Convergence Rates for Relaxation Schemes Approximating Conservation Laws∗

SIAM J. NUMER. ANAL. c 2000 Society for Industrial and Applied Mathematics Vol. 37, No. 4, pp. 1316–1337

CONVERGENCE RATES FOR RELAXATION SCHEMES APPROXIMATING CONSERVATION LAWS∗

HAILIANG LIU† AND GERALD WARNECKE‡ Abstract. 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 ω() we obtain the rate of convergence of in L1 for the relaxation step. The estimate√ here is independent of the type of nonlinearity. In the discretization step a convergence rate of ∆x in L1 is obtained. These rates are independent of the choice of initial error ω(). Thereby, we obtain the order 1/2 for the total error.

Key words. relaxation scheme, relaxation model, convergence rate

AMS subject classiﬁcations. 35L65, 65M06, 65M15

PII. S0036142998346882

1. Introduction. Let u : R × R+ → R be the unique global entropy solution in the sense of Kruˇzkov [11] to the Cauchy problem for the conservation law

(1.1) ut + f(u)x =0

with initial data

(1.2) u(x, 0) = u0(x),

1 ∞ 1 where f ∈ C (R) and u0 ∈ L (R)∩L (R)∩BV (R). The solution u satisﬁes Kruˇzkov’s entropy conditions

(1.3) ∂t|u − k| + ∂x[sgn(u − k)(f(u) − f(k))] ≤ 0inD for all k ∈ R.

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 . Our main purpose is to study the convergence rate of the relaxation scheme to the conservation laws as both the relaxation rate and the mesh length ∆x tend to zero. The relaxation model takes the form ∂tu + ∂xv =0, (1.4) 1 ∂tv + a∂xu = − [v − f(u )]. The variables u and v are the unknowns, >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.

∗Received by the editors November 6, 1998; accepted for publication (in revised form) July 2, 1999; published electronically April 11, 2000. http://www.siam.org/journals/sinum/37-4/34688.html †Department of Mathematics, University of California, Los Angeles, CA 90095 ([email protected]). This author was supported by an Alexander von Humboldt Fellowship at the Otto-von-Guericke-Universit¨at Magdeburg. ‡IAN, Otto-von-Guericke-Universit¨at Magdeburg, PSF 4120, D-39016 Magdeburg, Germany ([email protected]). This author was supported by the Deutsche Forschungsgemeinschaft, Wa 633 4/2, 7/1. 1316 CONVERGENCE RATES FOR RELAXATION SCHEMES 1317

In fact, for small , using the Chapman–Enskog expansion [4], one may deduce from (1.4) the following convection-diﬀusion equation (see [10]): 2 ut + f(u )x = [a − f (u ) ]ux x,

which gives a viscosity solution to the conservation law (1.1) if the well-known subcharacteristic condition (cf. Liu [16]) holds:

(1.5) a>f(u)2.

Natalini [20] proved that the solutions to (1.4) converge strongly to the unique entropy solution of (1.1) as → 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 ∆x, ∆t in space and time as well as for n ∈ N,j∈ Z a n n numerical approximate solution (uj ,vj )=(u, v)(j∆x, n∆t). The relaxation scheme associated with the relaxation system (1.4) is given as   λ µ  un+1 − un + (vn − vn ) − (un − 2un + un )=0, j j 2 j+1 j−1 2 j+1 j j−1 (1.6)  aλ µ  vn+1 − vn + (un − un ) − (vn − 2vn + vn )=−k(vn+1 − f(un+1)), j j 2 j+1 j−1 2 j+1 j j−1 j j

∆t √ ∆t where λ = ∆x is the mesh ratio, µ = aλ, and k = ε . 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 κ = exp(∆t/ ) − 1, 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. Throughout this paper we assume the usual Courant–Friedrichs– Lewy (CFL) condition √ (1.7) 0 <µ= aλ < 1.

