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\centerline{\bf Convergence of Difference Schemes with High Resolution}
\centerline{\bf  For Conservation Laws}   	   
\footnote[]{1980 {\it Mathematics Subject Classification\/} 
         (1985 {\it Revision\/}). Primary 65M12; Secondary 35L65
         \newline\indent
         {\it Key words and phrases.\/} 
     Conservation laws, convergence, entropy solution, Lax-Wendroff scheme}
\bigskip
\centerline {Gui-Qiang \, Chen}
\smallskip
\centerline {\it Department of Mathematics, Northwestern University}
\centerline {\it Evanston, Illinois \rm 60208}
\bigskip
\centerline {Jian-Guo \, Liu}
\smallskip
\centerline {\it Department of Mathematics, Temple University}
\centerline {\it Philadelphia, \rm PA 19122}
\bigskip
\bigskip
\bigskip

\noindent
{\bf Descriptive Title:} 
 Convergence of Difference Schemes with High Resolution

\bigskip
\noindent {\bf Abstract} \quad We are concerned with the 
convergence of Lax-Wendroff type schemes with high resolution to the
entropy solutions for conservation laws.  These schemes include
the original Lax-Wendroff scheme proposed by Lax and Wendroff in~1960
and its two step versions--the Richtmyer scheme and the MacCormack
scheme.  For the convex scalar conservation laws with algebraic growth
flux functions, we prove the convergence of these schemes to the weak
solutions satisfying appropriate entropy inequalities.  The proof is
based on detailed $L^p$ 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.

\bigskip
\bigskip
\centerline {\bf 1. Introduction}
\smallskip

We are interested in the convergence of finite difference schemes
with high resolution for conservation laws
$$
  \del_t u + \del_x f(u) = 0 \,, \qquad
  u(x,0) = u_0(x) \in L^2(\RR)\cap L^\infty(\RR) \,.
  \tag 1.1
$$
One of the most fundamental and important second-order finite
difference schemes is the Lax-Wendroff scheme~[11]. It is defined by
$$
  u^{n+1}_j = u^n_j - \tfrac{\lambda_n}{2} \Delta_{0} f(u^n_j)
  + \tfrac{\lambda_n^{\,2}}{2} \Delta_{-} 
    \biggl( f'\bigl( \frac{u^n_j+u^n_{j+1}}2 \bigr) 
            \Delta_{+} f(u^n_j) \biggr)
  + \lambda_n \Delta_{-}\bigl(\beta_{j+1/2}^n |\Delta_{+}f'(u^n_j)| 
    \Delta_{+} u^n_j \bigr) \,,
  \tag 1.2
$$
where $u_j^n = u(x_j, t_n)$, $t_n=\sum_{k=1}^n \tau_k$, $x_j=jh$,
$\lambda_n = \tau_n/h$, $\Delta_{+} u_j= u_{j+1}-u_j$, 
$\Delta_0 u_j=u_{j+1}-u_{j-1}$, $\Delta_{-}u_j= u_j-u_{j-1}$, and
$\beta_{j+1/2}^n, 0\le\beta_0\le\beta_{j+1/2}^n\le\beta_1< +\infty$, 
is a smooth function of $u_j^n$ and $u_{j+1}^n$ 
(e.g.  $\beta_{j+1/2}^n = $ const. in [11]).  Moreover, this scheme 
has to satisfy the Courant-Friedrichs-Lewy condition
$$
  \eps_0 = \max_n \bigl( \lambda_n \max_j |f'(u_j^n)| \bigr) \le 1
  \tag 1.3
$$
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 regions of solutions, and
stabilization for the convex case for $\beta_0>0$ and $\eps_0\le 1$.  
When $\beta_{j+1/2}^n=0$, the Lax-Wendroff 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 $\beta_{j+1/2}^n\ge \beta_0 > 0$ 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 $|\Delta_+f'(u_j^n)|$ in the third term of the scheme (1.3) is
to reduce effectively 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-Wendroff approximate solutions to the entropy solutions and to
provide an analytical approach for such a convergence analysis for
high-order finite difference schemes, which do not preserve BV (and even
$L^\infty$ bound).  This is motivated by the fact that any
high-resolution difference 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 $\beta_0$ is
suitably large and $\epsilon_0$ 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 $\beta_0$ and $\epsilon_0$ from
this approach for some concrete equations.  Our analysis is based on careful
$L^p$ 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 finite
difference 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 $L^p$ 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-Wendroff approximate
solutions to the convex scalar conservation laws.  We obtain 
the uniform $L^p$
estimates of the approximate solutions and the $H^{-1}$ compactness
estimates of the corresponding entropy dissipation measures by
carefully analyzing 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 flux functions.  This is because we do not require the
$L^\infty$ bound or the BV bound for the Lax-Wendroff approximate
solutions to achieve our goal.  One difficulty in the analysis is the
fact that the dissipation is third-order in the Lax-Wendroff scheme,
comparing with the second-order dissipation in the first-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 differences
with a favorable sign.  Moreover, the factor in each combined
third-order difference needs to be bounded by the growth factor in the
dissipative term in each cell.  Some special treatment is also made since
the Lax-Wendroff scheme is a three-point scheme.  A requirement of
small CFL number is made here to use the grid ratio $\lambda$ to
control some growth factors. 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-Wendroff 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-Wendroff schemes in
Section~6, which are widely used in industry and engineering.

  In connection with earlier work on the Lax-Wendroff type schemes, we
recall that Majda and Osher~[14,15] showed the $L^2$-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 time-independent
cases of general systems endowed with a convex entropy, the efficient
choices of artificial 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],
~[32], and the references cited therein for the existence and
stability of the local discrete shock profile for the Lax-Wendroff
scheme.

  Regarding work on the convergence analysis of full discrete
high-resolution finite difference schemes, we refer to [4] and the
references cited therein for the flux-limit schemes with slope
modification or antidiffusive flux approach, which preserve $L^\infty$
bound.  Since the Lax-Wendroff type schemes do not have a TVD property,
our analysis consists of two steps: One is to prove the convergence,
which is an essential difficulty 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]. 


\bigskip
\bigskip

\centerline {\bf 2. Compactness Frameworks}
\smallskip

  In this section we discuss some compactness frameworks for the
approximate solutions for subsequent developments.  A pair of
functions $\bigl(\eta(u), q(u)\bigr)$ is called an entropy-entropy
flux pair if they satisfy $q'(u)=\eta'(u)f'(u)\,$. 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
$\eta_0(u)=\frac 12 u^2$ for the convex case and $\eta_0(u)=f(u)$ for
the general case.

\proclaim{Theorem 2.1} Consider the scalar conservation laws~$(1.1)$ 
satisfying $\text{meas}\bigl\{u \, : \, f''(u) = 0 \bigr\} = 0$.
Let $u_h(x,t)$ be numerical approximate solutions of $(1.1)$ satisfying 
the following conditions:
\roster
\item $u_h$ is bounded in $L^p$ for some $p \ge 2$ and 
$\bigl(f(u_h),\eta_0(u_h),q_0(u_h)\bigr)$ is bounded in $L^2_{\text{loc}}$; 

\item The dissipation measures $\del_t u_h + \del_x f(u_h)$ and 
$\del_t \eta_0(u_h) + \del_x q_0(u_h)$ are compact in $H^{-1}_{\text{loc}}$;

\item For any $C^2$ convex entropy pair $\bigl(\eta(u), q(u)\bigr)$, 
$\del_t \eta(u_h) + \del_x q(u_h) \le o(1)$ in ${\Cal D}'$, provided that 
$|\eta(u)|+|q(u)|\le M(1+|u|^r), r<p$.
\endroster
Then there is a subsequence\/ $($still denoted as\/$)$ $u_h$ such that
$u_h(x,t) \to u(x,t) \quad\text{a.e.} \quad\text{as } h \to 0$ 
and $u$ is the entropy solution of $(1.1)$ satisfying 
$\del_t \eta(u)+\del_xq(u)\le 0 \quad\text{in }{\Cal D}' $ 
for any $C^2$ convex entropy pair $(\eta,q)$.
\endproclaim

  We remark that, for general conservation laws, the conditions~(1)
and~(2) imply that 
$\text{w-}\lim f(u_h)=f(\text{w-}\lim u_h)\quad\text{in } L^2$ 
and the condition~(3) implies the following entropy
condition for the Young measures $\nu_{x,t}$, determined by the
Lax-Wendroff approximate solutions $u_h(x,t)$,
$$ 
\del_t <\nu_{x,t},\eta>+ \del_x <\nu_{x,t}, q> \le 0,
$$
in the sense of distributions.
We refer the reader to Chen-Lu~[3]
for a detailed discussion. The proof of the $L^\infty$ version of
Theorem~2.1 can be found in~[2,3,28] using the div-curl lemma of
Tartar and Murat~[28].

  For a $2\times 2$ system of conservation laws endowed with
global Riemann invariants, we have the following similar framework.

\proclaim{Theorem 2.2} Let $u_h(x,t)$ be numerical approximate 
solutions of the $2\times 2$ genuinely nonlinear and strictly hyperbolic
system of conservation laws~$(1.1)$ satisfying the following
conditions:
\roster
\item $u_h$ is bounded in $L^\infty$;

\item For any $C^2$ entropy pair $(\eta,q)$, $\del_t \eta(u_h) + \del_x q(u_h)$
is compact in $H^{-1}_{\text{loc}}$;

\item For any $C^2$ convex entropy pair $(\eta,q)$,
$\del_t \eta(u_h) + \del_x q(u_h) \le o(1)$ in ${\Cal D}'$. 
\endroster
Then there is a subsequence\/ $($still denoted as\/$)$ $u_h$ such that
$u_h(x,t) \to u(x,t) \quad\text{a.e.} \quad\text{as } h \to 0$ and
$u$ is the entropy solution of  $(1.1)$ satisfying 
$\del_t\eta(u)+\del_xq(u)\le 0\quad \text{in } {\Cal D}'$ 
for any $C^2$ convex entropy $(\eta,q)$.
\endproclaim

  This theorem is proved by DiPerna~[5] by estimating the entropy
dissipation measures and by using the Lax entropy pairs and the
compensated compactness methods. 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.

\proclaim{Lemma 2.1} The embedding of the positive cone of 
$W^{-1,p}$ in $W^{-1,q}$ is completely continuous for all $q < p$.
\endproclaim

\proclaim{Lemma 2.2} Let $\Omega \subset R^n$ be a bounded open set.
Then
$$ 
  \bigl(\text{compact set of } W^{-1,q} (\Omega) \bigr) \cap 
  \bigl(\text{bounded set of } W^{-1,r} (\Omega) \bigr)
  \subset \bigl(\text{compact set of } H_{\text{loc}}^{-1} (\Omega)\bigr)\,,  
$$
where $q$ and $r$ are constants, $1 < q \le 2 < r < \infty$. 
\endproclaim

\bigskip
\bigskip
\centerline {\bf 3. Convex Scalar Conservation Laws}
\smallskip

  In this section we are concerned with the Lax-Wendroff approximate
solutions of the scalar conservation laws (1.1) with flux function $f(u)$
satisfying
$$
  f''(u) \ge c_0 > 0 \,, \quad
  f^{(k)}(u) \sim O(|u|^{m-k}) \,, \quad 
  \text{for } |u| \gg 1, \ k=0,1,2 \,.
  \tag 3.1
$$
For convenience, we write the Lax-Wendroff scheme~(1.2) into the form
$$
  u^{n+1}_j = u^n_j + F^n_j + H^n_j + J^n_j \,,
  \tag 3.2
$$
where
$$
  F^n_j = - \tfrac12 \lambda_n \Delta_{0} f(u^n_j) \,, \quad
  H^n_j = \tfrac12 \lambda_n^{\,2} \Delta_{-}
  \biggl(\frac{\bigl(\Delta_{+}f(u^n_j)\bigr)^2}{\Delta_{+}u^n_j}\biggr)
  \,, \quad
  J^n_j = \lambda_n \Delta_{-}\bigl( \tilde \beta_{j+1/2}^n 
  |\Delta_{+} a(u^n_j)| \Delta_{+} u^n_j \bigr) \,,
  \tag 3.3
$$
with 
$$
  \tilde \beta_{j+1/2}^n = \beta_{j+1/2}^n + \lambda_n 
  \frac{a\bigl(\frac{u_j^n+u_{j+1}^n}{2}\bigr)
        -\Delta_{+}f(u_j^n)/\Delta_{+} u_j^n}{|\Delta_{+} a(u_j^n)|}
  = \beta_{j+1/2}^n + O(\lambda_n |a'|).
$$
\ From $u_j^n$, we construct the Lax-Wendroff approximate
solutions on $\RR^2_+$:
$$
  u_h(x,t)\equiv u_j^n, \quad \text{for } (x,t)
  \in [x_{j-1/2}, x_{j+1/2})\times [t_n, t_{n+1}) \,.
  \tag 3.4
$$
Then we have the following main convergence theorem of this section.

