A PRIORI ERROR ESTIMATES FOR NUMERICAL METHODS FOR SCALAR CONSERVATION LAWS
PART III: MULTIDIMENSIONAL FLUX-SPLITTING
MONOTONE SCHEMES ON NON-CARTESIAN GRIDS∗
BERNARDO COCKBURN†, PIERRE-ALAIN GREMAUD‡,AND JIMMY XIANGRONG YANG†
Abstract. This paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we consider multidimensional flux-splitting monotone schemes defined on non-Cartesian grids. We identify those schemes which areconsistent and prove that theL∞(0, T;L1(Rd))-norm of the error goes to zero as (∆x)1/2when the discretization
parameter ∆xgoes to zero. Moreover, we show thatnonconsistentschemes can converge at optimal rates of (∆x)1/2 because (i) theconservation formof the schemes and (ii) the so-calledconsistency
of the numerical fluxesallow the regularity properties of the approximate solution to compensate for their lack ofconsistency.
Key words. a priori error estimates, irregular grids, monotone schemes, conservation laws, supraconvergence
AMS subject classifications. 65M60, 65N30, 35L65 PII.S0036142997316165
1. Introduction. This is the third of a series of papers in which we develop a theory of a priori error estimates, that is, estimates given solely in terms of the exact solution, for numerical methods for the scalar conservation law [16]
vt+∇ ·f(v) = 0, in (0, T)×Rd,
(1.1a)
v(0) =v0, onRd.
(1.1b)
In the first paper of this series [6], a new general approach for obtaining a priori error estimates for numerical methods for scalar conservation laws was introduced. The approach was then used to obtain optimal a priori error estimates for the Engquist– Osher scheme [9] on one-dimensional uniform grids; in contrast with previous work, [4], [5], [18], [19], [20], [21], [23], [24], [25], [27], [29], [31], (see also [28]), no regularity properties of the approximate solution were explicitly used. In the second paper of this series [7], an extension of the above result to the case of flux-splitting monotone schemes defined on irregular Cartesian grids was undertaken. As it is well known, see, for example, [13], [26], and [30], if the numerical fluxes do not properly take into account the irregularity of the grids, aloss of consistencyis generated and, as a consequence, a new term in the truncation error appears. In [7] it was shown that this new term could be controlled without using regularity properties of the approximate solutiononly for a subset of the schemes which were called consistent schemes. For
∗Received by the editors February 7, 1997; accepted for publication October 30, 1997; published electronically July 9, 1998.
http://www.siam.org/journals/sinum/35-5/31616.html
†School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455 ([email protected]). The first author was partially supported by the National Science Foundation grant DMS-9407952 and by the University of Minnesota Supercomputer Institute.
‡Center for Research in Scientific Computation and Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205. This author was partially supported by the Army Research Office through grant DAAH04-95-1-0419.
Fig. 1.Initial conditionv0 and triangulation with∆x= 1/8.
thisnewclass of schemes, optimal a priori error estimates were proven. In this paper, we consider the case of flux-splitting monotone schemes of the form
un+1
K =unK−|∆tK|
X
e∈∂K
|e|fe,K(unK, unKe), (1.2)
defined on non-Cartesian grids (see section 2.a), we identify those schemes which are consistent and we prove an optimal a priori error estimate for them; more precisely, we show that the L∞(0, T;L1(Rd))-norm of the error goes to zero as (∆x)1/2 where
∆x is the maximum diameter of the finite volumes K. Moreover, we uncover the mechanism that allows optimal a priori error estimates to hold even fornonconsistent schemes. Extending a similar result obtained in the one-dimensional case [7], we show that the regularity properties of the approximate solution can compensate for the loss ofconsistency of the scheme thanks to (i) theconservation form of the schemes and to (ii) the so-called consistency of the numerical fluxes; the nonlinear nature of the equations does not play any role in this mechanism.
The error estimate presented in this paper is the first a priori error estimate and the first optimal error estimate for numerical schemes for scalar conservation laws defined on non-Cartesian grids. As pointed out above, see also the discussion in [5], all previous work in error estimation for nonlinear conservation laws has been devoted to a posteriori error estimates. As a consequence, upper bounds of the total variation of the approximate solution had to be obtained. Since it is not a trivial matter to obtain such bounds even for simple schemes, this requirement has significantly slowed down the progress in this area. Indeed, there are only a few upper bounds of the total variation available in the literature. In [17] and [8], for monotone schemes in uniform Cartesian grids, and in [27], for monotone schemes in nonuniform Cartesian grids, it was proven that the total variation of the approximate solution does not increase in time. Unfortunately, the techniques used in the above mentioned papers are of no use in the non-Cartesian grids case since in this case the total variation of the approximate solutionsdoesincrease with time, even when the grids are uniform. For example, consider the simple case of the linear conservation law in two dimensions withf(v) = (0, b)v with initial conditionv0 equal to one in the shaded area shown in
Figure 1 and equal to zero elsewhere.
101 102 103 104 1.68
1.70 1.72 1.74
R(N)
1/ ∆ x = N
Fig. 2.The ratioR(N) = |uh(∆t)|T V(R2)/|uh(0)|T V(R2).
and grids of equilateral triangles as shown in Figure 1, a computation shows that the ratioRof the total variation of the approximate solutionuh(∆t) after a single
time-step to the total variation of the initial datauh(0) =v0 as a function of ∆x= 1/N
is bigger than one, as is shown in Figure 2; therein, we have taken λ b = .9 and cfl= 2√3 ∆t
λ∆x =.9.
The other technique to obtain upper bounds for the total variation of approxi-mate solutions of numerical schemes for nonlinear conservation laws is an extension, obtained in [4] (see also [15]) of an argument introduced in [2] to obtain the conver-gence of finite difference methods [3] in the framework of measure-valued solutions. This technique, which is essentially an a priori L2-estimate, was used in [4], [31], [25],
and [5] to establish that the total variation of the approximate solution blows up like (∆x)−1/2 as ∆x goes to zero. This upper bound results in a posteriori error
esti-mates with a rate of convergence of (∆x)1/4for finite volume schemes [4], [31], [25], of
(∆x)1/4for the discontinuous Galerkin method [5], and of (∆x)1/8for the streamline
diffusion method [5]. Unfortunately, this technique does not lead to optimal error estimates since the total variation is bounded only in terms of the L2-norm of the
initial data instead of in terms of its total variation. The advantage of our approach is that no estimate of the total variation is required to obtain optimal error estimates for the so-calledconsistentschemes.
