A New Convergence Proof for Finite Volume Schemes Using the Kinetic Formulation of Conservation Laws.
We give a new convergence proof for finite  volume schemes approximating scalar conservation  laws. The main ingredients of the proof are the kinetic formulation of scalar conservation  laws, a discrete entropy inequality, and the velocity averaging technique.
Introduction
. We consider the Cauchy problem for nonlinear hyperbolic
scalar conservation laws in several space dimensions.
#t
on the slab # := [0, T
for compactly supported initial data
(R d-1 ). We assume the
flux function f in C 1,1
loc (R) and As is well known, solutions of nonlinear
conservation laws may become discontinuous in finite time, so weak solutions must
be considered, i.e., functions
# L #) such that
#t
for all # D(#), where
x). As usual, we require an entropy condition (cf. Lax
[La'73]). For any entropy U # C 2 (R) we define the entropy flux
ds.
Then the entropy condition reads as follows: For all convex U and # D(# 0,
#t
A function
# L #) such that (1.2) and (1.4) hold for all convex entropies U
will be called a weak entropy solution of the Cauchy problem (1.1).
We are concerned with the convergence of approximations of
u by finite volume
schemes. This question has a history going back to the 1950s. Let us point out two
modern developments: The first is Kuznetsov's [Kz'76] approximation theory, which
was generalized by Vila [Vi'94] to first-order finite volume methods on unstructured
# Received by the editors September 30, 1997; accepted for publication (in revised form) February
26, 1999; published electronically February 1, 2000. This work was supported by Deutsche
Forschungsgemeinschaft, SFB 256 at Bonn University.
http://www.siam.org/journals/sinum/37-3/32806.html
Institut fur Angewandte Mathematik, Wegelerstrasse 10, 53115 Bonn, Germany (mwest@
iam.uni-bonn.de, noelle@iam.uni-bonn.de).
CONVERGENCE OF FINITE VOLUME SCHEMES 743
grids and by Cockburn, Coquel, and LeFloch [CCL'94] to higher-order schemes. Further
generalizations can be found in Cockburn and Gremaud [CG'96] and Noelle
[No'96]. The second approach is based on a uniqueness result for measure-valued
solutions due to DiPerna [Di'85], which was first applied to the analysis of numerical
schemes by Szepessy [Sz'89] and Coquel and LeFloch [CL'91, CL'93]. Cockburn,
Coquel, and LeFloch [CCL'95] and Kroner and Rokyta [KR'94] applied this theory
to first-order finite volume schemes, and Kroner, Noelle, and Rokyta [KNR'95] to
higher-order schemes. Noelle [No'95] extended these results to irregular grids, where
cells may become flat as h # 0, and to general E-fluxes, which include Godunov's
flux. Both Kuznetsov's and DiPerna's approaches rely on Kruzkov's existence and
uniqueness result [Kr'70].
In this paper, we give a convergence proof for finite volume schemes which does not
rely on [Kr'70]. Instead, our approach is built upon the recent kinetic formulation for
scalar conservation laws and the velocity averaging technique. As in all convergence
proofs, a discrete entropy inequality plays a crucial role.
The kinetic formulation was introduced by Lions, Perthame, and Tadmor [LPT'94].
They show that there is a one-to-one correspondence between the entropy solutions
of a scalar conservation law and solutions of a linear transport equation for which
a certain nonlinear constraint holds true. More precisely, one considers functions
depending on space-time and an additional v # R that solve the following equation:
#R
#t
R
for all # D(# R). Here
m is a bounded nonnegative measure defined on #R
and  # 0 are the initial data. This equation is supplemented with an assumption on the
structure of  #. If the function # is defined by
for all
should have the form
for some scalar function
u defined on #. (An analogous statement should hold for the
initial data.) Then we have the following equivalence (shown in [LPT'94]).
Theorem 1.1.
(i) Let
u be a weak entropy solution of problem (1.1). Then there is a bounded
nonnegative measure
m such that
solves the transport equation (1.5) for appropriate initial data. The measure
m is supported in #  [-
. Furthermore, we
have
744 MICHAEL WESTDICKENBERG AND SEBASTIAN NOELLE
Conversely, let
and a bounded nonnegative
measure
m be given, that solve the transport problem (1.5). Assume that
# can
be written as in (1.7) for some function  u. Then
u is a weak entropy solution
of the Cauchy problem (1.1).
Note that by definition
for su#ciently smooth #.