\proclaim{Theorem 3.1} Let $u_h(x,t)$ be the Lax-Wendroff approximate 
solutions~$(3.4)$ of the scalar conservation laws~$(1.1)$ and $(3.1)$. 
Assume that the coefficient $\beta_0>0$ is suitably large and the CFL number
$\eps_0$ 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 
$\del_t\eta(u) +\del_x q(u)\le 0 \quad \text{in } {\Cal D}'$ 
for $C^2$ convex entropy pair $(\eta,q)$ satisfying 
$\eta^{(k)}(u) \sim O(|u|^{r-k})$, for $|u| \gg 1$, $0\le k \le r < 4m$.
\endproclaim

  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
$$
  f_j = f(u^n_j) \,, \quad
  a_j = a(u^n_j) \,, \quad
  a_{j+1/2} = a\bigl(\frac{u^n_j+u^n_{j+1}}2 \bigr)\,, \quad
  \eta_j = \eta(u^n_j) \,, \quad
  \eta_j' = \eta'(u^n_j) \,, \quad
  \eta_j'' = \eta''(u^n_j) \,,
$$
where $\eta$ is an entropy function
of~(1.1).  We first introduce the following three technical lemmas.

\proclaim{Lemma 3.1} 
If $g'(u)\ge c_0>0$ and 
$g^{(k)}(u)=c_ku^{m-k}\bigl(1+o(1)\bigr)\,,|u|\gg 1\,,$ 
for some constants $c_k, k=0,1,2$. 
Then there is $\bar u\gg 1$ and $\alpha>0$ such that
$$
\frac{g(u)-g(v)}{u-v} \ge \Phi(\max(|u|,|v|)) \,,
\quad \text{with}\quad
\Phi(s) = \cases \alpha g'(s)\,, & s \ge \bar u \,, \\ 
                            c_0\,, & s < \bar u \,.\endcases
$$
\endproclaim
%   The proof is straight forward and hence is omitted.

Summing by parts, one has

\proclaim{Lemma 3.2} If $\{a_j\}$ and $\{b_j\}$ are two arbitrary 
sequences, then
$$
  \Delta_{+} (a_j \Delta_{0} b_j) =
  \Delta_{0} (a_j \Delta_{+} b_j) 
  + \Delta_{-} (\Delta_{+} a_j \Delta_{+} b_j)
  \tag 3.5
$$
and
$$
  2 \sum_j a_j b_j \Delta_{0} b_j
  = \sum_j (\Delta_{+} b_j)^2 \Delta_{+} a_j
  - \sum_j b_j^{\,2} \Delta_{0} a_j 
  \tag 3.6
$$
provided that the above sums make sense.  
\endproclaim

\demo{Proof} The formula~(3.6) comes from a direct computation.
Notice that
$$
\Delta_{0} (a_j b_j) = a_j \Delta_{0} b_j +  b_j \Delta_{0} a_j
+ \Delta_{-} (\Delta_{+} a_j \Delta_{+} b_j) \,.
$$ 
Multiplying both sides of the above equation by $b_j$, taking sum for
$j$, and summing by parts, one arrives at~(3.7).
\enddemo

  Using the relation $\eta'f'=q'$, expanding $\eta$ at $u_j$ in the
intervals $[u_{j-1},u_j]$ and $[u_j,u_{j+1}]$, and using the
integration by parts, one has

\proclaim{Lemma 3.3} If $(\eta, q)$ is an entropy pair of~$(1.1)$,
then 
$$
\aligned
  \eta'_j\Delta_0 f_j 
  & = \Delta_{0} q_j
  - \tfrac12 \eta''_j (\Delta_{+} u_j)^2\Delta_{+}a_j
  - \tfrac12 \eta''_j \Delta_{-}\bigl((\Delta_{+} u_j)^2 a_j\bigr)
  + \tfrac12 \eta''_j \int_{u_{j-1}}^{u_{j+1}} (u_j-s)^2 a'(s)\,ds
\\
  & \quad
  + \tfrac12 \eta'''(\xi_j) \int_{u_j}^{u_{j+1}} (u_j-s)^2 a(s)\,ds
  + \tfrac12 \eta'''(\xi_{j-1}) \int_{u_{j-1}}^{u_j} (u_j-s)^2 a(s)\,ds \,,
\endaligned
$$
where $\xi_{j-1}$ and $\xi_j$ are some values in $[u_{j-1}, u_j]$ and
$[u_j, u_{j+1}]$, respectively.
\endproclaim

\demo{Proof} Using the relation $\eta'f'=q'$ and expanding 
$\eta$ at $u_j$ in the intervals $[u_{j-1},u_j]$ and $[u_j,u_{j+1}]$,
one has
$$
\align
  \eta'_j\Delta_0 f_j 
  & = \Delta_{0} q_j
  + \eta''_j \int_{u_{j-1}}^{u_{j+1}} (u_j-s) a(s)\,ds \\
  & \quad
  + \tfrac12 \eta'''(\xi_j) \int_{u_{j}}^{u_{j+1}} (u_j-s)^2 a(s)\,ds
  + \tfrac12 \eta'''(\xi_{j-1}) \int_{u_{j-1}}^{u_{j}} (u_j-s)^2 a(s)\,ds,
\endalign
$$
for some $\xi_j\in [u_j, u_{j+1}]$ and $\xi_{j-1}\in [u_{j-1}, u_j]$.
Now using the integration by parts to the second term on the right
hand side of the above equation, we obtain the lemma.
\enddemo

  We remark here that we split the remainders in Lemma~3.3 into two
cells $[u_{j-1}, u_j]$ and $[u_j, u_{j+1}]$. A basic reason for this
is that we do not require the $L^\infty$ 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.

\bigskip
\noindent
{\bf 3.1. $L^p$ Estimate}
\smallskip

  We now verify the first condition of Theorem~2.1 for the Lax-Wendroff
scheme~(3.2)-(3.3). Since the flux function has the algebraic growth
rate $m$, we only need to show that $u_h$ is bounded in $L^2$ and
$L^{4m-2}$. The method used here is to estimate the entropy function
$\eta$ 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
$$
  \eta(u) = \frac1{4m(4m-1)}u^{4m} + \tfrac12 u^2
  \tag 3.7
$$
and estimate the bound of $\sum\eta(u^n_j)$ for the Lax-Wendroff
scheme $u^n_j$.

  We first expand the time increment of $\sum\eta(u^n_j)$ and
split it into three terms:
$$
\aligned
  \sum_j \bigl(\eta(u^{n+1}_j)-\eta(u^n_j)\bigr) 
  & = \sum_j \eta'(u^n_j) (u^{n+1}_j-u^n_j)
      + \tfrac12 \sum_j \eta''(u^n_j) (u^{n+1}_j-u^n_j)^2 
\\
  & \quad + \sum_{k=3}^{4m}\tfrac1{k!} 
    \sum_j \eta^{(k)}(u^n_j)(u^{n+1}_j-u^n_j)^{k} 
  \equiv I_1 + I_2 + I_3,
\endaligned
\tag 3.8
$$
which will be estimated in this subsection,
where we used the exact expansion to the highest order to avoid
the remainder in $[u^n_j,u^{n+1}_j]$, which is difficult to control.

\smallskip
\noindent
{\it Estimate of $I_1$.} \quad Plugging~(3.2) into $I_1$, one has
$$
  I_1 = \sum_j \eta'_j J_j + \sum_j \eta'_j H_j + \sum_j \eta'_j F_j \,.
\tag 3.9
$$
To estimate the first term, one uses the expression of $J_j$ in~(3.3), 
the summation by parts, and Lemma~3.1 to get
$$
  \sum_j \eta_j' J_j  
  = - \lambda \sum_j \tilde\beta_{j+1/2}^n \Delta_{+} \eta_j' 
        |\Delta_{+} a_j| \Delta_{+} u_j
  \le - \lambda \beta_0 \sum_j \Phi(s_j) |\Delta_{+} u_j|^3 \,,
  \tag 3.10
$$
where 
$$
  s_j = \max( |u_j|, |u_{j+1}|),
  \quad \qquad
  \Phi(s) = \cases c s^{5m-4}, & s \ge \bar u \,, \\ 
                            c_0, & s < \bar u \,,\endcases
  \tag 3.11
$$
for some constant $c>0$. Note that~(3.10) is the sum of
products of three first-order differences due to the fact that the
viscosity term is of 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 $H_j$ in~(3.3) and substitutes the expansion
$$
  \Delta_{+} \eta'_j 
  = \eta''_j \Delta_{+} u_j + \tfrac12 \eta'''(\xi_j) (\Delta_{+} u_j)^2 \,,
  \tag 3.12
$$
in the resulting terms to get
$$
  \sum_j \eta'_j H_j  
  = - \tfrac12 \lambda^2 \sum_j \frac{(\Delta_{+}f_j)^2}{\Delta_{+}u_j}
   \Delta_{+} \eta'_j 
   = - \tfrac12 \lambda^2 \sum_j \eta''_j (\Delta_{+} f_j)^2 
  -\tfrac14\lambda^2 \sum_j \eta'''(\xi_j)(\Delta_{+} f_j)^2\Delta_{+}u_j.
  \tag 3.13
$$
We use the term of right hand side in~(3.10) to control the second
term in~(3.13). Notice that there is a factor $(\Delta_{+} u_j)^3$ in
the second term that can be clearly controlled by the term of right hand side
in~(3.10). The factor in front of the difference 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 $\lambda^2$ in front of this
factor, therefore, we can use the fact 
$\lambda|\Delta_{+}f_j/\Delta_{+}u_j|\le\eps_0$ to control certain 
powers of the growth rate to obtain
$$
  \sum_j \eta'_j H_j  
  \le - \tfrac12 \lambda^2 \sum_j \eta''_j (\Delta_{+} f_j)^2 
  + \lambda \eps_0 C \sum_j \Phi(s_j) |\Delta_{+} u_j|^3 \,, 
  \tag 3.14
$$
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 $\xi_j$ and $\bar\xi_j$ as
some values in $[u_j, u_{j+1}]$, and $C$ is a positive constant,
independent of the grid size $h$, the CFL number, the viscosity
constant $\beta$, and the numerical solution $u_h$.  In different
contexts, they may have different values.

  Now we estimate the term $\sum_j\eta'_j F_j$. Writing the expression 
of $F_j$ in~(3.3) in the integral form and using the conservative 
property of entropy, one has
$$
\aligned
  \sum_j \eta'_j F_j  
  & = - \tfrac12 \lambda \sum_j \eta'(u_j) \int_{u_{j-1}}^{u_{j+1}} a(s)\,ds
  = \tfrac12 \lambda \sum_j \int_{u_{j-1}}^{u_{j+1}} 
   \bigl(\eta'(s)-\eta'(u_j)\bigr) a(s)\,ds \,
\\
  & = \tfrac12 \lambda \sum_j \eta''(u_j)
   \int_{u_{j-1}}^{u_{j+1}} (s-u_j) a(s)\,ds
    + \tfrac14 \lambda \sum_j \eta'''(\xi_{j-1})
      \int_{u_{j-1}}^{u_j} (s-u_j)^2 a(s)\,ds 
\\
  & \quad
    + \tfrac14 \lambda \sum_j \eta'''(\xi_j)
      \int_{u_j}^{u_{j+1}} (s-u_j)^2 a(s)\,ds \,.
\endaligned
  \tag 3.15
$$
Applying the following integration by parts to the first term in the
right hand side of the above equality
$$
  \int_{u_{j-1}}^{u_{j+1}} (s-u_j) a(s)\,ds
  = \tfrac12 (\Delta_{+} u_j)^2 \Delta_{+} a_j
  - \tfrac12 \int_{u_{j-1}}^{u_{j+1}} (s-u_j)^2 a'(s)\,ds \,,
$$
and plugging the above two estimates back into~(3.15), we obtain
$$
  \sum_j \eta'_j F_j  
  \le \lambda C \sum_j \Phi(s_j) |\Delta_{+} u_j|^3 \,.
  \tag 3.16
$$

  Finally, one uses~(3.10),~(3.14), and~(3.16) to get the following
estimate:
$$
  I_1 \le - \lambda \beta_0 \sum_j \Phi(s_j) |\Delta_{+} u_j|^3
  - \tfrac12 \lambda^2 \sum_j \eta''_j (\Delta_{+} f_j)^2 
  + \lambda (1+\eps_0) C \sum_j \Phi(s_j) |\Delta_{+} u_j|^3 \,.
  \tag 3.17
$$