Convergence theory for this kind of relaxation scheme 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 , convergence of n n a subsequence of (uj ,vj )j∈Z,n∈N to the unique weak solution of (1.1) was established by standard compactness arguments. Currently, there are very few computational results for relaxation schemes avail- able in the literature; see 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 require neither the use of Riemann solvers nor the computation of nonlinear flux Jacobians. This seems to become an important advantage when considering fluids with nonstandard 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 descrip- tion of the hydrodynamic equations by the detailed microscopic evolution of gases in kinetic theory. The rigorous theory of kinetic approximation for solutions with shocks 1318 HAILIANG LIU AND GERALD WARNECKE

is well developed when the limit equation is scalar. For works using the continuous velocity kinetic approximation, 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] and 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 λ2 = −λ1 = a>0 and defining the Riemann invariants 1 v 1 v R = u − √ ,R = u + √ 1 2 a 2 2 a and the Maxwellians 1 f(u) 1 f(u) M (u):= u − √ ,M(u):= u + √ . 1 2 a 2 2 a The relaxation model (1.4) becomes 1 (1.8) ∂ R + λ ∂ R = [M (u) − R)],i=1, 2. t i i x i i i 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 1 u−u∆ between the exact solution u and the numerical solution u∆ measured in the L norm. The parameters and ∆x 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 (H1) u0(x):=u0(x),v0 = f(u0(x)) + K(x)ω( ), where K ∈ L∞ ∩ L1(R) ∩ BV (R),ω:[0, ∞[→ [0, ∞[ is continuous, and ω(0)=0. Here we allow for an initial error K(x)ω( ) instead of v0 = f(u0) because 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 u0 − u0 L1(R) would persist in time and may prevent the convergence of u 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 , and therefore it does not prevent the convergence of u. We initialize the relaxation scheme (1.6) by cell averaging the initial data (u0,v0) in the usual way: 1 (1.9) (u0,v0)= (u (x),v(x))χ (x)dx. j j ∆x 0 0 j Here and elsewhere without the integral limit denotes the integral on the whole of R, and χj(x) denotes the indicator function χj(x):=1{|x−j∆x|≤∆x/2}. CONVERGENCE RATES FOR RELAXATION SCHEMES 1319

1 Let us now introduce some notations. The L -norm is denoted by · 1, and TV(·) denotes the total variation, deﬁned on a subset Ω ⊆ R by |u(x + h) − u(x)| TV(u):=sup dx. h =0 Ω |h| The BV-norm is deﬁned as

u BV = u 1 + TV(u). For grid functions the total variation is deﬁned by n n n TV(u )= |ui − ui−1|, i∈Z

1 and · 1 denotes the discrete l -norm n n u 1 =∆x |ui |. i∈Z Taking initial data (1.9), we summarize our main convergence rate result by stating the following. Theorem 1.1. Take any T>0 and let the relation T = N∆t be satisﬁed for a suitable N ∈ N and time step ∆t. Further, let u be the entropy solution of ∞ N N (1.1)–(1.2) with initial data u0 in L (R) ∩ BV (R), and let (u ,v ) be a piecewise n n constant representation on R × [0,T] of the approximate solution (ui ,vi )i∈Z,0≤n≤N generated by the relaxation scheme (1.6) with initial data satisfying (H1) and (1.9). ∆t Then for ﬁxed λ = ∆x satisfying the CFL condition (1.7) there exists a constant CT , independent of ∆x, ∆t, and , such that √ √ N (1.10) u − u(·,T) 1 ≤ CT + ∆x .

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 (u,v) to the relaxed system (1.4), as well as of 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 = u(·,T) − u (·,t ) into a relaxation error e with e ≤ √ ∆ ∆ N √ 1 CT and a discretization error e∆ with e∆ 1 ≤ CT ∆x, i.e., we have the decom- position

e∆ = e + e∆ with

e = u(·,T) − u (·,T) and e∆ =u(·, T) − u∆(·, tN). 1320 HAILIANG LIU AND GERALD WARNECKE

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-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 disconti- nuities 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 L1 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 , i.e., ω( )= . Our result shows that the uniform estimate does not depend on ω( ), which is more natural since in the discretization step is kept a constant. Taking = 0 in the 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 multispeeds 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 (1.4) 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.