We have to point out, however, thatconsistentschemes of the form (1.2), (i) can only be defined on fairly uniform grids and (ii) have to have fairly restrictive numerical fluxes. For example, upwinding-like schemes are notconsistent; see a related result in [12]. This unfortunate situation is, of course, related to the fact that the schemes of the form (1.2) update the value at the volumeKbyonlyusing the information stored in those volumesKesharing a common faceewithK. We believe that the consistency
of the L∞-like truncation error is studied. By a careful consideration of thestructure of that truncation error, it is shown that this degeneracy does not actually take place provided that the scheme is stable and the exact solution is smooth enough. Our ap-proach is similar in that we show that thestructureof the truncation error allows the stability properties of the scheme to compensate for its lack of consistency. However, we study the general nonlinear conservation laws, work with an L1-like truncation
er-ror and half-order accurate schemes, require minimal regularity of the exact solution, and consider general non-Cartesian grids.
Finally, let us emphasize that the results presented in this paper are not a straight-forward extension of those obtained in our second paper of this series [7] for irregular Cartesian grids. Although the technique to obtain a priori error estimates introduced in [6] is independent of the space dimension, the irregularity of the grids plays a greater role in the multidimensional case and renders the manipulation of the truncation error a much more delicate undertaking.
The paper is organized as follows. In section 2, the numerical schemes under consideration and related technical assumptions are presented, and the main results are stated and discussed. In section 3, we give a proof of our main a priori error estimate, Theorem 2.1. In section 4, we give an explanation of thesupraconvergence phenomenon and the proof of the corresponding a priori error estimate, Theorem 2.3. We end with some concluding remarks in section 5.
2. The numerical schemes and main results.
2a. The numerical schemes. We consider, for the sake of simplicity, uniform partition of R+ of the form{tn =n∆t}n∈N and triangulations of Rd, T∆x ={K},
made of nonoverlapping polyhedra. We require that for any two elementsK andK0, K∩K0 is either a facee of bothK and K0 with nonzero (d−1)-Lebesgue measure
|e| or has Hausdorff dimension less thand−1. The maximum diameter of the sets K of the triangulationT∆x is denoted by ∆x.
Given these partitions, we define an approximationuto the entropy solutionvof (1.1) as the piecewise-constant function
u(t, x) =un
K, for (t, x)∈[tn, tn+1)×K
(2.1)
constructed as follows. Att= 0, the degrees of freedom ofuare given by
u0
K= |K1 |
Z
K u0(s)ds,
(2.2a)
where|K|is the measure of the finite volumeK. The remaining degrees of freedom are defined by the following flux-splitting scheme:
un+1
K =unK−|∆tK|
X
e∈∂K
|e|fe,K(unK, unKe), (2.2b)
whereKe denotes the finite volume sharing the faceewith the finite volumeK and
the numerical fluxfe,K(v, w) has the form
fe,K(v, w) =fcent;e,K(v, w)−fvisc;e,K(v, w),
(2.3a) with
fcent;e,K(v, w) =ae,Kf(v)·ne,K+be,Kf(w)·ne,K,
Table 1
Values of the mesh related parameters on simple grids.
mesh type aspect ratio d D∆x ι∆x
equilateral N/A 2 4 3
Cartesian triangular λ 2 2√4 +λ2 3
Cartesian rectangular λ 2 √4 +λ2 4
Cartesian rectangular λ, µ 3 p4 +λ2+µ2 6
wherene,K is the outward unit normal at the faceeof the finite volume Kand
fvisc;e,K(v, w) =αe(Ne(w)−Ne(v)).
(2.3c)
We assume that the flux fe,K is consistent with the nonlinearity f ·ne,K and
con-servative, i.e., thatfe,K(u, u) =f(u)·ne,K andfe,K(uK, uKe) +fe,Ke(uKe, uK) = 0, respectively. For the schemes under consideration, these properties are equivalent to the following conditions:
ae,K+be,K= 1, (consistency),
(2.3d)
ae,Ke =be,K, be,Ke =ae,K, (conservativity). (2.3e)
We ask that
0≤ae,K, be,K ≤1.
(2.3f)
Finally, we require that for every finite volumeK and each of its facesewe have αeNe0(·)≥be,K|f0(·)·ne,K|,
(2.3g)
and normalizeαeandNe0 by asking that for all bounded intervalsI:
αe≤1, sup c∈IN
0
e(c)≤sup c∈I kf
0(c)k
l∞.
(2.3h)
As it is well known, this condition ensures themonotonicity of the scheme under the following condition on the size of the time-step ∆t:
∆t
|K| X
e∈∂K
|e|¡αeNe0(unK)−be,Kf0(unK)·ne,K≤1, ∀K∈ T∆x, n≥0.
(2.4)
Our results also involve two grid-related quantities which we introduce next. Given a face e ∈ E∆x ≡ S{e : e ∈ ∂K for allK ∈ T∆x}, we define Ce to be the
convex hull ofK+
e ∪Ke−,deto be the diameter ofCe, and we set D∆x= sup
e∈E∆x
|e|de
max{ |K+
e |,|Ke−| }.
(2.5)
Also, given the finite volume K, we denote by ιK the number of sets Ce for which
|K∩Ce| 6= 0 for somee∈ E∆x and we set
ι∆x= sup K∈T∆xιK. (2.6)
2b. Consistency and a priori error estimates.
The generalization of the notion ofconsistency introduced in [7] for schemes on irregular Cartesian grids will be based on the following two quantities:
∆K= max
1≤j≤d d
X
i=1
X
e∈∂K
|e|(δe)j(ne,K)i,
(2.7a)
AK= sup I∈R
1
supc∈Ikf0(c)ksupc∈I
d
X
i=1
X
e∈∂K
|e|αeNe0(c) (xKe−xK)i
, (2.7b)
where
δe= (xe−xK)−be,Ke(xKe−xK), (2.7c)
wherexK denotes the barycenter of the finite volumeKandxethe barycenter of the
face e. We say that the scheme (2.2b) with fluxes satisfying the conditions (2.3) is consistentwith respect to the family of triangulationsT∆xª∆x>0 if ∆K=AK ≡0.
If we consider, for instance, a grid of equilateral triangles or any affine transformation of it and choose
ae,K =be,K = 1/2, αe= 1/2, Ne0(c) =kf0(c)k or Ne0(c) = 1/λ,
then the corresponding scheme is consistent. Note that the first choice of N0
e
corre-sponds to an Engquist–Osher-like scheme with isotropic artificial viscosity whereas the second choice corresponds to the Lax–Friedrichs like scheme described in the in-troduction.
The conditions forconsistencyfor the schemes under consideration impose heavy restrictions on the grids and on the numerical fluxes. Indeed, ∆K ≡0 ifδe≡0, that
is, if
(xc−xK) =be,Ke(xKe−xK), or, equivalently, if
xe=be,KexKe+be,KxK.
