A priori error estimates for numerical methods for scalar conservation laws. Part II, Flux-splitting monotone schemes on irregular Cartesian grids

Volume 66, Number 218, April 1997, Pages 547–572 S 0025-5718(97)00838-7

A PRIORI ERROR ESTIMATES FOR NUMERICAL METHODS FOR SCALAR CONSERVATION LAWS. PART II:

FLUX-SPLITTING MONOTONE SCHEMES ON IRREGULAR CARTESIAN GRIDS

BERNARDO COCKBURN AND PIERRE-ALAIN GREMAUD

Abstract. _{This paper is the second of a series in which a general theory ofa} priorierror estimates for scalar conservation laws is constructed. In this pa-per, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of (∆x)1/2in L∞(L1_{) for consistent schemes in arbitrary gridswithoutthe use of any}

regu-larity property of the approximate solution. We then extend this result to less consistent schemes, calledp−consistent schemes, and prove that they converge to the entropy solution with the rate of (∆x)min{1/2,p}_inL∞_(L1_{); again, no}

regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of (∆x)1/2 _inL∞_(L1_{). We show that this well-knownsupraconvergence}

phenomenon takes place because theconsistencyof the numerical flux and the fact that the scheme is written inconservation formallows the regularity prop-erties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

1. Introduction

This is the second of a series of papers in which we develop a theory ofa priori

error estimates, that is, estimates given solely in terms on the exact solution, for numerical methods for the scalar conservation law [11]

vt+∇ ·f(v) = 0, in (0, T)×Rd,

(1.1a)

v(0) =v0, onRd.

(1.1b)

In the first paper of this series [4], we constructed a general approach aimed at obtaininga priorierror estimates for numerical methods for scalar conservation laws

Received by the editor November 27, 1995 and, in revised form, May 6, 1996. 1991Mathematics Subject Classification. Primary 65M60, 65N30, 35L65 .

Key words and phrases. A priori error estimates, irregular grids, monotone schemes, conser-vation laws, supraconvergence .

The first author was partially supported by the National Science Foundation (Grant DMS-9407952) and by the University of Minnesota Supercomputer Institute.

The second author was partially supported by the Army Research Office through grant DAAH04-95-1-0419.

c

1997 American Mathematical Society

by a suitable modification of Kuznetsov approximation theory [12]. We illustrated the approach by establishing optimal error estimates for the Engquist-Osher scheme [5] on one-dimensional uniform grids without using any smoothness property of the approximate solution generated by the scheme; in previous work, [2], [3], [13]–[16], [18], [19], [20], [22], [24], [26], regularity properties of the approximate solution were always used (see also [23]). The extension of this result to the case of nonuniform grids is by no means trivial since the nonuniformity of the grids introduces a “loss” of consistency (see, for example, Hoffman [9], Pike [21], and Turkel [25]) which, nevertheless,does notdeteriorate the rate of convergence of the global error. This paper is devoted to the study of thissupraconvergencephenomenon, that is, to the study of the relation between the part of the truncation error generated by the lack of consistency of the scheme and the global error.

Supraconvergenceof numerical schemes has been analyzed in a variety of cases. For example, Manteuffel and White [17] studiedsupraconvergencefor linear, second-order boundary value problems, Kreisset al. [11] for high-order linear differential equations, B. Wendroff and A.B. White [27], [28] for nonlinear hyperbolic systems, Archilla and Sanz-Serna [7] for third-order finite differences, and Garc´ıa-Archilla [6] for the Korteweg-de Vries equation. To illustrate thissupraconvergence

phenomenon in our setting, let us consider the standard Engquist-Osher scheme on nonuniform grids, i.e.,

(un_j+1−u_jn)/∆t+ (fEO(un_j, un_j+1)−fEO(un_j−1, un_j))/∆j = 0, n∈N, j∈Z,

with numerical flux fEO_{(a, b) =f}+_{(a) +f}−_{(b), f}+ _andf− _{being respectively the}

increasing and decreasing part off. As usual, ∆j =xj+1/2−xj−1/2denotes the cell

centered around the nodexj. Assuming that the solutionvis smooth, the (formal)

truncation error is given byT Ef_(tn_{, x}

j) =T E f

visc+T E f

cons+T E f

h.o.t, where

T E_viscf = ∆t 2 ∂x(f

02

(vn_j)∂xvjn)−

∆2

j+1/2+ ∆ 2 j−1/2

4∆j

∂x(|f0(vjn)|∂xvjn)

+∆

2

j+1/2−∆ 2 j−1/2

4∆j

∂x(f0(vjn)∂xvjn),

T E_consf = (∆j+1/2+ ∆j−1/2 2∆j −

1)f0(vn_j)∂xvnj +

−∆j+1/2+ ∆j−1/2

2∆j |

f0(v_jn)|∂xvnj,

T E_h.o.tf = O(∆

3 j+1/2

∆j

) +O(∆

3 j−1/2

∆j

) +O(∆t2),

where vn

j stands for v(tn, xj) and ∆j+1/2 = (∆j + ∆j+1)/2. The above terms

correspond respectively to the numerical viscosity of the scheme, to the consistency of the scheme, and to some “high-order” terms; note that the termT Ef

consvanishes

if uniform grids are considered. It is easy to see that the (formal) truncation error tends to zero upon refinement if ∆jvaries smoothly with respect toj. Convergence

can thus reasonably be expected in this case. On the other hand, if nonsmooth grids are considered, the scheme is not consistent. Indeed if, for instance, the grids

. . . ,∆x/2,∆x,∆x/2,∆x, . . . are considered, the termT Ef

consdoesnottend to zero,

and thus neither does the (formal) truncation errorT Ef_(tn_{, x} j).

Figure 1. L1-error vs. ∆xfor a continuous (top), and

discontin-uous (bottom) solution, on random grids

we display the performance of the Engquist-Osher scheme on the classical example of the Burgers’ equation with periodic boundary conditions and a sinusoidal initial condition (see [8] for details).

About 400 randomly generated—and thus non smooth—grids were considered. The globalL1-error at the final time is represented with respect to ∆x, size of the largest element. In Figure 1 (top), the exact solution is smooth; the convergence rate is one. In Figure 1 (bottom), the exact solution exhibits a discontinuity but, interestingly enough, the scheme converges without any loss in the numerical rate of convergence. This shows that the (formal) truncation error is a poor indicator of the quality of a numerical algorithm.

point of significant importance, if one recalls that even the simplest schemes, the monotone schemes, have not been proven to generate approximate solutions with this kind of regularity, when defined on general triangulations. In this paper, to obtain our a priori error estimates, we do not use any regularity property of the approximate solution; as a consequence, we are forced to use suitable definitions of consistency. Thus, we obtain the optimal rate of convergence of (∆x)1/2 _in L∞(L1_{), for consistent schemes in arbitrary grids. We also consider a class of}

nu-merical schemes of varying degree of consistency calledp−consistent and prove that they converge to the entropy solution with the rate of (∆x)min{1/2,p} _{in L}∞_(L1_).

