2 [uj ?1; uj ]. Now using the integration by parts to the second term on the right-hand side of the above equation, we obtain the lemma. We remark here that we split the remainders in Lemma 3.3 into two cells [uj ?1; uj ] and [uj ; uj +1]. A basic reason for this is that we do not require the L1 bound of the approximate solutions, and thus need to control the growth rate. This kind of splitting techniques will be used in several places in this section.
3.1. Lp estimate. We now verify the rst condition of Theorem 2.1 for the Lax-
Wendro scheme (3.2)-(3.3). Since the ux function has the algebraic growth rate m, we only need to show that uh is bounded in L2 and L4m?2 . The method used here is to estimate the entropy function with certain required growth rate. In order to control the growth rate in the compactness analysis in Subsection 3.3, we estimate the entropy with a growth rate slightly more than that required in this subsection. For this reason, we choose a strictly convex entropy function 1 4m 1 2 (3.7) (u) = 4m(4m ? 1) u + 2 u P and estimate the bound of (unj ) for thePLax-Wendro scheme unj . We rst expand the time increment of (unj) and split it into three terms:
X?
(3.8)
j
X 0(un)(un+1 ? un) + 1 X 00(un)(un+1 ? un)2 j j j j j j 2
(unj +1) ? (unj ) =
j
+
4m X
j
1 X (k)(un )(un+1 ? un )k I1 + I2 + I3 ; j j j k! k=3 j
which will be estimated in this subsection, where we used the exact expansion to the highest order to avoid the remainder in [unj; unj +1], which is dicult to control.
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
Estimate of I1 .
Plugging (3.2) into I1, one has
(3.9)
I1 =
X j
j0 Jj +
X j
j0 Hj +
X j
1033
j0 Fj :
To estimate the rst term, one uses the expression of Jj in (3.3), the summation by parts, and Lemma 3.1 to get (3.10)
X j
j0 Jj = ?
X ~n j
j +1=2 + j0 j+ aj j+ uj ? 0
where (3.11)
X j
(sj )j+ uj j3 ;
(
5m?4 ; s u ; (s) = cs c0 ; s < u ;
sj = max(juj j; juj +1j);
for some constant c > 0. Note that (3.10) is the sum of products of three rst-order di erences due to the fact that the viscosity term is second-order accurate. This is the main dissipative term that is used to control all of the error terms. In estimating the second term, one uses the summation by parts and the expression of Hj in (3.3), and substitutes the expansion + j0 = j00 + uj + 12 000(j )(+ uj )2 ;
(3.12)
in the resulting terms to get
X
(3.13)
j
X (+fj )2
+ j0 u + j j X X 2 1 j00 (+ fj )2 ? 14 2 000 (j )(+ fj )2+ uj : = ?2
j0 Hj = ? 12 2
j
j
We use the term of the right-hand side in (3.10) to control the second term in (3.13). Notice that there is a factor (+ uj )3 in the second term that can be clearly controlled by the term of the right-hand side in (3.10). The factor in front of the di erence is of growth rate 2(m ? 1)+4m ? 4 = 6(m ? 1) that is larger than 5m ? 4 in (3.11). Noting that there is also a parameter 2 in front of this factor, therefore, we can use the fact j+ fj =+ uj j "0 to control certain powers of the growth rate to obtain (3.14)
X j
j0 Hj ? 21 2
X j
j00 (+ fj )2 + "0 C
X j
(sj )j+ uj j3 ;
for some positive constant C > 0. This kind of simple techniques will be used in several places in the rest of this section and we will omit the detailed explanations. Here and henceforth, we use the notations j and j as some values in [uj ; uj +1], and C is a positive constant, independent of the grid size h, the CFL number, the viscosity constant , and the numerical solution uh . In di erent contexts, they may have di erent values.
1034
GUI-QIANG CHEN AND JIAN-GUO LIU
P
Now we estimate the term j j0 Fj . Writing the expression of Fj in (3.3) in the integral form and using the conservative property of entropy, one has
Z uj
X
X j0 Fj = ? 21 0 (uj ) a(s) ds uj?1 j j X Z uj+1 ? 0 1 = 2 (s) ? 0 (uj ) a(s) ds u j?1
j
= 21
(3.15)
X j
+ 14 + 14
00 (uj )
X j
X j
Z uj
+1
uj?1
000(j ?1 ) 000(j )
+1
(s ? uj )a(s) ds
Z uj
uj?1
Z uj uj
+1
(s ? uj )2 a(s) ds
(s ? uj )2 a(s) ds :
Applying the following integration by parts to the rst term in the right-hand side of the above equality
Z uj
+1
uj?1
(s ? uj )a(s) ds = 12 (+ uj )2+ aj ? 12
Z uj
+1
uj?1
(s ? uj )2 a0(s) ds ;
and plugging the above two estimates back into (3.15), we obtain
X
(3.16)
j
j0 Fj C
X j
(sj )j+ uj j3 :
Finally, one uses (3.10), (3.14), and (3.16) to get the following estimate: (3.17) X X X I1 ? 0 (sj )j+ uj j3 ? 21 2 j00 (+ fj )2 + (1 + "0 )C (sj )j+ uj j3 : Estimate of I2 .
(3.18)
j
j
j
Substituting (3.2) into I2 in (3.8), one has the expansion
I2 = 12
X j
00 (uj )(Fj2 + Hj2 + 2Fj Hj + Jj2 + 2Jj (Fj + Hj )) :
We now estimate I2 term by term. In estimating the rst term in (3.18), one uses the identity (3.19) (0 fj )2 = 2(+ fj )2 + 2(+ fj ?1)2 ? (? + fj )2 ; and a direct estimate of Fj from its expression in (3.3) to get
X j
X 00? X j (+ fj )2 + (+ fj ?1 )2 ? 14 2 00 (uj )(? + fj )2 j j 2 1 2X 00 2 1 2 X 00 2 X 00
j00 Fj2 = 21 2 =
j
j (+ fj ) ? 4
j
j (? + fj ) + 2
j
+ j (+ fj )2 :
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1035
In the rst and third terms on the right-hand side of the above inequality, one uses the Taylor expansion for + fj and some simple calculations used in estimating (3.13) to obtain
X j
(3.20)
j00 Fj2 2
X j
j00aj2+1=2(+ uj )2 ? 14 2
+ "0 C
X j
X
(sj )j+ uj j3 :
j
j00 (? + fj )2
Notice that the rst term on the right-hand side of (3.20) is exactly the same as the second term on that of (3.17) except with a di erent sign, which cancel each other. The second term on the right-hand side of (3.20) is a sum of squares of second-order di erences in a favorable sign. We will use it to control a similar term below. To estimate the second term in (3.18), we use the identity
2 + fj ?1 + f + fj ; ? (+ fuj ) = ? + fj + j ? u + j + uj ?1 + j
(3.21)
and directly estimate the expression of Hj in (3.3) to get
+fj?1 2 X 00 00 2 2 4 1 j Hj 2 j (? + fj ) u + j ?1 j j + fj 2 X 00 4 2 1 j (+ fj ) ? u + 2 : + j j
X
The rst term on the right-hand side of the above inequality can be controlled by the second term on the right-hand side of (3.20) in view of the fact that the CFL number is less than 1. The last term of the above inequality involves three points. Hence we split it into the following two terms by using the Taylor expansion for a in the intervals [uj ?1; uj ] and [uj ; uj +1]: 4
f + j j00 (+ fj )2 ? uj + j
X
2
f 2 + j j00(+ uj )2 ? uj + j X "03 j00 ja0(j ?1)jj?uj j(+ uj )2 2"03
X
j X 3 + "0 j00ja0 (j )jj+ uj j(+ uj )2 : j
The last term only involves two points and hence can be directly estimated. In the last second term, we use the Holder inequality to split it again to obtain
X j
j00 ja0(j ?1)jj?uj j(+ uj )2
X j
C
juj j4m?2jsj ?1jm?2 j?uj j(+ uj )2
X j
(sj )j+ uj j3 :
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GUI-QIANG CHEN AND JIAN-GUO LIU
The methods used here will be applied to estimating the similar terms. With the above three estimates, we arrive at (3.22) +fj?1 2 X 00 2 1 4 X 00 3 C X (sj )j+ uj j3 : j Hj 2 + " j (? + fj )2 0 u j
+ j ?1
j
j
The third term in (3.18) consists of products of a factor of rst-order di erence and a factor of second-order di erence, we utilize a symmetric property to transform it into a sum of products of three di erences. In doing this, we rst split the positive function 00 , and sum by parts to get ( f )2 X 00 3 1 ? 4 j 0 fj ? + uj + j j q q 2 X 00 q 00 (+ fj )2 = ? 14 3 j 0fj ? j u ? ? j00 (+ fuj ?1 ) + j + j ?1 j (3.23) X?q 00 q 00 +fj j + fj + j 0fj u = 41 3 + j j q 2 X q 00 j 0 fj ? j00 (+ fuj ?1) : + 14 3 + j ?1 j Now the last term consists of products of three di erences and hence can be directly estimated. The rst term has some symmetric property after switching the di erence operator + with 0. This can be done by using Lemma 3.2 as follows.
X?q 00
?q00 f + fj = X?q00 f ?q00 f +fj j 0 j + uj j + j 0 j + j + uj j X?q 00 ? q 00 +fj f f : +
j + fj +
j
j + j
j
?
+
j + j
+ uj
Again the last term in the above equality is a sum of products of three di erences in uj and hence can be estimated directly. The rst term can now be transformed into a sum q 00of products of three di erences by using Lemma 3.3 where bj are replaced by j + fj . (3.24)
X?q 00 j
?q00 f + fj = 1 X ?q00 f 2 +fj + u + j + j + uj 2 j + j + j j X?q 00 2 +fj f ?1 :
j + fj 0
2
j
j + j
Finally, combining all the estimates (3.23)-(3.24), we arrive at X X 00 (3.25) j Fj Hj "02 C (sj )j+ uj j3 : j
j
0 u + j
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1037
Estimating the remainder three terms in (3.18) is rather simple since each of them consists of more than three di erence factors. We estimate them as follows: (3.26) X X 00 2 2 X 00? ~n j Jj = j ? ( j +1=2 j+aj j+uj ) 2 "0 12 C (sj )j+ uj j3 ; j
j
j
and (3.27) X X 2 j00 Jj (Fj + Hj ) = ?2 j00 ?( ~jn+1=2 j+ aj j+ uj )0 fj
j X 00 ~n 3 j ? ( j +1=2j+ aj j+ uj )? (aj +1=2+ fj ) + j X ("0 + "02) 1 C (sj )j+ uj j3 : j
j
The estimate of I2 is now completed by combining (3.20) and (3.22) with (3.25)(3.27). X X I2 12 2 j00aj2+1=2(+ uj )2 ? 18 2 (1 ? 22 aj2?1=2)j00 (? + fj )2 j j (3.28) X + "0 (1 + "0 + "02 + 1 + 12 )C (sj )j+uj j3 : j
Estimate of I3 .
The estimate of I3 is rather easy. We rst substitute the expansion of unj +1 ? unj in (3.2)-(3.3) into I3 to get 4m 3k X X
?
j(k)(unj )j jFj jk + jHj jk + jJj jk : k! j k=3 Using the expressions in (3.3), we have (3.30) 4m k X X jI3 j 3k! j(k)(unj)j k j+ fj jk + k j?fj jk + 2k jaj +1=2+ fj jk k=3 j jI3j
(3.29)
+ 2k jaj ?1=2?fj jk + (2 1 )k j+aj + uj jk + (2 1 )k j?aj ? uj jk
C
4m ? X k=3
"0k?1 + "02k?1 + "0k?1 1k
X (s )j u j3 : j + j j
We have now estimated all I1 , I2 , and I3 . Using (3.17), (3.28), and (3.30), we obtain X n+1 n X X 00( f )2 ? (j ? j ) ? 2 0 (sj )j+ aj j3 ? 18 2 1 ? 2"20 j ? + j j j j + C 1 + "0 1 +
4m X
k=2
"0k?1 1k
!X j
(sj )j+ uj j3 :
1038
GUI-QIANG CHEN AND JIAN-GUO LIU
Choosing 0 large and "0 1 small, one has X n+1 n X X (j ? j ) ?C1 (sj )j+aj j3 ? 2 C2 (? + fj )2 : j
j
j
This gives the following proposition.
Proposition 3.1. Under the same assumptions of Theorem 3.1, we have X k X? X n (uj )h + (3.31) (sj )j+ukj j3h + 2 ?+ f(ukj ) 2h C; kn;j
j
where is the entropy function given by (3:7).
kn;j
As an immediate consequence, we have
Corollary 3.1. Under the same assumptions of Theorem 3.1, we have
? kuh kL m + f(uh ); 0(uh ); q0(uh ) L m= m? C ; C independent of h; 4 loc
4 loc
(2
1)
where (0; q0) is the entropy pair given by (2:2).
