Convergence Rates for Relaxation Schemes Approximating Conservation Laws.
In this paper, we prove a global error estimate for a relaxation scheme approximating scalar conservation laws. To this end, we decompose the error into a relaxation error and a discretization error.  Including an initial error $\omega(\ep)$ we obtain the rate of convergence of $\sqrt{\ep}$ in L1 for the relaxation step. The estimate here is independent of the type of nonlinearity. In the discretization step a convergence rate of $\sqrt{\Del x} $ in L1 is obtained. These rates are independent of the choice of initial error $\omega(\ep)$. Thereby, we obtain the order 1/2 for the total error.
Introduction
be the unique global entropy solution in the sense of Kru-zkov [11] to
the Cauchy problem for the conservation law
with initial data
(IR). The solution u satisfies Kru-zkov's
entropy conditions
IAN, Otto-von-Guericke-Universit?t Magdeburg, PSF 4120, D-39016 Magdeburg, Germany. Supported
by an Alexander von Humboldt Fellowship at the Otto-von-Guericke-Universit?t Magdeburg. Email:
hailiang.liu@mathematik.uni-magdeburg.de
y IAN, Otto-von-Guericke-Universit?t Magdeburg, PSF 4120, D-39016 Magdeburg, Ger-
many. Supported by the Deutsche Forschungsgemeinschaft (DFG), Wa 633 4/2, 7/1. Email:
gerald.warnecke@mathematik.uni-magdeburg.de
We are considering here a relaxation scheme, proposed by Jin and Xin [10] to compute
the entropy solution of (1.1) using a small relaxation rate ffl. Our main purpose is to
study the convergence rate of the relaxation scheme to the conservation laws as both the
relaxation rate ffl and the mesh length \Deltax tend to zero.
The relaxation model takes the form
The variables u ffl and v ffl are the unknowns, ffl ? 0 is referred to as the relaxation rate and a
is a positive constant. The system (1.4) was introduced by Jin and Xin [10] as a new way
of regularizing hyperbolic systems of the same kind as the scalar equation (1.1). It is also
the basis for the construction of relaxation schemes.
In fact, for small ffl, using the Chapman-Enskog expansion [4], one may deduce from
(1.4) the following convection-diffusion equation
x
see [10], which gives a viscosity solution to the conservation law (1.1) if the well known
subcharacteristic condition (cf. Liu [16]) holds:
a
Natalini [20] proved that the solutions to (1.4) converge strongly to the unique entropy
solution of (1.1) as ffl ! 0. Thus the system (1.4) provides a natural way to regularize the
scalar equation (1.1). This is in analogy to the regularization of the Euler equations by the
Boltzmann equation, see Cercignani [5].
Consider the grid sizes \Deltax, \Deltat in space and time as well as for n 2 IN, j 2 ZZ a numerical
approximate solution (u n
n\Deltat). The relaxation scheme associated with
the relaxation system (1.4) is given as
\Deltax is the mesh ratio,
a- and
" . We refer to Aregba-Driollet and
Natalini [1] for a class of fractional-step relaxation schemes for the relaxation system (1.4).
The first order scheme proposed there can be easily rewritten in the form of (1.6) with
where the homogeneous (linear) part is treated by some monotone
scheme and then the source term can be solved exactly due to its particular structure.
Throught this paper we assume the usual CFL condition
Convergence theory for this kind of relaxation schemes can be found in Aregba-Driollet
and Natalini [1], Wang and Warnecke [31] and Yong [32]. Based on proper total variation
bounds on the approximate solutions, independent of ffl, convergence of a subsequence of
to the unique week solution of (1.1) was established by standard compactness
arguments.
Currently, there are only very few computational results for relaxation schemes available
in the literature, see e.g. Jin and Xin [10], who introduced these schemes, as well as Aregba-
Driollet and Natalini [2]. Therefore, it is very hard to tell how useful they may be for
practical computations in the future. The main advantages of these schemes are that they
neither require the use of Riemann solvers nor the computation of nonlinear flux Jacobians.