This means not only that the value of the scalar be,Ke is determined, but that the barycentersxKe andxKe and the barycenter of the facee,xe, must lie on the same line! This condition is trivially satisfied for Cartesian or affine transformations of Cartesian grids, but that is not the case for general non-Cartesian grids. Moreover,
AK≡0 if and only if
xK =
X
e∈∂K
ζc,K(c)xKE, where ζc,K(c) =
|c|αeNe0(c)
X
e∈∂K
|e|acNe0(c)
.
This means, not only that the quantities ζe,K(c) should be independent of c, but
that the barycenter xK has to be equal to a very precise convex combination of the
barycenters xKe. This condition is impossible to satisfy for upwinding-like fluxes
unless the grids are Cartesian or affine transformations of Cartesian grids.
For example, consider the linear casef(v) = (0, b)v in two-space dimensions and triangulations made of acute triangles. For the upwinding scheme N0
ne,K|. If for some faceeof the finite volumeK the velocity (0, b) is perpendicular to
ne,K, we haveNe0(c) = 0 and the previous condition is impossible to satisfy. This lack
of consistency of upwinding-like schemes has been pointed out in [12] in the framework of a formal L∞-like truncation error analysis.
Fortunately, the consistency condition can be relaxed. We say that a scheme is p-consistent if there are two nonnegative constantsCδ andCαsuch that
|δ|var,1/2≤Cδ(∆x)p, |α|var,1/2≤Cα(∆x)p,
(2.8) where
|δ|var,1/2= sup
x∈Rd
X
x:|x−xK|≤(∆x)1/2 ∆K
(∆x)d/2 ,
(2.9)
|α|var,1/2= sup
x∈Rd
X
x:|x−xK|≤(∆x)1/2
AK
(∆x)d/2 .
(2.10)
For instance, if we consider the two-dimensional case and define be,K= (xe−kxxKe)·(xK−xKe)
K−xKek2
, ae,K = 1−be,K,
αe= max{ae,K, ae,Ke}, Ne0(c) =kf0(c)k or Ne0(c) = 1/λ,
then the conservativity condition (2.3e) is satisfied. Moreover, (2.3f) is verified pro-vided that
kxe−xKek ≤ kxK−xKek and |θe| ≤ π 2,
whereθe=](xe−xKe, xK−xKe). In particular, those conditions are satisfied if the triangulation is acute. We observe that
∆K ≤2ι∆xD∆x sup e∈E∆x
kδekl∞
de max{K
+
e, Ke−}
,
AK ≤dι∆x∆x sup e∈E∆x
°° °° °xK−
X
e⊂∂K
ζe,K(c)xKe
°° °° °
`∞
.
Assuming now the triangulation to be uniformly regular, i.e., ∆x ≤ κhK for any
K⊂ T∆x(hK= diameter ofK), gives
|δ|var,1/2≤2κ2ι∆xD∆x sup e∈E∆x
kδckl∞
de ,
|α|var,1/2≤2dι∆xκ
2
∆xe∈E∆xsup
°° °° °xK−
X
e⊂∂K
ζc,K(c)xKx
°° °° °
`∞
.
Without further assumptions on the grid, we have|δ|var,1/2=O(1) and|α|var,1/2= O(1) and the above scheme is 0-consistent, i.e., inconsistent (case treated in Theorem 2.3). However, if we assume, for instance,
kδek`∞ =O(∆x1+p), °°
°° °
X
e∈∂K
(xKe−xK)
°° °° °
`∞
then the scheme isp-consistent (case treated in Corollary 2.2). The first of the above assumptions amounts to having, fore∈ E∆x, the three nodesxK,xe,xKe lining up as ∆xgoes to zero. The second assumption means, for instance, that the grid becomes asymptotically equilateral, if simplicial elements are used.
We are now ready to state our error estimate which, following [6], is expressed in terms of the numerical viscosity associated to the scheme under consideration and in terms of the measure of consistency introduced above.
Theorem 2.1. Let the Courant–Friedrichs–Levy condition(2.4)be satisfied. Let ube the piecewise-constant solution given by the scheme (2.2)with coefficients satis-fying(2.3), letv be the entropy solution. Then,
ku(tN)−v(tN)k
L1(Rd)≤2ku0−v0kL1(Rd)+ 8|v0|T V(Rd)
q
2tNkνvk(∆x)1/2
+C|v0|T V(Rd)¡|δ|var,1/2+|α|var,1/2 +|v0|T V(Rd)(b1(∆x)3/4+b2∆x),
wherekνvk= supKsupt∈(0,tN)
x∈Rd
max1≤i,j≤d|νKij(v(t, x))/∆x|and the local viscosity
co-efficientνKij is given by
νKij(c) = ∆t2 f0
i(c)fj0(c)−|K1 |
X
e∈∂K
|e|be,Kef0(c)·ne,K+αeNe0(c)
ξe,Kij ,
ξe,Kij =12(xKe−xK)i(xKe−xK)j+IijKe−I
ij K
ª
,
IijΩ = 1
|Ω| Z
Ω(x
0−x
Ω)i(x0−xΩ)jdx0.
The constantC is given by
C= 4kf0(v)k tN +q2tN/kνvk
(1 +b0(∆x)1/4),
andkf0(v)k is given by
kf0(v)k= sup
t∈(0,tN)
x∈Rd
kf0(v(t, x))k
`∞.
The constantsb0,b1, and b2 are locally bounded functions that depend solely on the
quantitieskf0(v)k∆t/∆x,kf0(v)k/kν
vk,{tNkνvk}1/2,D∆x, andι∆.
An immediate consequence of this result follows.
Corollary 2.2 (p-consistent schemes). With the notation and under the as-sumptions of Theorem 2.1, if the scheme isp-consistent, we have
ku(tN)−v(tN)k
L1(Rd)≤ 2ku0−v0kL1(Rd)+ 8|v0|T V(Rd)
q
2tNkνvk(∆x)1/2
+O((∆x)min{p,3/4}).
schemes mentioned in the previous section, it is possible to obtain an error estimate in terms of the following quantities:
kδk`∞(E∆x)/Rd= inf
ˆ
δ∈Rdesup∈E∆x1max≤j≤d|(δe− ˆ δ)j|,
(2.11a)
kαk`∞(E∆x)= sup
e∈E∆x|αe|1max≤j≤d|(xKe−xK)j|, (2.11b)
which are always of order ∆x, by (2.7c), (2.3f), and (2.3h), instead of in terms of the quantities|δ|var,1/2 and|α|var,1/2. The price to pay is that the error estimate now
must depend on the smoothness properties of the approximate solution; see Lemma 4.1.