3.2. Entropy inequality. We now estimate the entropy inequalities as stated in
the third condition in Theorem 2.1 for the entropy functions of form (3.32) (k) (u) O(juj2`?k) ; for juj 1; k = 0; 1; ; 2`; where ` 2m is a positive integer. In the next subsection, we will use the integer ` (13m ? 3)=6 in the compactness arguments. We will always take ` (3m+1)=2 for using Theorem 2.1 to ensure the entropy inequalities. For any (x; t) 2 C01 , 0, denote nj = (xj ; tn) and X ? X ? Ih ( ) h nj (unj +1 ) ? (unj) + 21 h nj q(unj+1) ? q(unj?1) : j;n
j;n
We decompose Ih into the following three terms that are similar to (3.8). X X X ? Ih ( ) = h njj0 Jj + h njj0 Hj + h nj j0 Fj + 21 0qj
j;n j;n j;n X ? 2 2 n 00 + 21 h j j Fj + Hj + 2Fj Hj + Jj2 + 2Jj (Fj + Hj ) j;n 2X `?1 X (3.33) 1 njj(k) (Fj + Hj + Jj )k +h k! k=3 j;n X + h (21`)! nj (2`)( )(Fj + Hj + Jj )2` j;n = I1h ( ) + I2h ( ) + I3h ( ) : Notice that there is a remainder term in I3 with 2 [unj; unj +1]. This term can be estimated by using the fact that (2`) is uniformly bounded for the entropy
functions of form (3.32). We now estimate all these terms in the same order as those in Subsection 3.1 to keep consistency. We will skip some parts of the estimates, which are similar to the corresponding parts in the previous subsection. Estimate of I1h . Similar to the estimate (3.10), one can use the identity (3.34) + ( nj j0 ) = 21 ( nj + nj+1)+ j0 + 12 (j0 + j0 +1)+ nj
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1039
and the expression of Jj in (3.3) to get X n0 X j j Jj = ? 12 ~jn+1=2( nj + nj+1 )+ j0 j+ aj j+ uj j;n
? 12
j;n X ~n j;n
j +1=2(j0 + j0 +1)+ nj j+aj j+ uj :
Applying Lemma 3.1 to the rst term on the right-hand side and estimating the second term directly, one has X n0 X j j Jj ? 12 0 ( nj + nj+1 )(sj )j+ uj j3 j;n j;n (3.35) X + 1 C sj (sj )j+ uj j2j+ njj ; j;n
where sj and (s) are determined by (3.11). With this good term in our hands, we now estimate other terms. Notice that 2 2 X n0 X X j j Hj = ? 12 2 nj+ j0 (+ fuj ) ? 12 2 j0 +1+ nj (+ fuj ) : + j + j j;n j;n j;n
As in estimating (3.13)-(3.14), expanding + j0 and + fj in the rst term on the right-hand side of the above equality and estimating the resulting terms directly, we have X X X n0 j j Hj ? 21 2 njj00(+ fj )2 + "0 C nj(sj )j+ uj j3 j;n j;n j;n (3.36) X + "0 C sj (sj )j+ uj j2 j+ nj j : j;n
Using Lemma 3.4, one has
X n? 0 1 X ? j j Fj + 2 0 qj = ? 12 nj j0 0 fj ? 0qj j;n j;n Zu X n 00 2 00 j 2 0 1 = 4
j;n
? 14 ? 14
j j (+ uj ) + aj ? j
X
njaj + j00 (+ uj )2 ? 41
j;n X n j;n
j 000 (j )
Z uj uj
+1
+1
uj?1
X
(uj ? s) a (s) ds
j00+1aj (+ uj )2 + nj
j;n Z uj (uj ? s)2 a(s) ds : (uj ? s)2 a(s) ds + 000 (j ?1) uj?1
Each term on the right-hand side of the above equality is the sum of products of three di erences and, therefore, can be directly estimated by similar arguments as in (3.15)-(3.16). X n? 0 1 j j Fj + 2 0qj j;n (3.37) X C ( nj + nj+1 )(sj )j+ uj j3 + sj (sj )j+ uj j2j+ njj : j;n
1040
GUI-QIANG CHEN AND JIAN-GUO LIU
Using (3.35)-(3.37), we have I1h ( ) ? 21 0 h
X j;n
( nj + nj+1)(sj )j+ uj j3 ? 12 2 h
+ (1 + "0 + 1 )Ch
(3.38)
X j;n
X j;n
nj j00 (+ fj )2
( nj + nj+1)(sj )j+ uj j3
+ sj (sj )j+ uj j2j+ njj : Estimate of I2h .
Using the identity (3.19), one can decompose the rst term in I2h and estimate similar to that of (3.20) to obtain (3.39) X X n 00 2 1 2 X n 00 00 j (j + j +1 )(+ fj )2 ? 14 2 nj j00 (? + fj )2 j j Fj = 2 j;n
j;n j;n X 00 1 2 2 n + 2 j +1(+ fj ) + j j;n X X 2 nj aj2+1=2j00 (+ uj )2 ? 41 2 nj j00 (? + fj )2 j;n j;n X n n 3 2 n ( j + j +1 )(sj )j+ aj j + sj (sj )j+ uj j j+ j j : + "0 C j;n
Estimating the second term of I2h is similar to that of (3.21)-(3.22) by
f 2 X n 00 4 2 n 00 2 1 j j (? + fj ) +uj ?1 j j Hj 2 + j ?1 j;n j;n X + "03 C ( nj + nj+1)(sj )j+ uj j3 : j;n
X
(3.40)
The main idea in estimating the third term in I2h is to utilize the symmetric property. It involves some similar techniques as in estimating (3.23) and (3.24). We omit detailed calculations here and write down the resulting terms as follows: 2 X X n 00 j j Fj Hj = ? 41 3 nj j00 0fj ? (+ fuj ) + j j;n j;n X ?q 00 2 f + j n 3 1 j + fj 0 j u = ?8 + j j;n + fj ?q 2 ? + nj u + j00 + fj ( f )2 +j ? q q X + 41 3 j00 + uj ? + nj j00 + fj j;n
+ j
+ nj0fj (+ fuj ?1 ) + j ?1
2
q 00
? j :
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1041
Clearly the terms on the right-hand side of the above equality are the sums of three di erence products of unj and hence can be estimated directly.
X
njj00Fj Hj "02 C
X
( nj + nj+1)(sj )j+ uj j3
j;n X ? 2 + "0 C sj (sj )j+ uj j2 j+ njj + j? njj : j;n h The remainder three terms in I2 can be similarlyestimated as in the estimates (3.26) j;n
(3.41)
and (3.27).
(3.42)
X j;n
nj j00 Jj2 + 2
X
njj00Jj (Fj + Hj )
j;n X "0 ( 1 + 12 )C ( nj + nj+1 )(sj )j+ uj j3 : j;n
Therefore, combining (3.39)-(3.40) with (3.41)-(3.42), we have the following estimate for I2h : X X I2h ( ) 12 2 h nj j00 (+ fj )2 ? 81 2 h nj(1 ? 2"20)j00 (? + fj )2 j;n
(3.43)
j;n X 2 2 + C"0 (1 + "0 + "0 + + )h ( nj + nj+1)(sj )j+ uj j3 j;n X ? + "0 (1 + "0 )Ch sj (sj )j+ uj j2 j+ nj j + j? njj : j;n
Estimating I3h is straightforward as in that of (3.29) and (3.30). One can obtain (3.44)
4m ? X X h I3 ( ) Ch "0k?1 + "02k?1 + "0k?1 k ( nj + nj+1)(sj )j+ uj j3 : j;n k=3
Finally, we have the following estimate of Ih after combining (3.38) with (3.43)(3.44): X Ih ( ) ? 41 0 h ( nj + nj+1)(sj )j+ uj j3 j;n X 2 1 ? 8 (1 ? 2"02)h njj00 (?+ fj )2 j;n
+ (1 + "0 + 1 )Ch
+ C 1 + "0 1 +
X
j;n 4X m
k=2
?
sj (sj )j+ uj j2 j+ njj + j? nj j
X
"0k?1 1k h
j;n
( nj + nj+1)(sj )j+ uj j3 :
Choose 0 large and "0 1 small. Then we have X Ih ( ) ? 18 0 h ( nj + nj+1 )(sj )j+ uj j3 + 1 Ch
j;n
X j;n
?
sj (sj )j+ uj j2 j+ njj + j? njj :
1042
GUI-QIANG CHEN AND JIAN-GUO LIU
Now we deal with the last term in the above inequality. From the Holder inequality, one has h
X j;n
?
X
sj (sj )j+ uj j2 j+ njj + j? njj h1+ k kC
(xj ;tn )2
(sj )sj (+ u)2
12=3 11=3 0 0 X X (sj )j+ uj3hA : sj6`?9m+3 n hA @ Ch ?1=3k kC @ (xj ;tn )2
(xj ;tn )2
Let 6` ? 9m + 3 4m. Then X ? h sj (sj )j+uj j2 j+ njj + j? nj j Ch ?1=3k kC : j;n
Substituting it back to (3.33), we have the following proposition.
Proposition 3.2. Under the same assumptions of Theorem 3.1, (3.45) X j;n
?
nj (unj +1) ? (unj ) h + 21
X n? n j q(uj +1) ? q(unj?1) h Ch ?1=3k kC j;n
holds for any 2 C01, 0, where (; q) is the entropy pair given by (3:32).
3.3. H ?1 compactness of entropy dissipation measures. For the entropy pair (; q), we de ne the following functional as the entropy dissipation measures Mh ( ) =
(3.46)
ZZ?
(uh )@t + q(uh )@x dxdt ;
2 C01 ;
for the approximate solutions uh (x; t) de ned by (3.4). In this subsection, we will show that Mh is compact in H ?1 for the entropy pairs (u; f(u)) and (0; q0) with 0(u) = u2 . From Proposition 3.1, we can easily get from the Holder inequality that jMh ( )j ?1;q for q= 2m?1 >2. Lemma 2.2 C k kW ; m = m . Hence Mh is bounded in Wloc 4m ? 1 tells us that Mh is compact in Hloc as long as we can prove that Mh is compact in ?1;p for some p < 2. We de ne the following two functionals: Wloc 1 (2
(3.47)
+1) (4
Ih ( ) =
)
X n? n+1 X ? j (uj ) ? (unj ) h + 21 nj q(unj+1) ? q(unj?1) h ; j;n
j;n
and h ( ) = Ih ( ) + Mh ( ) for any strictly convex entropy (; q) and 2 C01. It is easy to show that (3.48) Z xj = ? X (x; tn+1) ? (xj ; tn) dx h ( ) = ? (jn+1 ? jn) +1 2
j;n
xj?1=2
Z tn ? X 2 (xj +1=2; t) ? (xj ; tn ) ? (xj +1 ; tn) dt : ? 21 (qjn+1 ? qjn ) +1
j;n
tn
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1043
Estimating directly and using the Holder inequality and Proposition 3.1, one obtains (3.49) X ? n+1 n 1+ n n jj ? j jh + jqj +1 ? qj j nh jh ( )j k kC (xj ;tn )2
h1+ k kC
X
2 3k X
(xj ;tn )2 k=1
Ch1+ k kC
X
(k)j?jF jk + jH jk + jJ jk + jq0 ( )jj unj j j j j j + j j k!
!
(sj )1=3j+ unjj
(xj ;tn )2
1 = 3 ? 2 = 3 Ch k kC ! 0 ; for 32 < < 1 : ?1;p1 for p1 < 2 < 2. We know from This implies that h is compact in Wloc 1+ Proposition 3.2 that Ih can be split as Ih ( ) = I1h ( ) + I2h ( ) such that (3.50) I1h 0 ; jI2h ( )j Ch ?1=3k kC : ?1;p2 for 1 < p < 2. The remainder Consequently, we have that I2h is compact in Wloc 2 ? 1 ;p h is to show that I1 is also compact in Wloc for some p < 2. Since I1h is negative due to (3.50), we know from Lemma 2.1 that we only need to show that I1h is bounded ?1;p for some p < 2. This is a direct consequence of (3.46), (3.49), and (3.50). in Wloc
Thus, we have the following proposition.
Proposition 3.3. Under the same assumptions of Theorem 3.1,
?1 @t uh + @x f(uh ) ; @t0 (uh ) + @x q0(uh ) are compact in Hloc where (0; q0) is the entropy pair given by (2:2). As a consequence of Theorem 2.1, Corollary 3.1, Proposition 3.2, and Proposition 3.3, we conclude Theorem 3.1 for the convex scalar conservation laws.
(3.51)
4. Extension to the nonconvex case
For the nonconvex scalar conservation laws, the second-order numerical viscosity term ? ? jn+1=2j+ f 0 (unj )j+ unj is degenerate at the points fu j f 00(u) = 0g. Typically such a degenerate point occurs near the contact discontinuities, which is quite sensitive to the stability of the discontinuities. To assume the stability, we consider the following second-order numerical viscosity in the Lax-Wendro schemes: ? ? jn+1=2 j+unj j+unj (4.1) ? with jn+1=2 2 0 + O(ja0j); 1 + O(ja0j) , 0 > 0, for the nonconvex scalar case. This covers both the convex case (the convexity implies 0 > 0) and the nonconvex case. With such a numerical viscosity we can use the same arguments as in Section 3, when C 1 and 0 1, to obtain the same viscosity estimate X (4.2) (sj )junj ? unj+1j3h C j;n
1044
GUI-QIANG CHEN AND JIAN-GUO LIU
and entropy estimate (4.3)
X j
(unj +1)
X j
(unj)
for 00 = u4m . Using these estimates, we can similarly verify that the corresponding
approximate solutions uh (x; t) satisfy the three conditions in Theorem 2.1 for any convex entropy pair. Therefore we have the following convergence theorem for the scalar conservation laws (1.1) whose ux functions satisfy (4.4) meas u j f 00(u) = 0 = 0; f (k) (u) O(jujm?k ) ; for juj 1; 0 k m : Theorem 4.1. Let uh(x; t) be numerical approximate solutions, generated from the Lax-Wendro scheme (1:2) endowed with the numerical viscosity (4:1), of the scalar conservation laws (1:1) satisfying (4:4). Assume that the coecient 0 in (4:1) is suitably large and the CFL number "0 given by (1:4) is suitably small. Then the sequence uh has a strong convergent subsequence and the limit function is the entropy solution of (1:1) satisfying the entropy inequalities @t (u) + @x q(u) 0 in D0 ; for any convex entropy pair (; q) satisfying (k) (u) O(jujr?k ), for juj 1, 0 k r < 4m. Proof. Only one thing we should check is that, for any such a C 2 entropy pair ?1. This can be achieved as follows. ((u); q(u)), @t(uh )+@xq(uh ) is compact in Hloc 2 For any C entropy pair (; q), there exists C0 > 0 such that j00j C0 j00j where is a strictly convex entropy. De ne = C0 ? . Then is a convex entropy. Noting that both @t (uh ) + @x q(uh ) and @t (uh ) + @x q (uh ) are compact sets in ?1, one obtains that @t (uh ) + @x q(uh ) is compact in H ?1. This completes the Hloc loc proof. Remark. The assumption (4.4) can be relaxed to more general ux functions for the Lax-Wendro scheme. The strong convergence becomes weak one in Lr , r < 4m, and the limit function is the weak solution satisfying the entropy condition for the Young measures x;t determined by the Lax-Wendro approximate solutions uh (x; t): @t hx;t; i + @x hx;t; qi 0; for any convex entropy pair (; q) in the above theorem. 5. Extension to hyperbolic systems
We write the Lax-Wendro scheme in the following form unj +1 = unj + Fjn + Hjn + Jjn : (5.1) Here ? un + u n Fjn = ? 12 0 f(unj ) ; Hjn = 21 2 ? a j 2 j +1 + f(unj ) ; (5.2) ? Jjn = ? jn+1=2j+ unj j+unj ; where the scalar function jn+1=2 0 > 0, for some 0 , is smooth with respect to unj and unj+1 and is bounded from below. We assume that (5.3) sup junjj M : j;n
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1045
In this section we show the convergence of such a scheme for 2 2 systems and the entropy consistency of the boundedly convergent Lax-Wendro approximate solutions with general hyperbolic systems of conservation laws with a convex entropy. This depends on the following estimates. 5.1. Dissipation estimates. We rst start with the entropy estimate. Let be a C 2 convex entropy function. As in the scalar case, we take the Taylor expansion to the time increment of the entropy to get X? n+1 X (uj ) ? (unj ) = 0 (unj )(unj +1 ? unj) j
j
+ 21
(5.4)
X
(unj +1 ? unj)> 00 (unj )(unj +1 ? unj)
Xj ? n+1 n 3 + O juj ? uj j I1 + I2 + I3 : j
We will keep the same order of the estimates as in Section 3 and will only emphasize the new features, which are di erent from the scalar case. The arguments similar to those in Section 3 will be omitted. We point out here that some of the estimates in this section are simpler than the ones in Section 3 since we assumed the uniform boundedness of the approximate solutions. Estimate of I1 . As in (3.11), we directly estimate that X 0> X X (j ) Jj = ? j +1=2 j+ uj j+ (j0 )> + uj ? 0 j+ uj j3 ; (5.5) j
j
j
using the expression of Jj in (5.2), the summation by parts, and the convexity of . This gives us the main dissipative term. Here we used the notation j +1=2 = (uj ; uj +1), which is a scalar function. We remark that (5.5) is the only place we restrict the viscosity (uj ; uj +1) to be a scalar function. As in estimating (3.13) and (3.14), we have X X 0> (j ) Hj = ? 12 2 (+ j0 )> aj +1=2+ fj j j (5.6) X X 2 1 = ? 2 (+ uj )> j00aj + fj + "0 O(j+ uj j3) : j
j
Notice that the necessary and sucient condition for a function to be an entropy is that 00 f 0 is symmetric, that is, j00 aj = (aj )> j00. One has from (5.6) X 0> X X (j ) Hj = ? 21 2 (+ fj )> j00 + fj + "0 O(j+ uj j3) : (5.7) j
j
j
In estimating the rst term of I1, we use the conservative property of entropy. Denote uj = 12 (uj ?1 + uj +1). Similar to (3.16)-(3.17), one can obtain (5.8) X X Z 1=2 X 0> a(uj + 0 uj ) d 0 uj = "0 O(j+ uj j3) : (j ) Fj = ? 12 (j0 )> j
?1=2
j
j
Combining (5.5) with (5.7)-(5.8), we have the following estimate of I1 . X X I1 ? 0 j+ uj j3 + "0 C j+ uj j3 ? 21 2 (+ fj )> j00 + fj : (5.9) j
j
1046
GUI-QIANG CHEN AND JIAN-GUO LIU
Estimate of I2. As in (3.19), we substitute (5.1)-(5.2) into I2 and have the follow-
ing expansion.