In both cases, no regularity property of the approximate solution is used.

To explain the supraconvergence of the numerical schemes under consideration (which was proven by Sanders [22]) we allow ourselves to use the total variation boundedness of the approximate solution butonlyto estimate the term that appears in thepropertruncation error due to the inconsistency introduced by the nonuni-formity of the grids. We show that the optimal rate of convergence of (∆x)1/2 in

L∞(L1) can be obtained, even for inconsistent schemes, because the consistency

of the numerical flux and the fact that the scheme is written inconservation form

allow the regularity properties of the numerical approximation to compensate for the lack of consistency of the scheme; the nonlinearity of the problem does not play any role in this mechanism. To the knowledge of the authors, this is the first rigor-ous explanation of a supraconvergence phenomenon for hyperbolic problems with low regularity; the study of B. Wendroff and A.B. White [27], [28] on hyperbolic systems is formal and applies to smooth solutions only.

Finally, we strongly emphasize that, although all our results are stated and proved in a one-dimensional framework, they can be immediately extended to the case of multidimensional problems, provided the grids are Cartesian products of nonuniform one-dimensional grids. The case of time-varying meshes will not be considered in this paper since it would add a great deal of complexity to the already very technical analysis presented. To the authors’ knowledge, no such result is available in the present context.

The paper is organized as follows. In§2, the numerical schemes under considera-tion are presented, related technical assumpconsidera-tions are discussed, and the main results are stated and discussed. In §3, we give a proof of our main result. Concluding remarks are offered in§4.

2. The numerical schemes and the main results

a. The numerical schemes. Given a partition of R+_{, {t}n _{= n∆t}}

n_∈N, and a

grid or partition of R,{xj+1/2}j_∈Z, we define an approximationuto the entropy

solutionvof (1.1) (with d= 1) as the piecewise-constant function

u(t, x) =u_jn, for (t, x)∈[tn, tn+1)×(xj−1/2, xj+1/2),

(2.1)

constructed as follows. Att= 0, the degrees of freedom ofuare given by

u0_j = 1 ∆j

Z xj+1/2

xj−1/2

u0(s)ds.

(2.2a)

The remaining degrees of freedom are defined by the following flux-splitting scheme inconservation form:

(un_j+1−u_jn)/∆t+ (f_jn_+1/2−f_jn_−1/2)/∆j = 0, n∈N, j∈Z,

where ∆j =xj+1/2−xj₋1/2 and the numerical fluxfjn+1/2=fj+1/2(unj, u n j+1) has

the form

f_jn_+1/2=f_cent,jn _+1/2−f_visc,jn _+1/2,

(2.3a)

with

f_cent,jn _+1/2=aj+1/2 ∆j+1/2

f(u_jn) + bj+1/2 ∆j+1/2

f(u_jn₊₁),

(2.3b)

and

f_visc,jn _+1/2=αj+1/2 ∆j+1/2

N(u_jn₊₁)−N(u_jn),

(2.3c)

where ∆j+1/2= (∆j+ ∆j+1)/2. We assume that the fluxfjn+1/2isconsistentwith

the nonlinearityf, i.e., thatfj+1/2(u, u) =f(u); this is equivalent to assume

aj+1/2+bj+1/2= ∆j+1/2.

(2.3d)

We also require

max{aj+1/2, bj+1/2,0} ≤αj+1/2≤ ∆x≡sup j∈Z

∆j,

(2.3e)

and

N0(s)≥ |f0(s)|.

(2.3f)

Two standard examples of viscosityN areN(u) =Ru|f0(s)|ds (Engquist-Osher flux) andN(u) =Cu (Lax-Friedrichs flux), whereC is chosen as to satisfy (2.3f). As is well-known, condition (2.3f) ensures the monotonicityof the scheme under a suitable condition on the size of ∆twhich in our case turns out to be the following:

∆t

∆jk

N0(u)k ≤cfl(j), j∈Z,

(2.4a)

where

cf l−1(j) = aj+1/2 ∆j+1/2 −

bj−1/2

∆j−1/2

+ αj+1/2 ∆j+1/2

+ αj−1/2 ∆j−1/2

,

(2.4b)

kN0(u)k= sup

t∈(0,T) x∈R

N0(u(t, x)).

(2.4c)

Note that (for the Engquist-Osher scheme, for example) the above stability condi-tion (2.4) boils down to the usual cfl(j)≡1 in the case of uniform grids. It should also be noted that (2.4) is essentially a condition of the type ∆t ≤ kinfj_∈Z∆j,

wherek is a positive constant independent of the grid. It is therefore the smallest element ∆0which limits the size of the time step ∆t. Although customary, such a

b. Consistency and a priori error estimates. Before stating our error esti-mates, we need to elaborate on the notion of consistency of the schemes.

The (formal) truncation errorT Ef_(tn_{, x}

j) for the schemes under consideration

can be split into three terms,T E_viscf +T Econsf +T E f

h.o.t., defined as follows:

T E_viscf =∆t 2 ∂x(f

02

(v_jn)∂xvnj)−

∆j−1/2αj−1/2+ ∆j+1/2αj+1/2

2∆j

∂x(N0(vjn)∂xvjn)

+bj+1/2∆j+1/2−aj−1/2∆j−1/2 2∆j

∂x(f0(vjn)∂xvjn),

T E_consf =bj+1/2+aj−1/2−∆j ∆j

∂xf(vjn)−

αj+1/2−αj−1/2

∆j

∂xN(vnj),

T E_h.o.t.f =O(∆t2) +rjO(∆x2),

whererj = max{∆_∆j−1

j ,

∆j+1

∆j }. Note that we can use theconsistencyof the numerical

flux (2.3) to write

T E_consf =− ˆ

δj+1/2−ˆδj−1/2

∆j

∂xf(vjn)−

αj+1/2−αj−1/2

∆j

∂xN(vnj),

where

ˆ

δj+1/2=aj+1/2−

1 2∆j+1.

Although it is not clear at this point, it is this structureof the consistency error

T Ef

cons which allows the phenomenon of supraconvergence to take place. What is

clear, however, is that the consistency error is identically zero if both ˆδj+1/2 and αj+1/2 are constant. It is thus reasonable to measure the degree ofconsistency of

the scheme by some seminorm related to the variation of ˆδandα.