The second important ingredient of our proof is a discrete entropy inequality (cf.
Theorem 2.6 below). Here, we estimate the rate of entropy dissipation over each cell
in terms of the local oscillation of the numerical flux function. We refer to [KNR'95]
and [No'95]. It turns out that this result fits very neatly into the kinetic formulation
stated above.
Finally, our analysis relies on so-called velocity averaging lemmas first introduced
by Golse, Lions, Perthame, and Sentis [GLPS'88] and further developed by DiPerna,
Lions, and Meyer [DLM'91] and others. We refer to the survey article of Bouchut
[Bo'98] for more references and recent results.
The velocity averaging technique allows us to prove the strong compactness of a
sequence of approximate solutions u h of problems (1.2)-(1.4). The principal idea is
that the macroscopic quantity  u has more regularity than
# whose v-average it is.
The following result is a variant of Theorem B in [LPT'94].
Theorem 1.2. Let 1 < p # 2 and 0 < # < 1. Choose some test function
# D(R) and define # := spt . Assume there are sequences (# h ), (m h ), and
uniformly bounded in L p (R d
-# (R d )), respectively,
such that
#t
(R d
R).
If the following nondegeneracy condition now holds:
sup (#R d meas #
then the sequence z h := # R
belongs to a compact subset of L 1
loc (R d ).
Remark 1.3. Here stands for the space of strongly measurable, integrable
functions on # taking values in X, where X is some Banach space (cf. [DU'77]),
M(R d ) is the space of bounded Radon measures, and B 1,1
-# (R d ) is a Besov space
below). The assumptions on (# h ), (m h ), and are precisely
adapted to the estimates which we will derive in section 3.
Note that the nondegeneracy condition (1.10) (which we will assume throughout)
restricts the class of admissible flux functions: f should be nonlinear. Theorem 1.2 is
another instance of the fact that the nonlinearity of a problem can have a regularizing
e#ect on the solutions. Think of the transport operator as a directional
derivative along the vector (1, f # ). Then the partial regularity information contained
in (1.9) is transformed into compactness of the moments of # h , that is, of z h , as long
as a condition on the distribution of the directions (1, f # ) holds true. This is the heart
of the matter.
Condition (1.10) or some variant appears in many papers dealing with averaging
lemmas (see [Ger'90, LPT'94, Li'95, Bo'98] and the references therein). In our context,
CONVERGENCE OF FINITE VOLUME SCHEMES 745
it can be seen as a generalization of an assumption formulated by Tartar [Tar'83] in
his existence proof for scalar conservation laws in one spatial dimension.
The outline of the paper is as follows. In the next section we define a class of finite
volume schemes for the scalar conservation law (1.1) and state the main convergence
theorem. This theorem is proved in section 3. In the last section we outline the proof
of the velocity averaging result, Theorem 1.2.
2. A class of finite volume schemes. Let I be a countable index set and (T i ),
a family of closed convex polygons T i # R d-1 . We assume that the T i cover the
whole space, and that the intersection of two di#erent polygons consists of common
faces and vertices only. Define the mesh parameter h as sup i diam T i . Let (S ij ) be
the faces of be the corresponding outer unit normal vectors, and J i be their
number. Then we have
By definition, for every T i there is exactly one T k with T We denote
that polygon by T ij . Next choose
. Now the family of space-time prisms T n
I gives an unstructured mesh on #. We write S n
for faces normal to the time direction, while faces in spatial directions are called
ij .
Finally, we denote the polygon neighboring T n
i at the face S n
ij by T n
ij .
The finite volume approximation u h of the entropy solution  u will be piecewise
constant on the cells of an unstructured mesh with mesh parameter h. To keep the
notation simpler, we omit the index h in what follows. We write u(x) =: u n
almost all
ij ). The update formula is given by
for approximate fluxes g n
ij to be defined in a moment. The numbers
are taken as numerical initial data. It is well known that in this case the sequence of
approximate initial data converges strongly in L 1
loc (R d-1 ) to
The class of approximate fluxes, to which the convergence result given below
applies, is the class of so-called E-fluxes as introduced by Osher [Os'84]. An E-flux is
a family (g ij
-# R such that
(i) (consistency). For all v # R
(ii) (conservativity). If
(iii) (Osher's condition). For
One example of an E-flux is Godunov's flux
min
another one is the Lax-Friedrichs flux. Every E-flux can be obtained from these two as
a convex combination (cf. Tadmor [Tad'84]). We will restrict ourselves to Godunov's
flux in all that follows. Godunov's flux can be rewritten
is a family of piecewise continuous functions
-# R such that
R. Note that
Lipschitz-continuous and monotone, i.e.,
nondecreasing in the first and nonincreasing in the second argument.