(5.10)
X
I2 = 21
+
F>j j00 Fj + 12
X
Hj> j00 Hj +
X
Fj>j00Hj
j j j 1 X J > 00 Jj + X J > 00 (Fj + Hj ) : j j j j 2 j j
The estimate of the rst term in (5.10) is similar to that in (3.20)-(3.22) by the identity (3.20). Putting all products of three di erences together in O(j+ uj j3 ), one has X X > 00 X Fj j Fj =2 (+ fj )> j00+ fj ? 14 2 (? + fj )> j00 ?+ fj j j (5.11) j X 3 + "0 O(j+ uj j ): j
The estimate of the second term in (5.10) is similar to that in (3.23)-(3.24). After putting all products of three di erences in O(j+ unjj), the remainder term is the sum of products of two second di erences, which can be controlled by the last term in (5.11). (5.12) X X > 00 Hj j Hj = 41 4 ? (aj +1=2+ fj )> j00 ?(aj +1=2+ fj ) j
j X 4
X
j
j
= 14
(? + fj )> a>j j00 aj ? + fj + "03
O(j+ uj j3) :
The main idea in estimating the third term in (5.10) is to utilize the symmetric property so that every term can be combined into the product of three di erences. For any symmetric matrix A, we have as in Lemma 3.1 that X X X (5.13) ? (0bj )> Abj = 21 b>j 0Abj ? 21 (+ bj )> + A+ bj : j
j
j
Similar to (3.25)-(3.27), we have X X > 00 Fj j Hj = ? 14 3 (0fj )> j00 ? (aj +1=2+ uj )
X Xj ?q 00 >q 00 3 1 0 j + fj j aj + uj + "02 O(j+ uj j3) : = 4 j j q 00 ?q 00 ?1 We know from (5.6) that j aj j is symmetric. Hence we can apply the j
identity (5.13) to rst term on the right-hand side of the above equality with q the 00 bj replaced by j + fj . As a result that every term now is decomposed as the product of three di erences, we obtain X > 00 X (5.14) Fj j Hj = "02 O(j+uj j3 ) : j
j
The estimate of the last three terms in (5.10) is rather straightforward since every term is the sum of products of more than three di erences. One has from a
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1047
direct estimate that X X X > 00 (5.15) Jj j Jj + 2 Jj> j00 (Fj + Hj ) "0 O(j+uj j3) : j
j
j
Finally, combining (5.11)-(5.12) with (5.14)-(5.15), we have X X I2 12 2 (+ fj )> j00 + fj ? 81 2 (?+ fj )> j00? + fj j j (5.16) X X > > 00 4 1 + 4 (? + fj ) aj j aj ?+ fj + "0 O(j+ uj j3) : j
j
Estimating I3 simply follows from the fact that each term is the sum of products of more than three di erence factors X X (5.17) jI3 j C (jFj j3 + jHj j3 + jJj j3) "02 C j+ uj j3 : j
j
Combining (5.9) with (5.16), we have
X
X
j
j
(jn+1 ? jn) ? 0
j+ uj j3 ? 18 2(1 ? "02 )
+ (1 + "0 + "02)C
X j
j+ uj j3 :
X j
(?+ fj )> j00? + fj
For 0 large and "0 small, we have X X X n+1 n (j ? j ) ?C1 j+ uj j3 ? 2 C2 j?+ fj j2 : This gives
j
j
j
Proposition 5.1. Let unj be the Lax-Wendro approximate solutions (5:1)-(5:2) of the hyperbolic systems of conservation laws (1:1) with a convex entropy . Assume that uh is uniformly bound and the coecient in (5:2) is suitably large and the CFL number "0 given by (1:3) is suitably small. Then
(5.18)
X
(unj )h +
X
kn;j
j+ ukj j3 h + 2
X
j? + f(ukj )j2h C :
kn;j 5.2. H ?1 compactness estimates. For any C 2 convex entropy pair (; q) any (x; t) 2 C01, 0, we denote nj = (xj ; tn) and X ? X ? Ih ( ) h nj (unj +1) ? (unj ) + 21 nj q(unj+1) ? q(unj?1) : j;n j;n j
As (3.37), we have from the Taylor expansion that X ? Ih ( ) = h nj (j0 )> Fj + 21 0 qj + (j0 )> Hj + (j0 )> Jj (5.19)
j;n X + 21 h nj(Fj + Hj + Jj )> j00 (Fj + Hj + Jj ) j;n X 1 + 2 h nj000()(Fj + Hj + Jj )3 I1h ( ) + I2h ( ) + I3h ( ) : j;n
and
1048
GUI-QIANG CHEN AND JIAN-GUO LIU
Estimate of I1h .
The estimate of the last term in I1h is similar to that in (3.38)
and (3.39). X X X n 0> (5.20) j (j ) Jj ? 12 0 ( nj + nj+1)j+ aj j3 + C j+ uj j2j+ njj : j;n
j;n
j;n
Similar to that in (3.41), one can obtain (5.21) X n 0> X X j (j ) Hj ? 21 2 nj(+ fj )> j00 + fj + "0 C ( nj + nj+1)j+ uj j3 j;n
j;n
+ "0 C
X j;n
j;n
j+ uj j2j+ njj :
The estimate of the rst term in I1h is similar to that in (3.40). One has
X n? 0 > 1 X ? j (j ) Fj + 2 0 qj = ? 12 nj (j0 )> 0fj ? 0qj j;n Xj;n n n X 3 C
The estimates (5.20)-(5.21) yield I1h ( ) ? 21 0 h
X
( nj + nj+1)j+ uj j3 ? 21 2 h
X
j;n
j+ uj j2j+ njj :
nj(+ fj )> j00 + fj
j;n X n n 3 + (1 + "0 )Ch ( j + j +1)j+ uj j + (1 + "0 )Ch j+ uj j2j+ njj : j;n j;n h h We can similarly estimate I2 and I3 . Finally we have
Ih ( ) ? 21 0 h
j;n
j;n
( j + j +1)j+ uj j + C
X
X
Xj;n
( nj + nj+1)j+ uj j3 ? 2 Ch
X
nj(? + fj )2
j;n X ? 2 n n + Ch j+ uj j j+ j j + j? j j + C"02 h ( nj + nj+1)j+ uj j3 : j;n j;n We choose 0 large and "0 small and then have X ? Ih ( ) Ch j+ uj j2 j+ njj + j? njj h ?1=3k kC : j;n
Therefore, we obtain
Proposition 5.2. Under the same assumptions of Proposition 5.1,
(5.22) X ? X n? n+1 j (uj ) ? (unj ) h + 21 nj q(unj+1) ? q(unj?1) h Ch ?1=3k kC ; j;n
j;n
for C 2 convex entropy pair (; q) and 2 C01 , 0.
?1 Similar to Section 3.3, we can show that @t (uh ) + @x q(uh ) is compact in Hloc for any convex entropy pair (; q). For a general entropy function we write it as a linear combination of two convex entropy functions as we have explained in Section 4. Hence we have
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1049
Proposition 5.3. Under?1the same assumptions of Proposition 5.1, @t(uh ) +
@x q(uh ) is compact in Hloc for any C 2 entropy pair (; q). 5.3. Convergence and entropy consistency. Now we are concerned with the convergence of uniformly bounded approximate solutions of 2 2 systems of conservation laws endowed with global Riemann invariants and the entropy consistency of boundedly convergent Lax-Wendro numerical solutions of general hyperbolic systems of conservation laws with a convex entropy. One important example of the 2 2 systems is the elasticity equations (cf. [5]): @tv ? @x u = 0 ; @tu ? @x (v) = 0 ; where u is the speci c volume, v is the strain, and is the stress-strain relation satisfying 0 (v) > 0. The numerical experiments indicate that the Lax-Wendro approximate solutions (vh ; uh) are uniformly bounded. It is important from the viewpoint of numerical analysis whether uniformly bounded Lax-Wendro numerical solutions may generate some oscillations. The following theorem shows that, if (vh ; uh ) is uniformly bounded, then it indeed strongly converges to the corresponding entropy solution. The proof follows from Theorem 2.2, the estimates in Sections 5.1 and 5.2, and similar compactness arguments as in the scalar cases. Theorem 5.1. Consider a 2 2 genuinely nonlinear and strictly hyperbolic system of conservation laws (1:1) endowed with global Riemann invariants. Let uh (x; t) be the Lax-Wendro approximate solutions. Assume that uh is uniformly bounded and the coecient in (5:2) is suitably large and the CFL number "0 given by (1:3) is suitably small. Then there is a subsequence (still denoted as ) uh such that uh (x; t) ! u(x; t) a.e. as h ! 0 and u is the entropy solution of (1:1) satisfying @t(u) + @x q(u) 0 in D0 for any convex entropy (; q). For a general system, we conclude that the numerical solutions of the LaxWendro scheme, if they are boundedly convergent, then converge to the entropy solutions. A similar result has been obtained by Majda and Osher [15] for the semi-discrete case of time-dependent systems and for the fully discrete case of timeindependent systems.
Theorem 5.2. Consider a general system of genuinely nonlinear and strictly hy-
perbolic conservation laws (1:1) endowed with a convex entropy. Let uh (x; t) be the uniformly bounded Lax-Wendro approximate solutions converging boundedly a.e. to u(x; t). Assume that the coecient in (5:2) is suitably large and the CFL number "0 given by (1:3) is suitably small. Then u is the entropy solution of (1:1) satisfying @t (u) + @x q(u) 0 in D0 for any convex entropy pair (; q).