Theorem 2.3. Let the Courant–Friedrichs–Levy condition(2.4)be satisfied. Let ube the piecewise-constant solution given by the scheme (2.2)with coefficients satis-fying(2.3); letv be the entropy solution. Then,
ku(tN)−v(tN)k
L1(Rd)≤ 2ku0−v0kL1(Rd)+ 8|v0|T V(Rd)
q
2tNkνvk(∆x)1/2
+ 2
√
2d
p
tNkνvk∆x
kδk`∞(E∆x)/Rd+kαk`∞(E∆x)ª
D∆xι∆xkf0(v)ktN|v0|T V(Rd)+ 1 ιinfkf
0(u)k kuk
L1(0,tN;T V(Rd))
+|v0|T V(Rd)(b1(∆x)3/4+b2∆x),
whereιinf = infK∈T∆xιK. The constantsb1 andb2 are locally bounded functions that
depend solely on the quantities kf0(v)k∆t/∆x, kf0(v)k/kν
vk, {tNkνvk}1/2, D∆x,
andι∆x, wherekνvk is defined in Theorem2.1.
Beside the estimate itself, which is new, an explanation of why it is at all possible is worth considering. As will be shown in section 4, it is a combination of both consistency and conservativity of the numerical fluxes, which makes it possible by forcing the quantityDe,K = (xe−xK)−be,Ke(xKe−xK) to be such thatDe,K = De,Ke≡δe. Indeed, we have
De,K = (xe−xK)−be,Ke(xKe−xK)
= (xe−xK)−ae,K(xKe−xK), by conservativity (2.3e), = (xe−xK)−(1−be,K) (xKe−xK), by consistency (2.3d), = (xe−xKe)−be,K(xK−xKe)
=De,Ke.
3. Proof of Theorem 2.1. This section is entirely devoted to the proof of Theorem 2.1. It has exactly the same structure as section 3 of [7]. In all the proofs, we assume that the entropy solutionvis smooth since the general case can be obtained by a standard density argument; see the proof of Proposition 5.5 in [6]. The case in whichv is piecewise smooth is treated in [35].
3a. The approximation inequality. Along the lines of [6], we consider the form
E(u, v;tN) =Z t
N
0 Z
Rd
NX−1
n=0 X
K∈T∆x Ψn
whereU(·) is the absolute value function| · |, and Ψn
K is given by
Ψn
K(c) =U0(unK−c)
(
un+1
K −unK
∆t + 1
|K| X
e∈∂K
|e|fe,K(unK, unKe)
)
.
The termφ(t, x, tn+1, xK) is an averaged test function defined by
φ(t, x, tn+1, x
K) =|K1 |
Z
K ϕ(t, x, t
n+1, x0)dx0, (3.1)
where xK denotes the barycenter of the finite volume K, and where the function
ϕ(t, x, t0, x0) is defined as follows:
ϕ(t, x, t0, x0) =w
t(t−t0)
d
Y
i=1
ηx(xi−x0i), (3.2a)
x= (x1, . . . , xd),x0= (x01, . . . , x0d), andt,t0being, respectively, points inRdandR+.
Finally, the functionswt andηx are constructed as follows:
wt(s) = 1 tw
s t
, ηx(s) = 1 xη
s x
, ∀s∈R, (3.2b)
wheretandxare two arbitrary positive numbers. Bothwandηsatisfy the following
conditions: (i)η(t)≥0, fort >0, (ii)η(t) =η(−t), fort >0, (iii) the support of η is [−1,1], and (iv)R01η(r)dr= 1/2. For future reference, we set
W(t) =
Z t
0 wt(s)ds.
(3.2c)
Furthermore, the functionswandηare taken to be smooth approximations toχ≡χ0
andχ, where
χ(x) =
(1 +)/2 for|x| ≤(1−)/(1 +), (1 +)2(1− |x|)/4 for|x| ∈[(1−)/(1 +),1],
0 elsewhere.
It is easy to verify that we can find a sequence of functionsη such that lim
η→χ|η|T V(R)=|χ|T V(R)= 1 +, (3.3a)
lim
η→χ|η 0|
T V(R)=|χ0 |T V(R)= 2 ++ 1/,
(3.3b)
lim
η→χkηkL∞(R)=kχkL∞(R)= (1 +)/2. (3.3c)
We now describe the main ideas on which the derivation of our estimates is based. From the identity
0 =E(u, v;tN) =−E?
div(u, v;tN) +Ediss(u, v;tN) +Terr(u, v;tN),
and the structure of the term Terr(u, v;tN), the following approximation inequality
can be derived [6, Proposition 7.6]: e(tN)≤2e(0) + 8¡
x+tkf0(v)k|v0|T V(Rd)+ 2kf0(v)k |v0|T V(Rd)∆t + 2 lim
w→χ1≤supn≤N
E?
wheree(t) denotes the errorku(t)−v(t)kL1(Rd). Now, to obtain the error estimate, it is enough to estimate the forms E?
div(u, v;tn) andEdiss(u, v;tn). The remainder
of this section is devoted to identifying these forms and estimating them. We start by rewriting Ψn
K(c),
Ψn
K(c) =U0(unK−c)∆t1
(
un+1
K −unK+|∆tK|
X
e∈∂K
|e|fe,K(unK, unKe)
)
=U0(un
K−c)∆t1
(
unK+1−un
K+|∆tK|
X
e∈∂K
(pe,K(unK)−pe,K(unKe))
)
,
where
pe,K(s) =−|e|be,Kf(s)·ne,K−αeNe(s),
and proceed as in [6]. Here, the formEdiss(u, v;tN) is
Ediss(u, v;tN) =
Z tN
0 Z
Rd
NX−1
n=0 X
K∈T∆§
LREDn
K(v(t, x))φ(t, x, tn+1, xK)|K|∆t dx dt,
where the local rate of entropy dissipationLREDn
K(c) is given by
LREDn
K(c) =∆t1
Z un K
un+1 K
(pK(unK)−pK(s))U00(s−c)ds
+|K1 | X
e∈∂K
Z un Ke
un+1 K
(pe,K(unKe)−pe,K(s))U00(s−c)ds,
andpK(s) =s−|∆Kt|Pe∈∂Kpe,K(s).