Our analysis shows that the correct quantity to consider is not ˆδj+1/2 but

δj+1/2=aj+1/2−

1 2∆j, (2.5)

and that the consistency of the error should be measured with the following semi-norm:

|ζ|var,1/2= sup x∈R

P

|xj−x|≤(∆x)1/2|ζj+1/2−ζj−1/2|

(∆x)1/2 .

(2.6a)

This motivates the following concepts of consistency. We say that the scheme is

p−consistent with respect to the family of grids{xj+1/2}j∈Z _∆x>0 if there are

two nonnegative constantsCδ andCα such that

|δ|var,1/2≤Cδ(∆x)p, |α|var,1/2≤Cα(∆x)p.

(2.6b)

If Cδ = Cα = 0, we say that the scheme is consistent. For example, for the

one-parameter family of schemes

aj+1/2=

1

2 (1−θ)∆j+θ∆j+1

,

bj+1/2=

1

2 (1−θ)∆j+1+θ∆j

, αj+1/2=

for θ ∈ [0,1], we have δj+1/2 = θ(∆j+1−∆j)/2. Moreover, it is clear that the

schemes are consistent for θ = 0 regardless of the family of grids. For θ ∈(0,1], these schemes arep−consistent if the grids are such that

X

|xj−x|≤(∆x)1/2

|∆j−1−2∆j+ ∆j+1| ≤(∆x)1/2·Cδ(∆x)p.

This property holds if the grids arep−smooth, that is, if there is a constantκsuch that

rh−1≤κ∆xp, rh= sup j_∈Z

∆j±1

∆j

.

(2.7)

Note that for 0−smooth grids like. . . ,∆x/2,∆x,∆x/2,∆x, . . ., the schemes above are 0−consistent and clearly inconsistent, except for the scheme obtained with

θ= 0.

We are now ready to state our error estimate which, following [4], 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. Letu be the piecewise-constant solution given by the scheme(2.2)with coefficients satisfy-ing(2.3), letvbe the entropy solution, and setR(v0) = [infx∈Rv0(x),supx∈Rv0(x)].

Then

ku(tN)−v(tN)kL1_(R)≤2ku₀−v₀kL1_(R)+ 8|v₀|T V_(R) q

2tNk_ν

vk(∆x)1/2

+C|v0|T V(R) |δ|var,1/2+|α|var,1/2

+|v0|T V(_R)(b1(∆x)3/4+b2∆x),

wherekνvk= supj∈Zsupw∈R(v0)νj(w)and the local viscosity coefficientνjis given

by

νj(w) =

1 2

αj+1/2

∆x

∆j+ 2∆j+1

3∆j

+αj−1/2 ∆x

∆j+ 2∆j₋1

3∆j

N0(w)

+ _a

j+1/2

∆x

∆j+ 2∆j+1

3∆j −

bj−1/2

∆x

∆j+ 2∆j−1

3∆j

f0(w)− ∆t ∆x(f

0_(w))2

.

The constantC is given by

C= 4kN0(v)k(tN + q

2tN_/kν

vk) (1 +b0(∆x)1/4),

and the constants b0, b1, and b2 are locally bounded functions that depend solely

on the quantitieskf0(v)k∆t/∆x,kf0(v)k/kνvk, and{tNkνvk}1/2. Moreover, if

the entropy solution has a finite number of discontinuities on each compact set of (0, T)×R, we can take kνvk = supt∈(0,τ)

x∈R

νj(v−, v+)≡supu∈[v−∧v+_,v−_∨v+_]|νj(u;v−, v+)| and

νj(u;v−, v+) =

1 2

αj+1/2

∆j+ 2∆j+1

3∆j

+αj−1/2

∆j+ 2∆j₋1

3∆j

[N(v)]

[v]

+

aj+1/2

∆j+ 2∆j+1

3∆j −

bj₋1/2

∆j+ 2∆j₋1

3∆j

_[F(v)]

[v]

−∆t[F(v)]

[v]

[f(v)] [v]

.

In the above expression, [v] = v+ _−v−_{, [F(v)] =} Rv+

v− f0(s)sgn(s−u)ds, and

[N(v)] =R_vv₋+ N0(s)sgn(s−u)ds.

An immediate consequence of this result is the following.

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)kL1_(R)≤2ku0−v0kL1_(R)+ 8|v0|T V(R)

q 2tNk_ν

vk(∆x)1/2

+O((∆x)min{p,3/4}).

For consistent schemes, we have thatδj+1/2≡δ, αj+1/2≡α, and we can write

νj(w)≤

α+|δ|

∆x N

0_(w)− ∆t 2 ∆x(f

0_(w))2 _{+ Θ(w),}

where

Θ(w)≤

2 3

α

∆x+

|δ|

∆x rh−1+

10 3 +

8

3rh−1rh−1

N0(w)

≤C N0(w) (∆x)q,

forq−smooth grids. Thus, Theorem 2.1 gives the following result.

Corollary 2.3 (Consistent schemes). With the notation and under the assump-tions of Theorem2.1, if the grids areq−smooth and if the scheme is consistent, we have

ku(tN)−v(tN)kL1_(R)≤2ku0−v0kL1_(R)+ 8|v0|T V(R)

q 2tNk_ν

vk(∆x)1/2

+O((∆x)min{1/2+q,3/4}),

where

kνvk= sup w_∈R(v0)

α+∆x|δ|N

0_(w)− ∆t 2∆x(f

0_(w))2

.

Note that even for 0−smooth grids, the optimal rate of convergence ofO((∆x)1/2) is achieved by the above schemes.

inequality [4, Proposition 7.6]. Ife(tn_{) denotes the errorku(t}n_)−v(tn_)k

L1_(R), then

e(tN)≤2e(0) + 8 x+tkf0(v)k

|v0|T V(_R)+ 2kf0(v)k |v0|T V(_R)∆t

+ 2 lim

w_→χ_1≤sup_n≤N

E_div? (u, v;tn)/W(tn)−Ediss(uh, v;tn)/W(tn) ,

where the so-called dual form E?

div(u, v;tn) is, in this case, nothing but the

trun-cation error and the form Ediss(uh, v;tn) contains the information on the entropy

dissipation (or “hyperbolic coercivity”) of the numerical scheme. The third term in the right-hand side reflects the fact that the scheme is first-order accurate in time. The parametersxandtare auxiliary positive numbers that will be suitably

chosen after obtaining the estimates of the formsE_div? (u, v;tn) andEdiss(uh, v;tn).