In case of a first-order scheme, the approximate flux is now given by
It is also possible to consider higher-order schemes, but we will not do this here. The
approximate entropy flux corresponding to the entropy U is defined as
for all v 1 , v 2 # R. Obviously, G ij is consistent and conservative, too. Moreover, we
have the compatibility relation
Here # k stands for the partial derivative with respect to the kth argument, 2.
We use the notation
Update formula (2.1) can be recast in a somewhat di#erent form. We assume that
we are given numbers #x n
0 such that
Then we define
ij and # n
Now we can write
ij
ij , with u n+1
We refer to [No'95] for the optimal choice of the numbers # n
ij .
Theorem 2.1. Let (u h ) be a sequence of approximate solutions of (1.1) built
from the finite volume scheme described above. Assume that
CONVERGENCE OF FINITE VOLUME SCHEMES 747
(i) The sequence (u h ) is uniformly bounded for su#ciently small h
and has uniformly compact support in #.
(ii) There exists s # 1
such that for #t := inf n #t n
lim
(iii) There exists a constant # > 0 such that for #h/ #t and all i, j, n
(iv) The nondegeneracy condition (1.10) holds.
Then a subsequence of (u h ) converges strongly in L 1
loc (#) to a weak entropy solution
of the Cauchy problem (1.1).
Remark 2.2. We first remark that convergence can be shown not only for Go-
dunov's scheme but for the whole class of E-schemes. It is also possible to treat
higher-order schemes (see [NW'97]). Higher-order means that on each cell a polynomial
reconstruction of the solution is built using the values u n
i at a given time level.
In order to avoid oscillations near discontinuities, the reconstructions are stabilized
using h-dependent limiters. Then the values of these reconstructions at fixed quadrature
points on the cell faces are used in definition (2.4) of the approximate fluxes.
This approach goes back to the monotone upstream-centered scheme for conservation
laws (MUSCL) schemes of van Leer [VL'77]. It is shown in [CHS'90, KNR'95, Gei'93]
that such schemes may indeed be higher-order accurate in space.
Remark 2.3. Uniform boundedness (for spatially higher-order schemes) was shown
in Cockburn, Hou, and Shu [CHS'90] and Geiben [Gei'93]. We will not reprove this
here. Since we assume compactly supported initial data, u h will live on a bounded
set for all schemes with a finite speed of propagation, e.g., for standard finite volume
schemes with #t # Ch for some constant C not depending on h.
Remark 2.4. Theorem 2.1 contains no explicit assumption on the regularity of the
triangulation. Combining conditions (2.10) and (2.11) one obtains a mild restriction
on the geometry of the cells, which allows them to become flat in the limit as h tends
to zero. For a detailed analysis, see [No'95, No'96].
Remark 2.5. The convergence result stated in Theorem 2.1 is not new. A similar,
somewhat more general theorem was shown in [No'95] using DiPerna's theory of measure-valued
solution. Compare also [Sz'89, CL'91, CL'93, Vi'94, CCL'94, CCL'95,
KR'94, KNR'95, No'96] for related results. What is new and presented here is the
proof given below, using the kinetic formulation of conservation laws. Note that the
nonlinearity of the flux, in the form of assumption (1.10), is required.
In the proof of Theorem 2.1 the following discrete entropy inequality, which holds
for Godunov's flux as well as for other E-schemes, plays a prominent role in the
following theorem.
Theorem 2.6. For all convex entropies U # C 2 (R) and all i, j, n
Here # := min-M#v#M U # (v).
This entropy inequality was derived in [No'95]. Analogous estimates can be found
in [KR'94] for the Lax-Friedrichs and Engquist-Osher schemes and in [KNR'95, No'95]
for (spatially) higher-order schemes.
748 MICHAEL WESTDICKENBERG AND SEBASTIAN NOELLE
3. Proof of Theorem 2.1 (convergence). The proof consists of two steps.
First we construct an approximate distribution function # h from the numerical solution
u h and apply the transport operator to it. We split the resulting term into three
parts and give bounds for them in various norms. In the second step we use the velocity
averaging result (1.2) to show strong compactness of the approximate solution
u h and complete the proof.