The dual formE?
div(u, v;tN) reads
E?
div(u, v;tN) =−
Z tN
0 Z
Rd
Z tN
0 Z
RdU(u(t
0, x0)−v(t, x))ϕ
t(t, x, t0, x0)dx dt dx0dt0
+
Z tN
0 Z
Rd
Z
RdU(u(t
0, x0)−v(tN, x))ϕ(tN, x, t0, x0)dx dx0dt0
− Z tN
0 Z
Rd
Z
RdU(u(t
0, x0)−v
0(x))ϕ(0, x, t0, x0)dx dx0dt0 −
NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd
X
e∈∂K
|e|be,Kφ(t, x, tn+1, xK)
−φ(t, x, tn+1, x
Ke)
F(un
K, v)·ne,K∆t dx dt
+NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd
X
e∈∂K
|e|αeφ(t, x, tn+1, xKe)
−φ(t, x, tn+1, x
K)Ne(unK, v) ∆t dx dt,
where the functionsF(u, c) andNe(u, c) are defined as follows:
F(u, c) =
Z u
c f
0(s)U0(s−c)ds, N
e(u, c) =
Z u
c N
0
e(s)U0(s−c)ds.
3b. Estimate of Ediss(u, v;tn).
Proposition 3.1. Under condition (2.3g) on the viscosity term N, and if the Courant–Friedrichs–Levy(CFL)condition(2.4) is satisfied, the local rate of entropy dissipationLREDn
j (c)is nonnegative. Hence,
−Ediss(u, v;tn)≤0.
Sketch of the proof. The above conditions ensure that the functionspe,K(s) and
pK(s) are nondecreasing ins. The result follows from this fact and the definition of
Ediss(uh, v;tn).
3c. Estimate of E?
div(u, v;tN).
Proposition 3.2. We have
lim
w→χ1≤supn≤N
E?
div(u, v;tn)/W(tn)
ª
≤T EWvisc+T EWcons+T EWh.o.t.,
where
T EWvisc(u, v;tn)≤ C0tNd
2|η|T V(R) x
1 + ∆t
t kνvk∆x,
T EWcons(u, v;tn)≤ 2C1tN
1 +∆t
t 1 +
(∆x)d/2
d x
kηkd
L∞(R) ¡
|δ|var,1/2+|α|var,1/2,
T EWh.o.t.(u, v;tn)≤C1
1
2kf0(v)k
(∆t)2d|η|
T V(R)
tx t N + 4
1 + ∆t
t
∆t
+ 32D∆xι∆x(∆x)2
d2|η|2
T V(R)
2
x +
d|η0|
T V(R)
2
x 1 +
∆t t
tN.
whereC0=|v0|T V(Rd)andC1=C0kf0(v)k.
To prove this result, we proceed in several steps. First step: Relating the dual formE?
div(u, v;t) to the truncation error. We have the following upper bound forE?
div(uh, v;tN).
Lemma 3.3. We have
E?
div(uh, v;tN)≤ NX−1
n=0 X
K∈T∆x
Z tN
0 Z
RdF(u
n
K, v(t, x))· ∇xφ(t, x, tn+1, xK)|K|∆t dx dt
−NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd
X
e∈∂K
|e|be,Ke
φ(t, x, tn+1, x
K)
−φ(t, x, tn+1, x
Ke)
F(un
K, v(t, x))·ne,K∆t dx dt
+
NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd
X
e∈∂K
|e|αeφ(t, x, tn+1, xKe)
−φ(t, x, tn+1, x
K)Ne(unK, v(t, x)) ∆t dx dt.
To prove this result, we use the fact thatvis the entropy solution and make some algebraic manipulations; see the proof of the similar result [6, Lemma 7.9].
clarity, we drop the dependence ontandt0. This simple result is a cornerstone of the proof of Theorem 2.1.
Lemma 3.4. We have
φ(x, xKe)−φ(x, xK) =−(xKe−xK)· ∇xφ(x, xK) +He,K(x, xK) +He,K(x), where
He,K(x, xK) = d
X
i,j=1
ξije,K∂2
xixjφ(x, xK),
ξe,Kij =12(xKe−xK)i(xKe−xK)j+IijKe−I
ij K
ª
,
IijΩ = 1
|Ω| Z
Ω(x
0−x
Ω)i(x0−xΩ)jdx0,
and
He,K(x) = |K1 e|
Z
Ke 1
|K|3 Z
K3
Z 1
0 Z 1
0 Z 1
0 Ψ(x)dλ dµ dν dz
0dy0dw0dx0,
Ψ(x) =
d
X
i,j,k=1
ζijk∂3
xixjxkϕ(x, λ(µ(νx0+ (1−ν)w0) + (1−µ)y0) + (1−λ)z0), ζijk = (x0−w0)
i(νx0+ (1−ν)w0−y0)j(µ(νx0+ (1−ν)w0) + (1−µ)y0−z0)k.
Proof of Lemma3.4. Since
ϕ(x, x0) =ϕ(x, w0)−Z 1
0 (x
0−w0)· ∇
xϕ(x, νx0+ (1−ν)w0)dν,
by the definition of ϕ(3.2), averaging the above equality over K with respect tow0 yields
ϕ(x, x0) =φ(x, x
K)−|K1 |
Z
K
Z 1
0 (x
0−w0)· ∇
xϕ(x, νx0+ (1−ν)w0)dν dw0,
by the definition of φ, (3.1). Using the above relation recursively twice, we easily obtain
ϕ(x, x0) =φ(x, x
K) + d
X
i=1
Ai(x0)∂
xiφ(x, xK) +
d
X
i,j=1
Bij(x0)∂2
xixjφ(x, xK) +Res(x, x 0),
where
Ai(x0) =− 1
|K| Z
K(x
0−w0)
idw0,
Bij(x0) = 1
|K| Z
K
Z 1
0 (x
0−w0)
i
1
|K| Z
K(νx
0+ (1−ν)w0−y0)
jdy0
dν dw0,
Res(x, x0) = 1
|K|3 Z
K3
Z 1
0 Z 1
0 Z 1
0 Ψ(x)dλ dµ dν dz
Averaging the above equation Ke with respect to x0 and using the definition of φ,
(3.1), yields
φ(x, xKe) =φ(x, xK) +
d
X
i=1
Ai∂
xiφ(x, xK) +
d
X
i,j=1
Bij∂2
xixjφ(x, xK) +He,K(x),
where
Ai= 1
|Ke|
Z
Ke
Ai(x0)dx0, Bij = 1
|Ke|
Z
Ke
Bij(x0)dx0.