A typical example for such a system is the system of gas dynamics in Lagrangian coordinates: 0 v 1 0?u1 @t @ u A + @ x @ p A = 0 ; E pu where v > 0 denotes the speci c volume of the gas, u the velocity, p > 0 the pressure, E = 21 u2 + e the total speci c energy and e the speci c internal energy. The additional conservation law assumes the form @t (?S) 0 for physical solutions. For polytropic gases, ?S(v; u; E) = ? log(E ? 21 u2) ? ( ? 1) logv is a strictly convex function of the state variables v, u, and E for > 1, a xed constant,
1050
GUI-QIANG CHEN AND JIAN-GUO LIU 1
and p = ( ? 1) E ?v2 u = A(S)v . Theorem 5.2 indicates that uniformly bounded convergent Lax-Wendro approximation is consistent with this system. 2
6. The MacCormack scheme and Richtmyer scheme
An ecient implementation of the Lax-Wendro scheme is to use two step splitting methods. One particularly popular version is the following MacCormack scheme: u~nj = unj ? + f(unj ) ; ? (6.1) unj +1 = 12 (unj + u~nj) ? ? f(~unj ) + ? jn+1=2 j+f 0 (unj)j+ unj : The convergence of this scheme is very much similar to what we did for the LaxWendro scheme in Sections 3-5. We only state some convergence theorems and show those arguments that are di erent from ones of the previous sections. We rst rewrite the MacCormack scheme (6.1) as follows (6.2) unj +1 = unj + Fjn + Hjn + H~ jn + Jjn ; where Fjn = ? 21 n 0f(unj ) ; ? ( f(un))2 ; Hjn = 21 n2 ? + unj + j ? n n 1 ~ (6.3) Hj = 2 n ? f(uj ) ? f(~unj ) ? f 0 (unj )(unj ? u~nj ) ; ? Jjn = n? ~jn+1=2 j+a(unj )j+ unj ; a(un ) ? f = u ~jn+1=2 = jn+1=2 + n j j +a jj + j : + j Clearly, we can carry out the analysis without any di erence in the absence of the term H~ jn. We only point out that H~ jn can only produce third-order terms and can be controlled by the viscosity term based on the following facts. First, we notice that u~n in (6.1) is always in the interval [unj; unj+1] provided the CFL number is less than 1=2. Using the Taylor expansion and (6.1), one has (6.4) f(unj ) ? f(~unj ) ? f 0 (unj)(unj ? u~nj) = 21 f 00 (jn )(unj ? u~nj )2 = 12 2 f 00 (jn )(+ f(unj ))4: Substituting (6.4) into H~ j and using the summation by parts, one has (6.5) X X 0 n ~ n 1 3 X 0 00 n (uj )Hj = ? 4 + j f (j )(+ f(unj ))4 "02 C (snj )j+ unjj3 ; j
X (6.6)
j
j
j
00(unj )(H~ jn ) 2 = 14 6
X
2
? 00 (unj) ? a0(jn )(+ f(unj ))2
j X 5 "0 C (snj )j+ unjj3 ; j
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
and
X
(6.7)
j
00 (unj )FjnH~ jn = ? 18 4
"03 C
X j X j
1051
? 00(unj )0f(unj )? a0 (jn )(+ f(unj ))2 (snj )j+ unjj3 :
Now we treat another two step version of the Lax-Wendro scheme|the Richtmyer scheme, u~nj+1=2 = 12 (unj + unj+1) ? 12 + f(unj ) ; (6.8) ? unj +1 = unj ? ?f(~unj+1=2 ) + ? jn+1=2j+ f 0 (unj)j+ unj : We can rewrite it into the form unj +1 = unj + Fjn + Hjn + H~ jn + Jjn ; (6.9) where (6.10) Fjn = ? 12 n0 f(unj ) ; ? ( f n)2 Hjn = 12 n2 ? + ujn ; + j ? n n 1 ~ Hj = 2 n ? f(uj ) + f(unj+1 ) ? 2f(~unj+1=2 ) ? f 0 (unj+1=2)(unj + u2j +1 ? 2~unj) ; ? an ? f n = un ~jn+1=2 + n j j+ uj nj + j : Jjn = n ? ~jn+1=2 j+ a(unj)j+ unj ; + j It is also easy to verify that H~ jn can produce only third-order terms and can be controlled by the viscosity term. Therefore we have the following theorems corresponding to those of the LaxWendro scheme in Sections 3-5 under the same assumptions. Theorem 6.1. Let uh(x; t) be approximate solutions of the convex scalar conservation laws (1:1) and (3:1) by the the MacCormack scheme (6:1) or the Richtmyer scheme (6:8). Then there is a subsequence strongly converging to the weak solution of (1:1) satisfying the entropy inequalities. Theorem 6.2. Let uh(x; t) be approximate solutions of the scalar conservation laws (1:1) or (4:4) by the MacCormack scheme (6:1) or by the Richtmyer scheme (6:8) with the arti cial viscosity term (4:1). Then there is a subsequence strongly converging to the weak solution of (1:1) satisfying the entropy inequalities. Theorem 6.3. Consider a general 2 2 genuinely nonlinear and strictly hyperbolic system of conservation laws (1:1) endowed with global Riemann invariants. Let uh (x; t) be the uniformly bounded approximate solutions of the MacCormack scheme (6:1) or the Richtmyer scheme (6:8). Then there is a subsequence (still denoted as ) uh such that uh (x; t) ! u(x; t) a.e. as h ! 0 and u is the entropy solution of (1:1) satisfying @t (u) + @x q(u) 0 in D0 for any C 2 convex entropy pair (; q). Theorem 6.4. Consider a general system of hyperbolic conservation laws (1:1) endowed with a convex entropy. Let uh (x; t) be the approximate solutions of the MacCormack scheme (6:1) or the Richtmyer scheme (6:8). If uh converges boundedly
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GUI-QIANG CHEN AND JIAN-GUO LIU
a.e. u(x; t), then u(x; t) is the entropy solution of (1:1) satisfying @t(u)+@x q(u) 0 in D0 for any C 2 convex entropy pair (; q). Acknowledgments
The authors would like to thank Professor P. D. Lax for his encouragements and Professors J. Goodman, T.-P. Liu, A. Majda, S. Osher, and E. Tadmor for their interest in this work. G.-Q. Chen was supported in part by ONR grant N0001491-J-1384, NSF grant DMS-9623203, and Alfred P. Sloan Foundation Fellowship. J.-G. Liu was supported in part by NSF Grant DMS-9114456, DOE grant DEFG02-88ER25053 and ARO Grant DAAL03-92-G-0143. This work was initiated while the authors visited Stanford University and was supported in part through Army grant DAAL 03-91-G-0017. References 1. J.P. Boris and D.L. Book, Flux corrected transport. I. SHASTA, a uid transport algorithm that works, J. Comp. Phys. 11 (1973), 38-69. 2. G.-Q. Chen, The compensated compactness method and the system of isentropic gas dynamics, MSRI Preprint 00527-91, Berkeley (1990). 3. G.-Q. Chen and Y.-G. Lu, A study on application approaches of the theory of compensated compactness, Chinese Science Bulletin 34 (1989), 15{19. MR 90h:35024 4. F. Coquel and P. Le Floch, Convergence of nite di erence schemes for conservation laws in several space variables: the corrected antidi usive ux approach, Math. Comp. 57 (1991), 169-210. MR 91m:65229 5. R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. 82 (1983), 27{70. MR 84k:35091 6. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high order accurate nonoscillatory schemes, III, J. Comp. Phys. 71 (1987), 231{303. MR 90a:65199 7. A. Harten, J.M. Hyman, and P.D. Lax, On nite-di erence approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297{322. MR 54:1640 8. A. Harten, P. D. Lax, and B. van Leer, On upstream di erencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review 25 (1983), 35. MR 85h:65188 9. S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Preprint (1993). 10. P.D. Lax and C.D. Levermore and S. Venakides, The generation and propagation of oscillations in dispersive IVPs and their limiting behavior, Importantdevelopments in soliton theory 1980{1990, T. Fokas and V.E. Zakharov eds, Springer-Verlag, Berlin, 1992. MR 95c:35245 11. P.D. Lax and B. Wendro , Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217{237. MR 22:11523 12. P.L. Lions and P. Souganidis, Convergence of MUSCL type methods for scalar conservation laws, C. R. Acad. Sci. Paris, Serie I 311 (1990), 259{264. MR 91i:65168 13. R.W. MacCormack, The e ect of viscosity in hyperbolicity impact cratering, AIAA Paper (1969), 69{354. 14. A. Majda and S. Osher, A systematic approach for correcting nonlinear instability: the Lax-Wendro scheme for scalar conservation laws, Num. Math. 30 (1978), 429{452. MR 80g:65101 15. A. Majda and S. Osher, Numerical viscosity and the entropy condition, Comm. Pure Appl. Math. 32 (1979), 797{838. MR 80j:65031 16. C.S. Morawetz, An alternative proof of DiPerna's theorem, Comm. Pure Appl. Math. 44 (1991), 1081{1090. MR 92m:35165 17. F. Murat, L'injection du cone positif de H ?1 dans W ?1 est compacte pour tout q < 2, J. Math. Pures Appl. 60 (1981), 309-322. MR 83b:46045 18. H. Nessyahu and E. Tadmor, Non-oscillatory central di erencing for hyperbolic conservation laws, J. Comp. Phys. 87 (1990), 408{463. MR 91i:65157 19. S. Osher, On convergence of generalized MUSCL schemes, SIAM J. Numer. Anal. 22 (1985), 947-961. MR 87b:65147 ;q
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
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20. S. Osher and E. Tadmor, On the convergence of di erence approximations to scalar conservation laws, Math. Comp. 50 (1988), 19{51. MR 89m:65086 21. B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal. 29 (1992), 1-19. MR 92m:76111 22. R.D. Richtmyer and K.W. Morton, Di erence Methods for Initial Value Problems, 2nd ed., Wiley{Interscience, New York, 1967. MR 36:3515 23. D. Serre, La compacite par compensation pour les systemes hyperboliques non lineaires de deux equations a une dimension d'espace, J. Math. Pures Appl. 65 (1986), 423-468. MR 88d:35123 24. C.-W. Shu and S. Osher, Ecient implementation of essentially non-oscillatory shock capturing schemes, J. Comp. Phys. 83 (1989), 32{51. MR 90i:65167 25. Y.S. Smyrlis, Existence and stability of stationary pro les of the LW scheme, Comm. Pure Appl. Math. 43 (1990), 509{545. MR 91d:65143 26. P.R. Sweby, High resolution schemes using ux limiters for hyperbolic conservation laws, SIAM J Num. Anal. 21 (1984), 995{1011. MR 85m:65085 27. T. Tang, On three-point second-order accurate conservative di erence schemes, J. Comp. Math. 5 (1987), 105{118. MR 89g:65115 28. L. Tartar, Compensated compactness and applications to partial di erential equations, Research Notes in Mathematics, Nonlinear Analysis and Mechanics, ed. R.J. Knops, vol. 4, Pitman Press, New York, 1979. MR 81m:35014 29. B. van Leer, Towards the ultimate conservative di erence schemes, V, A second order sequel to Godunov's method, J. Comp. Phys. 43 (1981), 357{372. 30. P. Woodward and P. Colella, The numerical simulation of two dimensional uid ow with strong shock, J. Comp. Phys. 54 (1984), 115{173. MR 85e:76004 31. H. Yang, Nonlinear wave analysis and convergence of MUSCL schemes, IMA Preprint 697 (1990). 32. S.-H. Yu, Existence of the local discrete shock pro le for the Lax-Wendro scheme, Preprint (1992). Department of Mathematics, Northwestern University, Evanston, Illinois 60208
E-mail address :
[email protected]
Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
E-mail address :
[email protected]
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION FOR CONSERVATION LAWS GUI-QIANG CHEN AND JIAN-GUO LIU Abstract. We are concerned with the convergence of Lax-Wendro type
schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendro scheme proposed by Lax and Wendro in 1960 and its two step versions{the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth ux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed L estimates of the approximate solutions, H ?1 compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws. p
1. Introduction
We are interested in the convergence of nite di erence schemes with high resolution for conservation laws (1.1) @t u + @x f(u) = 0 ; u(x; 0) = u0(x) 2 L2 (R) \ L1 (R) : One of the most fundamental and important second-order nite di erence schemes is the Lax-Wendro scheme [11]. It is de ned by ? un + u n unj +1 =unj ? 2n 0 f(unj ) + 2n ? f 0 j 2 j +1 + f(unj ) (1.2) ? + n ? jn+1=2 j+ f 0 (unj )j+ unj ; P where unj = u(xj ; tn), tn = nk=1 k , xj = jh, n = n =h, + uj = uj +1 ? uj , 0uj = uj +1 ? uj ?1, ? uj = uj ? uj ?1, and jn+1=2 ; 0 0 jn+1=2 1 < +1, is a smooth function of unj and unj+1 (e.g. jn+1=2 = const. in [11]). Moreover, this scheme has to satisfy the Courant-Friedrichs-Lewy condition ? max jf 0(un)j 1 "0 = max (1.3) j n n j 2
for stabilization in general. This scheme is designed to have the following desirable computational features: conservation form, three-point dependence, second-order accuracy on the smooth
Received by the editor April 1, 1996. 1991 Mathematics Subject Classi cation. Primary 65M12; Secondary 35L65. Key words and phrases. Conservation laws, convergence, entropy solution, Lax-Wendro scheme. 1027
c 1997 American Mathematical Society
1028
GUI-QIANG CHEN AND JIAN-GUO LIU
regions of solutions, and stabilization for the convex case for 0 > 0 and "0 1. When jn+1=2 = 0, the Lax-Wendro scheme serves as not only a simple mode for the physical phenomenon of the dissipation-dispersion couplings, but also an example of dispersive schemes that do not converge in the sense of strong topology (cf. [10]). Indeed, as observed by Harten-Hyman-Lax [7] and Majda-Osher [14], [15], the second-order numerical viscosity jn+1=2 0 > 0 in this scheme is essential to guarantee that the numerical solutions are nonlinearly stable and converge to the physical solutions. The main role of the factor j+ f 0 (unj )j in the third term of the scheme (1.3) is to reduce e ectively viscosity in the smooth regions of solutions to produce sharp discontinuities in numerical computations. Such a scheme does not have a TVD property. The main objective of this paper is to prove the convergence of the Lax-Wendro approximate solutions to the entropy solutions and to provide an analytical approach for such a convergence analysis for high-order nite di erence schemes, which do not preserve BV (and even L1 bound). This is motivated by the fact that any high-resolution di erence scheme cannot, in general, preserve BV for the hyperbolic systems of conservation laws, especially for the nonstrictly hyperbolic case and the multidimensional case. Since our analysis is qualitative, we will always assume that 0 is suitably large and 0 is suitably small to make our analysis more convenient without loss of our purpose. One can follow our analysis to get the optimal constants for 0 and 0 from this approach for some concrete equations. Our analysis is based on careful Lp estimates of the approximate solutions, H ?1 compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. In Section 2 we describe some compensated compactness theorems for conservation laws, which guarantee the convergence of the nite di erence schemes provided that the corresponding approximate solutions satisfy these frameworks. Since these compactness frameworks do not need the BV estimates, this enables us to carry through our analysis by using the Lp estimates and the H ?1 compactness estimates, which are weaker than the BV estimates in general. Section 3 is devoted to the estimates of the Lax-Wendro approximate solutions to the convex scalar conservation laws. We obtain the uniform Lp estimates of the approximate solutions and the H ?1 compactness estimates of the corresponding entropy dissipation measures by analyzing carefully the properties of this scheme and by developing some useful estimate techniques. A global entropy error estimate is also obtained to ensure the consistency of this scheme with the scalar conservation laws. In our analysis we need a technical assumption of the algebraic growth of ux functions. This is because we do not require the L1 bound or the BV bound for the Lax-Wendro approximate solutions to achieve our goal. One diculty in the analysis is the fact that the dissipation is third-order in the Lax-Wendro scheme, comparing with the second-order dissipation in the rst-order schemes. All these terms need to be carefully combined into a third-order term or a term consisting of square products of second-order di erences with a favorable sign. Moreover, the factor in each combined third-order di erence needs to be bounded by the growth factor in the dissipative term in each cell. Some special treatment is also made since the Lax-Wendro scheme is a three-point scheme. A requirement of small CFL number is made here to use the grid ratio to control some growth factors.