The functionsω, χ, andW are auxiliary functions to be precisely defined in §3.a. Since the numerical schemes under consideration are monotone, it can be easily proven that

−Ediss(uh, v;tN)≤0,

under the condition (2.4) on the size of ∆t. To estimate the dual formE?

div(u, v;tn), we first show that it is bounded by the

truncation error

E_div? (u, v;tn)≤T E(u, v;tn),

and then we obtain the corresponding estimate.

To illustrate the estimate ofE_div? (u, v;tn), let us consider that both the entropy solutionv and the “approximate solution”uare smooth. We also assume that the functionsa,b,δ, andαdefining the coefficients of the numerical scheme are smooth functions. In this case, the truncation error T E =T E(u, v;T) can be written as the sum of the following three terms

T Evisc=

Z T

0

Z

R

Z T

0

Z

R

∆x V(u, v;x0)ϕxxdx0dt0dx dt,

T Econs=−

Z T

0

Z

R

Z T

0

Z

R

δx0(x0)F(u, v) +αx0(x0)N(u, v) ϕxdx0dt0dx dt,

T Eh.o.t=

Z T

0

Z

R

Z T

0

Z

R

(∆t)2

6 F(u, v)ϕxxtdx

0_dt0_{dx dt}

+ Z T

0

Z

R

Z T

0

Z

R

P(x0)F(u, v) +Q(x0)N(u, v) ϕxxxdx0dt0dx dt,

where, in order to render as clear as possible the manipulations that will be per-formed, we abbreviated v(t, x) by v, u(t0, x0) by u, and the auxiliary function

ϕ(t, x, t0, x0) by ϕ. The functions V, F, N and F are related to the numerical viscosity coefficient

ν(w;x0) =α(x 0₎ ∆x N

0_{(w) +}b(x0)−a(x0) 2 ∆x f

0_(w)− ∆t 2 ∆x(f

and to the functionsf andN as follows:

V(u, v;x0) = Z u

v

ν(s;x0)U0(u−s)ds, F(u, v) = Z u

v

f0(s)U0(u−s)ds,

N(u, v) = Z u

v

N0(s)U0(u−s)ds, F(u, v) = Z u

v

(f0(s))2U0(u−s)ds,

whereU(w) =|w|. The functionsP andQsatisfy

k P kL∞(_R),k Q kL∞(_R)≤(∆x)2/2.

Before estimating the truncation error T E, let us compare it with the (formal) truncation errorT Ef_{, which is the sum of the following terms:}

T E_viscf (t, x) =1

2∆t ∂x(f 02

(v(t, x))∂xv(t, x))−α(x)∂xx(N(v(t, x)))

+1

2(b(x)−a(x))∂xx(f(v(t, x))),

T Ef_cons(t, x) =−δx(x)∂xf(v(t, x)) +αx(x)∂xN(v(t, x)) ,

T E_h.o.t.f = O(∆t2) +O(∆x2).

We see that the definition of the (formal) truncation error T Ef _{collapses when}

v is a nonsmooth function. However, the truncation error T E remains defined even if v and u are only bounded and measurable. Moreover, in the expression of the truncation error T E, it is possible to integrate by parts very easily due to the fact that the functions v = v(t, x) and u= u(t0, x0) are always evaluated at different points; this key feature was introduced by Kruˇzkov [11]. In order to compensate for this “doubling of the variables,” the auxiliary function ϕ is introduced and is defined to be an approximation of the product of the Dirac delta functions with support {t = t0}and {x =x0} respectively; more precisely,

ϕ(t, x, t0, x0) ={w((t−t0)/t)/t} {η((x−x0)/x)/x}, wherew andη are positive,

even, smooth functions of unit mass and support in [−1,1].

We are now ready to estimateT E. To estimateT Evisc, we integrate by parts in

the variablexand use the definition of the functionV(u, v;x0) to obtain

T Evisc=

Z T

0

Z

R

Z T

0

Z

R

∆x ν(v;x0)vxU0(u−v)ϕxdx0dt0dx dt

≤∆xkνvk

Z T

0

Z

R |

vx|

Z T

0

Z

R |

ϕx|dx0dt0

dx dt

≤2C0|

η|T V(R)

x k

νvk∆x,

whereC0=T|v0|T V(R)W(T), and whereW(s) is the antiderivative ofw(s/t)/t,

since

Z T

0

Z

R |

ϕx|dx0dt0dx dt≤2

|η|T V(R) x

W(T),

Z T

0

Z

R|

To estimate the consistency truncation error,T Econs, we integrate once again by

parts in the variablexand use the definitions of the functionsF(u, v) andN(u, v):

T Econs=−

Z T

0

Z

R

Z T

0

Z

R

δx0(x0)F(u, v) +αx0(x0)N(u, v) ϕxdx0dt0dx dt

=− Z T

0

Z

R

Z T

0

Z

R

δx0(x0)f0(v) +αx0(x0)N0(v) U0(u−v)vxϕ dx0dt0dx dt

≤Z T 0

Z

R |

f0(v)| |vx|

Z T

0

Z

R

ϕ|δx0(x0)|dx0dt0

dx dt

+ Z T

0

Z

R|

N0(v)| |vx|

Z T

0

Z

R

ϕ|αx0(x0)|dx0dt0

dx dt

≤2C1 1 +

(∆x)1/2

x

kηkL∞(R) |δ|var,1/2+|α|var,1/2

,

whereC1=C0kN0(v)k, since

Z T

0

Z

R

ϕ|ζx0(x0)|dx0dt0≤2W(T) 1 +

(∆x)1/2

x

kηkL∞(R)|ζ|var,1/2.

Finally, to estimateT Eh.o.t, we integrate by parts inx, use the definitions ofF(u, v), N(u, v), andF and proceed as before to get

T Eh.o.t≤C1

_(∆t)2_|η| T V(R)

tx k

f0k+ 2(∆x)

2_|η0_| T V(R)

2 x

.