2. 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 (u0,v0), which will be of use in our error analysis. Let us make the following assumptions: 1 (H2) the flux function f is a C function with f(0) = f (0)=0; 1 ∞ (H3) the initial data satisfy (u0,v0) ∈ L (R) ∩ L (R) ∩ BV (R) and there exist constants ρ0 > 0,M>0 not depending on such that ρ0 = max sup v0 ∞, sup u0 ∞ , (u0,v0) BV := u0 BV + v0 BV ≤ M, >0 >0

and for the ﬂux function f as well as K given in (H1), f(x) − f(y) Lip(f):=sup ≤ M, K 1 ≤ M. x =y x − y CONVERGENCE RATES FOR RELAXATION SCHEMES 1321

Let us deﬁne, for any ρ>0, the notations F (ρ) := sup |f(ξ)|, |ξ|≤ρ

B(ρ):=2ρ + F (2ρ), and M(ρ) := sup |f (ξ)|. |ξ|≤B(ρ)

∞ 2 Theorem 2.1 (see Natalini [20]). Assume (H2) and (u0,v0) ∈ L (R) for any ρ0 > 0 and >0;if √ (2.1) a>M(ρ0),

1 2 then the system (1.4) admits a unique, global solution (u ,v ) in C([0, ∞[; [Lloc(R)] ) satisfying √ √ (2.2) v ± au L∞(R×]0,∞[) ≤ aB(ρ0), √ (2.3) |f (u(x, t))| < a for all >0 and for almost every (x, t) ∈ R×]0, ∞[. 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 (H1)–(H3), it has been proved that, as → 0 , the 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 ↓ 0 is summarized in the following theorem. Theorem 2.2. Consider the system (1.4), subject to L∞(R) ∩ BV (R)-perturbed initial data satisfying (H1)–(H3). Then the global solution (u ,v ) converges to (u, f(u)) as ↓ 0 and the following error estimates hold: √ (2.4) u (·,t) − u(·,t) 1 ≤ CT ,

− t − t (2.5) v (·,t) − f(u (·,t)) 1 ≤ CT [e ω( )+ (1 − e )], 0 ≤ t ≤ T.

Thus, (2.5) reflects two sources of error: the initial contribution of size ω( ) and the relaxation error of order . However, it should be mentioned that the effect of the initial contribution persists only for a short time of order , and 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 (u0,v0) satisfying (H1) and (H3). To make this more precise, we denote a family of approximate solutions given as piecewise constant functions, dropping the superscript for notational convenience, by n n n n (2.6) (u∆,v∆):=(u∆,v∆)(·,n∆t):= (uj ,vj )χj(x). j∈Z 1322 HAILIANG LIU AND GERALD WARNECKE

Here, n ∈ N represents the number of time steps performed. For the initial conditions 0 0 (u∆,v∆) we take the orthogonal projection of the initial data (u0,v0) onto the space of piecewise constant functions on the given grid 1 u0 = P u0 = u0χ (x),u0 := u (x)χ (x)dx, ∆ ∆ j j j ∆x 0 j j∈Z 1 (2.7) v0 = P v0 = v0χ (x),v0 := v(x)χ (x)dx. ∆ ∆ j j j ∆x 0 j j∈Z

Thus, it follows from (H1), (H3), and (2.7) that the discrete initial data satisfy 0 0 (2.8) (a) max u∆ ∞, v∆ ∞ ≤ ρ0, (2.9) (b) TV(u0 )+TV(v0 ) ≤ M, ∆ ∆ (2.10) (c) v0 − f(u0 ) ≤ Mω( ). ∆ ∆ 1

∆t The grid parameters ∆x and ∆t are assumed to satisfy ∆x = const. We note that, since λ =∆t/∆x is assumed constant, ∆x → 0 implies ∆t → 0 as well. It is well known that the projection error is of order ∆x. More precisely, we have