The result follows if we show that Ai = −(xK
e−xK)i and Bij = ξe,Kij . But, since Ai(x0) =−(x0−xK)i, we do have thatAi=−(xK
e−xK)i, and since
Bij(x0) = 1
|K| Z
K
Z 1
0 (x
0−w0)
i(νx0+ (1−ν)w0−xK)jdν dw0
= |K1 |
Z
K(x
0−w0)
i((x0+w0)/2−xK)jdw0
= 1 2|K|
Z
K((x
0−x
K)−(w0−xK))i((x0−xK) + (w0−xK))jdw0
= 12(x0−x
K)i(x0−xK)j−IijKª,
we have
Bij = 1
2
1
|Ke|
Z
Ke (x0−x
K)i(x0−xK)jdx0−IijK
= 12
(xKe−xK)i(xKe−xK)j+ 1
|Ke|
Z
Ke (x0−x
Ke)i(x0−xKe)jdx0−IijK
= 12(xKe−xK)i(xKe−xK)j+IijKe−I
ij K
ª
=ξe,Kij .
This completes the proof.
With the above lemma, we can now rewrite the upper bound of E?
div(uh, v;tN)
as follows.
Lemma 3.5. We have
E?
where
T Evisc(u, v;tN) = NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd V ISC
n
K(v(t, x);t, x))dx dt∆t
−NX−1
n=0 X
K∈T∆x
Z
Rd F(u
n
K, v(tN, x))· ∆t2 ∇xφ(tN, x, tn+1, xK)dx|K|∆t
+
NX−1
n=0 X
K∈T∆x
Z
Rd F(u
n
K, v(t0, x))·∆t2 ∇xφ(0, x, tn+1, xK)dx|K|∆t,
T Econs(u, v;tN) = NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd CONS
n
K(v(t, x);t, x))dx dt∆t,
T Eh.o.t.(u, v;tN) = NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd HOT
n
K(v(t, x);t, x))dx dt∆t
+NX−1
n=0 X
K∈T∆x
Z
Rd F(u
n
K, v(tN, x))· ∆t2 ∇xφ(tN, x, tn+1, xK)dx|K|∆t
−NX−1
n=0 X
K∈T∆x
Z
Rd F(u
n
K, v(t0, x))·∆t2 ∇xφ(0, x, tn+1, xK)dx|K|∆t.
The ‘viscosity’ termV ISCn
K(c;t, x)is given by
V ISCn
K(c;t, x) =time→ V ISCKn(c;t, x)+cent→ V ISCKn(c;t, x)+visc→ V ISCKn(c;t, x),
where
time→ V ISCn
K(c;t, x) =F(uKn, c)·∆t2 |K| ∇xφt(t, x, tn+1, xK), cent→ V ISCn
K(c;t, x) =F(uKn, c)·
X
e∈∂K
|e|be,KeHe,K(t, x, tn+1, xK)ne,K,
visc→ V ISCn
K(c;t, x) =
X
e∈∂K
|e|αeHe,K(t, x, tn+1, xK)Ne(unK, c).
The ‘consistency’ termCONSn
K(c;t, x) is given by
CONSn
K(c;t, x) =cent→ CONSKn(c;t, x)+visc→ CONSKn(c;t, x),
where
cent→ CONSn
K(c;t, x) =F(uKn, c)·
X
e∈∂K
|e| ∇xφ(t, x, tn+1, xK)·δene,K,
visc→ CONSn
K(c;t, x) =−
X
e∈∂K
|e|αe(xKe−xK)· ∇xφ(t, x, tn+1, xK)Ne(unK, c).
Finally, the ‘high-order’ termHOTn
K(c;t, x)is given by
HOTn
where
time→ HOTn
K(c;t, x) =F(uKn, c)·
|K|∇xφ(t, x, tn+1, xK)− |K| ∇xφ(t, x, tn+1, xK)
−∆t2 |K| ∇xφt(t, x, tn+1, xK)
,
cent→ HOTn
K(c;t, x) =F(unK, c)·
X
e∈∂K
|e|be,KeHe,K(t, x, tn+1)ne,K,
visc→ HOTn
K(c;t, x) =
X
e∈∂K
|e|αeHe,K(t, x, tn+1)Ne(unK, c).
Proof of Lemma 3.5. The result can be easily obtained by inserting the expres-sion for φ(x, xKe)−φ(x, xK) of Lemma 3.4 into the upper bound for the dual form E?(u, v;tN) in Lemma 3.3 and rearranging terms.
The only manipulation that deserves mentioning is the following. After the above mentioned rearrangement, we get forcent→ CONSn
K(c;t, x) the expression
F(un K, c)·
"
|K| ∇xφ(t, x, tn+1, xK)−
X
e∈∂K
|e|be,Ke(xKe−xK)· ∇xφ(t, x, tn+1, xK)ne,K
#
.
This expression can be rewritten as F(un
K, c)·
X
e∈∂K
|e| ∇xφ(t, x, tn+1, xK)·[(xe−xK)−be,Ke(xKe−xK)]ne,K,
or, equivalently, see the definition ofδe, (2.7c), as
F(un K, c)·
X
e∈∂K
|e| ∇xφ(t, x, tn+1, xK)·δene,K,
by using the identity
|K|m= X
e∈∂K
|e|m·(xe−xK)ne,K,
withm=∇xφ(t, x, tn+1, xK). The above identity can be proved as follows:
|K|m=
Z
K m dx
0=Z
K∇x
0m·(x0−xK)dx0
= X
e∈∂K
Z
e
m·(x0−x
K)ne,KdΓ(x0)
= X
e∈∂K
|e|m·(xe−xK)ne,K.
This completes the proof.
Second step: Estimating T Evisc(u, v;tN). In this section, we prove the following result.
Lemma 3.6. We have
T Evisc(u, v;tN) ≤ 2C0 d|η|T V(R)
x
1 +∆t
t
whereC0= tN|v0|T V(Rd)W(tN).
Proof of Lemma3.6. We have, by the definition ofT Evisc(u, v;tN) in Lemma 3.5,
T Evisc(u, v;tN) = NX−1
n=0 X
K∈T∆x
Ξ(tn+1, x
K)|K|∆t,
where
Ξ(tn+1, x
K) =
Z tN
0 Z
Rd
"
F(un
K, v(t, x))· ∆t2 ∇xφt(t, x, tn+1, xK)
+F(un
K, v(t, x))·|K1 |
X
e∈∂K
|e|be,KeHe,K(t, x, tn+1, xK)ne,K
+ 1
|K| X
e∈∂K
|e|αeHe,K(t, x, tn+1, xK)Ne(unK, v(t, x))
#
dx dt
− Z
Rd F(u
n
K, v(tN, x))·∆t2 ∇xφ(tN, x, tn+1, xK)dx
+
Z
Rd F(u
n
K, v(t0, x))·∆t2 ∇xφ(0, x, tn+1, xK)dx.