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1029
In order to get the compactness of entropy dissipation measures, we estimate certain entropy inequalities with a higher growth rate. These techniques are generalized to study the convergence of the Lax-Wendro scheme for the nonconvex scalar case in Section 4 and for the hyperbolic systems of conservation laws in Section 5, and to prove the convergence of the Richtmyer scheme [22] and the MacCormack scheme [13]{two step versions of the Lax-Wendro schemes in Section 6, which are widely used in industry and engineering. In connection with earlier work on the Lax-Wendro type schemes, we recall that Majda and Osher [14], [15] showed the L2 -stability for general scalar conservation laws, the entropy consistency for the boundedly convergent approximate solutions for the semi-discrete cases as well as the complete-discrete scheme for the timeindependent cases of general systems endowed with a convex entropy, the ecient choices of arti cial viscosity such as the switching techniques, and the validity of the CFL number in their analysis. Many of their techniques have been melted into our analysis. We also refer to [25] for the stability of the local discrete shock pro le for the Lax-Wendro scheme. Regarding work on the convergence analysis of full discrete high-resolution nite di erence schemes, we refer to [4] and the references cited therein for the ux-limit schemes with slope modi cation or antidi usive ux approach, which preserve L1 bound. Since the Lax-Wendro type schemes do not have a TVD property, our analysis consists of two steps: One is to prove the convergence, which is an essential diculty here and is, however, automatically ensured by the Helly principle for the TVD or TVB schemes, and the other is to verify the entropy consistency. The convergence analysis of TVD or TVB schemes focus mainly upon the second step for the scalar case, that is, the consistency proof. The convergence of a class of semi-discrete generalized MUSCL schemes for the strictly convex case was obtained in [19]. For the semi-discrete MUSCL scheme for the convex scalar conservation laws, some consistency results were announced in [12], [31]. 2. Compactness frameworks
In this section we discuss some compactness frameworks ? for the approximate solutions for subsequent developments. A pair of functions (u); q(u) is called an entropy-entropy ux pair if they satisfy q0 (u) = 0 (u)f 0 (u) . For the scalar case, any function is an entropy function. In the following theorem and the analysis in Section 3 and Section 4, we will denote entropy 0 (u) = 21 u2 for the convex case and 0(u) = f(u) for the general case. Theorem 2.1. Consider the scalar conservation laws (1:1) satisfying measu : f 00 (u) = 0 = 0. Let uh (x; t) be numerical approximate solutions of (1:1) satisfying the following conditions: ? (1) uh is bounded in Lp for some p 2 and f(uh ); 0(uh ); q0(uh ) is bounded in L2loc ; (2) The dissipation measures @tuh +@x f(uh ) and @t 0(uh )+@x q0(uh ) are compact ?1; in Hloc ? (3) For any C 2 convex entropy pair (u); q(u) , @t (uh )+@x q(uh ) o(1) in D0, provided that j(u)j + jq(u)j M(1 + jujr ); r < p. Then there is a subsequence (still denoted as ) uh such that uh (x; t) ! u(x; t) a.e. as h ! 0 and u is the entropy solution of (1:1) satisfying @t(u)+@x q(u) 0 in D0 for any C 2 convex entropy pair (; q).
1030
GUI-QIANG CHEN AND JIAN-GUO LIU
We remark that, for general conservation laws, the conditions (1) and (2) imply that w- limf(uh ) = f(w- limuh ) in L2 and the condition (3) implies the following entropy condition for the Young measures x;t, determined by the Lax-Wendro approximate solutions uh (x; t), @t hx;t; i + @x hx;t; qi 0 in the sense of distributions. We refer the reader to Chen-Lu [3] for a detailed discussion. The proof of the L1 version of Theorem 2.1 can be found in [2], [3], [28] using the div-curl lemma of Tartar and Murat [28]. For a 2 2 system of conservation laws endowed with global Riemann invariants, we have the following similar framework. Theorem 2.2. Let uh (x; t) be numerical approximate solutions of the 2 2 genuinely nonlinear and strictly hyperbolic system of conservation laws (1:1) satisfying the following conditions: (1) uh is bounded in L1 ; ?1; (2) For any C 2 entropy pair (; q), @t(uh ) + @x q(uh ) is compact in Hloc (3) For any C 2 convex entropy pair (; q), @t(uh ) + @x q(uh ) o(1) in D0 . Then there is a subsequence (still denoted as ) uh such that uh (x; t) ! u(x; t) a.e. as h ! 0 and u is the entropy solution of (1:1) satisfying @t(u)+@x q(u) 0 in D0 for any C 2 convex entropy (; q).
This theorem is proved by DiPerna [5] by estimating the entropy dissipation measures and by using the Lax entropy pairs and the compensated compactness method. An alternative extension proof can be found in [16], [23]. Finally we state the following two lemmas, which can be found in [17] and [2], respectively. ?1;q is completely Lemma 2.1. The embedding of the positive cone of W ?1;p in Wloc continuous for all q < p. Lemma 2.2. Let Rn be a bounded open set. Then ?compact set of W ?1;q ( ) \ ?bounded set of W ?1;r( ) loc loc ? ? 1 compact set of Hloc ( ) ; where q and r are constants, 1 < q 2 < r < 1. 3. Convex scalar conservation laws
In this section we are concerned with the Lax-Wendro approximate solutions of the scalar conservation laws (1.1) with ux function f(u) satisfying (3.1) f 00 (u) c0 > 0 ; f (k) (u) O(jujm?k ) ; for juj 1; k = 0; 1; 2 : For convenience, we write the Lax-Wendro scheme (1.2) into the form unj +1 = unj + Fjn + Hjn + Jjn ; (3.2) where ?+f(un )2 j n n 2 n 1 1 Fj = ? 2 n 0f(uj ) ; Hj = 2 n ? ; n + uj (3.3) ? Jjn = n? ~jn+1=2 j+a(unj )j+unj ;
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
with
? unj+unj a 2 ~jn+1=2 = jn+1=2 + n
+1
? f(un )= un + j + j = n
1031
0
j +1=2 + O(nja j):
j+ a(unj)j
From unj, we construct the Lax-Wendro approximate solutions on R2+: (3.4) uh (x; t) unj; for (x; t) 2 [xj ?1=2; xj +1=2) [tn; tn+1) : Then we have the following main convergence theorem of this section. Theorem 3.1. Let uh(x; t) be the Lax-Wendro approximate solutions (3:4) of the scalar conservation laws (1:1) and (3:1). Assume that the coecient 0 > 0 is suitably large and the CFL number "0 given by (1:3) is suitably small. Then there exists a subsequence strongly converging to the entropy solution of (1:1) satisfying the following entropy condition @t (u) + @x q(u) 0 in D0 for any C 2 convex entropy pair (; q) satisfying (k)(u) O(jujr?k ), for juj 1, 0 k r < 4m. The proof consists of the following three subsections in which we check the three conditions in Theorem 2.1, respectively. For simplicity of our proof, we drop the subscript n and use the following notations: ? un + u n fj = f(unj ) ; aj = a(unj) ; aj +1=2 = a j 2 j +1 ; j = (unj ) ; j0 = 0(unj ) ; j00 = 00 (unj) ; where is an entropy function of (1.1). We rst introduce the following three technical lemmas. Lemma 3.1. If g0 (u) c0 > 0 and g(k)(u) = ck um?k ?1+o(1) ; juj 1 ; for some constants ck ; k = 0; 1; 2, then there is u 1 and > 0 such that ( g(u) ? g(v) (max(juj; jvj)) ; with (s) = g0(s) ; s u ; u?v c0 ; s < u : The proof is straightforward and hence is omitted. Summing by parts, one has Lemma 3.2. If faj g and fbj g are two arbitrary sequences, then (3.5) + (aj 0bj ) = 0(aj + bj ) + ? (+ aj + bj ); and
(3.6)
2
X j
aj bj 0 bj =
X j
(+ bj )2 + aj ?
X j
bj20aj ;
provided that the above sums make sense. Proof. The formula (3.6) comes from a direct computation. Notice that
0 (aj bj ) = aj 0 bj + bj 0aj + ? (+ aj + bj ) : Multiplying both sides of the above equation by bj , taking sum for j, and summing by parts, one arrives at (3.7). Using the relation 0 f 0 = q0, expanding at uj in the intervals [uj ?1; uj ] and [uj ; uj +1], and using the integration by parts, one has
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GUI-QIANG CHEN AND JIAN-GUO LIU
Lemma 3.3. If (; q) is an entropy pair of (1:1), then
?
j0 0 fj = 0qj ? 12 j00(+ uj )2 + aj ? 12 j00 ? (+ uj )2 aj Z uj + 21 j00 (uj ? s)2 a0(s) ds
+1
uj?1
+ 21 000 (j )
Z uj
+1
uj
(uj ? s)2 a(s) ds + 12 000 (j ?1)
Z uj uj?1
(uj ? s)2 a(s) ds ;
where j ?1 and j are some values in [uj ?1; uj ] and [uj ; uj +1], respectively.
Proof. Using the relation 0 f 0 = q0 and expanding at uj in the intervals [uj ?1; uj ]
and [uj ; uj +1], one has j0 0fj = 0qj + j00
+ 12 000 (j )
Z uj
+1
Zuuj?j+11 uj
(uj ? s)a(s) ds (uj ? s)2 a(s) ds + 21 000 (j ?1)
Z uj uj?1
(uj ? s)2 a(s) ds;
for some j 2 [uj ; uj +1] and j ?1 2 [uj ?1; uj ]. Now using the integration by parts to the second term on the right-hand side of the above equation, we obtain the lemma. We remark here that we split the remainders in Lemma 3.3 into two cells [uj ?1; uj ] and [uj ; uj +1]. A basic reason for this is that we do not require the L1 bound of the approximate solutions, and thus need to control the growth rate. This kind of splitting techniques will be used in several places in this section.
3.1. Lp estimate. We now verify the rst condition of Theorem 2.1 for the Lax-
Wendro scheme (3.2)-(3.3). Since the ux function has the algebraic growth rate m, we only need to show that uh is bounded in L2 and L4m?2 . The method used here is to estimate the entropy function with certain required growth rate. In order to control the growth rate in the compactness analysis in Subsection 3.3, we estimate the entropy with a growth rate slightly more than that required in this subsection. For this reason, we choose a strictly convex entropy function 1 4m 1 2 (3.7) (u) = 4m(4m ? 1) u + 2 u P and estimate the bound of (unj ) for thePLax-Wendro scheme unj . We rst expand the time increment of (unj) and split it into three terms:
X?
(3.8)
j
X 0(un)(un+1 ? un) + 1 X 00(un)(un+1 ? un)2 j j j j j j 2
(unj +1) ? (unj ) =
j
+
4m X
j
1 X (k)(un )(un+1 ? un )k I1 + I2 + I3 ; j j j k! k=3 j
which will be estimated in this subsection, where we used the exact expansion to the highest order to avoid the remainder in [unj; unj +1], which is dicult to control.
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
Estimate of I1 .
Plugging (3.2) into I1, one has
(3.9)
I1 =
X j
j0 Jj +
X j
j0 Hj +
X j
1033
j0 Fj :
To estimate the rst term, one uses the expression of Jj in (3.3), the summation by parts, and Lemma 3.1 to get (3.10)
X j
j0 Jj = ?
X ~n j
j +1=2 + j0 j+ aj j+ uj ? 0
where (3.11)
X j
(sj )j+ uj j3 ;
(
5m?4 ; s u ; (s) = cs c0 ; s < u ;
sj = max(juj j; juj +1j);
for some constant c > 0. Note that (3.10) is the sum of products of three rst-order di erences due to the fact that the viscosity term is second-order accurate. This is the main dissipative term that is used to control all of the error terms. In estimating the second term, one uses the summation by parts and the expression of Hj in (3.3), and substitutes the expansion + j0 = j00 + uj + 12 000(j )(+ uj )2 ;
(3.12)
in the resulting terms to get
X
(3.13)
j
X (+fj )2
+ j0 u + j j X X 2 1 j00 (+ fj )2 ? 14 2 000 (j )(+ fj )2+ uj : = ?2
j0 Hj = ? 12 2
j
j
We use the term of the right-hand side in (3.10) to control the second term in (3.13). Notice that there is a factor (+ uj )3 in the second term that can be clearly controlled by the term of the right-hand side in (3.10). The factor in front of the di erence is of growth rate 2(m ? 1)+4m ? 4 = 6(m ? 1) that is larger than 5m ? 4 in (3.11). Noting that there is also a parameter 2 in front of this factor, therefore, we can use the fact j+ fj =+ uj j "0 to control certain powers of the growth rate to obtain (3.14)
X j
j0 Hj ? 21 2
X j
j00 (+ fj )2 + "0 C
X j
(sj )j+ uj j3 ;
for some positive constant C > 0. This kind of simple techniques will be used in several places in the rest of this section and we will omit the detailed explanations. Here and henceforth, we use the notations j and j as some values in [uj ; uj +1], and C is a positive constant, independent of the grid size h, the CFL number, the viscosity constant , and the numerical solution uh . In di erent contexts, they may have di erent values.
1034
GUI-QIANG CHEN AND JIAN-GUO LIU
P
Now we estimate the term j j0 Fj . Writing the expression of Fj in (3.3) in the integral form and using the conservative property of entropy, one has
Z uj
X
X j0 Fj = ? 21 0 (uj ) a(s) ds uj?1 j j X Z uj+1 ? 0 1 = 2 (s) ? 0 (uj ) a(s) ds u j?1
j
= 21
(3.15)
X j
+ 14 + 14
00 (uj )
X j
X j
Z uj
+1
uj?1
000(j ?1 ) 000(j )
+1
(s ? uj )a(s) ds
Z uj
uj?1
Z uj uj
+1
(s ? uj )2 a(s) ds
(s ? uj )2 a(s) ds :
Applying the following integration by parts to the rst term in the right-hand side of the above equality
Z uj
+1
uj?1
(s ? uj )a(s) ds = 12 (+ uj )2+ aj ? 12
Z uj
+1
uj?1
(s ? uj )2 a0(s) ds ;
and plugging the above two estimates back into (3.15), we obtain
X
(3.16)
j
j0 Fj C
X j
(sj )j+ uj j3 :
Finally, one uses (3.10), (3.14), and (3.16) to get the following estimate: (3.17) X X X I1 ? 0 (sj )j+ uj j3 ? 21 2 j00 (+ fj )2 + (1 + "0 )C (sj )j+ uj j3 : Estimate of I2 .