\smallskip
\noindent
{\it Estimate of $I_2$.} \quad Substituting~(3.2) into $I_2$ in~(3.8), one
has the expansion
$$
  I_2 = \tfrac12 \sum_j \eta''(u_j) 
   (F_j^{\,2} + H_j^{\,2} + 2 F_j H_j + J_j^{\,2} + 2J_j(F_j+H_j)) \,.
  \tag 3.18
$$
We now estimate $I_2$ term by term. In estimating the first term in~(3.18), 
one uses the identity
$$
  (\Delta_{0}f_j)^2 = 2(\Delta_{+}f_j)^2 + 2(\Delta_{+}f_{j-1})^2
   - (\Delta_{-}\Delta_{+}f_j)^2 \,, 
\tag 3.19
$$
and a direct estimate of $F_j$ from its expression in~(3.3) to get
$$
\aligned
  \sum_j \eta''_j F_j^{\,2}
  & =\tfrac12 \lambda^2 \sum_j \eta''_j
   \bigl((\Delta_{+}f_j)^2 + (\Delta_{+}f_{j-1})^2\bigr)
   -\tfrac14 \lambda^2 \sum_j \eta''(u_j) (\Delta_{-}\Delta_{+}f_j)^2 
\\
  & = \lambda^2 \sum_j \eta''_j (\Delta_{+}f_j)^2 
   - \tfrac14 \lambda^2 \sum_j \eta''_j (\Delta_{-}\Delta_{+}f_j)^2
   + \tfrac12 \lambda^2 \sum_j \Delta_{+} \eta''_j (\Delta_{+}f_j)^2 \,. 
\endaligned
$$
In the first and third terms on the right hand side of the above inequality, 
one uses the Taylor expansion for $\Delta_{+}f_j$ and some simple
calculations used in estimating~(3.13) to obtain
$$
  \sum_j \eta''_j F_j^{\,2}
  \le \lambda^2 \sum_j \eta''_j  a_{j+1/2}^{\,2} (\Delta_{+} u_j)^2
  - \tfrac14 \lambda^2 \sum_j \eta''_j (\Delta_{-}\Delta_{+}f_j)^2
  + \lambda \eps_0 C \sum_j \Phi(s_j) |\Delta_{+}u_j|^3 \,.
  \tag 3.20
$$
Notice that the first 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 different sign, which cancel each other. The second term on the right
hand side of~(3.20) is a sum of squares of second-order
differences 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
$$
  \Delta_{-}\biggl(\frac{(\Delta_{+}f_j)^2}{\Delta_{+}u_j}\biggr)
  = \Delta_{-}\Delta_{+}f_j \, \frac{\Delta_{+}f_{j-1}}{\Delta_{+}u_{j-1}}
    + \Delta_{+}f_j \, \Delta_{-}\biggl(\frac{\Delta_{+}f_j}
                                      {\Delta_{+}u_j}\biggr) \,, 
  \tag 3.21
$$
and directly estimate the expression of $H_j$ in~(3.3) to get
$$
  \sum_j \eta''_j H_j^{\,2} 
  \le\tfrac12\lambda^4\sum_j\eta''_j (\Delta_{-}\Delta_{+} f_j)^2 
  \left(\frac{\Delta_{+}f_{j-1}}{\Delta_{+}u_{j-1}}\right)^2
  + \tfrac12 \lambda^4 \sum_j \eta''_j (\Delta_{+}f_j)^2 
\biggl(\Delta_{-}\left(\frac{\Delta_{+}f_j}{\Delta_{+}u_j}\right)\biggr)^2 \,.
$$
The first 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
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
$[u_{j-1},u_j]$ and $[u_j,u_{j+1}]$:
$$
\aligned
  \lambda^4 \sum_j \eta''_j (\Delta_{+}f_j)^2 
  \biggl(\Delta_{-}\left(\frac{\Delta_{+}f_j}{\Delta_{+}u_j}\right)\biggr)^2 
  &\le 2 \lambda \eps_0^{\,3} \sum_j \eta''_j (\Delta_{+}u_j)^2
  \biggl(\Delta_{-}\left(\frac{\Delta_{+}f_j}{\Delta_{+}u_j}\right)\biggr)^2 
\\
  &\le \lambda \eps_0^{\,3} \sum_j
     \eta''_j |a'(\xi_{j-1})||\Delta_{-}u_j|(\Delta_{+}u_j)^2
    + \lambda \eps_0^{\,3} \sum_j
     \eta''_j |a'(\xi_j)||\Delta_{+}u_j|(\Delta_{+}u_j)^2 \,.
\endaligned
$$
The last term only involves two points and hence can be directly
estimated.  In the last second term, we use the H\"{o}lder
inequality to split it again to obtain
$$
  \sum_j \eta''_j |a'(\xi_{j-1})||\Delta_{-}u_j|(\Delta_{+}u_j)^2
   \le \sum_j |u_j|^{4m-2} |s_{j-1}|^{m-2}|\Delta_{-}u_j|(\Delta_{+}u_j)^2
   \le C \sum_j \Phi(s_j) |\Delta_{+}u_j|^3 \,.
$$
The methods used here will be applied to estimating the similar terms.
With the above three estimates, we arrive at
$$
  \sum_j \eta''_j H_j^{\,2} 
  \le \tfrac12 \lambda^4 \sum_j \eta''_j (\Delta_{-}\Delta_{+} f_j)^2
  \left(\frac{\Delta_{+}f_{j-1}}{\Delta_{+}u_{j-1}}\right)^2
  + \lambda \eps_0^{\,3} C \sum_j \Phi(s_j) |\Delta_{+}u_j|^3 \,.
  \tag 3.22
$$
The third term in~(3.18) consists of products of a factor of first-order
difference and a factor of second-order difference, we
utilize a symmetric property to transform it into a sum of
products of three differences.  In doing this, we first split the
positive function~$\eta''$, and sum by parts to get
$$
\aligned
  & - \tfrac14 \lambda^3 \sum_j \eta''_j \Delta_{0} f_j \Delta_{-} 
  \biggl(\frac{(\Delta_{+} f_j)^2}{\Delta_{+} u_j}\biggr)
\\
  &  = - \tfrac14 \lambda^3 \sum_j \sqrt{\eta''_j} \Delta_{0} f_j 
    \left[ \Delta_{-} \biggl(\sqrt{\eta''_j} \, 
   \frac{(\Delta_{+} f_j)^2}{\Delta_{+} u_j}\biggr)
 -\Delta_{-}\sqrt{\eta''_j}\, \frac{(\Delta_{+} f_{j-1})^2}{\Delta_{+} u_{j-1}}
   \,\,\right]
\\
  &=\tfrac14 \lambda^3 \sum_j 
     \bigl(\sqrt{\eta''_j} \Delta_{+} f_j\bigr) 
                         \Delta_{+} \bigl(\sqrt{\eta''_j} \Delta_{0} f_j
   \frac{\Delta_{+} f_j}{\Delta_{+} u_j}\biggr)
  + \tfrac14 \lambda^3 \sum_j \sqrt{\eta''_j} 
   \Delta_{0} f_j \Delta_{-} \sqrt{\eta''_j} \,
   \frac{(\Delta_{+} f_{j-1})^2}{\Delta_{+} u_{j-1}}\,.
\endaligned
\tag 3.23
$$
Now the last term consists of products of three differences and hence
can be directly estimated. The first term has some
symmetric property after switching the difference operator
$\Delta_{+}$ with $\Delta_{0}$. This can be done by using Lemma~3.2 as
follows.
$$
\aligned
  \sum_j \bigl(\sqrt{\eta''_j} \Delta_{+} f_j \bigr) 
    \Delta_{+} \bigl(\sqrt{\eta''_j} \Delta_{0} f_j \bigr)   
    \frac{\Delta_{+} f_j}{\Delta_{+} u_j}
  & = \sum_j \bigl(\sqrt{\eta''_j} \Delta_{+} f_j \bigr) 
              \Delta_{0} \bigl(\sqrt{\eta''_j} \Delta_{+} f_j \bigr) 
    \frac{\Delta_{+} f_j}{\Delta_{+} u_j}
\\
  & + \sum_j \bigl(\sqrt{\eta''_j} \Delta_{+} f_j \bigr) 
  \Delta_{-} \bigl(\Delta_{+} \sqrt{\eta''_j} \Delta_{+} f_j \bigr) 
    \frac{\Delta_{+} f_j}{\Delta_{+} u_j}.
\endaligned
$$
Again the last term in the above equality is a sum of products of
three differences in $u_j$ and hence can be estimated directly. The
first term now can be transformed into a sum of products of
three differences by using Lemma~3.3 in where $b_j$ are replaced 
by~$\sqrt{\eta''_j} \Delta_{+} f_j$.
$$
\aligned
  \sum_j \bigl(\sqrt{\eta''_j} \Delta_{+} f_j \bigr) 
              \Delta_{0} \bigl(\sqrt{\eta''_j} \Delta_{+} f_j \bigr) 
    \frac{\Delta_{+} f_j}{\Delta_{+} u_j}
  & =\tfrac12\sum_j\biggl(\Delta_{+}\bigl(\sqrt{\eta''_j}
     \Delta_{+} f_j \bigr) \biggr)^2
    \Delta_{+} \biggl(\frac{\Delta_{+} f_j}{\Delta_{+} u_j}\biggr)
\\
  &
  - \tfrac12 \sum_j \bigl(\sqrt{\eta''_j} \Delta_{+} f_j\bigr)^2
   \Delta_{0}  \biggl( \frac{\Delta_{+} f_j}{\Delta_{+} u_j}\biggr) \,.
\endaligned
\tag 3.24
$$
Finally, combining all the estimates (3.23)-(3.24), we arrive at
$$
  \sum_j \eta''_j F_j H_j 
  \le \lambda \eps_0^{\,2} C \sum_j \Phi(s_j) |\Delta_{+}u_j|^3 \,.
  \tag 3.25
$$

  Estimating the rest three terms in~(3.18) is rather simple since
each of them consists of more than three difference factors. We
estimate them as follows:
$$
  \sum_j \eta''_j J_j^{\,2}
  = \lambda^2 \sum_j \eta''_j \bigl(\Delta_{-}(\tilde\beta_{j+1/2}^n 
     |\Delta_{+} a_j| \Delta_{+} u_j)\bigr)^2  
  \le \lambda \eps_0 \beta_1^2 C \sum_j \Phi(s_j) |\Delta_{+}u_j|^3 \,,
\tag 3.26
$$
and
$$
\aligned
  2 \sum_j \eta''_j J_j (F_j + H_j) & = - \lambda^2 \sum_j \eta''_j
  \Delta_{-}(\tilde\beta_{j+1/2}^n |\Delta_{+} a_j| 
  \Delta_{+} u_j)\Delta_{0} f_j
\\
  & \qquad
  + \lambda^3 \sum_j \eta''_j \Delta_{-}(\tilde\beta_{j+1/2}^n 
    |\Delta_{+} a_j| \Delta_{+} u_j)
    \Delta_{-} ( a_{j+1/2} \Delta_{+} f_j)
\\
  & \le \lambda (\eps_0+\eps_0^{\,2})
        \beta_1 C \sum_j \Phi(s_j) |\Delta_{+}u_j|^3 \,.
\endaligned
\tag 3.27
$$

  The estimate of $I_2$ is now completed by combining~(3.20) and~(3.22)
with~(3.25)-(3.27).
$$
\aligned
  I_2 & \le \tfrac12 \lambda^2 \sum_j 
     \eta''_j  a_{j+1/2}^{\,2} (\Delta_{+} u_j)^2
  - \tfrac18 \lambda^2 \sum_j (1-2 \lambda^2 a_{j-1/2}^{\,2})
    \eta''_j (\Delta_{-}\Delta_{+}f_j)^2
\\
  & \quad
  + \lambda \eps_0 (1+\eps_0+\eps_0^{\,2}+\beta_1+\beta_1^2) C 
    \sum_j \Phi(s_j) |\Delta_{+}u_j|^3 \,.
\endaligned
\tag 3.28
$$

\smallskip
\noindent
{\it Estimate of $I_3$.} \quad The estimate of $I_3$ is rather easy. We
first substitute the expansion of $u^{n+1}_j-u^n_j$ in~(3.2)-(3.3) into
$I_3$ to get
$$
  |I_3| \le  \sum_{k=3}^{4m}\frac{3^k}{k!} \sum_j |\eta^{(k)}(u^n_j)|
  \bigl( |F_j|^k + |H_j|^k + |J_j|^k \bigr) \,.
  \tag 3.29
$$
Using the expressions in~(3.3), we have
$$
\aligned
  |I_3| & \le \sum_{k=3}^{4m}\frac{3^k}{k!} \sum_j |\eta^{(k)}(u^n_j)| 
  \biggl( \lambda^k |\Delta_{+} f_j|^k + \lambda^k |\Delta_{-} f_j|^k
  + \lambda^{2k} |a_{j+1/2} \Delta_{+} f_j|^k
\\
  & \qquad \quad
  + \lambda^{2k} |a_{j-1/2} \Delta_{-} f_j|^k
  + (2\lambda \beta_1)^k |\Delta_{+}a_j \Delta_{+} u_j|^k
  + (2\lambda \beta)1)^k |\Delta_{-}a_j \Delta_{-} u_j|^k \biggr) 
\\
  & \le \lambda C \sum_{k=3}^{4m}\bigl(\eps_0^{\,k-1} + \eps_0^{\,2k-1} 
    + \eps_0^{\,k-1} \beta_1^k \bigr)
   \sum_j \Phi(s_j) |\Delta_{+} u_j|^3 \,.
\endaligned
\tag 3.30
$$

  We have now estimated all $I_1$, $I_2$, and $I_3$. Using~(3.17),
(3.28), and~(3.30), we obtain
$$
\aligned
  \sum_j (\eta^{n+1}_j - \eta^n_j)
  & \le
   - \frac{\lambda}{2} \beta_0 \sum_j \Phi(s_j) |\Delta_{+} a_j|^3
   - \tfrac18 \lambda^2 \bigl(1 - 2 \eps_0^2\bigr)
        \sum_j \eta''_j (\Delta_{-}\Delta_{+}f_j)^2
\\
  & \quad
  + \lambda C \left( 1+ \eps_0 \beta_1
             + \sum_{k=2}^{4m} \eps_0^{\,k-1} \beta_1^k\right) 
   \sum_j \Phi(s_j) |\Delta_{+} u_j|^3 \,.
\endaligned
$$
Choosing $\beta_0$ large and $\eps_0 \beta_1$ small, one has
$$
  \sum_j (\eta^{n+1}_j - \eta^n_j)
  \le - \lambda C_1 \sum_j \Phi(s_j) |\Delta_{+} a_j|^3
   - \lambda^2 C_2 \sum_j (\Delta_{-}\Delta_{+}f_j)^2 \,.
$$
This gives the following proposition.