A couple of integrations by parts yield
Ξ(tn+1, x
K) =−
Z tN
0 Z
Rd
Ft(uKn, v(t, x))·∆t2 ∇xφ(t, x, tn+1, xK)
+ 1
|K| X
e∈∂K
|e|be,Ke
d
X
i,j=1
∂xi
F(un
K, v(t, x))·ne,Kξije,K∂xjφ(t, x, tn+1, xK)
+|K1 | X
e∈∂K
|e|αe d
X
i,j=1
∂xiNe(unK, v(t, x))ξe,Kij ∂xjφ(t, x, tn+1, xK)
dx dt .
Then, by using the definition ofF andN, (3.4), we get
Ξ(tn+1, x
K) =− d
X
i,j=1 Z tN
0 Z
Rd U 0(un
K−v)∂xiv νKij(v(t, x))∂xjφ(t, x, tn+1, xK)dx dt , where the entries of the matrixν are
νKij(c) =∆t2 f0
i(c)fj0(c)−|K1 |
X
e∈∂K
|e|be,Kef0(c)·ne,K+αeNe0(c)
ξe,Kij .
We have now
T Evisc(u, v;tN)≤ Taux|v0|T V(Rd)tN, where
Taux= sup t∈(0,tn)
x∈Rd
NX−1
n=0 X
K∈T∆x max
1≤i≤d d
X
j=1
νKij(v(t, x))∂xjφ(t, x, tn+1, xK)|K|∆t
≤ d|η|T V(R)
x 2
1 + ∆t
t
where we have made use of Lemma 3.7 of [7], the definition of φ, (3.1), and the definition ofkνvk. This completes the proof.
Third step: EstimatingT Econs(u, v;tN).
Lemma 3.7. We have
T Econs(u, v;tN)≤2C1
1 + ∆t
t 1 +
(∆x)d/2
d x
kηkd
L∞(R) ¡
|δ|var,1/2+|α|var,1/2,
whereC1= tN|v0|T V(Rd)kf0(v)kW(tN).
Proof of Lemma3.7. By definition of the termT Econs(u, v;tN) in Lemma 3.5, we
can write
T Econs(u, v;tN) = NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd ∇xφ(t, x, t
n+1, x
K)·AK(unK, v(t, x))dx dt∆t,
where
AK(u, c) =
X
e∈∂K
|e|¡F(u, c)·ne,Kδe−αeNe(u, c) (xKe−xK)
.
After a simple integration by parts, we obtain
T Econs(u, v;tN) =− NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd φ(t, x, t
n+1, x
K)∇xv(t, x)·∂vAK(unK, v(t, x))dx dt∆t
≤NX−1
n=0 X
K∈T∆x
Z tN
0 Z
Rd φ(t, x, t
n+1, x
K) d
X
i=1
|∂xiv(t, x)|BK(v(t, x))dx dt∆t,
where
BK(c) = max1≤i≤d
X
e∈∂K
|e|¡f0(c)·n
e,Kδe−αeNe0(c) (xKe−xK)
.
Hence,
T Econs(u, v;tN)≤TauxtN|v0|T V(Rd), where
Taux = sup t∈(0,tn)
x∈Rd
NX−1
n=0 X
K∈T∆x
φ(t, x, tn+1, x
K)BK(v(t, x)) ∆t
= sup
t∈(0,tn)
x∈Rd
NX−1
n=0
X
K:|x−xK|≤x
φ(t, x, tn+1, x
K)BK(v(t, x)) ∆t
≤ sup
t∈(0,tn)
x∈Rd
NX−1
n=0
sup
x∈Rdφ(t, x, t
n+1, x
K) ∆t
with
Λ(x) = sup t∈(0,tn)
x∈Rd
X
K:|x−xK|≤x
BK(v(t, x))
.
Now, by Lemma 3.7 of [7], we have
sup
t∈(0,tn)
x∈Rd
NX−1
n=0
sup
x∈Rdφ(t, x, t
n+1, x
K) ∆t
≤ kηkdLd∞(R)
x 2
1 + ∆t
t
W(tN);
observing also
Λ(x)≤ dx
1 +(∆x)d/2 d
x
Λ((∆x)1/2)
≤ d x
1 +(∆x)dd/2
x
kf0(v)k¡|δ|
var,1/2+|α|var,1/2,
and using the definition of|δ|var,1/2and|α|var,1/2, (2.9) and (2.10), we get
Taux ≤ 2
1 + ∆t
t
W(tN)1 + (∆x)d/2
d x
kηkd
L∞(R) ¡
|δ|var,1/2+|α|var,1/2,
and the results follow.
Fourth step: EstimatingT Eh.o.t.(u, v;tN).
Lemma 3.8. Suppose that the conditions (2.3)are satisfied. Then,
T Eh.o.t.(u, v;tN)≤C0
1
2kf0(v)k
(∆t)2d|η|
T V(R)
tx t
N + 41 +∆t
t
∆t
+ 32D∆xι∆x(∆x)2
d2|η|2
T V(R)
2
x +
d|η0|
T V(R)
2
x 1 +
∆t t
tN,
whereC0=|v0|T V(Rd)kf0(v)kW(tN).
To prove the above result, we rewrite T Eh.o.t.(u, v;tN) as the sum time→ T Eh.o.t.
(u, v;tN)+ cent→ T E
h.o.t.(u, v;tN)+ visc→ T Eh.o.t.(u, v;tN), with the obvious notation,
see Lemma 3.5, and estimate each of the above three terms.
Lemma 3.9. We have
time→ T E
h.o.t.(u, v;tN)≤C0
1
2kf0(v)k
(∆t)2d|η|
T V(R)
tx t
N + 41 +∆t
t
∆t
,
whereC0=|v0|T V(Rd)kf0(v)kW(tN).
Proof. Integrating by parts in time and inserting the definition ofφ, (3.1), leads to
time→ T E
h.o.t.(u, v;tN) = NX−1
n=0 Z
Rd Θ(t
where
Θ(tn+1, x0) =Z t N
0 Z
Rd F(u(t
n, x0), v(t, x))· ∇
x¡ϕ(t, x, t¯ n+1, x0)−ϕ(t, x, tn+1, x0)dx dt
+∆t2
Z tN
0 Z
Rd Ft(u(t
n, x0), v(t, x))· ∇
xϕ(t, x, tn+1, x0)dx dt.
Noticing that ¯ϕ(t, tn+1)−ϕ(t, tn+1) = 1 ∆t
Rtn+1
tn (s−tn)ϕt(t, s)ds, one gets after integrating by parts and some elementary manipulations,
Θ(tn+1, x0) =Z t N
0 Z
RdFt(u(t
n, x0), v(t, x))· ∇
x
Z tn+1
tn
(s−tn)2
2 ϕt0(t, x, s, x0)ds dx dt
− Z
Rd∇x·F(u(t
n, x0), v(tN, x))Z t
n+1
tn (s−t
n)ϕ(tN, x, s, x0)ds dx
+
Z
Rd∇x·F(u(t
n, x0), v
0(x)) Z tn+1
tn (s−t
n)ϕ(0, x, s, x0)ds dx.