Now, we pickη such that:

|η|T V(R)= 1 +, |η0|T V(R)= 2 ++ 1/, kηkL∞(R)= (1 +)/2,

and insert the above estimates into the right-hand side of the approximation in-equality. To prove Theorem 2.1, we simply have to minimize the right-hand side of the approximation inequality with respect to the parameters x, t, and . It

turns out that the optimal parameters arex = O((∆x)1/2), t = O((∆x)3/4),

and= O((∆x)1/4). The estimates of the truncation errors then take the form

T Evisc(u, v;T)/W(T)≤C00(∆x) 1/2_,

T Econs(u, v;T)/W(T)≤C10 |δ|var,1/2+|α|var,1/2

,

T Eh.o.t.(u, v;T)/W(T)≤C20(∆x)3/4,

where the constantsC_i0,i= 0,1,2, are independent of ∆xfor ∆xsmall enough.

d. An explanation of thesupraconvergence. To illustrate the idea that allows the supraconvergence phenomenon to take place, we only need to show how to exploit the structure of the termT Econs to obtain a better estimate. Since both

terms ofT Econsare similar in structure, we concentrate only on the first:

Θ =− Z T

0

Z

R

Z T

0

Z

R

δx0(x0)F(u, v)ϕxdx0dt0dx dt.

ofsupraconvergenceto take place. The price to pay, however, is that we must give up the restriction of not using regularity properties of the approximate solutionu, as we show next.

Thus, to estimate Θ, we integrate by parts in the variablex0and use the definition of the functionF(u, v) to obtain

Θ =− Z T

0

Z

R

Z T

0

Z

R

F(u, v)ϕx(δ(x0)−δ¯)x0dx0dt0dx dt

= Z T

0

Z

R

Z T

0

Z

R

U0(u−v)f0(u)ux0ϕx+F(u, v)ϕx x0

δ(x0)−¯δ dx0dt0dx dt

= Z T

0

Z

R

Z T

0

Z

R

U0(u−v)f0(u)ux0ϕx+f0(v)vxϕx0 δ(x0)−δ¯ dx0dt0dx dt

≤

Z T

0

Z

R |

f0(u)| |ux0|

Z T

0

Z

R |

ϕx|

δ(x0)−δ¯ dx dt

dx0dt0

+ Z T

0

Z

R|

f0(v)| |vx|

Z T

0

Z

R |

ϕx0|

δ(x0)−¯δ dx0dt0

dx dt.

At this point, it becomes clear that in order to estimate Θ, we must obtain a bound on theL∞-norm ofuand on its total variation. Sinceuis the “approximate solution” of a monotone scheme in one-space dimension, it is well-known that we have

kf0(u)k kukL∞(0,tN_{;T V(R))}≤kf0(v)k |v0|T V(R),

With the above regularity property of the “approximate solution”, we can obtain

Θ≤4C1k

δkL∞(R)/R

x |

η|T V(R),

wherekδkL∞(R)/R= inf¯_δ∈Rkδ−δ¯kL∞(R),since

Z T

0

Z

R |

ϕx0|dx0dt0=

Z T

0

Z

R|

ϕx|dx dt≤ 2|

η|T V(_R) x

W(T).

This implies the following upper bound for the consistency error:

T Econs(u, v;T)≤4C1|η|T V(R)

kδkL∞(R)/R+kαkL∞(R)/R

x

.

The error estimate follows as in the previous section. A discrete version of the above argument can be easily obtained which, under the notation and assumptions of Theorem 2.1, leads to the error estimate:

ku(tN)−v(tN)kL1_(R)≤ 2ku0−v0kL1_(R)

+ 8|v0|T V(R)

q

2tN_(kν

vk∆x+ 2kN0(v)k(kδkL∞(R)/R+kαkL∞(R)/R))

which gives the optimal rate of convergence of (∆x)1/2_{, as expected. Although the}

above error estimate is new, we are more interested in the technique to obtain it since it sheds light into thesupraconvergencephenomenon. As we have just shown, the optimal rate of convergence of (∆x)1/2_{can be obtained even though the scheme}

is not consistent because theconsistencyof its numerical flux and itsconservativity

makes possible for the lack of consistency of the scheme to be compensated by the regularity of its approximate solution u. The fact that the problem is nonlinear does not play any fundamental role in this mechanism.

3. Proof of Theorem 2.1

In this section, we prove our Theorem 2.1. This section is closely related to section 7 of [4] in which we establish the same result for the Engquist-Osher scheme defined in uniform grids. Thus, we shall use the same notation and omit detailed proofs when those are variations of similar proofs in [4].

a. The approximation inequality. We start by displaying the following inequal-ity proven in [4, Proposition 7.6]. Ife(tn) denotes the errorku(tn)−v(tn)kL1_(R), then

e(tN)≤2e(0) + 8 x+tkf0(v)k

|v0|T V(R)+ 2kf0(v)k |v0|T V(R)∆t

+ 2 lim

w→χ_1≤sup_n≤N

E_div? (u, v;tn)/W(tn)−Ediss(uh, v;tn)/W(tn) ,

where, in this case, the formEdiss(uh, v;tN) is given by

Ediss(uh, v;tN) =

Z tN

0

Z

R N_X−1

n=0

X

j∈Z

LRED_jn(v(t, x))φ(t, x, tn+1, xj) ∆j∆t dx dt,

where the local rate of entropy dissipationLREDn

j (c) is given by

LREDjn(c) =

1 ∆t

Z un j

un_j+1

(p1j(u n j)−p

1

j(s))U00(s−c)ds

+ 1 ∆j

Z un j−1

un_j+1

(p2_j(un_j−1)−p2_j(s))U00(s−c)ds

+ 1 ∆j

Z un j+1

un_j+1

(p3_j(un_j+1)−p3_j(s))U00(s−c)ds,

p1_j =s− ∆t

∆j

(aj+1/2 ∆j+1/2 −

bj−1/2

∆j−1/2

)f(s)−∆t ∆j

(αj+1/2 ∆j+1/2

+ αj−1/2 ∆j−1/2

)N(s),

p2_j = aj−1/2 ∆j₋1/2

f(s) + αj−1/2 ∆j₋1/2

N(s),

p3_j =−bj+1/2 ∆j+1/2

f(s) + αj+1/2 ∆j+1/2

and the dual formE?

div(uh, v;tN) is given by

E_div? (uh, v;tN) =−

Z tN

0

Z

R

Z tN

0

Z

R

U(u(t0, x0)−v(t, x))ϕt(t, x, t0, x0)dx dt dx0dt0

+ Z tN

0

Z

R

Z

R

U(u(t0, x0)−v(tN, x))ϕ(tN, x, t0, x0)dx dt dx0

−Z tN

0

Z

R

Z

R

U(u(t0, x0)−v0(x))ϕ(0, x, t0, x0)dx dt dx0

− N_X−1

n=0

X

j∈Z

Z tN

0

Z

R

aj+1/2

φ(t, x, tn+1_{, x}

j)−φ(t, x, tn+1, xj+1)

∆j+1/2

+bj₋1/2

φ(t, x, tn+1, xj−1)−φ(t, x, tn+1, xj)

∆j−1/2

·F(u_jn, v(t, x))dx dt∆t

+

N_X−1

n=0

X

j∈Z

Z tN

0

Z

R

−αj+1/2

φ(t, x, tn+1, xj)−φ(t, x, tn+1, xj+1)

∆j+1/2

+αj₋1/2

φ(t, x, tn+1_{, x}

j−1)−φ(t, x, tn+1, xj)

∆j−1/2

· N(u_jn, v(t, x))dx dt∆t.