\proclaim{Proposition 3.1} Under the same assumptions of Theorem~3.1, 
we have
$$
  \sum_j \eta(u^n_j) h 
  + \lambda \sum_{k\le n;j} \Phi(s^k_j) |\Delta_{+} u^k_j|^3 h
   + \lambda^2 \sum_{k\le n;j} 
  \bigl(\Delta_{-}\Delta_{+}f(u^k_j)\bigr)^2 h
  \le C, 
  \tag 3.31
$$
where $\eta$ is the entropy function given by $(3.7)$.
\endproclaim

  As an immediate consequence, we have

\proclaim{Corollary 3.1} Under the same assumptions of Theorem~3.1, 
we have
$$
  \|u_h\|_{L^{4m}_{\text{loc}}} 
  + \bigl\|\bigl(f(u_h),\eta_0(u_h), q_0(u_h)\bigr)
               \bigr\|_{L^{4m/(2m-1)}_{\text{loc}}}
  \le C \,, \quad C \text{ independent of }h, 
$$
where $(\eta_0,q_0)$ is the entropy pair given by $(2.2)$. 
\endproclaim

\bigskip
\noindent
{\bf 3.2. Entropy Inequality}
\smallskip

  We now estimate the entropy inequalities as stated in the third
condition in Theorem~2.1 for the entropy functions of form
$$
  \eta^{(k)}(u) \sim O(|u|^{2\ell-k}) \,, \quad 
  \text{for } |u| \gg 1, \ k=0,1,\cdots, 2\ell, 
  \tag 3.32
$$
where $\ell \le 2m$ is a positive integer. In the next subsection, we will
use the integer $\ell \le (13m-3)/6$ in the compactness arguments. We
will always take $\ell \le (3m+1)/2$ for using Theorem 2.1 to ensure the
entropy inequalities. For any $\phi(x,t) \in C_0^1$, $\phi\ge0$, 
denote $\phi^n_j = \phi(x_j,t_n)$ and 
$$
  I_h(\phi) \equiv h \sum_{j,n} \phi^n_j 
     \bigl(\eta(u^{n+1}_j) - \eta(u^n_j) \bigr)
     + \tfrac12 \lambda h \sum_{j,n} \phi^n_j
               \bigl(q(u^n_{j+1}) - q(u^n_{j-1}) \bigr).
$$
We decompose $I_h$ into the following three terms that are similar to
(3.8).
$$
\aligned
  I_h(\phi) 
  & = h \sum_{j,n} \phi_j^n \eta'_j J_j + h \sum_{j,n} \phi_j^n \eta'_j H_j 
  + h \sum_{j,n} \phi_j^n 
   \bigl(\eta'_j F_j + \tfrac12 \lambda \Delta_{0} q_j \bigr)
\\
  &\quad
  + \tfrac12 h \sum_{j,n} \phi_j^n \eta''_j
  \bigl(F_j^{\,2} + H_j^{\,2} + 2 F_j H_j + J_j^{\,2} + 2J_j (F_j+H_j)\bigr)
\\
  & \quad
  + h \sum_{k=3}^{2\ell-1}\tfrac1{k!} 
    \sum_{j,n} \phi_j^n \eta^{(k)}_j (F_j+H_j+J_j)^k
  + h \tfrac1{(2\ell)!} \sum_{j,n} \phi_j^n \eta^{(2\ell)}(\xi^*)
    (F_j+H_j+J_j)^{2\ell}
\\
  & = I_1^h(\phi) + I_2^h(\phi) + I_3^h(\phi) \,. 
\endaligned
\tag 3.33
$$
Notice that there is a remainder term in $I_3$ with 
$\xi^*\in [u^n_j,u^{n+1}_j]$. This term can be estimated by
using the fact that $\eta^{(2\ell)}$ 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.

\smallskip
\noindent
{\it Estimate of $I_1^h$.} \quad Similar to the estimate~(3.10), one 
can use the identity
$$
  \Delta_{+} (\phi^n_j \eta_j')
  = \tfrac12 (\phi^n_j+\phi^n_{j+1}) \Delta_{+} \eta_j'
  + \tfrac12 (\eta'_j+\eta_{j+1}') \Delta_{+} \phi^n_j \,
  \tag 3.34
$$
and the expression of $J_j$ in~(3.3) to get
$$
  \sum_{j,n} \phi^n_j \eta_j' J_j  
  =-\tfrac12 \lambda\sum_{j,n}\tilde{\beta}_{j+1/2}^n (\phi^n_j+\phi^n_{j+1}) 
  \Delta_{+} \eta_j' |\Delta_{+} a_j| \Delta_{+} u_j
   - \tfrac12\lambda\sum_{j,n}\tilde{\beta}_{j+1/2}^n 
  (\eta'_j+\eta_{j+1}') \Delta_{+} \phi^n_j
  |\Delta_{+} a_j| \Delta_{+} u_j.
$$
Applying Lemma~3.1 to the first term on the right hand side and
estimating the second term directly, one has
$$
  \sum_{j,n} \phi^n_j \eta_j' J_j  
  \le - \tfrac12 \lambda \beta_0 \sum_{j,n} (\phi^n_j+\phi^n_{j+1})
        \Phi(s_j) |\Delta_{+} u_j|^3
   + \lambda \beta_1 C \sum_{j,n} s_j\Phi(s_j)
     |\Delta_{+} u_j|^2 |\Delta_{+} \phi^n_j| \,,
  \tag 3.35
$$
where $s_j$ and $\Phi(s)$ are determined by (3.11).
With this good term in our hands, we 
now estimate other terms.  Notice that
$$
  \sum_{j,n} \phi^n_j \eta'_j H_j  
  = - \tfrac12 \lambda^2 \sum_{j,n} 
   \phi^n_j \Delta_{+} \eta'_j \frac{(\Delta_{+} f_j)^2}{\Delta_{+}u_j} 
  - \tfrac12 \lambda^2 \sum_{j,n}
   \eta'_{j+1} \Delta_{+}\phi^n_j \frac{(\Delta_{+} f_j)^2}{\Delta_{+}u_j}. 
$$
As in estimating~(3.13)-(3.14), expanding $\Delta_{+} \eta'_j$
and $\Delta_{+} f_j$ in the first term on the right hand side of the
above equality and estimating the resulting terms directly, we have
$$
  \sum_{j,n} \phi^n_j \eta'_j H_j  
  \le - \tfrac12 \lambda^2 \sum_{j,n} \phi^n_j \eta''_j (\Delta_{+} f_j)^2 
  + \lambda \eps_0 C \sum_{j,n} \phi^n_j \Phi(s_j) |\Delta_{+} u_j|^3
  + \lambda \eps_0 C \sum_{j,n} s_j\Phi(s_j) 
    |\Delta_{+} u_j|^2 |\Delta_{+} \phi^n_j| \,.
\tag 3.36
$$
Using Lemma~3.4, one has
$$
\aligned
  & \sum_{j,n} \phi^n_j \bigl(\eta'_j F_j
    + \tfrac12 \lambda \Delta_{0} q_j \bigr)
  = - \tfrac12 \lambda \sum_{j,n} \phi^n_j 
  \bigl( \eta'_j \Delta_{0}f_j -\Delta_{0} q_j \bigr)
\\
  & \quad
  = \tfrac14 \lambda \sum_{j,n} \phi^n_j 
  \biggl( \eta''_j (\Delta_{+} u_j)^2\Delta_{+}a_j
  - \eta''_j \int_{u_{j-1}}^{u_{j+1}} (u_j-s)^2 a'(s)\,ds \biggr)
\\
  & \quad \quad
  - \tfrac14 \lambda \sum_{j,n} \phi^n_j a_j 
      \Delta_{+} \eta''_j (\Delta_{+} u_j)^2
  - \tfrac14 \lambda \sum_{j,n} \eta''_{j+1} a_j 
    (\Delta_{+} u_j)^2 \Delta_{+}\phi^n_j 
\\
  & \quad \quad
  - \tfrac14 \lambda \sum_{j,n} \phi^n_j 
    \biggl[ \eta'''(\xi_j) \int_{u_j}^{u_{j+1}} (u_j-s)^2 a(s)\,ds 
    + \eta'''(\xi_{j-1}) \int_{u_{j-1}}^{u_j} (u_j-s)^2 a(s)\,ds \biggr]. 
\endaligned
$$
Each term on the right hand side of the above equality is the sum
of products of three differences and, therefore, can be directly
estimated by similar arguments as in~(3.15)-(3.16).
$$
   \sum_{j,n} \phi^n_j \bigl(\eta'_j F_j
    + \tfrac12 \lambda \Delta_{0} q_j \bigr)
   \le \lambda C \sum_{j,n} \biggl[ (\phi^n_j+\phi^n_{j+1}) 
      \Phi(s_j) |\Delta_{+} u_j|^3
  + s_j \Phi(s_j) |\Delta_{+} u_j|^2 |\Delta_{+}\phi^n_j| \biggr]\,.
  \tag 3.37
$$
Using~(3.35)-(3.37), we have
$$
\aligned
  I_1^h(\phi) & \le - \tfrac12 \lambda \beta_0 h \sum_{j,n}
    (\phi^n_j+\phi^n_{j+1}) \Phi(s_j) |\Delta_{+} u_j|^3
   - \tfrac12 \lambda^2 h \sum_{j,n} \phi^n_j \eta''_j (\Delta_{+} f_j)^2 
\\
  & \quad
  + \lambda (1+\eps_0+\beta_1) C h \sum_{j,n} \biggl[ 
  (\phi^n_j+\phi^n_{j+1})\Phi(s_j) |\Delta_{+} u_j|^3
   + s_j\Phi(s_j) |\Delta_{+} u_j|^2 |\Delta_{+} \phi^n_j| \biggr]\,.
\endaligned
  \tag 3.38
$$