This seems to become an important advantage when considering fluids with non-standard
equations of state, e.g. in multiphase mixtures. Note also that an extension of our results
to second order schemes seems easily feasible since we are only making use of the TVD
property which can be achieved using flux limiters, see Jin and Xin [10].
The relaxation approximation to conservation laws is in spirit close to the description of
the hydrodynamic equations by the detailed microscopic evolution of gases in kinetic theory.
The rigorous theory of kinetic approximation for solutions with shocks is well developed
when the limit equation is scalar. For works using the continuous velocity kinetic approxi-
mation, see Giga and Miyakawa [9], Lions, Perthame and Tadmor [17] and Perthame and
Tadmor [19], for discrete velocity approximation of entropy solutions to multidimensional
scalar conservation laws see Natalini [21], also Katsoulakis and Tzavaras [13]. Based on a
discrete kinetic approximation for multidimensional systems of conservation laws [21], the
authors in [2] constructed a class of relaxation schemes approximating the scalar conservation
laws.
It was pointed out by Natalini [21] that the relaxation system (1.4) can be rewritten into
the two velocities "kinetic" formulation by just setting defining
the Riemann invariants
a
a
and the Maxwellians
a
a
The relaxation model (1.4) becomes
@ t R ffl
2: (1.8)
The relaxation rate plays the role of the mean free path in kinetic theories. Indeed the
system (1.8) provides more insight into the properties of the relaxation system. In our
investigation of the convergence rates we will use this formulation for the relaxation model
as well as for the corresponding relaxation scheme.
The main goal of this paper is to improve on the previous convergence results, see
Aregba-Driollet and Natalini [1], Wang and Warnecke [31] and Yong [32] for the relaxation
scheme (1.6) by looking at the accuracy of the relaxation scheme for solving the conservation
law (1.1). We do this here by studying the error of approximation
between the exact
solution u and the numerical solution u ffl
measured in the L 1 norm. The parameters ffl and
\Deltax determine the scale of approximation and converge to zero as the scales become finer.
We shall call the order of this error in these parameters the convergence rate of the numerical
solution generated by the relaxation scheme.
To make this point precise, we choose the initial data for (1.4) as
continuous and
we allow for an initial error K(x)!(ffl) instead of v ffl
we want to see the
contribution of this initial error to the global error. We mention that it is possible to
consider perturbed data in the u-component, then in the final result an initial error ku ffl
would persist in time and may prevent the convergence of u ffl to the entropy
solution, as shown in Theorem 3.4. However, the initial error in the v-component persists
only for a short time of order ffl, thereby it does not prevent the convergence of u ffl .
We initialize the relaxation scheme (1.6) by cell averaging the initial data (u ffl
in the
usual way
Z
Here and elsewhere
R without the integral limit denotes the integral on the whole of IR,
and - j (x) denotes the indicator function - j (x) := 1 fjx\Gammaj \Deltaxj-\Deltax=2g .
Let us now introduce some notations. The L 1 -norm is denoted by k
denotes the total variation, defined on a
subset\Omega ' IR by
dx:
The BV-norm is defined as
For grid functions the total variation is defined by
denotes the discrete l 1 -norm
Taking initial data (1.9), we summarize our main convergence rate result by stating
Theorem 1.1. Take any T ? 0 and let for a suitable N 2 IN and time step \Deltat the relation
\Deltat be satisfied. Further let u be the entropy solution of (1.1)-(1.2) with initial
data u 0 in L 1 constant representation on
IR \Theta [0; T ] of the approximate solution (u n
generated by the relaxation scheme
with initial data satisfying (H 1 ) and (1.9). Then for fixed
\Deltax satisfying the CFL
condition (1.7) there exists a constant C T , independent of \Deltax; \Deltat and ffl, such that
\Deltax
Theorem 1.1 suggests that the accumulation of error comes from two sources: the
relaxation error and the discretization error. The theorem will be a consequence of Theorem
2.2, giving a rate of convergence to the unique entropy solution of (1.1) in the relaxation
step of the solutions to the relaxed system (1.4), as well as Theorem 2.3, giving
a discretization error bound for the relaxation scheme (1.6) as an approximation to the
relaxation system (1.4).