0 u∆ − u0 1 ≤∆xT V (u0), 0 (2.11) v∆ − v0 1 ≤∆xT V (v0). As was shown by Aregba-Driollet and Natalini [1] as well as by 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(ρ0) such that if √ a>M(ρ0),

then the numerical solution satisﬁes v (2.12) (un,vn) ∈ K := (u, v) ∈ R2, u ± √ ≤ B(ρ ) , j j ρ0 a 0

where B(ρ0) is a constant depending only on ρ0. Starting with the discrete initial data satisfying (2.9)–(2.11), by using the Rie- mann 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 n n (u∆(x, tn),v∆(x, tn)) = j(uj ,vj )χj(x) converges strongly to the unique limit solution (u, v)(x, tn) as we reﬁne the grid taking ∆x ↓ 0. This, together with equi- continuity in time and the Lax–Wendroﬀ theorem, yields a weak solution (u,v)of the relaxation system (1.4). We note that the initial bounds (2.9)–(2.11) still hold for 0 0 the piecewise constant numerical data (u∆,v∆) for which

TV (u∆(·,t)) ≤ TV(u) and u∆(·,t) 1 ≤ u 1.

We refer to [1], [31], and [32] for the convergence of approximate solutions (u∆,v∆) to the unique weak solution of (1.4). Our goal here is to improve the previous convergence theory by establishing the following L1-error bound of the relaxation scheme. This theorem will be proved in section 4. CONVERGENCE RATES FOR RELAXATION SCHEMES 1323

Theorem 2.3. Let (u ,v ) be the weak solution of (1.4) with initial data (u0,v0), N N N N and let (u ,v ) be a piecewise constant representation of the data (ui ,vi )i∈Z gen- 0 0 erated by (1.6) starting with (u∆,v∆). Then, for any ﬁxed T = N∆t ≥ 0, there is a ﬁnite constant CT independent of ∆x, ∆t, and such that √ v(·,T) − vN + u(·,T) − uN ≤ C ∆x. 1 1 T Remark 2.4. Our uniform error bound is independent of the relaxation parameter and initial error ω( ). This is more general than the result obtained by [25] assuming the initial error ω( )tobe . As remarked earlier, the error is split into a relaxation error e and a discretization error e∆. 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

(3.1) ut + f(u)x =0 by solutions (u,v) of the relaxation system ut + vx =0, (3.2) 1 vt + aux = − [v − f(u )] . For the above purpose we have to show that u satisﬁes an approximate entropy condition. Let us begin with rewriting (3.2) in terms of the Riemann invariants (R1,R2) and Maxwellians (M1(u ),M2(u )) deﬁned in the introduction and recall some basic facts that will be used in our analysis. It is easy to see that √ (3.3) u = R1 + R2,v= a R2 − R1 . Then the system is rewritten as (1.8), i.e., 1 ∂ R + λ ∂ R = M (u) − R ,i=1, 2, t i i x i i i

where the functions Mi(u)(i =1, 2) have the following properties: 2 2 (3.4) Mi(u)=u, λiMi(u)=f(u). i=1 i=1 We note that the L∞ estimate obtained by Natalini [20] implies that √ v (3.5) (u,v) ∈ K := (u, v) ∈ R2||f (u)|≤ a, u ± √ ≤ B(ρ ) . ρ0 a 0 Further, we set

I ρ0 := {u ∈ R, (u, v) ∈ Kρ0 }. (u) are monotone (nondecreasing) functions of u ∈ I because due to the The Mi ρ0 subcharacteristic condition (1.5), d 1 f (u) M (u)= 1 ± √ ≥ 0. du i 2 a 1324 HAILIANG LIU AND GERALD WARNECKE

Thus, we have for any u andu ˜ in I ρ0 2 2 2 (3.6) |M (u) − M (˜u)| = M (u) − M (˜u) = |u − u˜|. i i i i i=1 i=1 i=1 Starting with the initial distribution 1 v 1 v (3.7) R (·, 0) = u − √0 ,R (·, 0) = u + √0 , 1 2 0 a 2 2 0 a the model evolves according to the system (1.8), which is well-posed. In fact, we rewrite (1.8) in the form 1 1 (3.8) ∂ R + λ ∂ R + R = M (u),i=1, 2. t i i x i i i From (3.8) one obtains the L1-contraction property using a Gronwall inequality (see [21]) which is

2 2 (3.9) Ri (·,t) − Si (·,t) 1 ≤ Ri (·, 0) − Si (·, 0) 1 ∀ t ≥ 0. i=1 i=1 As shown in [20], the above nice property for general data of bounded variation yields the following TV estimate:

(3.10) (u (·,t),v (·,t)) BV ≤ C (u0,v0) BV .