The function U(·) is nothing but | · |, and F(u, c) and N(u, c) are defined as follows:

F(u, c) = Z u

c

f0(s)U0(s−c)ds, N(u, c) = Z u

c

N0(s)U0(s−c)ds.

(3.1)

The functionφis given by

φ(t, x, t0, xj) =

1 ∆j

Z xj+1/2

xj−1/2

ϕ(t, x, t0, s)ds.

(3.2)

where the functionϕ=ϕ(t, x, t0, x0) is defined as follows:

ϕ=wt(t−t0)ηx(x−x0), (x, t),(x0, t0)∈R×R

+_,

(3.3a)

wheretand xare two arbitrary positive numbers and

wt(s) =

1

t

w(s

t

), ηx(s) =

1

x

η(s

x

),

(3.3b)

for anys∈R. For future reference, we also set

W(t) = Z t

0

wt(s)ds.

(3.3c)

Finally, the functionswandη are smooth approximations toχ≡χ0 andχ, where

χ(x) =

    

(1 +)/2, for|x| ≤(1−)/(1 +),

(1 +)2(1− |x|)/4, for|x| ∈[(1−)/(1 +),1],

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 +,

(3.4a)

lim

η_→χ|

η0|T V(R)=|χ0|T V(R)= 2 ++ 1/.

(3.4b)

lim

η→χk

ηkL∞(R)=kχkL∞(R)= (1 +)/2.

(3.4c)

b. Estimate ofEdiss(uh, v;tn). To estimateEdiss(uh, v;tn), it is enough to follow

the proof of the corresponding result in [4, Proposition 7.1].

Proposition 3.1. Under condition (2.3f) on the viscosity term N, and if the Courant-Friedrichs-Levy (CFL) condition (2.4) is satisfied, the local rate of en-tropy dissipation LRED_jn(c)is nonnegative. Hence

−Ediss(uh, v;tn)≤0.

Sketch of the proof. The above conditions ensure that the functionspi

j(s),i = 1,2,

3, are nondecreasing in s. The result follows from this fact and the definition of

Ediss(uh, v;tn).

c. Estimate of E_div? (uh, v;tN).

Proposition 3.2. We have

lim

w_→χ_1≤sup_n≤N

E?_div(u, v;tn)/W(tn) ≤T EWvisc+T EWcons+T EWh.o.t.,

where

T EWvisc(u, v;tn)≤ C0

2|η|T V(R)

x

1 +∆t

t

kνvk,

T EWcons(u, v;tn)≤2C1 1+

∆t t

1 +(∆x)

1/2

x

kηkL∞(R) |δ|var,1/2+|α|var,1/2

,

T EWh.o.t.(u, v;tn)≤C1

_(∆t)2_{|η|T V} (R)

tx k

f0k+ 2(∆x)

2_|η0_{|T V} (R)

2 x

1 + ∆t

t

,

whereC0= tn|v0|T V(_R) andC1=C0kf0(v)k.

To prove this result, we proceed in several steps.

First step: Relating the dual form E?

div(u, v;t) to the truncation error.

We start by suitably relating the dual form to the truncation error. To do that, we will need the following averages of the functionϕ:

φ(t, x, tn+1, xj) =

1 ∆t

Z tn+1

tn

φ(t, x, s, xj)ds,

(3.5a)

ˆ

φ(t, x, tn+1, xj+1/2) =

1 ∆j+1/2

Z 1/2

−1/2

Z xj+1+∆j+1ρ

xj+∆jρ

ϕ(t, x, tn+1, s)ds dρ,

(3.5b)

where ˆφ(t, x, tn+1, xj+1/2) has been defined in such a way that the following equality

holds:

ˆ

φx(t, x, tn+1, xj+1/2) =

φ(t, x, tn+1, xj)−φ(t, x, tn+1, xj+1)

∆j+1/2

.

(3.5c)

Lemma 3.3. We have

E_div? (uh, v;tN)≤ NX−1

n=0

X

j∈Z

Z tN

0

Z

R

∆t

2 ∆jφxt(t, x, t

n+1_{, x}

j)F(ujn, v(t, x))dx dt∆t

+

N_X−1

n=0

X

j∈Z

Z tN

0

Z

R

˜

Ψ_jn(t, x)N(u_jn, v(t, x))dx dt∆t

− N_X−1

n=0

X

j∈Z

Z tN

0

Z

R

˜

Φjn(t, x)F(ujn, v(t, x))dx dt∆t,

where

˜

Φ_jn(t, x) =aj+1/2φˆx(t, x, tn+1, xj+1/2) +bj−1/2φˆx(t, x, tn+1, xj−1/2)

−∆jφx(t, x, t n+1_{, x}

j) +

∆t

2 ∆jφxt(t, x, t

n+1_{, x} j),

and

˜

Ψ_jn(t, x) =−αj+1/2φˆx(t, x, tn+1, xj+1/2) +αj−1/2φˆx(t, x, tn+1, xj−1/2).

To prove this result, we use the fact that v is the entropy solution and make some algebraic manipulations; see the proof of the similar result [4, Proposition 7.9].

Next, we need to relate the functionsφand ˆφas defined in (3.5). The relations we need are displayed in the following result, which can be obtained by using simple Taylor expansions.

Lemma 3.4. We have

ˆ

φ(t, x, tn+1, xj+1/2) =φ(t, x, tn+1, xj)−

∆j+ 2∆j+1

6 φx(t, x, t

n+1_{, x} j)

+

Z xj+3/2

xj+1/2

P_j+(x0)ϕx0x0(t, x, tn+1, x0)dx0

+

Z xj+1/2

x_j−1/2

Q+_j(x0)ϕx0x0(t, x, tn+1, x0)dx0,

and

ˆ

φ(t, x, tn+1, xj−1/2) =φ(t, x, tn+1, xj) +

∆j+ 2∆j−1

6 φx(t, x, t

n+1_{, x} j)

+

Z xj+1/2

xj−1/2

Q−_j(x0)ϕx0x0(t, x, tn+1, x0)dx0

+

Z xj−1/2

xj−3/2

The polynomials P± andQ± are given by

P+(x0) = (xj+3/2−x 0₎3

6 ∆j+1/2∆j+1

, P−(x0) = (x 0_−x

j−3/2)3

6 ∆j₋1/2∆j−1 ,

Q+(x0) = (x 0_−x

j−1/2−∆j+1/2)3+ (∆j+1/2)3

6 ∆j+1/2∆j

+∆j+1−∆j 12 ∆j

(x0−xj−1/2),

Q−(x0) = (xj+1/2−x 0_−∆

j−1/2)3+ (∆j−1/2)3

6 ∆j−1/2∆j

+∆j−1−∆j 12 ∆j

(xj+1/2−x0).