It may be helpful, at the outset, to explain the structure of the proof. Consider that the
relaxation scheme was designed through two steps: the relaxation step and the discretization
step. Our basic idea is to investigate the error bound of the two steps separately and
then the total convergence rate by combining the relaxation error and discretization error.
The basic assumptions and the error bounds of the two steps will be given in detail in
Section 2.
We split the error e ffl
relaxation error e ffl with
ffl,
and a discretization error e \Delta with
\Deltax, i.e., we have the decomposition
with
In order to get the desired approximate entropy inequality, we work with the reformulated
system (1.8), in place of the original system (1.4).
We would like to mention that an analogous result for a class of relaxation systems was
already obtained by Kurganov and Tadmor [12] by using the Lip 0 -framework initiated by
Nessyahu and Tadmor [22]. But their argument uses the convexity of the flux function.
For the case of a possibly nonconvex flux function f , our work uses Kuznetzov-type error
estimates, see [15] and [3]. Recently, Teng [29] proved the first order convergence rate for
piecewise smooth solutions with finitely many discontinuities with the assumption of convex
fluxes f(u). Based on Teng's result, Tadmor and Tang [28] provided the optimal pointwise
convergence rate for the relaxation approximation to convex scalar conservation laws with
piecewise solutions. They use an innovative idea that they introduced in their paper [27]
which enables them to convert a global L 1 error estimate into a local error estimate.
In the discretization step the same error bound was obtained by Schroll, Tveito and
Winther [25] for a model that arises in chromatography, their argument is in the spirit of
Kuznetsov [15] and Lucier [18]. The results in [25] rely on the assumption that the initial
data are close to an equilibrium state of order ffl, i.e. Our result shows that the
uniform estimate does not depend on !(ffl), which is more natural since in the discretization
step ffl is kept a constant. Taking discretization step, we immediately recover
the optimal convergence rate of order 1=2 for monotone schemes, see Tang and Teng [26],
Sabac [23]. We thank a referee for pointing out the possibility of an extension of the present
arguments to multi-speeds kinetic schemes introduced in [2], even in the multidimensional
case [21].
The paper is organized as follows. In Section 2 we state the assumptions on the system
and recall properties of the solutions to (1.4) and the scheme (1.5)-(1.6). The main
results on the relaxation error and discretization error are given. Their proofs are presented
in Section 3 and Section 4, respectively. The authors thank a referee for bringing the paper
[14] to their attention. In that paper, a discrete version of the theorem by Bouchut and
Perthame [3], see Theorem 3.3 in Section 3, is established and convergence rates for some
relaxation schemes based on the relaxation approximation proposed in [13] were considered.
Preliminaries and Main Theorems
In this section we review some assumptions and the analytic results concerning the relaxation
model and relaxation schemes. After further preparing the initial data, we state our
main theorems.
Let us first recall some results obtained by Natalini [20] concerning the analytical properties
of the problem (1.4) with the specific initial data (u ffl
0 ), which will be of use in our
error analysis. Let us make the following assumptions:
the flux function f is a C 1 function with
data satisfy
there exist constants
not depending on ffl such that
sup
ffl?0
ffl?0
and for the flux function f as well as K given in (H 1 )
x6=y
us define, for any ae ? 0, the notations
F (ae) := sup
and
Theorem 2.1. (Natalini [20]) Assume
a
then the system (1.4) admits a unique, global solution
loc
isfying
for all ffl ? 0 and for almost every (x; t) 2 IR\Theta]0; 1[.
We refer to Natalini [20] for detailed discussions on the existence, uniqueness and convergence
of solutions to the relaxation model (1.4).
Equipped with assumptions in (H 1 )-(H 3 ), it has been proved that, as
solution sequence to (1.4) converges strongly to the unique entropy solution of (1.1)-(1.2),
see Natalini [20]. We will study the convergence rate in Section 3, our main result on the
limit ffl # 0 is summarized in the following theorem.