(3.18)
j
j
j
Substituting (3.2) into I2 in (3.8), one has the expansion
I2 = 12
X j
00 (uj )(Fj2 + Hj2 + 2Fj Hj + Jj2 + 2Jj (Fj + Hj )) :
We now estimate I2 term by term. In estimating the rst term in (3.18), one uses the identity (3.19) (0 fj )2 = 2(+ fj )2 + 2(+ fj ?1)2 ? (? + fj )2 ; and a direct estimate of Fj from its expression in (3.3) to get
X j
X 00? X j (+ fj )2 + (+ fj ?1 )2 ? 14 2 00 (uj )(? + fj )2 j j 2 1 2X 00 2 1 2 X 00 2 X 00
j00 Fj2 = 21 2 =
j
j (+ fj ) ? 4
j
j (? + fj ) + 2
j
+ j (+ fj )2 :
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1035
In the rst and third terms on the right-hand side of the above inequality, one uses the Taylor expansion for + fj and some simple calculations used in estimating (3.13) to obtain
X j
(3.20)
j00 Fj2 2
X j
j00aj2+1=2(+ uj )2 ? 14 2
+ "0 C
X j
X
(sj )j+ uj j3 :
j
j00 (? + fj )2
Notice that the rst term on the right-hand side of (3.20) is exactly the same as the second term on that of (3.17) except with a di erent sign, which cancel each other. The second term on the right-hand side of (3.20) is a sum of squares of second-order di erences in a favorable sign. We will use it to control a similar term below. To estimate the second term in (3.18), we use the identity
2 + fj ?1 + f + fj ; ? (+ fuj ) = ? + fj + j ? u + j + uj ?1 + j
(3.21)
and directly estimate the expression of Hj in (3.3) to get
+fj?1 2 X 00 00 2 2 4 1 j Hj 2 j (? + fj ) u + j ?1 j j + fj 2 X 00 4 2 1 j (+ fj ) ? u + 2 : + j j
X
The rst term on the right-hand side of the above inequality can be controlled by the second term on the right-hand side of (3.20) in view of the fact that the CFL number is less than 1. The last term of the above inequality involves three points. Hence we split it into the following two terms by using the Taylor expansion for a in the intervals [uj ?1; uj ] and [uj ; uj +1]: 4
f + j j00 (+ fj )2 ? uj + j
X
2
f 2 + j j00(+ uj )2 ? uj + j X "03 j00 ja0(j ?1)jj?uj j(+ uj )2 2"03
X
j X 3 + "0 j00ja0 (j )jj+ uj j(+ uj )2 : j
The last term only involves two points and hence can be directly estimated. In the last second term, we use the Holder inequality to split it again to obtain
X j
j00 ja0(j ?1)jj?uj j(+ uj )2
X j
C
juj j4m?2jsj ?1jm?2 j?uj j(+ uj )2
X j
(sj )j+ uj j3 :
1036
GUI-QIANG CHEN AND JIAN-GUO LIU
The methods used here will be applied to estimating the similar terms. With the above three estimates, we arrive at (3.22) +fj?1 2 X 00 2 1 4 X 00 3 C X (sj )j+ uj j3 : j Hj 2 + " j (? + fj )2 0 u j
+ j ?1
j
j
The third term in (3.18) consists of products of a factor of rst-order di erence and a factor of second-order di erence, we utilize a symmetric property to transform it into a sum of products of three di erences. In doing this, we rst split the positive function 00 , and sum by parts to get ( f )2 X 00 3 1 ? 4 j 0 fj ? + uj + j j q q 2 X 00 q 00 (+ fj )2 = ? 14 3 j 0fj ? j u ? ? j00 (+ fuj ?1 ) + j + j ?1 j (3.23) X?q 00 q 00 +fj j + fj + j 0fj u = 41 3 + j j q 2 X q 00 j 0 fj ? j00 (+ fuj ?1) : + 14 3 + j ?1 j Now the last term consists of products of three di erences and hence can be directly estimated. The rst term has some symmetric property after switching the di erence operator + with 0. This can be done by using Lemma 3.2 as follows.
X?q 00
?q00 f + fj = X?q00 f ?q00 f +fj j 0 j + uj j + j 0 j + j + uj j X?q 00 ? q 00 +fj f f : +
j + fj +
j
j + j
j
?
+
j + j
+ uj
Again the last term in the above equality is a sum of products of three di erences in uj and hence can be estimated directly. The rst term can now be transformed into a sum q 00of products of three di erences by using Lemma 3.3 where bj are replaced by j + fj . (3.24)
X?q 00 j
?q00 f + fj = 1 X ?q00 f 2 +fj + u + j + j + uj 2 j + j + j j X?q 00 2 +fj f ?1 :
j + fj 0
2
j
j + j
Finally, combining all the estimates (3.23)-(3.24), we arrive at X X 00 (3.25) j Fj Hj "02 C (sj )j+ uj j3 : j
j
0 u + j
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1037
Estimating the remainder three terms in (3.18) is rather simple since each of them consists of more than three di erence factors. We estimate them as follows: (3.26) X X 00 2 2 X 00? ~n j Jj = j ? ( j +1=2 j+aj j+uj ) 2 "0 12 C (sj )j+ uj j3 ; j
j
j
and (3.27) X X 2 j00 Jj (Fj + Hj ) = ?2 j00 ?( ~jn+1=2 j+ aj j+ uj )0 fj
j X 00 ~n 3 j ? ( j +1=2j+ aj j+ uj )? (aj +1=2+ fj ) + j X ("0 + "02) 1 C (sj )j+ uj j3 : j
j
The estimate of I2 is now completed by combining (3.20) and (3.22) with (3.25)(3.27). X X I2 12 2 j00aj2+1=2(+ uj )2 ? 18 2 (1 ? 22 aj2?1=2)j00 (? + fj )2 j j (3.28) X + "0 (1 + "0 + "02 + 1 + 12 )C (sj )j+uj j3 : j
Estimate of I3 .
The estimate of I3 is rather easy. We rst substitute the expansion of unj +1 ? unj in (3.2)-(3.3) into I3 to get 4m 3k X X
?
j(k)(unj )j jFj jk + jHj jk + jJj jk : k! j k=3 Using the expressions in (3.3), we have (3.30) 4m k X X jI3 j 3k! j(k)(unj)j k j+ fj jk + k j?fj jk + 2k jaj +1=2+ fj jk k=3 j jI3j
(3.29)
+ 2k jaj ?1=2?fj jk + (2 1 )k j+aj + uj jk + (2 1 )k j?aj ? uj jk
C
4m ? X k=3
"0k?1 + "02k?1 + "0k?1 1k
X (s )j u j3 : j + j j
We have now estimated all I1 , I2 , and I3 . Using (3.17), (3.28), and (3.30), we obtain X n+1 n X X 00( f )2 ? (j ? j ) ? 2 0 (sj )j+ aj j3 ? 18 2 1 ? 2"20 j ? + j j j j + C 1 + "0 1 +
4m X
k=2
"0k?1 1k
!X j
(sj )j+ uj j3 :
1038
GUI-QIANG CHEN AND JIAN-GUO LIU
Choosing 0 large and "0 1 small, one has X n+1 n X X (j ? j ) ?C1 (sj )j+aj j3 ? 2 C2 (? + fj )2 : j
j
j
This gives the following proposition.
Proposition 3.1. Under the same assumptions of Theorem 3.1, we have X k X? X n (uj )h + (3.31) (sj )j+ukj j3h + 2 ?+ f(ukj ) 2h C; kn;j
j
where is the entropy function given by (3:7).
kn;j
As an immediate consequence, we have
Corollary 3.1. Under the same assumptions of Theorem 3.1, we have
? kuh kL m + f(uh ); 0(uh ); q0(uh ) L m= m? C ; C independent of h; 4 loc
4 loc
(2
1)
where (0; q0) is the entropy pair given by (2:2).
3.2. Entropy inequality. We now estimate the entropy inequalities as stated in
the third condition in Theorem 2.1 for the entropy functions of form (3.32) (k) (u) O(juj2`?k) ; for juj 1; k = 0; 1; ; 2`; where ` 2m is a positive integer. In the next subsection, we will use the integer ` (13m ? 3)=6 in the compactness arguments. We will always take ` (3m+1)=2 for using Theorem 2.1 to ensure the entropy inequalities. For any (x; t) 2 C01 , 0, denote nj = (xj ; tn) and X ? X ? Ih ( ) h nj (unj +1 ) ? (unj) + 21 h nj q(unj+1) ? q(unj?1) : j;n
j;n
We decompose Ih into the following three terms that are similar to (3.8). X X X ? Ih ( ) = h njj0 Jj + h njj0 Hj + h nj j0 Fj + 21 0qj
j;n j;n j;n X ? 2 2 n 00 + 21 h j j Fj + Hj + 2Fj Hj + Jj2 + 2Jj (Fj + Hj ) j;n 2X `?1 X (3.33) 1 njj(k) (Fj + Hj + Jj )k +h k! k=3 j;n X + h (21`)! nj (2`)( )(Fj + Hj + Jj )2` j;n = I1h ( ) + I2h ( ) + I3h ( ) : Notice that there is a remainder term in I3 with 2 [unj; unj +1]. This term can be estimated by using the fact that (2`) is uniformly bounded for the entropy
functions of form (3.32). We now estimate all these terms in the same order as those in Subsection 3.1 to keep consistency. We will skip some parts of the estimates, which are similar to the corresponding parts in the previous subsection. Estimate of I1h . Similar to the estimate (3.10), one can use the identity (3.34) + ( nj j0 ) = 21 ( nj + nj+1)+ j0 + 12 (j0 + j0 +1)+ nj
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1039
and the expression of Jj in (3.3) to get X n0 X j j Jj = ? 12 ~jn+1=2( nj + nj+1 )+ j0 j+ aj j+ uj j;n
? 12
j;n X ~n j;n
j +1=2(j0 + j0 +1)+ nj j+aj j+ uj :
Applying Lemma 3.1 to the rst term on the right-hand side and estimating the second term directly, one has X n0 X j j Jj ? 12 0 ( nj + nj+1 )(sj )j+ uj j3 j;n j;n (3.35) X + 1 C sj (sj )j+ uj j2j+ njj ; j;n
where sj and (s) are determined by (3.11). With this good term in our hands, we now estimate other terms. Notice that 2 2 X n0 X X j j Hj = ? 12 2 nj+ j0 (+ fuj ) ? 12 2 j0 +1+ nj (+ fuj ) : + j + j j;n j;n j;n
As in estimating (3.13)-(3.14), expanding + j0 and + fj in the rst term on the right-hand side of the above equality and estimating the resulting terms directly, we have X X X n0 j j Hj ? 21 2 njj00(+ fj )2 + "0 C nj(sj )j+ uj j3 j;n j;n j;n (3.36) X + "0 C sj (sj )j+ uj j2 j+ nj j : j;n
Using Lemma 3.4, one has
X n? 0 1 X ? j j Fj + 2 0 qj = ? 12 nj j0 0 fj ? 0qj j;n j;n Zu X n 00 2 00 j 2 0 1 = 4
j;n
? 14 ? 14
j j (+ uj ) + aj ? j
X
njaj + j00 (+ uj )2 ? 41
j;n X n j;n
j 000 (j )
Z uj uj
+1
+1
uj?1
X
(uj ? s) a (s) ds
j00+1aj (+ uj )2 + nj
j;n Z uj (uj ? s)2 a(s) ds : (uj ? s)2 a(s) ds + 000 (j ?1) uj?1
Each term on the right-hand side of the above equality is the sum of products of three di erences and, therefore, can be directly estimated by similar arguments as in (3.15)-(3.16). X n? 0 1 j j Fj + 2 0qj j;n (3.37) X C ( nj + nj+1 )(sj )j+ uj j3 + sj (sj )j+ uj j2j+ njj : j;n
1040
GUI-QIANG CHEN AND JIAN-GUO LIU
Using (3.35)-(3.37), we have I1h ( ) ? 21 0 h
X j;n
( nj + nj+1)(sj )j+ uj j3 ? 12 2 h
+ (1 + "0 + 1 )Ch
(3.38)
X j;n
X j;n
nj j00 (+ fj )2
( nj + nj+1)(sj )j+ uj j3
+ sj (sj )j+ uj j2j+ njj : Estimate of I2h .
Using the identity (3.19), one can decompose the rst term in I2h and estimate similar to that of (3.20) to obtain (3.39) X X n 00 2 1 2 X n 00 00 j (j + j +1 )(+ fj )2 ? 14 2 nj j00 (? + fj )2 j j Fj = 2 j;n
j;n j;n X 00 1 2 2 n + 2 j +1(+ fj ) + j j;n X X 2 nj aj2+1=2j00 (+ uj )2 ? 41 2 nj j00 (? + fj )2 j;n j;n X n n 3 2 n ( j + j +1 )(sj )j+ aj j + sj (sj )j+ uj j j+ j j : + "0 C j;n
Estimating the second term of I2h is similar to that of (3.21)-(3.22) by
f 2 X n 00 4 2 n 00 2 1 j j (? + fj ) +uj ?1 j j Hj 2 + j ?1 j;n j;n X + "03 C ( nj + nj+1)(sj )j+ uj j3 : j;n
X
(3.40)
The main idea in estimating the third term in I2h is to utilize the symmetric property. It involves some similar techniques as in estimating (3.23) and (3.24). We omit detailed calculations here and write down the resulting terms as follows: 2 X X n 00 j j Fj Hj = ? 41 3 nj j00 0fj ? (+ fuj ) + j j;n j;n X ?q 00 2 f + j n 3 1 j + fj 0 j u = ?8 + j j;n + fj ?q 2 ? + nj u + j00 + fj ( f )2 +j ? q q X + 41 3 j00 + uj ? + nj j00 + fj j;n
+ j
+ nj0fj (+ fuj ?1 ) + j ?1
2
q 00
? j :
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1041
Clearly the terms on the right-hand side of the above equality are the sums of three di erence products of unj and hence can be estimated directly.
X
njj00Fj Hj "02 C
X
( nj + nj+1)(sj )j+ uj j3
j;n X ? 2 + "0 C sj (sj )j+ uj j2 j+ njj + j? njj : j;n h The remainder three terms in I2 can be similarlyestimated as in the estimates (3.26) j;n
(3.41)
and (3.27).
(3.42)
X j;n
nj j00 Jj2 + 2
X
njj00Jj (Fj + Hj )
j;n X "0 ( 1 + 12 )C ( nj + nj+1 )(sj )j+ uj j3 : j;n
Therefore, combining (3.39)-(3.40) with (3.41)-(3.42), we have the following estimate for I2h : X X I2h ( ) 12 2 h nj j00 (+ fj )2 ? 81 2 h nj(1 ? 2"20)j00 (? + fj )2 j;n
(3.43)
j;n X 2 2 + C"0 (1 + "0 + "0 + + )h ( nj + nj+1)(sj )j+ uj j3 j;n X ? + "0 (1 + "0 )Ch sj (sj )j+ uj j2 j+ nj j + j? njj : j;n
Estimating I3h is straightforward as in that of (3.29) and (3.30). One can obtain (3.44)
4m ? X X h I3 ( ) Ch "0k?1 + "02k?1 + "0k?1 k ( nj + nj+1)(sj )j+ uj j3 : j;n k=3
Finally, we have the following estimate of Ih after combining (3.38) with (3.43)(3.44): X Ih ( ) ? 41 0 h ( nj + nj+1)(sj )j+ uj j3 j;n X 2 1 ? 8 (1 ? 2"02)h njj00 (?+ fj )2 j;n
+ (1 + "0 + 1 )Ch
+ C 1 + "0 1 +
X
j;n 4X m
k=2
?
sj (sj )j+ uj j2 j+ njj + j? nj j
X
"0k?1 1k h
j;n
( nj + nj+1)(sj )j+ uj j3 :
Choose 0 large and "0 1 small. Then we have X Ih ( ) ? 18 0 h ( nj + nj+1 )(sj )j+ uj j3 + 1 Ch
j;n
X j;n
?
sj (sj )j+ uj j2 j+ njj + j? njj :
1042
GUI-QIANG CHEN AND JIAN-GUO LIU
Now we deal with the last term in the above inequality. From the Holder inequality, one has h
X j;n
?