Lemma 3.1. For any ρ0 > 0 and >0, if √ a>M(ρ0),

1 2 then the global solution (R1,R2) ∈ C([0, ∞[; Lloc(R) ) of the problem (1.8), (3.7) for any k ∈ R satisﬁes the entropy-type inequality 2 2 (3.11) ∂t |Ri − Mi(k)| + ∂x λi|Ri − Mi(k)| ≤ 0 in D . i=1 i=1

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 (R1,R2) 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 (H1)–(H3) hold. Let (u ,v ) be a solution of (1.4) with initial data (u0,v0). Then for every t>0, it holds that 2 − t − t A (t):= Ri (·,t) − Mi(u (·,t)) 1 ≤ C[e ω( )+(1− e ) ]. i=1

Proof. In view of the relations between (R1,R2) and (u ,v )wehave √ 2 (3.12) a |Ri − Mi(u )| = |v − f(u )|, i=1 CONVERGENCE RATES FOR RELAXATION SCHEMES 1325

which serves as a measure of the distance from equilibrium. Let ξ = v − f(u); then the function ξ satisﬁes

1 ξ + ξ = −au + f (u)v t x x for (x, t) ∈ R×]0, ∞[ and initial data

ξ (x, 0) = ξ0 := v0(x) − f(u0(x)) for x ∈ R.

t Then multiplying by sgn(ξ )e and integrating over R × [0,t], one gets

t − t −(t−s) |ξ |dx ≤ e |ξ0|dx + e |aux + f (u )vx|dxds. 0 Note that by (3.10) we have for suﬃciently regular initial data

aux + f (u )vx ≤C (∂xu0,∂xv0) 1.

For example, one could use C∞ data in combination with an approximation theorem in BV (R); see Theorem 1.17 in Giusti [8]. Combining the above facts, for general data of bounded variation, gives

− t − t ||v − f(u )||1 ≤ e ||v0 − f(u0)||1 + C ||(u0,v0)||BV (1 − e ).

Together with (H3) 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. ∞ 1 Theorem 3.3 (see Bouchut and Perthame [3]). Let u, v ∈ Lloc([0, ∞[,Lloc(R)) be 1 right continuous with values in Lloc(R). Assume that u satisfies the entropy condition (1.3) and that v satisfies, ∀ k ∈ R, (3.13) ∂t|v − k| + ∂x sgn(v − k) f(v) − f(k) ≤ ∂tJk + ∂xHk in D ,

where Jk,Hk are locally ﬁnite Radon measures such that for some nonnegative k- ∞ 1 independent Radon measures αJ and αH ∈ Lloc([0, ∞[,Lloc(R)),Jk and Hk satisfy in the sense of measures (3.14) Jk(x, t) ≤ αJ (x, t) and Hk(x, t) ≤ αH (x, t).

Then for any T>0,x0 ∈ R,ρ>0,δ >0, and the balls

Bt = B(x0,ρ+ M(T − t)+δ),

we have |u(x, T ) − v(x, T )| dx |x−x0|≤ρ ≤ |u(x, 0) − v(x, 0)| dx + C Et + Ex + EJ + EH ,

B0 1326 HAILIANG LIU AND GERALD WARNECKE where C is a uniform constant and

0 ≤ Et ≤ MδTV(v ), 0 ≤ Ex ≤ MδTV(u ), 0 0 (M +1) T EJ = 1+ sup α (x, t)dx, δ J t∈]0,2T [ Bt 1 T EH = α dxdt. δ H 0 Bt