Theorem 2.2. Consider the system (1.4), subject to L 1
data satisfying (H 1 )-(H 3 ). Then the global solution converges to (u; f(u)) as ffl # 0
and the following error estimates hold,
Thus, (2.5) reflects two sources of error: the initial contribution of size !(ffl) and the
relaxation error of order ffl. It should be mentioned that, however, the effect of initial
contribution persists only for a short time of order ffl, beyond this time the nonequilibrium
solution approaches a state close to equilibrium at an exponential rate. Note that the proof
of estimate (2.5) is included in the proof of Lemma 3.2 below. The proof of (2.4) will follow
from Theorem 3.4.
Now we turn to the formulation of the relaxation schemes.
In order to approximate the solutions of the initial value problem (1.4), we first discretize
the initial data (u ffl
To make this more precise, we
denote a family of approximate solutions given as piecewise constant functions, dropping
the superscript ffl for notational convenience, by
represents the number of time steps performed. For the initial conditions
we take the orthogonal projection of the initial data (u ffl
the space of
piecewise constant functions on the given grid
\Deltax
Z
\Deltax
Z
Thus it follows from (H 1 that the discrete initial data satisfy
(a)
(b)
(c)
The grid parameters \Deltax and \Deltat are assumed to satisfy \Deltat
\Deltax
Const. We note that, since
assumed constant, \Deltax ! 0 implies \Deltat ! 0 as well.
It is well known that the projection error is of order \Deltax. More precisely, we have
As was shown by Aregba-Driollet and Natalini [1] as well as Wang and Warnecke [31], for
a large enough constant a a uniform bound for the numerical approximations given by the
scheme (1.6) can be found. Precisely, there exists a positive constant M(ae 0 ) such that if
a
then the numerical solution satisfies
ae
a
oe
where B(ae 0 ) is a constant depending only on ae 0 .
Starting with the discrete initial data satisfying (2.9)-(2.11), by using the Riemann
invariants, the TVD bound of the approximate solutions of (1.6) was proved previously
by Aregba-Driollet and Natalini [1], Wang and Warnecke [31] and Yong [32]. By Helly's
compactness theorem, the piecewise constant approximate solution (u
converges strongly to the unique limit solution (u; v)(x; t n ) as we refine
the grid taking \Deltax # 0. This, together with equi-continuity in time and the Lax-Wendroff
theorem, yields a weak solution of the relaxation system (1.4). We note that the
initial bounds (2.9)-(2.11) still hold for the piecewise constant numerical data (u 0
which
We refer to [1], [31] and [32] for the convergence of approximate solutions (u
the unique weak solution of (1.4). Our goal here is to improve the previous convergence
theory by establishing the following L 1 -error bound of the relaxation scheme. This theorem
will be proved in Section 4.
Theorem 2.3. Let be the weak solution of (1.4) with initial data (u ffl
be a piecewise constant representation of the data (u N
generated by (1.6)
starting with (u 0
fixed there is a finite constant C T
independent of \Deltax, \Deltat and ffl such that
\Deltax:
Remark 1. Our uniform error bound is independent of the relaxation parameter ffl and
initial error !(ffl). This is more general than the result obtained by [25] assuming the initial
error !(ffl) to be ffl. 2
As remarked earlier, the error is split into a relaxation error e ffl and a discretization
error e \Delta , combining the two errors we arrive at the desired total error for the scheme (1.6)
as stated in Theorem 1.1.
3 Relaxation Error
In this section we establish Kuznetzov-type error estimates for the approximation of the
entropy solution of the scalar conservation law
by solutions of the relaxation system
For the above purpose we have to show that u ffl satisfies an approximate entropy con-
dition. Let us begin with rewriting (3.2) in terms of the Riemann invariants (R ffl
Maxwellians defined in the introduction and recall some basic facts that
will be used in our analysis. It is easy to see that
a
Then the system is rewritten as (1.8), i.e.,
@ t R ffl
\Theta

where the functions M i (u) have the following propertiesX
We note that the L 1 estimate obtained by Natalini [20] implies that
ae
a
oe
Further, we set
I ae 0
g:
The M i (u) are monotone (non-decreasing) functions of
because due to the subcharacteristic
condition
d
du
a
0:
Thus we have for any u ffl and ~
in I ae 0X
Starting with the initial distribution
a
a
the model evolves according to the system (1.8), which is well-posed. In fact, we rewrite
(1.8) in the form
@ t R ffl
2: (3.8)
From (3.8) one obtains the L 1 -contraction property using a Gronwall inequality, see [21],
which isX
As shown in [20], the above nice property for general data of bounded variation yields the
following
Lemma 3.1. For any ae 0 ? 0 and ffl ? 0, if
a
then the global solution (R ffl
loc of the problem (1.8), (3.7) for any
satisfies the entropy-type inequality
- 0; in D
Proof. See Natalini [21, Proposition 3.8].