X
sj (sj )j+ uj j2 j+ njj + j? njj h1+ k kC
(xj ;tn )2
(sj )sj (+ u)2
12=3 11=3 0 0 X X (sj )j+ uj3hA : sj6`?9m+3 n hA @ Ch ?1=3k kC @ (xj ;tn )2
(xj ;tn )2
Let 6` ? 9m + 3 4m. Then X ? h sj (sj )j+uj j2 j+ njj + j? nj j Ch ?1=3k kC : j;n
Substituting it back to (3.33), we have the following proposition.
Proposition 3.2. Under the same assumptions of Theorem 3.1, (3.45) X j;n
?
nj (unj +1) ? (unj ) h + 21
X n? n j q(uj +1) ? q(unj?1) h Ch ?1=3k kC j;n
holds for any 2 C01, 0, where (; q) is the entropy pair given by (3:32).
3.3. H ?1 compactness of entropy dissipation measures. For the entropy pair (; q), we de ne the following functional as the entropy dissipation measures Mh ( ) =
(3.46)
ZZ?
(uh )@t + q(uh )@x dxdt ;
2 C01 ;
for the approximate solutions uh (x; t) de ned by (3.4). In this subsection, we will show that Mh is compact in H ?1 for the entropy pairs (u; f(u)) and (0; q0) with 0(u) = u2 . From Proposition 3.1, we can easily get from the Holder inequality that jMh ( )j ?1;q for q= 2m?1 >2. Lemma 2.2 C k kW ; m = m . Hence Mh is bounded in Wloc 4m ? 1 tells us that Mh is compact in Hloc as long as we can prove that Mh is compact in ?1;p for some p < 2. We de ne the following two functionals: Wloc 1 (2
(3.47)
+1) (4
Ih ( ) =
)
X n? n+1 X ? j (uj ) ? (unj ) h + 21 nj q(unj+1) ? q(unj?1) h ; j;n
j;n
and h ( ) = Ih ( ) + Mh ( ) for any strictly convex entropy (; q) and 2 C01. It is easy to show that (3.48) Z xj = ? X (x; tn+1) ? (xj ; tn) dx h ( ) = ? (jn+1 ? jn) +1 2
j;n
xj?1=2
Z tn ? X 2 (xj +1=2; t) ? (xj ; tn ) ? (xj +1 ; tn) dt : ? 21 (qjn+1 ? qjn ) +1
j;n
tn
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1043
Estimating directly and using the Holder inequality and Proposition 3.1, one obtains (3.49) X ? n+1 n 1+ n n jj ? j jh + jqj +1 ? qj j nh jh ( )j k kC (xj ;tn )2
h1+ k kC
X
2 3k X
(xj ;tn )2 k=1
Ch1+ k kC
X
(k)j?jF jk + jH jk + jJ jk + jq0 ( )jj unj j j j j j + j j k!
!
(sj )1=3j+ unjj
(xj ;tn )2
1 = 3 ? 2 = 3 Ch k kC ! 0 ; for 32 < < 1 : ?1;p1 for p1 < 2 < 2. We know from This implies that h is compact in Wloc 1+ Proposition 3.2 that Ih can be split as Ih ( ) = I1h ( ) + I2h ( ) such that (3.50) I1h 0 ; jI2h ( )j Ch ?1=3k kC : ?1;p2 for 1 < p < 2. The remainder Consequently, we have that I2h is compact in Wloc 2 ? 1 ;p h is to show that I1 is also compact in Wloc for some p < 2. Since I1h is negative due to (3.50), we know from Lemma 2.1 that we only need to show that I1h is bounded ?1;p for some p < 2. This is a direct consequence of (3.46), (3.49), and (3.50). in Wloc
Thus, we have the following proposition.
Proposition 3.3. Under the same assumptions of Theorem 3.1,
?1 @t uh + @x f(uh ) ; @t0 (uh ) + @x q0(uh ) are compact in Hloc where (0; q0) is the entropy pair given by (2:2). As a consequence of Theorem 2.1, Corollary 3.1, Proposition 3.2, and Proposition 3.3, we conclude Theorem 3.1 for the convex scalar conservation laws.
(3.51)
4. Extension to the nonconvex case
For the nonconvex scalar conservation laws, the second-order numerical viscosity term ? ? jn+1=2j+ f 0 (unj )j+ unj is degenerate at the points fu j f 00(u) = 0g. Typically such a degenerate point occurs near the contact discontinuities, which is quite sensitive to the stability of the discontinuities. To assume the stability, we consider the following second-order numerical viscosity in the Lax-Wendro schemes: ? ? jn+1=2 j+unj j+unj (4.1) ? with jn+1=2 2 0 + O(ja0j); 1 + O(ja0j) , 0 > 0, for the nonconvex scalar case. This covers both the convex case (the convexity implies 0 > 0) and the nonconvex case. With such a numerical viscosity we can use the same arguments as in Section 3, when C 1 and 0 1, to obtain the same viscosity estimate X (4.2) (sj )junj ? unj+1j3h C j;n
1044
GUI-QIANG CHEN AND JIAN-GUO LIU
and entropy estimate (4.3)
X j
(unj +1)
X j
(unj)
for 00 = u4m . Using these estimates, we can similarly verify that the corresponding
approximate solutions uh (x; t) satisfy the three conditions in Theorem 2.1 for any convex entropy pair. Therefore we have the following convergence theorem for the scalar conservation laws (1.1) whose ux functions satisfy (4.4) meas u j f 00(u) = 0 = 0; f (k) (u) O(jujm?k ) ; for juj 1; 0 k m : Theorem 4.1. Let uh(x; t) be numerical approximate solutions, generated from the Lax-Wendro scheme (1:2) endowed with the numerical viscosity (4:1), of the scalar conservation laws (1:1) satisfying (4:4). Assume that the coecient 0 in (4:1) is suitably large and the CFL number "0 given by (1:4) is suitably small. Then the sequence uh has a strong convergent subsequence and the limit function is the entropy solution of (1:1) satisfying the entropy inequalities @t (u) + @x q(u) 0 in D0 ; for any convex entropy pair (; q) satisfying (k) (u) O(jujr?k ), for juj 1, 0 k r < 4m. Proof. Only one thing we should check is that, for any such a C 2 entropy pair ?1. This can be achieved as follows. ((u); q(u)), @t(uh )+@xq(uh ) is compact in Hloc 2 For any C entropy pair (; q), there exists C0 > 0 such that j00j C0 j00j where is a strictly convex entropy. De ne = C0 ? . Then is a convex entropy. Noting that both @t (uh ) + @x q(uh ) and @t (uh ) + @x q (uh ) are compact sets in ?1, one obtains that @t (uh ) + @x q(uh ) is compact in H ?1. This completes the Hloc loc proof. Remark. The assumption (4.4) can be relaxed to more general ux functions for the Lax-Wendro scheme. The strong convergence becomes weak one in Lr , r < 4m, and the limit function is the weak solution satisfying the entropy condition for the Young measures x;t determined by the Lax-Wendro approximate solutions uh (x; t): @t hx;t; i + @x hx;t; qi 0; for any convex entropy pair (; q) in the above theorem. 5. Extension to hyperbolic systems
We write the Lax-Wendro scheme in the following form unj +1 = unj + Fjn + Hjn + Jjn : (5.1) Here ? un + u n Fjn = ? 12 0 f(unj ) ; Hjn = 21 2 ? a j 2 j +1 + f(unj ) ; (5.2) ? Jjn = ? jn+1=2j+ unj j+unj ; where the scalar function jn+1=2 0 > 0, for some 0 , is smooth with respect to unj and unj+1 and is bounded from below. We assume that (5.3) sup junjj M : j;n
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1045
In this section we show the convergence of such a scheme for 2 2 systems and the entropy consistency of the boundedly convergent Lax-Wendro approximate solutions with general hyperbolic systems of conservation laws with a convex entropy. This depends on the following estimates. 5.1. Dissipation estimates. We rst start with the entropy estimate. Let be a C 2 convex entropy function. As in the scalar case, we take the Taylor expansion to the time increment of the entropy to get X? n+1 X (uj ) ? (unj ) = 0 (unj )(unj +1 ? unj) j
j
+ 21
(5.4)
X
(unj +1 ? unj)> 00 (unj )(unj +1 ? unj)
Xj ? n+1 n 3 + O juj ? uj j I1 + I2 + I3 : j
We will keep the same order of the estimates as in Section 3 and will only emphasize the new features, which are di erent from the scalar case. The arguments similar to those in Section 3 will be omitted. We point out here that some of the estimates in this section are simpler than the ones in Section 3 since we assumed the uniform boundedness of the approximate solutions. Estimate of I1 . As in (3.11), we directly estimate that X 0> X X (j ) Jj = ? j +1=2 j+ uj j+ (j0 )> + uj ? 0 j+ uj j3 ; (5.5) j
j
j
using the expression of Jj in (5.2), the summation by parts, and the convexity of . This gives us the main dissipative term. Here we used the notation j +1=2 = (uj ; uj +1), which is a scalar function. We remark that (5.5) is the only place we restrict the viscosity (uj ; uj +1) to be a scalar function. As in estimating (3.13) and (3.14), we have X X 0> (j ) Hj = ? 12 2 (+ j0 )> aj +1=2+ fj j j (5.6) X X 2 1 = ? 2 (+ uj )> j00aj + fj + "0 O(j+ uj j3) : j
j
Notice that the necessary and sucient condition for a function to be an entropy is that 00 f 0 is symmetric, that is, j00 aj = (aj )> j00. One has from (5.6) X 0> X X (j ) Hj = ? 21 2 (+ fj )> j00 + fj + "0 O(j+ uj j3) : (5.7) j
j
j
In estimating the rst term of I1, we use the conservative property of entropy. Denote uj = 12 (uj ?1 + uj +1). Similar to (3.16)-(3.17), one can obtain (5.8) X X Z 1=2 X 0> a(uj + 0 uj ) d 0 uj = "0 O(j+ uj j3) : (j ) Fj = ? 12 (j0 )> j
?1=2
j
j
Combining (5.5) with (5.7)-(5.8), we have the following estimate of I1 . X X I1 ? 0 j+ uj j3 + "0 C j+ uj j3 ? 21 2 (+ fj )> j00 + fj : (5.9) j
j
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GUI-QIANG CHEN AND JIAN-GUO LIU
Estimate of I2. As in (3.19), we substitute (5.1)-(5.2) into I2 and have the follow-
ing expansion.
(5.10)
X
I2 = 21
+
F>j j00 Fj + 12
X
Hj> j00 Hj +
X
Fj>j00Hj
j j j 1 X J > 00 Jj + X J > 00 (Fj + Hj ) : j j j j 2 j j
The estimate of the rst term in (5.10) is similar to that in (3.20)-(3.22) by the identity (3.20). Putting all products of three di erences together in O(j+ uj j3 ), one has X X > 00 X Fj j Fj =2 (+ fj )> j00+ fj ? 14 2 (? + fj )> j00 ?+ fj j j (5.11) j X 3 + "0 O(j+ uj j ): j
The estimate of the second term in (5.10) is similar to that in (3.23)-(3.24). After putting all products of three di erences in O(j+ unjj), the remainder term is the sum of products of two second di erences, which can be controlled by the last term in (5.11). (5.12) X X > 00 Hj j Hj = 41 4 ? (aj +1=2+ fj )> j00 ?(aj +1=2+ fj ) j
j X 4
X
j
j
= 14
(? + fj )> a>j j00 aj ? + fj + "03
O(j+ uj j3) :
The main idea in estimating the third term in (5.10) is to utilize the symmetric property so that every term can be combined into the product of three di erences. For any symmetric matrix A, we have as in Lemma 3.1 that X X X (5.13) ? (0bj )> Abj = 21 b>j 0Abj ? 21 (+ bj )> + A+ bj : j
j
j
Similar to (3.25)-(3.27), we have X X > 00 Fj j Hj = ? 14 3 (0fj )> j00 ? (aj +1=2+ uj )
X Xj ?q 00 >q 00 3 1 0 j + fj j aj + uj + "02 O(j+ uj j3) : = 4 j j q 00 ?q 00 ?1 We know from (5.6) that j aj j is symmetric. Hence we can apply the j
identity (5.13) to rst term on the right-hand side of the above equality with q the 00 bj replaced by j + fj . As a result that every term now is decomposed as the product of three di erences, we obtain X > 00 X (5.14) Fj j Hj = "02 O(j+uj j3 ) : j
j
The estimate of the last three terms in (5.10) is rather straightforward since every term is the sum of products of more than three di erences. One has from a
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1047
direct estimate that X X X > 00 (5.15) Jj j Jj + 2 Jj> j00 (Fj + Hj ) "0 O(j+uj j3) : j
j
j
Finally, combining (5.11)-(5.12) with (5.14)-(5.15), we have X X I2 12 2 (+ fj )> j00 + fj ? 81 2 (?+ fj )> j00? + fj j j (5.16) X X > > 00 4 1 + 4 (? + fj ) aj j aj ?+ fj + "0 O(j+ uj j3) : j
j
Estimating I3 simply follows from the fact that each term is the sum of products of more than three di erence factors X X (5.17) jI3 j C (jFj j3 + jHj j3 + jJj j3) "02 C j+ uj j3 : j
j
Combining (5.9) with (5.16), we have
X
X
j
j
(jn+1 ? jn) ? 0
j+ uj j3 ? 18 2(1 ? "02 )
+ (1 + "0 + "02)C
X j
j+ uj j3 :
X j
(?+ fj )> j00? + fj
For 0 large and "0 small, we have X X X n+1 n (j ? j ) ?C1 j+ uj j3 ? 2 C2 j?+ fj j2 : This gives
j
j
j
Proposition 5.1. Let unj be the Lax-Wendro approximate solutions (5:1)-(5:2) of the hyperbolic systems of conservation laws (1:1) with a convex entropy . Assume that uh is uniformly bound and the coecient in (5:2) is suitably large and the CFL number "0 given by (1:3) is suitably small. Then
(5.18)
X
(unj )h +
X
kn;j
j+ ukj j3 h + 2
X
j? + f(ukj )j2h C :
kn;j 5.2. H ?1 compactness estimates. For any C 2 convex entropy pair (; q) any (x; t) 2 C01, 0, we denote nj = (xj ; tn) and X ? X ? Ih ( ) h nj (unj +1) ? (unj ) + 21 nj q(unj+1) ? q(unj?1) : j;n j;n j
As (3.37), we have from the Taylor expansion that X ? Ih ( ) = h nj (j0 )> Fj + 21 0 qj + (j0 )> Hj + (j0 )> Jj (5.19)
j;n X + 21 h nj(Fj + Hj + Jj )> j00 (Fj + Hj + Jj ) j;n X 1 + 2 h nj000()(Fj + Hj + Jj )3 I1h ( ) + I2h ( ) + I3h ( ) : j;n
and
1048
GUI-QIANG CHEN AND JIAN-GUO LIU
Estimate of I1h .