3.1. Some estimates. Let us start with the definition of the distribution func-
tion. To simplify the notation we omit the index h most of the time. Extending u by
zero to R d we have
for almost all From the Gauss-Green theorem we get
#t
where
#(|0, u N
Here dS n
i is the (d-1)-dimensional Hausdor# measure restricted to S n
dH d-1
(same for S n
Note that in our notation the contribution from some cell face S n
is counted twice: S n
ij is the jth face of the cell T n
but also the lth face of some
neighboring cell T n
. We compensated that by the factor one half in (3.3). Now we
split R into three parts:
#(|0, u N
are defined in (2.8) and
To prove the identity R only have to check that2 # i#I
ij .
CONVERGENCE OF FINITE VOLUME SCHEMES 749
This follows easily from the properties of # and For fixed ij, let kl be the unique
index pair defined by S n
kl and i #= k. Then u
kl , u n
k , and
2 and definition (1.6)) we have
Using almost everywhere (a.e.) for all
we arrive at
This proves our claim. Let us take a closer look at the three parts of R. We
have R 0 because we extended u from # to R d . Note that the first summand in (3.4)
contains the numerical initial data. The second term R 1 is a measure for the entropy
production in the scalar conservation law. It corresponds to the right-hand side (RHS)
of (1.5). Finally R 2 is the residual. It measures the numerical error. In the following,
we will write # := [-M,M ].
Lemma 3.1. The R h
are uniformly bounded in L 1 (#, M(R d )).
Proof. Measurability follows from the tensor product structure of R h
and the boundedness is immediate from our assumptions on u h , e.g.,
Lemma 3.2. R h
1 can be written as
R h
in D # (R d
for some nonnegative uniformly bounded measure m h .
Proof. We suppress the mesh index h. Clearly, to obtain (3.5) we may simply
integrate R 1 in the kinetic variable. Using overbars to indicate primitives, as in
-#
we arrive at the representation
ij .
Note that R 1 vanishes outside the interval [-M,M ]. Therefore, m n+1
we have (using (1.8) and (2.3)-(2.4))
ds
which vanishes again because of (2.9). Note that # J i
1. We conclude that
m is compactly supported in R d
[-M,M ]. Now let us fix i, n for a moment. We
choose a test function U # C 2 (R) which is convex on [-M,M ] (a convex entropy)
and apply its second derivative to m n
. Integrating by parts and using compatibility
relation (1.3) (and (1.8) again) we find
(Remember that m n+1
i has compact support.) This quantity can be controlled using
the discrete entropy inequality in Theorem 2.6. In fact, from representation (2.9) and
Jensen's inequality we obtain
So, if we choose a sequence of convex entropies U k with
a.e.,
we find from the dominated convergence theorem
(v) dv
Since this holds for all i, n we conclude that m is a nonnegative measure as claimed.
To show the boundedness of m, we use (3.6) with U(v) := 1v 2 to obtain
But for all index pairs such that
|S kl | G n
kl
because the approximate entropy flux is conservative. Hence
CONVERGENCE OF FINITE VOLUME SCHEMES 751
Furthermore, we have
Therefore, the j-sum in (3.7) drops out if we sum over all cells. The remaining
)-terms, however, appear twice with alternating signs and therefore cancel out,
too, except for those with . Since the entropy U is nonnegative we
finally arrive at
(We used (2.2) and Jensen's inequality.) The lemma is proved.
Definition 3.3. Let # 0 # D(R d ) be a nonnegative radially symmetric test function
which equals 1 on the ball B(0, 1) and vanishes outside B(0, 2). Define
for 2. Introduce the dyadic operators
#}. Then the Besov
space B p,q
s (R d ) with s # R and 1 # p, q # contains all tempered distributions on
R d such that the norm
s (R d
(R d )
(modified if more details, see Triebel [Tr'83]).
Lemma 3.4. Let
lim
-# (R d
Remark 3.5. Note that the Besov space B 1,1
-# (R d ) can be identified with the
topological dual of the closure of D(R d ) in C # (R d ) (the space of Holder continuous
Proof. Again we suppress the index h. First we show that for all i, j, n
-# (R d
We apply a test function # D(R d ) and obtain by definition of # n
with # n+1
ij the averages of # over the cell faces S n+1
i and S n
ij . Then
(R d ) ,
where
ij is the evaluation of # in the center of mass of S n
ij .