Before establishing the desired convergence rate in Theorem 2.2, we first need a Lemma
giving a bound on the distance of (R ffl
2 ) from equilibrium. This bound is actually equivalent
to the estimate (2.5). The following lemma is a generalization of Natalini [20,
Proposition 4.7].
Lemma 3.2. Suppose that the assumptions (H 1 be a solution of
with initial data (u ffl
holds that
Proof. In view of the relations between (R ffl
which serves as a measure of the distance from equilibrium. Let - then the
function - ffl satisfies
x
for data
Then multiplying by sgn(- ffl )e t
ffl and integrating over IR \Theta [0; t] one gets
Z
Z
Z tZ
e
\Gamma(t\Gammas)
x jdxds:
Note that by (3.10) we have for sufficiently regular initial data
For example one could use C 1 data in combination with an approximation theorem in
BV (IR), see the Theorem 1.17 in Giusti [8]. Combining the above facts, for general data
of bounded variation, gives
Together with (H 3 ) and (3.12), this estimate implies the result as asserted.
Equipped with Lemmas 3.1-3.2, now we turn to the proof of our main result.
Our further analysis uses a result by Bouchut and Perthame [3, Theorem 2.1]. We first
state in a less general form their central result.
Theorem 3.3. (Bouchut and Perthame) Let u; v 2 L 1
loc
loc
(IR)) be right continuous
with values in L 1
loc (IR). Assume that u satisfies the entropy condition (1.3) and v
satisfies, for all k 2 IR,
are locally finite Radon measures such that for some nonnegative k-independent
Radon measures ff J and ff H 2 L 1
loc ([0; 1[; L 1
loc (IR)), satisfy in the sense of measures
Then for any T ? 0; x and the balls
we have
Z
Z
where C is a uniform constant and
sup
Z
ff J (x; t)dx;
Z TZ
Based on this general result we will prove
Theorem 3.4. Under the assumptions of Lemma 3.2. Let -
u be the entropy solution of
(1.1) with initial data - u 0 (x). Then, for any fixed T ? 0 and all t - T ,
ffl:
Here, C is a positive constant depending on the flux function f and the L 1 -norm of the
data
Proof. Since M i (u) is monotone for we have by (3.6)
where
The term J ffl is bounded from above by
due to the inverse triangle
inequality. Let
ff J =X
then we have jJ ffl j - ff J and by Lemma 3.2
loc ([0; 1[; L 1
loc (IR)) with kff J
Setting
we similarly get using (3.4)
Due to the monotonicity property of M i , we have
Using this fact and the inverse triangle inequality one obtains
Therefore, we also have
loc ([0; 1); L 1
loc
Combining the previous expressions (3.15) and (3.17) yields
\Theta
However by Lemma 3.1 the Riemann invariants (R ffl
the entropy-type inequalities
consequently one obtains
\Theta
The functions J are bounded by the L 1 -functions ff J , ff H respectively as required in
(3.14). Since the solutions
are bounded and the flux function f is assumed to be
Lipschitz, we may apply Lemma 3.3. Letting ae !1, Lemma 3.3 gives
Z
Z TZ
a
Z
to minimize the right hand side of (3.19), then
we have by choosing a suitably large constant C T , including k-u 0 k BV , that
r
sup
Note that the estimate in Lemma 3.2 yields
If !(ffl) - ffl, the above in combination with (3.20) yields Theorem 3.4.