∞ 1 (3.16) αJ ∈ Lloc([0, ∞[,Lloc(R)) with αJ 1 = A (t) < ∞. Setting 2 H = λi (sgn(u − k)(Mi(u ) − Mi(k)) −|Ri − Mi(k)|) , i=1 we similarly get using (3.4) 2 2 sgn(u − k)[f(u ) − f(k)] = sgn(u − k) λiMi(u ) − λiMi(k) i=1 i=1 2 (3.17) = λi |Ri − Mi(k)| + H . i=1 CONVERGENCE RATES FOR RELAXATION SCHEMES 1327

Due to the monotonicity property of Mi, we have

sgn(u − k)(Mi(u ) − Mi(k)) = |Mi(u ) − Mi(k)|. Using this fact and the inverse triangle inequality one obtains

√ 2 √ |H |≤αH := a |Ri − Mi(u )| = aαJ . i=1 Therefore, we also have √ ∞ 1 αH ∈ Lloc([0, ∞),Lloc(R)) with αH 1 = aA (t) < ∞. Combining the previous expressions (3.15) and (3.17) yields ∂t|u − k| + ∂x sgn(u − k) f(u ) − f(k) 2 2 = ∂t |Ri − Mi(k)| + ∂x λi|Ri − Mi(k)| + ∂tJ + ∂xH . i=1 i=1

However, by Lemma 3.1, the Riemann invariants (R1,R2) satisfy the entropy-type inequalities (3.11); consequently one obtains (3.18) ∂t|u − k| + ∂x sgn(u − k) f(u ) − f(k) ≤ ∂tJ + ∂xH in D .

1 The functions J ,H are bounded by the L -functions αJ ,αH , respectively, as re- quired in (3.14). Since the solutions u ,Ri are bounded and the ﬂux function f is assumed to be Lipschitz, we may apply Lemma 3.3. Letting ρ →∞, Lemma 3.3 gives t x J H u (·,T) − u¯(·,T) 1 ≤ u (·, 0) − u¯(·, 0) 1 + C E + E + E + E ≤||u (·, 0) − u¯(·, 0)||1 (M +1) T 1 T +C 2Mδ||u¯0||BV + 1+ supt∈]0,2T [ αJ dx + αH dxdt δ δ 0 ≤|u(·, 0) − u¯(·, 0)|| 1 √ T (M +1) T a (3.19) +C δ||u¯0||BV + 1+ supt∈]0,2T [A (t)+ A (t)dt δ δ 0 ∀ δ>0. Choosing δ = supt∈]0,2T [A to minimize the right-hand side of (3.19), we have by choosing a suitably large constant CT , including u¯0 BV , that (3.20) ||u (·,T) − u¯(·,T)||1 ≤||u (·, 0) − u¯(·, 0)||1 + CT sup A (t). t∈]0,2T [

Note that the estimate in Lemma 3.2 yields

A(t) ≤ C max{ , ω( )}.

If ω( ) ≤ , the above in combination with (3.20) yields Theorem 3.4. Next we treat the case ω( ) > . To obtain the desired estimate we again apply Lemma 3.3 on the interval [τ,T] with τ>0 to be determined; thus we have as above (3.21) ||u (·,T) − u¯(·,T)||1 ≤||u (·,τ) − u¯(·,τ)||1 + CT sup A (t). t∈]τ,2T [ 1328 HAILIANG LIU AND GERALD WARNECKE

Using the fact that both u andu ¯ lie in a bounded subset in Lip(R+,L1(R)) (see Katsoulakis and Tzavaras [13] and Smoller [24]) we have

||u (·,τ)−u¯(·,τ)||1 ≤ u (·, 0)−u¯(·, 0)||1 +||u (·,τ)−u (·, 0)||1 +||u¯(·,τ)−u¯(·, 0)||1 (3.22) ≤ u (·, 0) − u¯(·, 0)||1 + Cτ.