\smallskip
\noindent
{\it Estimate of $I_2^h$.} \quad Using the identity~(3.19), one can
decompose the first term in $I_2^h$ and estimate similar to that of 
(3.20) to obtain 
$$
\aligned
  \sum_{j,n} \phi^n_j \eta''_j F_j^2
  & =\tfrac12 \lambda^2 \sum_{j,n}\phi^n_j
  (\eta''_j + \eta''_{j+1}) (\Delta_{+}f_j)^2 
  - \tfrac14 \lambda^2 \sum_{j,n} \phi^n_j \eta''_j 
    (\Delta_{-}\Delta_{+}f_j)^2
  + \tfrac12 \lambda^2 \sum_{j,n} \eta''_{j+1} 
         (\Delta_{+}f_j)^2 \Delta_{+} \phi^n_j. 
\\
  & \le \lambda^2 \sum_{j,n}\phi^n_j a_{j+1/2}^{\,2}
        \eta''_j (\Delta_{+}u_j)^2 
  - \tfrac14 \lambda^2 \sum_{j,n} \phi^n_j\eta''_j 
    (\Delta_{-}\Delta_{+}f_j)^2
\\
  & \quad
  + \lambda \eps_0 C \sum_{j,n} \biggl[ (\phi^n_j+\phi^n_{j+1})
    \Phi(s_j) |\Delta_{+} a_j|^3
  + s_j\Phi(s_j) |\Delta_{+} u_j|^2 |\Delta_{+} \phi^n_j|\biggr] \,.
\endaligned
\tag 3.39
$$
Estimating the second term of $I^h_2$ is similar to that of (3.21)-(3.22) by 
$$
  \sum_{j,n} \phi^n_j \eta''_j H_j^2
  \le \tfrac12 \lambda^4 \sum_{j,n} \phi^n_j \eta''_j 
      (\Delta_{-}\Delta_{+} f_j)^2
     \biggl(\frac{\Delta_{+}f_{j-1}}{\Delta_{+}u_{j-1}} \biggr)^2
  + \lambda \eps_0^{\,3} C \sum_{j,n} (\phi^n_j+\phi^n_{j+1})
  \Phi(s_j) |\Delta_{+}u_j|^3 \,.
  \tag 3.40
$$
The main idea in estimating the third term in $I_2^h$ 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.
$$ 
\aligned
  \sum_{j,n} \phi^n_j \eta''_j F_j H_j 
  & = - \tfrac14 \lambda^3 \sum_{j,n} \phi^n_j \eta''_j \Delta_{0} f_j 
    \Delta_{-} \biggl(\frac{(\Delta_{+}f_j)^2}{\Delta_{+}u_j} \biggr)
\\
  & =-\tfrac18 \lambda^3 \sum_{j,n}
   \biggl[ \bigl(\sqrt{\eta''_j} \Delta_{+} f_j \bigr)^2 
      \Delta_{0}\left(\phi^n_j \, \frac{\Delta_+f_j}{\Delta_+ u_j}\right) 
   - \Delta_{+}\left(\phi^n_j \, \frac{\Delta_+f_j}{\Delta_+ u_j}\right) 
     \biggl(\Delta_{+}\bigl(\sqrt{\eta''_j}\Delta_{+}f_j\bigr)\biggr)^2\biggr]
\\
  & \quad
  + \tfrac14 \lambda^3 \sum_{j,n} \sqrt{\eta''_j} 
  \biggl[\frac{(\Delta_{+} f_j)^2}{\Delta_+u_j} 
   \Delta_{-} \biggl(\Delta_{+}\bigl(\phi^n_j \, 
   \sqrt{\eta''_j} \big) \Delta_{+} f_j \biggr)
  + \phi^n_j \Delta_{0} f_j \frac{(\Delta_{+}f_{j-1})^2}{\Delta_+u_{j-1}}
   \Delta_{-}\sqrt{\eta''_j} \biggr].
\endaligned
$$
Clearly the terms on the right hand side of the above
equality are the sums of three difference products of $u_j^n$ 
and hence can be estimated directly.
$$ 
  \sum_{j,n} \phi^n_j \eta''_j F_j H_j 
  \le \lambda \eps_0^{\,2} C \sum_{j,n}
   (\phi^n_j +\phi^n_{j+1})\Phi(s_j)|\Delta_{+}u_j|^3
  + \lambda \eps_0^{\,2} C \sum_{j,n} s_j \Phi(s_j)|\Delta_{+}u_j|^2 
   \bigl(|\Delta_{+} \phi^n_j| + |\Delta_{-} \phi^n_j|\bigr) \,.
  \tag 3.41
$$
The remaining three terms in $I^h_2$ can be similarly estimated as in the
estimates~(3.26) and~(3.27).
$$
  \sum_{j,n} \phi^n_j \eta''_j J_j^{\,2}
  + 2 \sum_{j,n} \phi^n_j \eta''_j J_j(F_j + H_j)
  \le \lambda \eps_0(\beta_1 + \beta_1^2)C \sum_{j,n}
   (\phi^n_j +\phi^n_{j+1})\Phi(s_j)|\Delta_{+}u_j|^3 \,.
  \tag 3.42
$$
Therefore, combining~(3.39)-(3.40) with~(3.41)-(3.42), we have the
following estimate for $I^h_2$.
$$
\aligned
  I^h_2(\phi) 
& \le \tfrac12 \lambda^2 h \sum_{j,n}\phi^n_j \eta''_j (\Delta_{+}f_j)^2 
  - \tfrac18 \lambda^2 h \sum_{j,n} 
    \phi^n_j (1-2 \eps_0^2) \eta''_j (\Delta_{-}\Delta_{+}f_j)^2
\\
  & \quad
  + \lambda C \eps_0 (1 + \eps_0 + \eps_0^{\,2} + \beta + \beta^2)
    h \sum_{j,n} (\phi^n_j+\phi^n_{j+1})\Phi(s_j) |\Delta_{+} u_j|^3
\\
  & \quad
  + \lambda \eps_0(1+\eps_0) C h \sum_{j,n} s_j \Phi(s_j)|\Delta_{+}u_j|^2 
           \bigl(|\Delta_{+} \phi^n_j| + |\Delta_{-} \phi^n_j|\bigr) \,.
\endaligned
\tag 3.43
$$

   Estimating $I^h_3$ is straightforward as in that of~(3.29) and
(3.30). One can obtain
$$
  I_3^h(\phi) \le \lambda C h \sum_{k=3}^{4m}
   \bigl(\eps_0^{\,k-1} + \eps_0^{\,2k-1} 
    + \eps_0^{\,k-1} \beta^k \bigr)
   \sum_{j,n} (\phi^n_j+\phi^n_{j+1})\Phi(s_j) |\Delta_{+} u_j|^3 \,.
  \tag 3.44
$$
Finally, we have the following estimate of $I_h$ after combining
(3.38) with~(3.43)-(3.44) 
$$
\aligned
  I_h(\phi) 
  & \le - \tfrac14 \lambda \beta_0 h \sum_{j,n} (\phi^n_j+\phi^n_{j+1})
        \Phi(s_j) |\Delta_{+} u_j|^3
  - \tfrac18 \lambda^2 (1-2\eps_0^{\,2})
  h \sum_{j,n} \phi^n_j\eta''_j (\Delta_{-}\Delta_{+}f_j)^2
\\
  & \quad
  + \lambda (1+\eps_0+\beta_1) C h \sum_{j,n} s_j\Phi(s_j) |\Delta_{+} u_j|^2 
  \bigl(|\Delta_{+} \phi^n_j| + |\Delta_{-} \phi^n_j|\bigr)
\\
  & \quad
  + \lambda C \biggl(1 + \eps_0 \beta_1
          + \sum_{k=2}^{4m}\eps_0^{\,k-1} \beta_1^k \biggr)
    h \sum_{j,n} (\phi^n_j+\phi^n_{j+1})\Phi(s_j) |\Delta_{+} u_j|^3 \,.
\endaligned
$$
Choose $\beta_0$ large and $\eps_0 \beta_1$ small. Then we have
$$
  I_h(\phi) 
  \le - \tfrac18 \lambda \beta_0 h \sum_{j,n} (\phi^n_j+\phi^n_{j+1})
        \Phi(s_j) |\Delta_{+} u_j|^3
  + \lambda \beta_1 C h \sum_{j,n} s_j\Phi(s_j) |\Delta_{+} u_j|^2 
  \bigl(|\Delta_{+} \phi^n_j| + |\Delta_{-} \phi^n_j|\bigr) \,.
$$

  Now we deal with the last term in the above inequality. From
the H\"older inequality, one has
$$
\aligned
  & \lambda h \sum_{j,n} s_j\Phi(s_j) |\Delta_{+} u_j|^2 
  \bigl(|\Delta_{+} \phi^n_j| + |\Delta_{-} \phi^n_j|\bigr)
 \le \lambda h^{1+\alpha}\|\phi\|_{C^\alpha} 
  \sum_{(x_j,t_n)\in\Omega} \Phi(s_j) s_j (\Delta_{+} u)^2
\\
  & \quad
  \le C h^{\alpha-1/3}\|\phi\|_{C^\alpha} 
  \left(\sum_{(x_j,t_n)\in\Omega} s_j^{\,6\ell-9m+3} \tau_n h \right)^{1/3}
  \left(\sum_{(x_j,t_n)\in\Omega} \Phi(s_j)|\Delta_{+} u|^3 h \right)^{2/3}\,.
\endaligned
$$
Let $6\ell-9m+3 \le 4m$. Then
$$
  \lambda h \sum_{j,n} s_j\Phi(s_j) |\Delta_{+} u_j|^2 
  \bigl(|\Delta_{+} \phi^n_j| + |\Delta_{-} \phi^n_j|\bigr)
  \le  C h^{\alpha-1/3}\|\phi\|_{C^\alpha} \,.
$$
Substituting it back to~(3.33), we have the following proposition.

\proclaim{Proposition 3.2} Under the same assumptions of Theorem~3.1, 
$$
  \sum_{j,n} \phi^n_j \bigl(\eta(u^{n+1}_j) - \eta(u^n_j) \bigr) h
  + \tfrac12 \lambda \sum_{j,n} \phi^n_j 
    \bigl(q(u^n_{j+1}) - q(u^n_{j-1})\bigr)h
  \le C h^{\alpha-1/3}\|\phi\|_{C^\alpha} \, 
  \tag 3.45
$$
holds for any $\phi\in C^1_0$, $\phi\ge0$, where $(\eta,q)$ is the 
entropy pair given by $(3.32)$. 
\endproclaim

\bigskip
\noindent
{\bf 3.3. $H^{-1}$ Compactness of Entropy Dissipation Measures}
\smallskip
  
  For the entropy pair $(\eta,q)$, we define the following functional
as the entropy dissipation measures
$$
  M_h(\phi) = 
  \int\!\int \bigl( \eta(u_h) \del_t \phi + q(u_h) \del_x \phi \bigr)\,dxdt\,,
  \qquad \phi \in C_0^1 \,,
  \tag 3.46
$$
for the approximate solutions $u_h(x,t)$ defined by~(3.4).  In this
subsection, we will show that $M_h$ is compact in $H^{-1}$ for the
entropy pairs $(u,f(u))$ and $(\eta_0,q_0)$ with $\eta_0(u) = u^2$.

  From Proposition~3.1, we can easily get from the H\"{o}lder
inequality that $|M_h(\phi)| \le C \|\phi\|_{W^{1, (2m+1)/(4m)}}$.
Hence $M_h$ is bounded in $W^{-1,q}_{\text{loc}}$ for 
$q=\tfrac{2m-1}{4m} > 2$.  Lemma~2.2 tells us that $M_h$ is compact in
$H^{-1}_{\text{loc}}$ as long as we can prove that $M_h$ is compact in
$W^{-1,p}_{\text{loc}}$ for some $p < 2$.  
We define the following two functionals:
$$
  I_h(\phi) =  \sum_{j,n} \phi^n_j 
  \bigl(\eta(u^{n+1}_j) - \eta(u^n_j) \bigr) h
  + \tfrac12 \lambda \sum_{j,n} \phi^n_j 
     \bigl(q(u^n_{j+1}) - q(u^n_{j-1})\bigr)h \,,
  \tag 3.47
$$
and $\Delta_h(\phi) = I_h(\phi) + M_h(\phi)$ for any strictly convex
entropy $(\eta,q)$ and $\phi \in C_0^1$.  It is easy to show that
$$
\aligned
  \Delta_h(\phi) & = - \sum_{j,n} (\eta^{n+1}_j - \eta^n_j)
  \int_{x_{j-1/2}}^{x_{j+1/2}}
       \bigl(\phi(x,t_{n+1}) - \phi(x_j,t_n) \bigr)\,dx
\\
  & \quad
   - \tfrac12 \sum_{j,n} (q^n_{j+1} - q^n_j)
  \int_{t_n}^{t_{n+1}}\bigl(2\phi(x_{j+1/2},t) 
        - \phi(x_j,t_n) - \phi(x_{j+1},t_n) \bigr)\,dt \,.
\endaligned
\tag 3.48
$$
Estimating directly and using the H\"older inequality and Proposition~3.1, 
one obtains
$$
\aligned
  |\Delta_h(\phi)| 
  & \le \|\phi\|_{C^\alpha} \sum_{(x_j,t_n)\in\Omega}
  \bigl(|\eta^{n+1}_j - \eta^n_j| h^{1+\alpha}
        + |q^n_{j+1} - q^n_j| \tau_n h^\alpha\bigr)
\\
  & \le h^{1+\alpha} \|\phi\|_{C^\alpha} \sum_{(x_j,t_n)\in\Omega}
  \left(\sum_{k=1}^2 \frac{3^k}{k!}|\eta^{(k)}_j|
          \bigl(|F_j|^k + |H_j|^k + |J_j|^k\bigr)
        + \lambda |q'(\xi_j)||\Delta_{+}u^n_j| \right)
\\
  & \le \lambda C h^{1+\alpha} \|\phi\|_{C^\alpha} \sum_{(x_j,t_n)\in\Omega}
   \Phi(s_j)^{1/3} |\Delta_{+}u^n_j|
\\
  & \le \lambda^{1/3} C h^{\alpha-2/3} \|\phi\|_{C^\alpha} \to 0 \,,
  \quad \text{for } \tfrac23 < \alpha < 1 \,.
\endaligned
  \tag 3.49
$$
This implies that $\Delta_h$ is compact in $W^{-1,p_1}_{\text{loc}}$
for $p_1 < \tfrac2{1+\alpha} < 2$. We know from Proposition~3.2 that
$I_h$ can be split as $I_h(\phi) = I_1^h(\phi) + I_2^h(\phi)$ such
that
$$
  I_1^h \le 0 \,, \quad
  |I_2^h(\phi)| \le C h^{\alpha-1/3} \|\phi\|_{C^\alpha} \,. 
  \tag 3.50
$$
Consequently, we have that $I_2^h$ is compact in $W^{-1,p_2}_{\text{loc}}$
for $1 < p_2 < 2$.  The remainder is to show that $I_1^h$ is also
compact in $W^{-1,p}_{\text{loc}}$ for some $p < 2$. Since $I_1^h$ is
negative due to~(3.50), we know from Lemma~2.1 that we only need to show
that $I_1^h$ is bounded in $W^{-1,p}_{\text{loc}}$ for some $p < 2$.
This is a direct consequence of~(3.46),~(3.49), and~(3.50). Thus, we
have the following proposition.