Next we treat the case !(ffl) ? ffl. To obtain the desired estimate we again apply Lemma
3.3 on the interval [-; T ] with - ? 0 to be determined, thus we have as above
r
sup
Using the fact that both u ffl and -
u lie in a bounded subset in Lip(IR
and Tzavaras [13] and Smoller [24], we have
It follows from the estimate in Lemma 3.2 that
Taking applying (3.22) and (3.23) into (3.21) one gets
choosing easily recover the order 1=2 estimate from (3.24) and
ffl:
This completes the proof.
4 Discretization Error
The purpose of this section is to derive the error estimate given in Theorem 2.3. Let us
define the computational cells as
I
We will prove that the error bound for the relaxation scheme (1.6) approximating the
relaxation system (1.4) is of order
\Deltax in L 1 , without requiring that the initial data be
close to equilibrium.
To this end, let us rewrite the relaxation scheme as a splitting or fractional step method
in terms of the Riemann invariants (R ffl
Dropping the superscript ffl and noting that
ffl , the scheme takes the form
R n+1
for the source terms while the intermediate states (R n+1=2
2;j are generated by the
following consistent monotone scheme in conservative form for the convections, namely the
upwind scheme,
R n+1=2
R n+1=2
The discrete values (R n
computed from (4.1) are considered as approximations of
i in the whole cell I j at time t n\Deltat, which can also be obtained through (u n
generated by (1.6). The discrete variables are related to each other by
a
a
and
a
a
Conversely
can also be expressed as
For the initial conditions R 0
i;j we take the orthogonal projection of the initial data R ffl
onto the space of piecewise constant functions on the given grid. It is given by the averages
\Deltax
Z
for each integer j.
In the previous studies concerning relaxation schemes, some important properties for
the numerical scheme were obtained through investigating the reformulated scheme using
the Riemann invariants. These properties include: the L 1 boundedness, the TVD property
and the L 1 continuity in time. Here we also rewrite the relaxation scheme in terms of the
Riemann invariants because they provide more insight into the convergence behavior of the
scheme. The above properties for our scheme are summarized as follows and will be used
in the later error analysis.
Lemma 4.1. Suppose that the initial conditions (u ffl
are bounded and of bounded vari-
ation, both uniformly with respect to ffl. Further, assume that the initial data also satisfy
i.e. as a consequence
then there exists a constant C 0 , independent of ffl and \Deltat such that for
ffl and K ae 0
as
defined by (3.5)
(a) (R n
; for all (j; n) 2 ZZ \Theta IN;
(b)
(R n
(c) kM i
a \Deltat
(d)
Proof. The proof of (a) can be found in Wang and Warnecke [31]. The proof of (b) and
(d) is straightforward, see Aregba-Driollet and Natalini [1] and Yong [32]. Since here we
choose initial data satisfying (H 1 ) instead of v 0
(c) as follows.
It follows from (4.3) and (4.4) that Z n
Summation of the scheme
and noting that u n
i;j , we obtain
Again let us consider (4.1) for
R n+1
Adding \GammaM 1 (u n+1
j ) on both sides gives
\GammaZ n+1
by which we obtain
Using the first equation of scheme (4.2) and the definition of Z n
Z n+1=2
Then by the Mean Value Theorem there exists a value ~
between u n+1=2
and u n
such that for 0 -
due to the monotonicity property of M 1 (u). Substituting (4.9) into (4.8), then using the
relation (4.5) and the scheme (4.2) gives
Z n+1=2
By summation over j and multiplying by \Deltax one obtains from (4.7) and using (b)
a-\DeltaxX
(R n
a\Deltat: (4.10)
From (4.10) it is easy to verify by iteration and the geometric sum that
a
\Deltat
\Gamman ]:
This completes the proof of Lemma 4.1.