It follows from the estimate in Lemma 3.2 that

− τ (3.23) supτ≤t≤2T A (t) ≤ C( + e ). Taking τ = α with α ∈ [0, 1] and applying (3.22) and (3.23) into (3.21), one gets α −α−1 (3.24) ||u (·,T) − u¯(·,T)||1 ≤||u (·, 0) − u¯(·, 0)||1 + CT + + e .

Now, choosing α =1/2, we easily recover the order 1/2 estimate from (3.24) and √ exp(− −1/2) ≤ .

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

Ij =[xj−1/2,xj+1/2),j∈ Z.

We will prove that the error bound√ for the relaxation scheme (1.6) approximating the relaxation system (1.4) is of order ∆x in L1, 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 (R1,R2). Dropping the superscript and ∆t noting that κ = , the scheme takes the form

n+1 n+1/2 n+1 n+1 (4.1) Ri,j = Ri,j + κ[Mi(uj ) − Ri,j ],i=1, 2,

n+1/2 n+1/2 for the source terms while the intermediate states (R1,j ,R2,j ) are generated by the following consistent monotone scheme in conservative form for the convections, namely the upwind scheme: √ n+1/2 n n n R1,j = R1,j + aλ(R1,j+1 − R1,j), √ n+1/2 n n n (4.2) R2,j = R2,j − aλ(R2,j − R2,j−1).

n n The discrete values (R1,j,R2,j) computed from (4.1) are considered as approximations of Ri in the whole cell Ij at time tn = n∆t, which can also be obtained through n n (uj ,vj ) generated by (1.6). The discrete variables are related to each other by 1 vn 1 vn (4.3) Rn = un − √j ,Rn = un + √j 1,j 2 j a 2,j 2 j a

and 1 f(un) 1 f(un) (4.4) M (un)= un − √ j ,M(un)= un + √ j . 1 j 2 j a 2 j 2 j a CONVERGENCE RATES FOR RELAXATION SCHEMES 1329

n n Conversely (uj ,vj ) can also be expressed as √ n n n n n n (4.5) uj = R1,j + R2,j,vj = a(R2,j − R1,j).

0 For the initial conditions Ri,j we take the orthogonal projection of the initial data Ri (x, 0) onto the space of piecewise constant functions on the given grid. It is given by the averages 1 (4.6) R0 = R(x, 0)χ (x)dx, i =1, 2, i,j ∆x i j 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∞ boundedness, the TVD property, and the L1 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 (u0,v0) are bounded and of bounded variation, both uniformly with respect to . Further, assume that the initial data also satisfy (H1), i.e., as a consequence

||v0 − f(u0)||1 ≤ Mω( );

∆t then there exists a constant C0, independent of and ∆t such that for κ = and K ρ0 as deﬁned by (3.5) (a) (Rn ,Rn ) ∈ K ∀(j, n) ∈ Z × N, 1,j 2,j ρ0 (b) 2 TV(Rn) ≤ C ∀n ∈ N, i=1 i 0 √ (c) M (un)−Rn|| ≤ (1+κ)−n M (u0)−R0 +C a ∆t [1−(1+κ)−n] ∀n ∈ N, i i 1 i i 1 0 κ 2 n m (d) i=1 Ri − Ri |1 ≤ C0∆t|n − m|∀n, m ∈ N. Proof. The proof of (a) can be found in Wang and Warnecke [31]. The proofs of (b) and (d) are straightforward; see Aregba-Driollet and Natalini [1] and Yong [32]. 0 0 Since here we choose initial data satisfying (H1) instead of vj = f(uj ), we prove (c) as follows. Set n n n Zj := M1(uj ) − R1,j. It follows from (4.3) and (4.4) that Zn := −[M (un) − Rn ]. Summing the scheme j 2 j 2,j n 2 n (4.1) over i =1, 2 and noting that uj = i=1 Ri,j, we obtain

n+1 n+1/2 uj = uj . Again let us consider (4.1) for i =1, i.e.,

n+1 n+1/2 n+1 R1,j = R1,j + κZj .