\proclaim{Proposition 3.3} Under the same assumptions of Theorem~3.1, 
$$
  \del_t u_h + \del_x f(u_h)\,, \quad 
  \del_t \eta_0(u_h) + \del_x q_0(u_h) \quad\text{is compact in } 
  H^{-1}_{\text{loc}} 
  \tag 3.51
$$
where $(\eta_0,q_0)$ is the entropy pair given by $(2.2)$.
\endproclaim

  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.

\bigskip
\bigskip
\centerline {\bf 4. Extension to the Nonconvex Case}
\smallskip

  For the nonconvex scalar conservation laws, the second-order numerical
viscosity term 
$$
  \lambda \Delta_{-} \bigl(\beta_{j+1/2}^n
|\Delta_{+}f'(u_j^n)| \Delta_{+} u_j^n \bigr)
$$
is degenerate at the points $\{u \, \, |\, \, f''(u)=0\}$.  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-Wendroff schemes:
$$
  \lambda \Delta_{-}\bigl(\beta_{j+1/2}^n |\Delta_{+} u_j^n| 
  \Delta_{+} u_j^n \bigr)
  \tag 4.1
$$
with $\beta_{j+1/2}^n \in \bigl(\beta_0 + O(|a'|), \beta_1 +
O(|a'|)\bigr)$, $\beta_0>0$ for the nonconvex scalar case.

  This covers both the convex case (the convexity implies $\beta_0>0$)
and the nonconvex case. With such a numerical viscosity we can use the
same arguments as in Section~3, when $C \gg 1$ and $\epsilon_0 \ll 1$,
to obtain the same viscosity estimate
$$
  \sum_{j,n} \Phi(s_j) |u^n_j-u^n_{j+1}|^3 h \le C
  \tag 4.2
$$
and entropy estimate
$$
  \sum_j \eta(u^{n+1}_j) \le \sum_j \eta(u^n_j)
  \tag 4.3
$$
for $\eta'' = u^{4m}$.  Using these estimates, we can similarly verify
that the corresponding approximate solutions $u_h(x,t)$ 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 flux functions satisfy
$$
 \text{meas} \bigl\{ u \,\,|\,\, f''(u) =0\bigr\} = 0, \qquad
  f^{(k)}(u) \sim O(|u|^{m-k}) \,, \quad 
  \text{for } |u| \gg 1, \ 0\le k\le m \,.
  \tag 4.4
$$

\proclaim{Theorem 4.1} Let $u_h(x,t)$ be numerical approximate 
solutions, generated from the Lax-Wendroff scheme~$(1.2)$ endowed with 
the numerical viscosity~$(4.1)$, of the scalar conservation laws $(1.1)$
satisfying~$(4.4)$.  Assume that the coefficient $\beta_0$ in~$(4.1)$
is suitably large and the CFL number $\eps_0$ given by~$(1.4)$ is
suitably small.  Then the sequence $u_h$ has a strong convergent subsequence
and the limit function is the entropy solution of~$(1.1)$ satisfying
the entropy inequalities 
$\del_t\eta(u)+\del_x q(u) \le 0 \quad \text{in }{\Cal D}' \,, $ 
for any convex entropy pair $(\eta,q)$
satisfying $\eta^{(k)}(u) \sim O(|u|^{r-k})$, 
for $|u| \gg 1$, $0\le k \le r < 4m$.
\endproclaim

\demo{Proof} Only one thing we should check is that, for any such 
a $C^2$ entropy pair $(\eta(u), q(u))$, $\del_t \eta(u_h) + \del_x
q(u_h)$ is compact in $H^{-1}_{\text{loc}}$.  This can be achieved as
follows.  For any $C^2$ entropy pair $(\eta,q)$, there exists $C_0>0$
such that $|\eta''|\le C_0 |\eta''_{*}|$ where $\eta_{*}$ is a
strictly convex entropy. Define $\bar \eta = C_0\eta_{*} - \eta$.
Then $\bar \eta$ is a convex entropy. Noting that both 
$\del_t \bar\eta(u_h) + \del_x \bar q(u_h)$ and 
$\del_t \eta_{*} (u_h) + \del_x q_{*} (u_h)$ are compact 
sets in $H^{-1}_{\text{loc}}$, one
obtains that $ \del_t \eta(u_h) + \del_x q(u_h)$ is compact in
$H^{-1}_{\text{loc}}$.  This completes the proof.
 
\demo{Remark} The assumption~(4.4) can be relaxed to more general 
flux functions for the Lax-Wendroff scheme.  The strong convergence
becomes weak one in $L^r$, $r < 4m$, and the limit function is the weak
solution satisfying the entropy condition for the Young
measures $\nu_{x,t}$ determined by the Lax-Wendroff approximate solutions
$u_h(x,t)$:
$$
\del_t <\nu_{x,t},\eta> + \del_x <\nu_{x,t}, q> \le 0,
$$
for any convex entropy pair $(\eta, q)$ in the above theorem.
\enddemo

\bigskip
\bigskip
\centerline {\bf 5. Extension to Hyperbolic Systems}
\smallskip

  We write the Lax-Wendroff scheme in the following form 
$$
  u^{n+1}_j = u^n_j + F^n_j + H^n_j + J^n_j \,.
  \tag 5.1
$$
Here
$$
  F^n_j = -\tfrac12 \lambda \Delta_{0} f(u^n_j) \,, \quad
  H^n_j = \tfrac12 \lambda^2 \Delta_{-}
  \biggl(a\bigl(\frac{u^n_j+u^n_{j+1}}2 \bigr)\Delta_{+}f(u^n_j)\biggr)
   \,,\quad
   J^n_j = \lambda \Delta_{-} 
 \bigl(\beta_{j+1/2}^n |\Delta_{+}u^n_j| \Delta_{+} u^n_j \bigr) 
\tag 5.2
$$
where the scalar function $\beta_{j+1/2}^n\ge \beta_0>0$, for
some $\beta_0$, is smooth with respect to $u_j^n$ and $u_{j+1}^n$
and is bounded from below.  We assume that
$$
  \sup_{j,n} |u^n_j| \le M \,.
  \tag 5.3
$$

  In this section we show the convergence of such a scheme for 
$2\times 2$ systems and the entropy consistency of the boundedly convergent
Lax-Wendroff approximate solutions with general hyperbolic systems of 
conservation laws with a convex entropy.  
This depends on the following estimates.

\bigskip
\noindent
{\bf 5.1. Dissipation Estimates}
\smallskip

  We first start with the entropy estimate. Let $\eta$ 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
$$
\aligned
  \sum_j \bigl(\eta(u^{n+1}_j)-\eta(u^n_j)\bigr)
  & = \sum_j \eta'(u^n_j) (u^{n+1}_j-u^n_j)
  + \tfrac12 \sum_j  (u^{n+1}_j-u^n_j)^\Tr \eta''(u^n_j) (u^{n+1}_j-u^n_j)
\\
  & \quad
  +  \sum_j O \left(|u^{n+1}_j-u^n_j|^3\right)
    \equiv I_1 + I_2 + I_3 \,.
\endaligned
\tag 5.4
$$
We will keep the same order of the estimates as in Section~3 and will
only emphasize the new features, which are different from the scalar
case.  The arguments similar to those in Section~3 will be omitted. 
We point
out here that some of estimates in this section are simpler than the
ones in Section~3 since we assumed the uniform boundedness of the
approximate solutions.

\smallskip
\noindent
{\it Estimate of $I_1$.} \quad As in~(3.11), we directly estimate 
that
$$
  \sum_j (\eta'_j)^\Tr J_j 
  = - \lambda \sum_j   \beta_{j+1/2} |\Delta_{+}u_j| 
        \Delta_{+} (\eta'_j)^\Tr \Delta_{+} u_j
  \le - \lambda \beta_0 \sum_j |\Delta_{+}u_j|^3 \,,
  \tag 5.5
$$
using the expression of $J_j$ in~(5.2), the summation by parts, and
the convexity of $\eta$. 
This gives us the main dissipative term. 
Here we used the notation $\beta_{j+1/2}= \beta(u_j,u_{j+1})$, 
which is a scalar function. We remark that~(5.5)
is the only place we restrict the viscosity $\beta(u_j,u_{j+1})$ to be a scalar
function.

As in estimating~(3.13) and~(3.14), we have
$$
  \sum_j (\eta'_j)^\Tr H_j 
  = - \tfrac12 \lambda^2 \sum_j (\Delta_{+} \eta'_j)^\Tr
  a_{j+1/2} \Delta_{+} f_j
  = - \tfrac12 \lambda^2 \sum_j (\Delta_{+} u_j)^\Tr \eta''_j 
   a_j \Delta_{+} f_j + \lambda\eps_0 \sum_j O(|\Delta_{+} u_j|^3) \,.
  \tag 5.6
$$
Notice that the necessary and sufficient condition for a function $\eta$
to be an entropy is that
$\eta''f'$ is symmetric, that is, $\eta''_j a_j=(a_j)^\Tr\eta''_j$.
One has from (5.6)
$$
  \sum_j (\eta'_j)^\Tr H_j 
  = - \tfrac12 \lambda^2 \sum_j (\Delta_{+} f_j)^\Tr \eta''_j 
  \Delta_{+} f_j + \lambda\eps_0 \sum_j O(|\Delta_{+} u_j|^3) \,.
  \tag 5.7
$$

  In estimating the first term of $I_1$, we use the conservative
property of entropy. Denote $\bar u_j = \tfrac12 (u_{j-1}+u_{j+1})$.
Similar to~(3.16)-(3.17), one can obtain
$$
\aligned
  \sum_j (\eta'_j)^\Tr F_j 
  & = - \tfrac12 \lambda \sum_j (\eta'_j)^\Tr
  \int_{-1/2}^{1/2} a(\bar u_j + \theta \Delta_0 u_j) \,d \theta \,
  \Delta_{0} u_j
  = \lambda \eps_0 \sum_j O(|\Delta_{+} u_j|^3) \,.
\endaligned
\tag 5.8
$$

  Combining~(5.5) with~(5.7)-(5.8), we have the following estimate
of $I_1$.
$$
  I_1 \le - \lambda \beta_0 \sum_j |\Delta_{+}u_j|^3
  + \lambda \eps_0 C|\Delta_{+} u_j|^3
  - \tfrac12 \lambda^2 
    \sum_j (\Delta_{+} f_j)^\Tr \eta''_j \Delta_{+} f_j \,. 
  \tag 5.9
$$