In the error analysis, we have to consider the finite difference solution as a function
defined in the whole upper-half plane, and t - 0, not only at the mesh points. There are
very simple ways of constructing a step function corresponding the grid function. For the
analysis we want to construct a different kind of approximate function that automatically
satisfies a convenient form of an entropy inequality. For this purpose we use the well known
interpretation of the upwind scheme (4.2) as the Godunov scheme. To accomplish this, we
construct a family of functions iteratively. Using piecewise constant data for the averages
R i this construction is actually equivalent to taking a characteristic scheme since we have
linear transport equations, see Childs and Morton [6], followed by an implicit step like (4.1)
for the source term. This is like a splitting method using the Godunov operator splitting.
We follow the approach used by Schroll et al. [25]. For notational clarity, we will use the
variables (y; -) instead of (x; t) in the context below.
The iteration is initialized by
Z
defined as the characteristic function for the interval I j .
(i) We use the notation t \Sigma
n to denote limits from above or below. In the interval (t
R i is the solution of the linear equation
with initial data R i
(ii) At t n+1 we project back onto the mesh by taking cell averages
2: (4.12)
(iii) The initial data for the next iteration are defined by the implicit formula treating the
source term
(4.13)From the definition of M i (u) it follows by (4.5) using (4.13) that
Thus the R i
n+1 ) can actually be obtained in the explicit form
Assuming R i
we conclude from the integral form of (4.11)
on the rectangle I j \Theta [t
Thus our step functions R respresenting the grid functions possess the following
property, due to -
2: (4.16)
The discrete estimates in Lemma 4.1 and the relations (4.16) yield the following properties
of R i .
Lemma 4.2. For all t, -
(a) (R 1
(b)
(c)
(d)
\Deltat).
Proof. The relations (4.16) imply directly that (c)-(d) hold. For the proof of (e), one has
to use the time-L 1 Lipschitz continuity of R i , we refer to [25] for a similar analysis and
omit the details.
The solution to the linear transport equations is unique. We do not need an entropy
inequality in order to impose uniqueness. Still, we may use a discrete version of the entropy
inequalities in order to study the convergence rate of R i to R i as \Deltax # 0.
is a weak solution (also an entropy solution due to its uniqueness) of the
system (4.11) in (t n ; t n+1 ), the Kru-zkov-type entropy formulation is valid. This means that
for all values (q
and all nonnegative C 1 -function OE with compact support in
On the other hand, from the step (iii) above we observe that for any (q 1
(R
\Theta M i
sgn
Inserting (4.18) into (4.17), summing over n from using R i
the following entropy inequality for the discrete solution
\Theta M i
Inspired by the papers [30] and [25] of Tveito and others, we use the following Kru-zkov-
type inequality for our problem since the exact solution R i (x; t) is the unique weak solution
of (1.8),
\Theta M i (u(x;
for any constants (q
and all
As in Kru-zkov [11] any weak
solution to (1.8) satisfies (4.20).
Equipped with the above variational inequalities, we return to estimate the error bound
of jjR
Proof of Theorem 2.3. Our proof in this section is inspired by the one given by
Schroll et al. [25]. They show an L 1 -error bound for a model arising in chromatography.
Their argument is in the spirit of the work by Kru-zkov[11], Kuznetsov[15] and Lucier[18].
To obtain the desired error bound we need to combine the inequality (4.19) with (4.20).
To this end we define a mollifier function
where j is any nonnegative smooth function with support in [\Gamma1; 1], even, i.e.
and unit mass. This mollifier therefore satisfies
Z
Z
Z
We proceed by selecting the constants (q 1
and the test functions OE; /. First taking q
and integrating in x and t, we obtain using R i
Z TZ Z TZ 2
dyd-dxdt
Z
\Theta

\Deltat
Z TZ N
sgn

In the inequality (4.20) we set q
over sum n from 1 to N
dxdtdyd-
sgn
\Delta\Theta

Adding this inequality to (4.21) and suitably grouping the terms we obtain an inequality
that we write in the short hand form
The individual expressions are
Z TZ N
\Theta

dyd-dxdt;
Z TZ N
Z
expressions we have the following bounds
and postpone their proof for the moment.