Next, we must control the L 1 -norm of # n
ij . For an arbitrary U # C 1 (R) we have
The first identity follows as above from the compatibility relation (1.3) and (1.8)
(consult also (2.5) and (2.7)). For the second we used the consistency and Lipschitz-
continuity of the approximate entropy flux G ij . To proceed we now replace the derivative
of G ij by (2.6). Since Godunov's flux is nonincreasing in the second argument,
the derivative of g ij has a sign and we can estimate
#
using the consistency of g ij and (2.4). Note that we do not assume convexity for U .
Since the measurability of # n
ij is obvious we learn that for all indices i, j, n
Now the norm of R 2 can be bounded by
-# (R d
and further, using the Cauchy-Schwarz inequality, by
i is the characteristic function of the set of indices i, n for which u n
i is nonva-
nishing. Note that by assumption, the support of the numerical solution is uniformly
bounded. These terms can be handled easily: First we have
#t
Moreover, from Theorem 2.6 with U(v) := 1v 2 we find
CONVERGENCE OF FINITE VOLUME SCHEMES 753
By definition,
Therefore (cf. (2.9))
ij
which is Jensen's inequality. We proceed as in the proof of Lemma 3.2 (cf. (3.7)) and
arrive at
-# (R d
#t#
(R d-1 ) .
Note that 1/# explodes as h # 0. But
#t#
finally we obtain
-# (R d
#t
Using assumption (2.10) we are finished.
Remark 3.6. We stop here for a moment to summarize what we have shown so
far. Since (# h ) is uniformly bounded in L # a subsequence converges weak* to some
function
#. Associated with there is a sequence (R h ) as defined above. Given
# D(#) and U # D(R) we have
#t
R h ,
The first term on the RHS goes to
For the second, we have shown in Lemma 3.2 that (m h ) is uniformly bounded and
nonnegative in the sense of measures. Extracting another subsequence, if necessary, we
have
The third term finally goes to zero in distributional
sense (even in a somewhat stronger topology) as shown in Lemma 3.4. Therefore the
m) solves the transport equation (1.5).
So far, the strategy of proof is similar to that of the first statement of Theorem
1.1 (see [LPT'94]), where it is shown that an entropy solution  u leads to a nonnegative
bounded measure
m such that
# defined by (1.7) satisfies the transport equation (1.5).
What remains to be done is to prove that the nonlinear constraint (1.7) holds for the
limit of the sequence (# h ). In this case, the limit defines a function  u which, according
to the second part of Theorem 1.1, is a weak entropy solution. For this we use the
velocity averaging technique and show that (some subsequence of)
strongly in L 1
loc .
3.2. End of proof. To apply Theorem 1.2, we choose a test function # which
equals 1 on the interval [-M,M ]. Then we have z using (1.8) and (3.1). Define
. Because of Lemmas 3.1 and 3.4, and since the space of measures
is continuously embedded into
-# (R d ) (cf. [Tr'83]), # h satisfies the assumptions of
Theorem 1.2. Moreover,
(R d #u h
which is uniformly bounded, too. But then Theorem 1.2 shows that u h belongs to a
compact subset of L 1
loc (R d ). Since
|#
- u h2
|,
the approximate distribution function # h converges strongly in L 1
loc
(R d
R) (up to
a subsequence). Hence, the nonlinear constraint (1.7) holds for the limit
#. From
Theorem 1.1 we conclude that  u is a weak entropy solution.
Remark 3.7. One classical approach to proving strong compactness for sequences
of approximate solutions consists in establishing a uniform bound on the total variation
and then making use of Helly's theorem. For the more modern approach relying
on measure-valued solutions, as introduced by DiPerna, no such control is neces-
sary. Once one has shown consistency with the entropy condition, the L 1 -contraction
ensures compactness. The result presented in this paper lies somewhere in between
these two cases. In fact, we do need some control over the residual, but this bound is
comparatively easy to obtain, since we can choose a very weak topology.
4. Proof of Theorem 1.2 (velocity averaging). For the sake of completeness,
we would like to give an outline of proof for Theorem 1.2. We will skip most details
since the arguments are technically involved and can be found in other papers on
velocity averaging.