\smallskip
\noindent
{\it Estimate of $I_2$.}\quad As in~(3.19), we substitute~(5.1)-(5.2)
into $I_2$ and have the following expansion.
$$
  I_2 = \tfrac12 \sum_j F_j^{\Tr} \eta''_j F_j
  + \tfrac12 \sum_j H_j^{\,\Tr} \eta''_j H_j
  + \sum_j F_j^{\,\Tr} \eta''_j H_j
  + \tfrac12 \sum_j J_j^{\,\Tr} \eta''_j J_j
  + \sum_j J_j^{\,\Tr} \eta''_j (F_j + H_j) \,.
  \tag 5.10
$$
The estimate of the first term in~(5.10) is similar to that in (3.20)-(3.22)
by the identity~(3.20). Putting all of products of three
differences together in $O(|\Delta_+u_j|^3)$, one has
$$
  \sum_j F_j^{\,\Tr} \eta''_j F_j
  = \lambda^2 \sum_j (\Delta_{+}f_j)^\Tr \eta''_j \Delta_{+}f_j 
  + \lambda \eps_0 \sum_j O(|\Delta_{+} u_j|^3)
  - \tfrac14 \lambda^2 \sum_j (\Delta_{-} \Delta_{+}f_j)^\Tr 
  \eta''_j \Delta_{-}\Delta_{+}f_j \,.
  \tag 5.11
$$
The estimate of the second term in~(5.10) is similar to that in (3.23)-(3.24).
After putting all of products of three differences in 
$O(|\Delta_+ u_j^n|)$, 
the remainder term is the sum of products of two second
differences, which can be controlled by the last term in~(5.11).
$$
\aligned
  \sum_j H_j^{\,\Tr} \eta''_j H_j
  & = \tfrac14 \lambda^4 \sum_j 
 \Delta_{-}(a_{j+1/2} \Delta_{+} f_j)^\Tr \eta''_j
 \Delta_{-}(a_{j+1/2} \Delta_{+} f_j)
\\
  & = \tfrac14 \lambda^4 \sum_j 
  (\Delta_{-}\Delta_{+} f_j)^\Tr 
  a_j^{\,\Tr} \eta''_j a_j
  \Delta_{-}\Delta_{+} f_j
  + \lambda \eps_0^{\,3}\sum_j O(|\Delta_{+} u_j|^3) \,.
\endaligned
\tag 5.12
$$

  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 differences. For any symmetric matrix $A$, we have as
in Lemma~3.1 that
$$
  - \sum_j (\Delta_{0} b_j)^\Tr A b_j 
  = \tfrac12 \sum_j b_j^{\,\Tr} \Delta_{0} A b_j 
  - \tfrac12 \sum_j (\Delta_{+}b_j)^\Tr \Delta_{+} A \Delta_{+} b_j \,.
  \tag 5.13
$$
Similar to~(3.25)-(3.27), we have
$$
\aligned
  \sum_j F_j^{\,\Tr} \eta''_j H_j
  & = - \tfrac14 \lambda^3 \sum_j 
  (\Delta_{0} f_j)^\Tr \eta''_j \Delta_{-}(a_{j+1/2} \Delta_{+} u_j)
\\
  & = \tfrac14 \lambda^3 \sum_j 
  \Delta_{0} \bigl(\sqrt{\eta''_j} \Delta_{+} f_j \bigr)^\Tr 
  \sqrt{\eta''_j} a_j \Delta_{+} u_j 
  + \lambda \eps_0^{\,2} \sum_j O(|\Delta_{+} u_j|^3) \,.
\endaligned
$$
We know from~(5.6) that 
$\sqrt{\eta_j''} a_j \bigl( \sqrt{\eta''_j} \,\, \bigr)^{-1}$ is symmetric. 
Hence we can apply the identity~(5.13) to 
the first term on the right hand side of the above equality with $b_j$
replaced by $\sqrt{\eta''_j} \Delta_{+} f_j$. As a result that every term
now is decomposed as the product of three differences, we obtain
$$
  \sum_j F_j^{\,\Tr} \eta''_j H_j
   = \lambda \eps_0^{\,2} \sum_j O(|\Delta_{+} u_j|^3) \,.
  \tag 5.14
$$

  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
differences. One has from a direct estimate that
$$
  \sum_j J_j^{\,\Tr} \eta''_j J_j
  + 2 \sum_j J_j^{\,\Tr} \eta''_j (F_j + H_j)
  \le \lambda \eps_0 \sum_j O(|\Delta_{+} u_j|^3) \,.
  \tag 5.15
$$
Finally, combining~(5.11)-(5.12) with~(5.14)-(5.15), we have
$$
\aligned
  I_2
  & \le  \tfrac12 \lambda^2 \sum_j (\Delta_{+}f_j)^\Tr \eta''_j \Delta_{+}f_j 
  - \tfrac18 \lambda^2 \sum_j (\Delta_{-} \Delta_{+}f_j)^\Tr 
  \eta''_j \Delta_{-}\Delta_{+}f_j
\\
  & \quad
  + \tfrac14 \lambda^4 \sum_j  (\Delta_{-}\Delta_{+} f_j)^\Tr 
    a_j^{\,\Tr} \eta''_j a_j \Delta_{-}\Delta_{+} f_j
  + \lambda \eps_0 \sum_j  O(|\Delta_{+} u_j|^3) \,.
\endaligned
\tag 5.16
$$

  Estimating $I_3$ simply follows from the fact that each term is
the sum of products of more than three difference factors
$$
  |I_3| \le  C \sum_j ( |F_j|^3 + |H_j|^3 + |J_j|^3 )
   \le  \lambda \eps_0^{\,2} C \sum_j |\Delta_{+} u_j|^3 \,.
  \tag 5.17
$$
Combining (5.9) with (5.16), we have
$$
  \sum_j (\eta^{n+1}_j - \eta^n_j)
  \le - \lambda \beta_0 \sum_j |\Delta_{+}u_j|^3
  - \tfrac18 \lambda^2(1-\eps_0^{\,2}) \sum_j (\Delta_{-} \Delta_{+}f_j)^\Tr 
  \eta''_j \Delta_{-}\Delta_{+}f_j
  + \lambda (1 + \eps_0 + \eps_0^{\,2}) C 
  \sum_j |\Delta_{+} u_j|^3 \,.
$$
For $\beta_0$ large and $\eps_0$ small, we have
$$
  \sum_j (\eta^{n+1}_j - \eta^n_j) 
  \le - \lambda C_1 \sum_j |\Delta_{+}u_j|^3
  - \lambda^2 C_2 \sum_j |\Delta_{-}\Delta_{+}f_j|^2 \,.
$$
This gives

\proclaim{Proposition 5.1} Let $u^n_j$ be the Lax-Wendroff approximate 
solutions~$(5.1)$-$(5.2)$ of the hyperbolic systems of conservation 
laws~$(1.1)$ with a convex entropy $\eta$. Assume that $u_h$ is uniformly
bound and the coefficient $\beta$ in~$(5.2)$ is suitably large and the
CFL number $\eps_0$ given by~$(1.3)$ is suitably small. Then 
$$
  \sum_j \eta(u^n_j) h 
  + \lambda \sum_{k\le n; j} |\Delta_{+} u^k_j|^3 h
  + \lambda^2 \sum_{k\le n;j} |\Delta_{-}\Delta_{+}f(u^k_j)|^2 h \le C \,.
  \tag 5.18
$$
\endproclaim

\bigskip
\noindent
{\bf 5.2. $H^{-1}$ Compactness Estimates}
\smallskip

  For any $C^2$ convex entropy pair $(\eta, q)$ and any $\phi(x,t) \in
C_0^1$, $\phi\ge0$, we denote $\phi^n_j = \phi(x_j,t_n)$ and
$$
  I_h(\phi) \equiv h \sum_{j,n} \phi^n_j 
  \bigl(\eta(u^{n+1}_j) - \eta(u^n_j) \bigr)
  + \tfrac12 \lambda
    \sum_{j,n} \phi^n_j \bigl(q(u^n_{j+1}) - q(u^n_{j-1}) \bigr) \,.
$$
As~(3.37), we have from the Taylor expansion that
$$
\aligned
  I_h(\phi) 
  & = h \sum_{j,n} \phi_j^n \bigl( (\eta'_j)^\Tr F_j 
  + \tfrac12 \lambda \Delta_{0} q_j + (\eta'_j)^\Tr H_j
  + (\eta'_j)^\Tr J_j \bigr)
\\
  &\quad
  + \tfrac12 h \sum_{j,n} \phi_j^n (F_j+H_j+J_j)^\Tr \eta''_j (F_j+H_j+J_j)
\\
  & \quad
  + \tfrac12 h \sum_{j,n} \phi_j^n \eta'''(\xi) (F_j+H_j+J_j)^3
  \equiv I^h_1(\phi) + I^h_2(\phi) + I^h_3(\phi) \,. 
\endaligned
\tag 5.19
$$

\smallskip
\noindent
{\it Estimate of $I^h_1$.} \quad The estimate of the last term in $I^h_1$ is 
similar to that in~(3.38) and~(3.39). 
$$
  \sum_{j,n} \phi^n_j (\eta_j')^\Tr J_j  
  \le - \tfrac12 \lambda \beta_0
       \sum_{j,n} (\phi^n_j+\phi^n_{j+1}) |\Delta_{+} a_j|^3
   + \lambda C \sum_{j,n} |\Delta_{+} u_j|^2 |\Delta_{+} \phi^n_j| \,.
\tag 5.20
$$
Similar to that in~(3.41), one can obtain
$$
  \sum_{j,n} \phi^n_j (\eta'_j)^\Tr H_j  
  \le - \tfrac12 \lambda^2 \sum_{j,n} \phi^n_j
   (\Delta_{+} f_j)^\Tr \eta''_j \Delta_{+} f_j
  + \lambda \eps_0 C \sum_{j,n} (\phi^n_j+\phi^n_{j+1})|\Delta_{+} u_j|^3
  + \lambda \eps_0 C 
    \sum_{j,n} |\Delta_{+} u_j|^2 |\Delta_{+} \phi^n_j|\,.
\tag 5.21
$$
The estimate of the first term in $I^h_1$ is similar to that in~(3.40). 
One has
$$
\aligned
  \sum_{j,n} \phi^n_j \bigl((\eta'_j)^\Tr F_j
    + \tfrac12 \lambda \Delta_{0} q_j \bigr)
  &= - \tfrac12 \lambda \sum_{j,n} \phi^n_j 
  \bigl( (\eta'_j)^\Tr \Delta_{0}f_j -\Delta_{0} q_j \bigr)
\\
  & \le \lambda C \sum_{j,n} (\phi^n_j+\phi^n_{j+1}) |\Delta_{+} u_j|^3
   + \lambda  C \sum_{j,n} |\Delta_{+} u_j|^2 |\Delta_{+} \phi^n_j|\,.
\endaligned
$$
The estimates~(5.20)-(5.21) yield 
$$
\aligned
  I_1^h(\phi)
  & \le - \tfrac12 \lambda \beta_0 h \sum_{j,n} (\phi^n_j+\phi^n_{j+1})
        |\Delta_{+} u_j|^3
   - \tfrac12 \lambda^2 h \sum_{j,n} \phi^n_j
   (\Delta_{+} f_j)^\Tr \eta''_j \Delta_{+} f_j
\\
  & \quad
  + \lambda (1+\eps_0) C h \sum_{j,n} (\phi^n_j+\phi^n_{j+1})
      |\Delta_{+} u_j|^3
  + \lambda (1+\eps_0) C h \sum_{j,n}
          |\Delta_{+} u_j|^2 |\Delta_{+} \phi^n_j| \,.
\endaligned
$$
We can similarly estimate $I_2^h$ and $I_3^h$. Finally we have
$$
\aligned
  I_h (\phi)
  & \le - \tfrac12 \lambda \beta_0 h \sum_{j,n} (\phi^n_j+\phi^n_{j+1})
       |\Delta_{+} u_j|^3
  - \lambda^2 C h \sum_{j,n} \phi^n_j (\Delta_{-}\Delta_{+}f_j)^2
\\
  & \quad
  + \lambda C h \sum_{j,n} |\Delta_{+} u_j|^2 
  \bigl(|\Delta_{+} \phi^n_j| + |\Delta_{-} \phi^n_j|\bigr)
  + \lambda C \eps_0^{\,2} 
    h \sum_{j,n} (\phi^n_j+\phi^n_{j+1})|\Delta_{+} u_j|^3 \,.
\endaligned
$$
We choose $\beta_0$ large and $\eps_0$ small and then have
$$
  I_h (\phi) \le \lambda C h \sum_{j,n} |\Delta_{+} u_j|^2 
  \bigl(|\Delta_{+} \phi^n_j| + |\Delta_{-} \phi^n_j|\bigr)
  \le h^{\alpha-1/3} \|\phi\|_{C^\alpha} \,.
$$
Therefore, we obtain

\proclaim{Proposition 5.2} Under the same assumptions of Proposition~5.1, 
$$
  \sum_{j,n} \phi^n_j \bigl(\eta(u^{n+1}_j) - \eta(u^n_j) \bigr) h
  + \tfrac12 \lambda \sum_{j,n} \phi^n_j 
     \bigl(q(u^n_{j+1}) - q(u^n_{j-1})\bigr)h 
  \le C h^{\alpha-1/3}\|\phi\|_{C^\alpha} \,. 
\tag 5.22
$$
for any $C^2$ convex entropy pair $(\eta,q)$ and $\phi\in C^1_0$, $\phi\ge0$.
\endproclaim

  Similar to Section~3.3, we can show that 
$\del_t \eta(u_h) + \del_x q(u_h)$ 
is compact in $H^{-1}_{\text{loc}}$ for any convex entropy
pair $(\eta,q)$. 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

\proclaim{Proposition 5.3}  Under the same assumptions of Proposition~5.1, 
$\del_t \eta(u_h) + \del_x q(u_h)$ is compact in
$H^{-1}_{\text{loc}}$ for any $C^2$ entropy pair $(\eta,q)$.
\endproclaim

\bigskip
\noindent
{\bf 5.3. Convergence and Entropy Consistency}
\smallskip

  Now we are concerned with the convergence of uniformly bounded
approximate solutions of $2\times 2$ systems of conservation laws
endowed with global Riemann invariants and the entropy consistency of
boundedly convergent Lax-Wendroff numerical solutions of general
hyperbolic systems of conservation laws with a convex entropy.