With the above lemma, we can now rewrite the upper bound ofE?

div(uh, v;t N₎

as the sum of three terms. The first,T Evisc(u, v;tN), is that part of the truncation

error which contains the information of the viscosity of the numerical scheme. The second term, T Econs(u, v;tN), contains information concerning theconsistency of

the numerical scheme; indeed, if the scheme is consistent, thenT Econs(u, v;tN) =

0. We emphasize that to define this term, the definition of δj+1/2, (2.5), must

be used. The third term, T Eh.o.t.(u, v;tN), contains the high-order terms in the

truncation error and, as expected, will be dominated by the termT Evisc(u, v;tN)

andT Econs(u, v;tN).

Lemma 3.5. We have

E_div? (uh, v;tN)≤T Evisc(u, v;tN) +T Econs(u, v;tN) +T Eh.o.t.(u, v;tN),

where

T Evisc(u, v;tN) = N_X−1

n=0

X

j∈Z

Z tN

0

Z

R

V ISC_jn(v(t, x);t, x)dx dt∆t

− N_X−1

n=0

X

j∈Z

Z

R

F(un_j, v(tN, x))∆t 2 φx(t

N_{, x, t}n+1_{, x}

j)dx∆j∆t

+

NX−1

n=0

X

j_∈Z

Z

R

F(un_j, v(t0, x))∆t

2 φx(0, x, t

n+1_{, x}

j)dx∆j∆t,

T Econs(u, v;tN) = NX−1

n=0

X

j∈Z

Z tN

0

Z

R

CON S_jn(v(t, x);t, x)dx dt∆t,

T Eh.o.t.(u, v;tN) = N_X−1

n=0

X

j∈Z

Z tN

0

Z

R

HOT_jn(v(t, x);t, x)dx dt∆t

+

N_X−1

n=0

X

j∈Z

Z

R

F(unj, v(tN, x))

∆t

2 φx(t

N_{, x, t}n+1_{, x}

j)dx∆j∆t

− NX−1

n=0

X

j∈Z

Z

R

F(un_j, v(t0, x))∆t

2 φx(0, x, t

n+1_{, x}

j)dx∆j∆t.

The ‘viscosity’ termV ISC_jn(c;t, x)is given by

V ISCjn(c;t, x) = time

V ISCjn(c;t, x) + cent

V ISCjn(c;t, x) + visc

where

time

V ISC_jn(c;t, x) =F(u_jn, c)∆t

2 ∆jφxt(t, x, t

n+1_{, x} j) , cent

V ISC_jn(c;t, x) =F(u_jn, c)

aj+1/2

∆j+ 2 ∆j+1

6

−bj−1/2

∆j+ 2 ∆j−1

6

φxx(t, x, tn+1, xj),

visc

V ISC_jn(c;t, x) =N(u_jn, c)

αj+1/2

∆j+ 2∆j+1

6

+αj₋1/2

∆j+ 2∆j−1

6

φxx(t, x, tn+1, xj).

The ‘consistency’ termCON Sn

j (c;t, x)is given by

CON Sjn(c;t, x) = cent

CON Sjn(c;t, x) + visc

CON Sjn(c;t, x),

where

cent

CON S

n

j(c;t, x) =F(u n

j, c) (−δj+1/2+δj−1/2)φx(t, x, tn+1, xj), visc

CON S

n

j(c;t, x) =N(u n

j, c) (−αj+1/2+αj−1/2)φx(t, x, tn+1, xj).

Finally, the ‘high-order’ termHOTn

j (c;t, x)is given by

HOT_jn(c;t, x) =

time

HOT_jn(c;t, x) +

cent

HOT_jn(c;t, x) +

visc

HOT_jn(c;t, x), where

time

HOT_jn(c;t, x) =F(u_jn, c)∆jφx(t, x, t n+1_{, x}

j)−∆jφx(t, x, tn+1, xj)

+∆t

2 ∆jφxt(t, x, t

n+1 , xj) , cent

HOT_jn(c;t, x) =

−aj+1/2

Z xj+3/2

xj+1/2

P_j+(x0)ϕx0x0x(t, x, tn+1, x0)dx0

−aj+1/2

Z xj+1/2

xj−1/2

Q+_j(x0)ϕx0x0x(t, x, tn+1, x0)dx0

−bj−1/2

Z xj+1/2

xj−1/2

Q−_j(x0)ϕx0x0x(t, x, tn+1, x0)dx0

−bj−1/2

Z xj−1/2

x_j−3/2

P_j−(x0)ϕx0x0x(t, x, tn+1, x0)dx0

F(u_jn, c),

visc

HOT_jn(c;t, x) =

−αj+1/2

Z xj+3/2

xj+1/2

P_j+(x0)ϕx0x0x(t, x, tn+1, x0)dx0

−αj+1/2

Z xj+1/2

xj−1/2

Q+_j(x0)ϕx0x0x(t, x, tn+1, x0)dx0

+αj−1/2

Z xj+1/2

xj−1/2

Q−_j(x0)ϕx0x0x(t, x, tn+1, x0)dx0

+αj−1/2

Z xj−1/2

xj−3/2

P_j−(x0)ϕx0x0x(t, x, tn+1, x0)dx0

Second step: EstimatingT Evisc(u, v;tN). In this section, we prove the following

result.

Lemma 3.6. If (2.3)and(2.4)hold, we have

T Evisc(u, v;tN) ≤ 2C0|

η|T V(R) x

1 +∆t

t

kνvk∆x,

whereC0= tN|v0|T V(R)W(tN).

We need the following auxiliary result whose proof can be found in the proof of [4, Lemma 7.13].

Lemma 3.7. We have

sup

t∈(0,tN₎

NX−1

n=0

wt(t−t

n+1_{) ∆t}

≤ 2 1 +∆t

t

W(tN).

Proof of Lemma 3.6 . As in the proof of [4, Lemma 7.11], it is enough to consider an entropy solution v that is everywhere smooth except on a single discontinuity curveC={(x(t), t) :t∈(0, tN_)}.