Let us fix some test function # D(R) and denote the RHS of (1.9) by R h . Then
we can recover # h from R h (formally) by inverting the transport operator
#)
# for all v # R, #,
(the Fourier transform is taken with respect to space-time only). But now we face
the problem that the symbol -i(#
#) becomes unbounded. We will need
a splitting. Let # D(R) be a nonnegative even test function, vanishing outside the
interval [-2, 2], with we define two operators:
|#|
#, v) (v) dv #
for some parameter # (0, #), and
|#|
#, v)
#)
(v) dv # .
Note that the inverse symbol
#) -1 appears in (4.1), but because of the
cut-o# function # it is e#ective only in the region
#,
CONVERGENCE OF FINITE VOLUME SCHEMES 755
i.e., outside a neighborhood around the singular set. Therefore, it is reasonable to
expect that B # has nice properties. Let # := spt #. Then we have the following
lemma.
Lemma 4.1. There exists a constant C not depending on # (0, #) such that
(R d
(R d
#)
for all # L p (R d
1. The function # is given by
#) := sup #,
# .
Remark 4.2. We assumed that the nondegeneracy condition (1.10) holds. It is
easy to show that in that case # 0 as # 0 [Ger'90]. As a consequence, the
function A #) for suitable # becomes small in L p -norm if we let # go to zero.
Definition 4.3. The generalized (fractional) Sobolev space H p
s (R d ) is defined
as the space of all tempered distributions such that the norm
s (R d ) := #(Id - #) s/2 # L p (R d )
stays finite. For more details consult [Tr'83].
Lemma 4.4. Let 1 < p # 2, # (0, #). Then we have for all # L p (R d
#)
(R d (R d # C # L 1 (#,L p (R d )) .
C # grows as # 0. The same estimate holds for the operator
These two lemmas are shown as in [DLM'91] (compare (4.2) and (4.3) with the estimates
(22) and (23) in that paper) with the modifications explained in the appendix
of [LPT'94]. Note that the operators B # , B #
are smoothing. We gain one derivative.
We will now prove Theorem 1.2 from these two results. First we note that the
dyadic operators S j (consult Definition 3.3) commute both with B # , B # and with
(Id - #) 1/2 . It is then a simple application of Minkowski's inequality to rewrite (4.3)
using Besov norms:
s (R d
s (R d
for all 1 < p # 2, 1 # q #, and s # R. But the operator (Id - # x ) 1/2 defines an
isomorphism (a lifting) between Besov spaces of di#erent regularity [Tr'83, 2.3.8]:
s (R d
1+s (R d ) .
So we conclude that B # maps
s (R d
1+s (R d ). The same holds for
# . Now for any # (0, #) we have the splitting
z
Denote by z #,h
0 the first term on the RHS of (4.6) and by z #,h
1 the terms in brackets.
As already pointed out in Remark 4.2
z #,h
0 can be made arbitrarily small in L 1
loc (R d ) uniformly
with respect to h by choosing # small enough.
Moreover, we have
z #,h
1 is strongly compact in L 1
loc (R d ) for all #.
To see this, we choose a p > 1, p near 1 such that the number #
than 1 (which is always possible since # < 1), and use the continuous embedding
-# (R d
-# (R d )
(cf. [Tr'83, 2.3.2 and 2.7.1]) to show that
are uniformly bounded in
-# (R d )).
We conclude from (4.4) and (4.5) that z #,h
1 is uniformly bounded in some Besov space
with strictly positive regularity and therefore relatively compact in L 1
loc (R d ). But
then the same is true for the sequence (z h ). This proves our claim.



--R

"Equation Cinetiques,"
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Convergence of the finite Volume
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The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case
Convergence of finite di
Convergence of finite di

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regularity of velocity averages
Convergence of MUSCL-Type Upwind Finite Volume <Volume>Schemes</Volume> on Unstructured Triangular Grids

Regularity of the moments of the solution of a transport equation
Convergence of upwind finite Volume
Convergence of higher order upwind finite Volume
First order quasilinear equations with several independent variables
Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation
Hyperbolic systems of conservation laws and the mathematical theory of shock waves

Kinetic formulation of multidimensional scalar conservation laws and related equations
Convergence of higher order finite Volume
A note on entropy inequalities and error estimates for higher order accurate finite Volume
A New Convergence Proof for Finite Volume
the entropy condition
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Numerical viscosity and the entropy condition for conservative di
The compensated compactness method applied to systems of conservation laws
Theory of Function Spaces

Convergence and error estimates in finite Volume
--TR
conservation laws;finite volume scheme;velocity averaging;kinetic formulation;convergence;entropy solutions;discrete entropy inequality