Lemma 4.3. For any T ? 0, there exists a positive constant C, independent of step sizes,
relaxation time ffl, and ffi such that
Equipped with these above estimates we continue the proof of the Theorem 2.3. Using
4.3 we find for a suitably large constant C ? 0X
Using an obvious bound for the initial error like (2.11), we haveX
(R i (\Delta; 0))\Deltax - C \Deltax (4.24)
Inserting this into (4.23) and picking
\Deltat to minimize the right hand side of the
relation (4.23), we haveX
Using the CFL condition \Deltat - \Deltax= p
a we haveX
\Deltax: (4.25)
Returning to the original variables (u; v) and noting the fact that we have
R N
we obtain
\Deltax:
This completes the proof of Theorem 2.3. 2
Finally, we return the proof of Lemma 4.3 in order to conclude this paper.
Proof of Lemma 4.3. The proof of (i)-(iii) and (v) can be done by an analogous
analysis as in the proof for the chromatography model given by Schroll et al.[25]. We next
estimate the term E 3 (ffi). Let us analogously as above use the notations
can be rewritten as
Z TZ N
Z TZ N
The first term in E 3 (ffi) is nonpositive due to the relations (3.3) and (3.6)X
uj \GammaX
Therefore
Z TN
Due to Lemma 3.2 the sum
ffl ]. The
integral of this sum over [0; T ] is clearly bounded by C ffl. Noting that
Z t\Gamma-
we find the estimates
\Deltat
\Deltat
which completes the proof. 2
Remark 2. The above proof uses the exponential rate
ffl without resorting to the
restriction on the initial error. To see this, note that fromX
we can easily deduce the boundffl
Z
Here, we do not use any specific choice of !(ffl). We mention that the authors in [25] obtained
the same error bound in the discretization step by assuming that



--R

Convergence of relaxation schemes for conservation laws
Discrete kinetic schemes for multidimensional conservation laws

The Mathematical Theory of Nonuniform Gases
The Boltzmann equation and its applications
Characteristic Galerkin methods for scalar conservation laws in one dimension
Hyperbolic conservation laws with stiff relaxation terms and entropy
Minimal surfaces and functions of bounded variation
A kinetic construction of global solutions of first order quasi-linear equations
The relaxation schemes for systems of conservation laws in arbitrary space dimensions

Stiff systems of hyperbolic conservation laws.
Contractive relaxation systems and scalar mul- tidimentional conservation laws

Accuracy of some approximate methods for computing the weak solutions of a first order quasilinear equation
Hyperbolic conservation laws with relaxation
A kinetic formulation of multidimensional scalar conservation laws and related equations
bounds for the methods of Glimm
A kinetic equation with kinetic entropy functions for scalar conservation laws
Convergence to equilibrium for the relaxation approximations of conservation laws
A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws
The convergence rate of approximate solutions for nonlinear scalar conservation laws
The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws
New York

The sharpness of Kuznetsov's O( p
convergence rate for scalar conservation laws with piecewise smooth solutions
Pointwise error estimates for relaxation approximations to conservation laws

On the rate of convergence to equilibrium for a ssystem of conservation laws with a relaxation term
Convergence of relaxing schemes for conservations laws
Numerical analysis of relaxation schemes for scalar conservation laws
--TR

--CTR
H. Joachim Schroll, High Resolution Relaxed Upwind Schemes in Gas Dynamics, Journal of Scientific Computing, v.17 n.1-4, p.599-607, December 2002
Tao Tang , Jinghua Wang, Convergence of MUSCL Relaxing Schemes to the Relaxed Schemes for Conservation Laws with Stiff Source Terms, Journal of Scientific Computing, v.15 n.2, p.173-195, June 2000
A. Chalabi , Y. Qiu, Relaxation Schemes for Hyperbolic Conservation Laws with Stiff Source Terms: Application to Reacting Euler Equations, Journal of Scientific Computing, v.15 n.4, p.395-416, December 2000
Mapundi Kondwani Banda, Variants of relaxed schemes and two-dimensional gas dynamics, Journal of Computational and Applied Mathematics, v.175 n.1, p.41-62, 1 March 2005
convergence rate;relaxation model;relaxation scheme