  One important example of the $2\times 2$ systems is the elasticity
equations (cf.~[5]):
$$
  \del_t v - \del_x u = 0 \,, \qquad \del_t u - \del_x \sigma(v) = 0 \,,
$$ 
where $u$ is the specific volume, $v$ is the strain, and $\sigma$ is the
stress-strain relation satisfying $\sigma'(v)>0$. The numerical
experiments indicate that the Lax-Wendroff approximate solutions
$(v_h,u_h)$ are uniformly bounded.  It is important from the viewpoint
of numerical analysis whether uniformly bounded Lax-Wendroff numerical
solutions may generate some oscillations.  The following theorem shows
that, if $(v_h,u_h)$ 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.

\proclaim{Theorem 5.1} Consider a $2\times 2$ genuinely 
nonlinear and strictly hyperbolic system of conservation laws~$(1.1)$
endowed with global Riemann invariants. Let $u_h(x,t)$ be the
Lax-Wendroff approximate solutions. Assume that $u_h$ is uniformly
bounded and the coefficient $\beta$ in~$(5.2)$ is suitably large and the
CFL number $\eps_0$ given by~$(1.3)$ is suitably small. Then there is
a subsequence\/ $($still denoted as\/$)$ $u_h$ such that 
$u_h(x,t) \to u(x,t)\quad\text{a.e.} \quad\text{as } h \to 0$ 
and $u$ is the entropy solution of~$(1.1)$ satisfying 
$\del_t \eta(u)+\del_x q(u)\le 0 \quad \text{in } {\Cal D}'$ 
for any convex entropy $(\eta,q)$.
\endproclaim

  For a general system, we conclude that the numerical solutions of the
Lax-Wendroff 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 time-independent systems.

\proclaim{Theorem 5.2} Consider a general system of genuinely 
nonlinear and strictly hyperbolic conservation laws~$(1.1)$ endowed
with a convex entropy. Let $u_h(x,t)$ be the uniformly bounded
Lax-Wendroff approximate solutions converging boundedly a.e.  to
$u(x,t)$. Assume that the coefficient $\beta$ in~$(5.2)$ is suitably
large and the CFL number $\eps_0$ given by~$(1.3)$ is suitably small.
Then $u$ is the entropy solution of~$(1.1)$ satisfying 
$\del_t \eta(u) + \del_x q(u) \le 0 \quad \text{in } {\Cal D}'$ 
for any convex entropy pair $(\eta,q)$.
\endproclaim

  A typical example for such a system is the system of gas dynamics in
Lagrangian coordinates:
$$
  \del_t \pmatrix v \\ u \\ E \endpmatrix
  + \del_x \pmatrix -u \\ p \\ pu \endpmatrix = 0 \,,
$$
where $v> 0$ denotes the specific volume of the gas, $u$ the velocity,
$p>0 $ the pressure, $E = \tfrac12 u^2 + e $ the total specific
energy and $e$ the specific internal energy. The additional
conservation law assumes the form $\del_t(-S) \le 0$ for
physical solutions. For polytropic gases, 
$-S(v,u,E)=-\log(E-\tfrac12u^2)-(\gamma-1)\log v$ is 
a strictly convex function of
the state variables $v$, $u$, and $E$ for $\gamma>1$, a fixed constant,
and $p=(\gamma-1)\frac{E-\tfrac12u^2}v= A(S)v^\gamma \,$.
Theorem 5.2 indicates that uniformly bounded convergent Lax-Wendroff
approximation is consistent with this system.

\bigskip
\bigskip
\centerline {\bf 6. The MacCormack Scheme and Richtmyer Scheme}
\smallskip
 
  An efficient implementation of the Lax-Wendroff scheme is to use two
step splitting methods. One particularly popular version is the
following MacCormack scheme:
$$
\aligned
  \tilde u_j^n & = u^n_j - \lambda \Delta_{+} f(u^n_j) \,,
\\
  u^{n+1}_j & = \tfrac12(u^n_j + \tilde u_j^n) 
  - \lambda \Delta_{-} f(\tilde u_j^n) 
  + \lambda \Delta_{-}
 \bigl(\beta_{j+1/2}^n |\Delta_{+}f'(u^n_j)| \Delta_{+} u^n_j \bigr) \,.
\endaligned
\tag 6.1
$$
The convergence of this scheme is very much similar to what we have
did for the Lax-Wendroff scheme in Sections~3-5. We only state 
some convergence theorems and show those arguments that are different
from ones of the previous sections.

  We first rewrite the MacCormack scheme~(6.1) as follows
$$
  u^{n+1}_j = u^n_j + F^n_j + H^n_j + {\tilde H}^n_j + J^n_j \,, 
  \tag 6.2
$$
where
$$
\aligned
  F^n_j &= - \tfrac12 \lambda_n \Delta_{0} f(u^n_j) \,, 
\\
  H^n_j &= \tfrac12 \lambda_n^{\,2} 
  \Delta_{-} \bigl(\frac{(\Delta_{+}f(u^n_j))^2}{\Delta_{+}u_j^n} \bigr) \,, 
\\
  {\tilde H}^n_j &= \tfrac12 \lambda_n \Delta_{-} 
  \bigl(f(u^n_j) - f({\tilde u}^n_j) 
     - f'(u^n_j)(u^n_j - {\tilde u}^n_j) \bigr) \,, 
\\
  J^n_j &= \lambda_n \Delta_{-}\bigl(\tilde\beta_{j+1/2}^n 
  |\Delta_{+} a(u^n_j)| \Delta_{+} u^n_j \bigr) \qquad
  \tilde\beta_{j+1/2}^n = \beta_{j+1/2}^n + \lambda_n 
  \frac{a(u_j^n) - \Delta_{+} f_j/\Delta_{+}u_j}{|\Delta_{+} a_j|} \,.
\endaligned
\tag 6.3
$$
Clearly, we can carry out the analysis without any difference in
the absence of the term ${\tilde H}^n_j$. We only point out that 
${\tilde H}^n_j$ can only produce third-order terms and can be
controlled by the viscosity term based on the following facts.

  First, we notice that $\tilde u^n$ in~(6.1) is always in the interval
$[u_j^n, u_{j+1}^n]$ provided the CFL number is less than $1/2$. 
Using the Taylor expansion and~(6.1), one has
$$
  f(u_j^n) - f({\tilde u}_j^n) - f'(u_j^n)(u_j^n - {\tilde u}_j^n)
  = \tfrac12 f''(\xi_j^n)(u_j^n - {\tilde u}_j^n)^2
  = \tfrac12 \lambda^2 f''(\xi_j^n)(\Delta_{+} f(u_j^n))^4
  \tag 6.4
$$
Substituting~(6.4) into ${\tilde H}_j$ and using the summation by parts,
one has
$$
  \sum_j \eta'(u_j^n){\tilde H}_j^n  
  =-\tfrac14\lambda^3\sum_j\Delta_{+}\eta'_jf''(\xi_j^n)(\Delta_{+}f(u_j^n))^4 
  \le \lambda \eps_0^{\,2} C \sum_j \Phi(s_j^n)|\Delta_{+} u_j^n|^3 \,, 
  \tag 6.5
$$
$$
  \sum_j \eta''(u_j^n) ({\tilde H}_j^n)^{\,2} 
  = \tfrac14 \lambda^6 \sum_j \eta''(u_j^n)
     \biggl(\Delta_{-}\bigl(a'(\xi_j^n)(\Delta_{+}f(u_j^n))^2\bigr)\biggr)^2
  \le\lambda\eps_0^{\,5}C\sum_j\Phi(s_j^n)|\Delta_{+}u_j^n|^3 \,,
  \tag 6.6
$$
and
$$
  \sum_j \eta''(u_j^n) F_j^n {\tilde H}_j^n
   = - \tfrac18 \lambda^4 \sum_j \eta''(u_j^n)\Delta_{0} f(u_j^n) 
        \Delta_{-} \bigl(a'(\xi_j^n)(\Delta_{+} f(u_j^n))^2 \bigr)
  \le \lambda \eps_0^{\,3} C \sum_j \Phi(s_j^n) |\Delta_{+} u_j^n|^3 \,.
  \tag 6.7
$$

  Now we treat another two step version of the Lax-Wendroff
scheme---the Richtmyer scheme,
$$   
\aligned
  \tilde u_{j+1/2}^n & = \tfrac12 (u^n_j + u^n_{j+1}) 
  - \tfrac12 \lambda \Delta_{+} f(u^n_j) \,,
\\
  u^{n+1}_j & = u^n_j - \lambda \Delta_{-} f(\tilde u_{j+1/2}^n) 
  + \lambda \Delta_{-}
    \bigl(\beta_{j+1/2}^n |\Delta_{+}f'(u^n_j)|
        \Delta_{+} u^n_j \bigr) \,.
\endaligned
\tag 6.8
$$
We can rewrite it into the form
$$
  u^{n+1}_j = u^n_j + F^n_j + H^n_j + {\tilde H}^n_j + J^n_j \,, 
  \tag 6.9
$$
where
$$
\aligned
  F^n_j &= - \tfrac12 \lambda_n \Delta_{0} f(u^n_j) \,, 
\\
  H^n_j &= \tfrac12 \lambda_n^{\,2} 
  \Delta_{-} \bigl(\frac{(\Delta_{+}f_j^n)^2}{\Delta_{+} u_j^n}\bigr) \,, 
\\
  {\tilde H}^n_j &= \tfrac12 \lambda_n \Delta_{-} 
  \bigl(f(u^n_j) + f(u^n_{j+1}) - 2 f({\tilde u}^n_{j+1/2}) 
     - f'(u^n_{j+1/2})(u^n_j + u^2_{j+1} - 2 {\tilde u}^n_j) \bigr) \,, 
\\
  J^n_j &= \lambda_n \Delta_{-}\bigl(\tilde\beta_{j+1/2}^n
  |\Delta_{+} a(u^n_j)| \Delta_{+} u^n_j \bigr) \,,\qquad
  \tilde\beta_{j+1/2}^n + \lambda_n 
  \frac{a_j^n - \Delta_{+}f^n_j/\Delta_{+} u_j^n}{|\Delta_{+} u_j^n|} \,.
\endaligned
  \tag 6.10
$$
It is also easy to verify that the ${\tilde H}^n_j$ 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
Lax-Wendroff scheme in Sections~3-5 under the same assumptions.

\proclaim{Theorem 6.1} Let $u_h(x,t)$ 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.
\endproclaim

\proclaim{Theorem 6.2} Let $u_h(x,t)$ 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 artificial 
viscosity term~$(4.1)$.  Then there is a subsequence strongly converging 
to the weak solution of~$(1.1)$ satisfying the entropy inequalities.
\endproclaim

\proclaim{Theorem 6.3} Consider a general $2\times 2$ genuinely 
nonlinear and strictly hyperbolic system of conservation laws~$(1.1)$
endowed with global Riemann invariants. Let $u_h(x,t)$ 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\/$)$ $u_h$ such that 
$u_h(x,t) \to u(x,t) \quad\text{a.e.} \quad\text{as } h \to 0 $
and $u$ is the entropy solution of~$(1.1)$ satisfying 
$\del_t\eta(u) +\del_x q(u)\le 0 \quad\text{in } {\Cal D}'$ 
for any $C^2$ convex entropy pair $(\eta,q)$.
\endproclaim

\proclaim{Theorem 6.4} Consider a general system of hyperbolic 
conservation laws~$(1.1)$ endowed with a convex entropy. Let
$u_h(x,t)$ be the approximate solutions of the MacCormack scheme~$(6.1)$
or the Richtmyer scheme~$(6.8)$. If $u_h$ converges boundedly a.e.
$u(x,t)$, then $u(x,t)$ is the entropy solution of~$(1.1)$
satisfying $\del_t \eta(u) + \del_x q(u) \le 0 \quad \text{in } {\Cal D}'$ 
for any $C^2$ convex entropy pair $(\eta,q)$.
\endproclaim

{\bf 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 N00014-91-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
DE-FG02-88ER25053 and ARO Grant DAAL03-92-G-0143. This work is
initiated when the authors visited Stanford University supported in
part through Army grant DAAL 03-91-G-0017.

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\end

\enddocument 