We have

T Evisc(u, v;tN) = NX−1

n=0

X

j∈Z

Ξ(tn+1, xj) ∆j∆t,

where

Ξ(tn+1, xj) =

Z tN

0

Z

R

F(un_j, v(t, x))∆t

2 φxt(t, x, t

n+1_{, x} j)

+F(un_j, v(t, x))

aj+1/2

∆j+ 2∆j+1

6∆j

−bj−1/2

∆j+ 2∆j−1

6∆j

φxx(t, x, tn+1, xj)

+N(un_j, v(t, x))

αj+1/2

∆j+ 2∆j+1

6∆j

+αj−1/2

∆j+ 2∆j−1

6∆j

φxx(t, x, tn+1, xj)

dxdt

−Z R

F(un_j, v(tN, x))∆t 2 φx(t

N_{, x, t}n+1_{, x} j)dx

+ Z

R

F(un_j, v(0, x))∆t

2 φx(0, x, t

A couple of integrations by parts yields

Ξ(tn+1, xj) =−

Z tN

0

Z x(t)

−∞

1 2∆tFt+

aj+1/2

∆j+ 2∆j+1

6∆j −

bj−1/2

∆j+ 2∆j−1

6∆j

Fx

+

αj+1/2

∆j+ 2∆j+1

6∆j

+αj−1/2

∆j+ 2∆j−1

6∆j

Nx

φxdxdt

−Z tN

0

Z ∞

x(t)

1 2∆tFt+

aj+1/2

∆j+ 2∆j+1

6∆j −

bj−1/2

∆j+ 2∆j₋1

6∆j

Fx

+

αj+1/2

∆j+ 2∆j+1

6∆j

+αj−1/2

∆j+ 2∆j−1

6∆j

Nx

φxdxdt

−

Z tN

0

−1₂∆t[F][f] [v]+

aj+1/2

∆j+ 2∆j+1

6∆j −

bj₋1/2

∆j+ 2∆j−1

6∆j

[F]

+

αj+1/2

∆j+ 2∆j+1

6∆j

+αj−1/2

∆j+ 2∆j−1

6∆j

[N]

φxdt,

where the last integral is understood to be along the discontinuity lineC.

The jump of a function G(v) across C at a point (x(t), t) is denoted [G] =

G(v(t, x(t) + 0))−G(v(t, x(t)−0)).

Setting ˜νj =νj(u;v(t, x(t)−0), v(t, x(t) + 0)), where νj(u;v−, v+) is defined in

Theorem 2.1, we get

Ξ(tn+1, xj) =−

Z tN

0

Z x(t)

−∞ ˜

νjvxφxdx+ ˜νj[v]φx+

Z ∞

x(t)

˜

νjvxφxdx

dt.

Proceeding again as in [4], we set

kν˜jk= sup t∈(0,tN₎

x∈R

sup

u∈R|

˜

νj|,

and thus

|Ξ(tn+1, xj)| ≤ kν˜jk

Z tN

0

Z x(t)

−∞ |

vx||φx|dx+|[v]||φx|+

Z ∞

x(t)

|vx||φx|dx

dt.

(3.6)

We would like to analyze further the dependence of ˜νj with respect to u. Setting

ν#_{(u) =ν}

j(u;v−, v+) and taking into account (3.1), direct calculations yield

∂uν#(u) =

1 2

U0(u−v+)−U0(u−v−)

v+−_v−

αj+1/2

∆j+ 2∆j+1

3∆j

+αj−1/2

∆j+ 2∆j−1

3∆j

N0(u)

+

aj+1/2

∆j+ 2∆j+1

3∆j −

bj−1/2

∆j+ 2∆j−1

3∆j

f0(u)

−∆tf(v

+_)−f(v−₎ v+−_v− f

Thereforeν#_{(u) is constant for any value ofuwhich does not lie betweenv− and} v+_{. This implies that}

sup

u_∈R|

ν#(u)|= sup

u∈[v−∧v+,v−∨v+]

|ν#(u)| ≡νj(v−, v+) ∆x.

Inserting νj(v−, v+) in the bound (3.6) and using the definition of kνvk, we get,

sincevis the entropy solution

T Evisc(u, v;tN)≤Tauxkνvk∆x

Z tN

0

Z x(t)

−∞ |

vx|dx+|[v]|+

Z ∞

x(t) |vx|dx

dt

≤TauxtN|v|L∞(0,tN_{;T V(R))}kνvk∆x ≤TauxtN|v0|T V(R)kνvk∆x,

where

Taux= sup t∈(0,tn)

x∈R

N_X−1

n=0

X

j∈Z

|φx(t, x, tn+1, xj)|∆j∆t

≤

Z

R

1

x|

η0(y)|dy

sup

t∈(0,tN₎

NX−1

n=0

wt(t−t

n+1_{) ∆t}

.

Taking Lemma 3.7 into account achieves the proof. Third step: Estimating T Econs(u, v;tN).

Lemma 3.8. We have T Econs(u, v;tN)≤2C1 1 +

∆t t

1 +(∆x)

1/2

x

kηkL∞(R) |δ|var,1/2+|α|var,1/2

,

whereC1= tN|v0|T V(R)kN0(v)kW(tN).

Note that if the scheme is consistent, the upper bound for T Econs(u, v;tN) is

equal to zero, as expected.

We will need the following simple auxiliary result. Lemma 3.9. We have

1

x

X

|x−xj|≤x

|δj+1/2−δj−1/2| ≤ 1 +

(∆x)1/2

x

|δ|var,1/2.

Proof of Lemma 3.8. The consistency errorT Econs(u, v;tN) is the sum of two terms

of the form

Θ =−

N_X−1

n=0

X

j∈Z

Z tN

0

Z

R

G(unj, v(t, x))

φ(t, x, tn+1, xj)x(ζj+1/2−ζj−1/2)dx dt∆t,

one of which has ζ=δand G=F, and the other ζ=αandG=N. Thus, it is enough to get an estimate for Θ.

To do that, we assume that the entropy solutionv is smooth; see the proof [4, Proposition 5.5]. First, we integrate by parts in thexvariable to obtain

Θ =

N_X−1

n=0

X

j∈Z

Z tN

0

Z

R

G(un_j, v(t, x))_xφ(t, x, tn+1, xj) (ζj+1/2−ζj−1/2)dx dt∆t

where

Taux= sup t∈(0,tn)

x∈R

N_X−1

n=0

X

j_∈Z

φ(t, x, tn+1, xj)|ζj+1/2−ζj−1/2|∆j<