This allows us to writetime→ T Eh.o.t.(u, v;tN) =I−II+III, with the obvious notation.
Now we only need to estimate termsI, II, and III. First, we have
|I| ≤ Tauxkf0(v)k2|v0|T V(Rd)tN, where
Taux=(∆t)
2
2 t∈sup(0,tn)
x∈Rd
Z tN
0 Z
Rd|∇xϕt
0(t, x, t0, x0)|dx0dt0
≤(∆t)2
2
d|η|T V(R)
tx W(t N),
where we follow the lines of [6, Lemma 7.12] and [6, Lemma 7.13]. Next we estimate termII:
|II| ≤ Tauxkf0(v)k |v0|T V(Rd)∆t , where
Taux= sup x∈Rd
Z tN
0 Z
Rd|φ(t
N, x, t0, x0)|dx0dt0
≤2
1 +∆t
t
W(tN),
by [7, Lemma 3.7]. The termIII can be estimated in exactly the same way. Com-bining the estimates of termsI, II, III completes the proof.
Before we estimate cent→ T Eh.o.t.(u, v;tN) and visc→ T Eh.o.t.(u, v;tN), we need to
prove a couple of simple auxiliary results.
Lemma 3.10. We have
X
K∈T∆x
X
e∈∂K
where
Ie,K(Φ) = |K1 e|
Z
Ke 1
|K|3 Z
K3
Z 1
0 Z 1
0 Z 1
0 Φdλ dµ dν dz
0dy0dw0dx0, Φ≡Φ(λ(µ(νx0+ (1−ν)w0) + (1−µ)y0) + (1−λ)z0).
Proof. Since the pointλ(µ(νx0+ (1−ν)w0) + (1−µ)y0) + (1−λ)z0 belongs to the convex hull ofK andKe, Ce, we have
Ie,K(Φ)≤ 1
max{|K|,|Ke|}kΦkL1(Ce).
Hence,
X
K∈T∆x
X
e∈∂K
|e|deIe,K(Φ)≤
X
K∈T∆x
X
e∈∂K
|e|de
max{|K|,|Ke|}kΦkL1(Ce)
≤D∆x
X
K∈T∆x
X
e∈∂K
kΦkL1(Ce), by (2.5),
≤2D∆x
X
e∈E∆x
kΦkL1(Ce)
≤2D∆x
X
K∈T∆x
ι∆xkΦkL1(K)
≤2D∆xι∆xkΦkL1(Rd), by (2.6). This completes the proof.
Lemma 3.11. We have
Θaux≤ 4tN(∆x)2D∆xι∆x
1 +∆t
t
W(tN)d2|η|2T V(R)
2
x +
d|η0|
T V(R)
2
x
,
where
Θaux =
X
1≤j,k≤d NX−1
n=0 X
K∈T∆x
X
e∈∂K
|e|be,KedeIe,K(∂xjxkϕ(t, x)) ∆t.
Proof. By the definition of the operatorIe,K in Lemma 3.10 and the definition
ofϕ, (3.2), we can write Θaux= Θaux,t·Θaux,x, where
Θaux,t = NX−1
n=0
ωt(t−tn+1) ∆t≤ 2
1 + ∆t
t
W(tN),
by [7, Lemma 3.7] and Θaux,x=
X
1≤j,k≤d
X
K∈T∆x
X
e∈∂K
|e|be,KedeIe,K(∂xjxkRjk(x,·)),
whereRjk(x, x0) =Qd
i=1ηx(xi−x0i). Since, by Lemma 3.10,
Θaux,x≤ 2D∆xι∆x
X
1≤j,k≤d
and since
X
1≤j,k≤d
∂xjxk
d
Y
i=1
ηx(xi−x0i)
= X
1≤j,k≤d
d Y i=1
i6=j6=k
ηx(xi−x0i)η0x(xk−x 0
k)η0x(xj−x 0 j) + d Y i=1
i6=j
ηx(xi−x0i)η00x(xj−x0j)δjk
,
we have that
Θaux,x≤ 2D∆xι∆x
d2|η|2
T V(R)
2
x +
d|η0|
T V(R)
2
x
.
This completes the proof.
We are now ready to estimate the remaining term inT Eh.o.t..
Lemma 3.12. Suppose that the conditions (2.3)are satisfied. Then,
cent→ T E
h.o.t.(u, v;tN)≤C2(∆x)2
d2|η|2
T V(R)
2
x +
d|η0|
T V(R)
2
x 1 +
∆t t
,
visc→ T E
h.o.t.(u, v;tN)≤C2(∆x)2
d2|η|2
T V(R)
2
x +
d|η0|
T V(R)
2
x 1 +
∆t t
kαk`∞(E∆x),
whereC2= 16tN|v0|T V(Rd)kf0(v)kD∆xι∆xW(tN).
Proof. We only have to prove the first estimate since the second is similar. We have
cent→ T E
h.o.t.(u, v;tN) = NX−1
n=0 X
K∈T∆x
X
e∈∂K
|e|be,KeΘne,K∆t,
where
Θn e,K =
X
1≤i,j,k≤d
Z tN
0 Z
Rd F(u
n
K, v(t, x))·ne,K ∂xiIe,K(Ξijk(t, x))dx dt.
Moreover, Ξijk(t, x)) =ζijk∂
xjxkφ, where ζijk andφare as in Lemma 3.4. Performing integrations by parts with respect toxi, we get
Θn e,K=−
X
1≤i,j,k≤d
Z tN
0 Z
Rd
¡
∂vF(unK, v(t, x))·ne,K∂xiv(t, x)Ie,K(Ξijk(t, x))dx dt,
and hence,
cent→ T E
h.o.t.(u, v;tN)≤ Tauxkf0(v)k
Z tN
0 Z
Rd
X
1≤i≤d
|∂xiv(t, x)|dx dt
where
Taux = max
1≤i≤dt∈sup(0,tN)
x∈Rd
X
1≤j,k≤d NX−1
n=0 X
K∈T∆x
X
e∈∂K
|e|be,KeIe,K(Ξijk(t, x)) ∆t.
Since
|ζijk| ≤4 (∆x)2d
e,
we have that
Ie,K(Ξijk(t, x))≤ 4 (∆x)2d
eIe,K(∂xjxkϕ(t, x)), and so,
Taux≤ 4 (∆x)2Θaux
≤