The estimate of the last term in I1h is similar to that in (3.38)
and (3.39). X X X n 0> (5.20) j (j ) Jj ? 12 0 ( nj + nj+1)j+ aj j3 + C j+ uj j2j+ njj : j;n
j;n
j;n
Similar to that in (3.41), one can obtain (5.21) X n 0> X X j (j ) Hj ? 21 2 nj(+ fj )> j00 + fj + "0 C ( nj + nj+1)j+ uj j3 j;n
j;n
+ "0 C
X j;n
j;n
j+ uj j2j+ njj :
The estimate of the rst term in I1h is similar to that in (3.40). One has
X n? 0 > 1 X ? j (j ) Fj + 2 0 qj = ? 12 nj (j0 )> 0fj ? 0qj j;n Xj;n n n X 3 C
The estimates (5.20)-(5.21) yield I1h ( ) ? 21 0 h
X
( nj + nj+1)j+ uj j3 ? 21 2 h
X
j;n
j+ uj j2j+ njj :
nj(+ fj )> j00 + fj
j;n X n n 3 + (1 + "0 )Ch ( j + j +1)j+ uj j + (1 + "0 )Ch j+ uj j2j+ njj : j;n j;n h h We can similarly estimate I2 and I3 . Finally we have
Ih ( ) ? 21 0 h
j;n
j;n
( j + j +1)j+ uj j + C
X
X
Xj;n
( nj + nj+1)j+ uj j3 ? 2 Ch
X
nj(? + fj )2
j;n X ? 2 n n + Ch j+ uj j j+ j j + j? j j + C"02 h ( nj + nj+1)j+ uj j3 : j;n j;n We choose 0 large and "0 small and then have X ? Ih ( ) Ch j+ uj j2 j+ njj + j? njj h ?1=3k kC : j;n
Therefore, we obtain
Proposition 5.2. Under the same assumptions of Proposition 5.1,
(5.22) X ? X n? n+1 j (uj ) ? (unj ) h + 21 nj q(unj+1) ? q(unj?1) h Ch ?1=3k kC ; j;n
j;n
for C 2 convex entropy pair (; q) and 2 C01 , 0.
?1 Similar to Section 3.3, we can show that @t (uh ) + @x q(uh ) is compact in Hloc for any convex entropy pair (; q). For a general entropy function we write it as a linear combination of two convex entropy functions as we have explained in Section 4. Hence we have
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
1049
Proposition 5.3. Under?1the same assumptions of Proposition 5.1, @t(uh ) +
@x q(uh ) is compact in Hloc for any C 2 entropy pair (; q). 5.3. Convergence and entropy consistency. Now we are concerned with the convergence of uniformly bounded approximate solutions of 2 2 systems of conservation laws endowed with global Riemann invariants and the entropy consistency of boundedly convergent Lax-Wendro numerical solutions of general hyperbolic systems of conservation laws with a convex entropy. One important example of the 2 2 systems is the elasticity equations (cf. [5]): @tv ? @x u = 0 ; @tu ? @x (v) = 0 ; where u is the speci c volume, v is the strain, and is the stress-strain relation satisfying 0 (v) > 0. The numerical experiments indicate that the Lax-Wendro approximate solutions (vh ; uh) are uniformly bounded. It is important from the viewpoint of numerical analysis whether uniformly bounded Lax-Wendro numerical solutions may generate some oscillations. The following theorem shows that, if (vh ; uh ) is uniformly bounded, then it indeed strongly converges to the corresponding entropy solution. The proof follows from Theorem 2.2, the estimates in Sections 5.1 and 5.2, and similar compactness arguments as in the scalar cases. Theorem 5.1. Consider a 2 2 genuinely nonlinear and strictly hyperbolic system of conservation laws (1:1) endowed with global Riemann invariants. Let uh (x; t) be the Lax-Wendro approximate solutions. Assume that uh is uniformly bounded and the coecient in (5:2) is suitably large and the CFL number "0 given by (1:3) is suitably small. Then there is a subsequence (still denoted as ) uh such that uh (x; t) ! u(x; t) a.e. as h ! 0 and u is the entropy solution of (1:1) satisfying @t(u) + @x q(u) 0 in D0 for any convex entropy (; q). For a general system, we conclude that the numerical solutions of the LaxWendro scheme, if they are boundedly convergent, then converge to the entropy solutions. A similar result has been obtained by Majda and Osher [15] for the semi-discrete case of time-dependent systems and for the fully discrete case of timeindependent systems.
Theorem 5.2. Consider a general system of genuinely nonlinear and strictly hy-
perbolic conservation laws (1:1) endowed with a convex entropy. Let uh (x; t) be the uniformly bounded Lax-Wendro approximate solutions converging boundedly a.e. to u(x; t). Assume that the coecient in (5:2) is suitably large and the CFL number "0 given by (1:3) is suitably small. Then u is the entropy solution of (1:1) satisfying @t (u) + @x q(u) 0 in D0 for any convex entropy pair (; q).
A typical example for such a system is the system of gas dynamics in Lagrangian coordinates: 0 v 1 0?u1 @t @ u A + @ x @ p A = 0 ; E pu where v > 0 denotes the speci c volume of the gas, u the velocity, p > 0 the pressure, E = 21 u2 + e the total speci c energy and e the speci c internal energy. The additional conservation law assumes the form @t (?S) 0 for physical solutions. For polytropic gases, ?S(v; u; E) = ? log(E ? 21 u2) ? ( ? 1) logv is a strictly convex function of the state variables v, u, and E for > 1, a xed constant,
1050
GUI-QIANG CHEN AND JIAN-GUO LIU 1
and p = ( ? 1) E ?v2 u = A(S)v . Theorem 5.2 indicates that uniformly bounded convergent Lax-Wendro approximation is consistent with this system. 2
6. The MacCormack scheme and Richtmyer scheme
An ecient implementation of the Lax-Wendro scheme is to use two step splitting methods. One particularly popular version is the following MacCormack scheme: u~nj = unj ? + f(unj ) ; ? (6.1) unj +1 = 12 (unj + u~nj) ? ? f(~unj ) + ? jn+1=2 j+f 0 (unj)j+ unj : The convergence of this scheme is very much similar to what we did for the LaxWendro scheme in Sections 3-5. We only state some convergence theorems and show those arguments that are di erent from ones of the previous sections. We rst rewrite the MacCormack scheme (6.1) as follows (6.2) unj +1 = unj + Fjn + Hjn + H~ jn + Jjn ; where Fjn = ? 21 n 0f(unj ) ; ? ( f(un))2 ; Hjn = 21 n2 ? + unj + j ? n n 1 ~ (6.3) Hj = 2 n ? f(uj ) ? f(~unj ) ? f 0 (unj )(unj ? u~nj ) ; ? Jjn = n? ~jn+1=2 j+a(unj )j+ unj ; a(un ) ? f = u ~jn+1=2 = jn+1=2 + n j j +a jj + j : + j Clearly, we can carry out the analysis without any di erence in the absence of the term H~ jn. We only point out that H~ jn can only produce third-order terms and can be controlled by the viscosity term based on the following facts. First, we notice that u~n in (6.1) is always in the interval [unj; unj+1] provided the CFL number is less than 1=2. Using the Taylor expansion and (6.1), one has (6.4) f(unj ) ? f(~unj ) ? f 0 (unj)(unj ? u~nj) = 21 f 00 (jn )(unj ? u~nj )2 = 12 2 f 00 (jn )(+ f(unj ))4: Substituting (6.4) into H~ j and using the summation by parts, one has (6.5) X X 0 n ~ n 1 3 X 0 00 n (uj )Hj = ? 4 + j f (j )(+ f(unj ))4 "02 C (snj )j+ unjj3 ; j
X (6.6)
j
j
j
00(unj )(H~ jn ) 2 = 14 6
X
2
? 00 (unj) ? a0(jn )(+ f(unj ))2
j X 5 "0 C (snj )j+ unjj3 ; j
CONVERGENCE OF DIFFERENCE SCHEMES WITH HIGH RESOLUTION
and
X
(6.7)
j
00 (unj )FjnH~ jn = ? 18 4
"03 C
X j X j
1051
? 00(unj )0f(unj )? a0 (jn )(+ f(unj ))2 (snj )j+ unjj3 :
Now we treat another two step version of the Lax-Wendro scheme|the Richtmyer scheme, u~nj+1=2 = 12 (unj + unj+1) ? 12 + f(unj ) ; (6.8) ? unj +1 = unj ? ?f(~unj+1=2 ) + ? jn+1=2j+ f 0 (unj)j+ unj : We can rewrite it into the form unj +1 = unj + Fjn + Hjn + H~ jn + Jjn ; (6.9) where (6.10) Fjn = ? 12 n0 f(unj ) ; ? ( f n)2 Hjn = 12 n2 ? + ujn ; + j ? n n 1 ~ Hj = 2 n ? f(uj ) + f(unj+1 ) ? 2f(~unj+1=2 ) ? f 0 (unj+1=2)(unj + u2j +1 ? 2~unj) ; ? an ? f n = un ~jn+1=2 + n j j+ uj nj + j : Jjn = n ? ~jn+1=2 j+ a(unj)j+ unj ; + j It is also easy to verify that H~ jn can produce only third-order terms and can be controlled by the viscosity term. Therefore we have the following theorems corresponding to those of the LaxWendro scheme in Sections 3-5 under the same assumptions. Theorem 6.1. Let uh(x; t) be approximate solutions of the convex scalar conservation laws (1:1) and (3:1) by the the MacCormack scheme (6:1) or the Richtmyer scheme (6:8). Then there is a subsequence strongly converging to the weak solution of (1:1) satisfying the entropy inequalities. Theorem 6.2. Let uh(x; t) be approximate solutions of the scalar conservation laws (1:1) or (4:4) by the MacCormack scheme (6:1) or by the Richtmyer scheme (6:8) with the arti cial viscosity term (4:1). Then there is a subsequence strongly converging to the weak solution of (1:1) satisfying the entropy inequalities. Theorem 6.3. Consider a general 2 2 genuinely nonlinear and strictly hyperbolic system of conservation laws (1:1) endowed with global Riemann invariants. Let uh (x; t) be the uniformly bounded approximate solutions of the MacCormack scheme (6:1) or the Richtmyer scheme (6:8). Then there is a subsequence (still denoted as ) uh such that uh (x; t) ! u(x; t) a.e. as h ! 0 and u is the entropy solution of (1:1) satisfying @t (u) + @x q(u) 0 in D0 for any C 2 convex entropy pair (; q). Theorem 6.4. Consider a general system of hyperbolic conservation laws (1:1) endowed with a convex entropy. Let uh (x; t) be the approximate solutions of the MacCormack scheme (6:1) or the Richtmyer scheme (6:8). If uh converges boundedly
1052
GUI-QIANG CHEN AND JIAN-GUO LIU
a.e. u(x; t), then u(x; t) is the entropy solution of (1:1) satisfying @t(u)+@x q(u) 0 in D0 for any C 2 convex entropy pair (; q). Acknowledgments
The authors would like to thank Professor P. D. Lax for his encouragements and Professors J. Goodman, T.-P. Liu, A. Majda, S. Osher, and E. Tadmor for their interest in this work. G.-Q. Chen was supported in part by ONR grant N0001491-J-1384, NSF grant DMS-9623203, and Alfred P. Sloan Foundation Fellowship. J.-G. Liu was supported in part by NSF Grant DMS-9114456, DOE grant DEFG02-88ER25053 and ARO Grant DAAL03-92-G-0143. This work was initiated while the authors visited Stanford University and was supported in part through Army grant DAAL 03-91-G-0017. References 1. J.P. Boris and D.L. Book, Flux corrected transport. I. SHASTA, a uid transport algorithm that works, J. Comp. Phys. 11 (1973), 38-69. 2. G.-Q. Chen, The compensated compactness method and the system of isentropic gas dynamics, MSRI Preprint 00527-91, Berkeley (1990). 3. G.-Q. Chen and Y.-G. Lu, A study on application approaches of the theory of compensated compactness, Chinese Science Bulletin 34 (1989), 15{19. MR 90h:35024 4. F. Coquel and P. Le Floch, Convergence of nite di erence schemes for conservation laws in several space variables: the corrected antidi usive ux approach, Math. Comp. 57 (1991), 169-210. MR 91m:65229 5. R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. 82 (1983), 27{70. MR 84k:35091 6. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high order accurate nonoscillatory schemes, III, J. Comp. Phys. 71 (1987), 231{303. MR 90a:65199 7. A. Harten, J.M. Hyman, and P.D. Lax, On nite-di erence approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297{322. MR 54:1640 8. A. Harten, P. D. Lax, and B. van Leer, On upstream di erencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review 25 (1983), 35. MR 85h:65188 9. S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Preprint (1993). 10. P.D. Lax and C.D. Levermore and S. Venakides, The generation and propagation of oscillations in dispersive IVPs and their limiting behavior, Importantdevelopments in soliton theory 1980{1990, T. Fokas and V.E. Zakharov eds, Springer-Verlag, Berlin, 1992. MR 95c:35245 11. P.D. Lax and B. Wendro , Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217{237. MR 22:11523 12. P.L. Lions and P. Souganidis, Convergence of MUSCL type methods for scalar conservation laws, C. R. Acad. Sci. Paris, Serie I 311 (1990), 259{264. MR 91i:65168 13. R.W. MacCormack, The e ect of viscosity in hyperbolicity impact cratering, AIAA Paper (1969), 69{354. 14. A. Majda and S. Osher, A systematic approach for correcting nonlinear instability: the Lax-Wendro scheme for scalar conservation laws, Num. Math. 30 (1978), 429{452. MR 80g:65101 15. A. Majda and S. Osher, Numerical viscosity and the entropy condition, Comm. Pure Appl. Math. 32 (1979), 797{838. MR 80j:65031 16. C.S. Morawetz, An alternative proof of DiPerna's theorem, Comm. Pure Appl. Math. 44 (1991), 1081{1090. MR 92m:35165 17. F. Murat, L'injection du cone positif de H ?1 dans W ?1 est compacte pour tout q < 2, J. Math. Pures Appl. 60 (1981), 309-322. MR 83b:46045 18. H. Nessyahu and E. Tadmor, Non-oscillatory central di erencing for hyperbolic conservation laws, J. Comp. Phys. 87 (1990), 408{463. MR 91i:65157 19. S. Osher, On convergence of generalized MUSCL schemes, SIAM J. Numer. Anal. 22 (1985), 947-961. MR 87b:65147 ;q
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Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
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