n+1 n+1/2 Adding −M1(uj )=−M1(uj ) on both sides gives

n+1 n+1/2 n+1 −Zj = −Zj + κZj 1330 HAILIANG LIU AND GERALD WARNECKE

by which we obtain

n+1 n+1/2 (4.7) (1 + κ)Zj = Zj .

n Using the ﬁrst equation of scheme (4.2) and the deﬁnition of Zj , one gets

n+1/2 n+1/2 n+1/2 Zj = M1(uj ) − R1,j √ n+1/2 n n n = M1(uj ) − R1,j − aλ(R1,j+1 − R1,j) √ n n+1/2 n n n (4.8) = Zj + M1(uj ) − M1(uj ) − aλ(R1,j+1 − R1,j).

n n+1/2 n+1 Then by the mean value theorem there exists a valueu ˜j between uj = uj and n n ∂M1 n uj such that for 0 ≤ γj = ∂u (˜uj ) we obtain

n+1/2 n n n+1/2 n (4.9) M1(uj ) − M1(uj )=γj (uj − uj )

due to the monotonicity property of M1(u). Substituting (4.9) into (4.8) and then using the relation (4.5) and the scheme (4.2) gives √ √ n+1/2 n n n n n n n Zj = Zj +(1 − γj ) aλ(R1,j − R1,j+1)+γj aλ(R2,j−1 − R2,j). By summation over j and multiplying by ∆x one obtains from (4.7) and using (b)

√ 2 √ n+1 n n n (4.10) (1 + κ) Z 1 ≤ Z 1 + aλ∆x TV(Ri ) ≤ Z 1 + C0 a∆t. i=1 From (4.10) it is easy to verify by iteration and the geometric sum that √ ∆t Zn ≤ (1 + κ)−n Z0 + C a [1 − (1 + κ)−n]. 1 1 0 κ 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 to 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 Ri, 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, Tveito, and Winther [25]. For notational clarity, we will use the variables (y, τ) instead of (x, t) in the context below. The iteration is initialized by

R (y, 0) = R0 ,y∈ I , i i,j j 1 R0 = P R (y, 0) := R (x, 0)χ (x)dx, i,j ∆ i ∆x j j

where χj was deﬁned as the characteristic function for the interval Ij. CONVERGENCE RATES FOR RELAXATION SCHEMES 1331

± (i) We use the notation tn to denote limits from above or below. In the interval + − (tn ,tn+1), Ri is the solution of the linear equation

(4.11) ∂τ Ri + λi∂yRi =0,i=1, 2, √ + with initial data Ri(y, tn ) and λ1 = − a = −λ2. (ii) At tn+1 we project back onto the mesh by taking cell averages

− (4.12) Ri(y, tn+1)=P∆(Ri(·,tn+1))(y),i=1, 2.

(iii) The initial data for the next iteration are deﬁned by the implicit formula treating the source term

+ + + (4.13) Ri(y, tn+1)=Ri(y, tn+1)+κ[Mi(¯u(y, tn+1)) − Ri(y, tn+1)].

From the deﬁnition of Mi(u) it follows by (4.5) using (4.13) that

+ u¯(y, tn+1)=¯u(y, tn+1).

+ Thus, the Ri(y, tn+1) can actually be obtained in the explicit form 1 (4.14) R (y, t+ )= R (y, t )+κM (¯u(y, t )) . i n+1 1+κ i n+1 i n+1

+ n Assuming Ri(y, tn )=Ri,j for y ∈ Ij,j ∈ Z, we conclude from the integral form of (4.11) on the rectangle Ij × [tn,tn+1[

n n n R1(y, tn+1)=R1,j − λ1λ[R1,j+1 − R1,j], n n n (4.15) R2(y, tn+1)=R2,j − λ2λ[R1,j − R1,j−1].

Thus our step functions Ri,i=1√, 2, representing the grid functions, possess the following property, due to λ1,2 = ± a:

1 n+ 2 + n+1 (4.16) Ri(y, tn+1)=Ri,j , Ri(y, tn+1)=Ri,j ,i=1, 2.

The discrete estimates in Lemma 4.1 and the relations (4.16) yield the following properties of Ri. Lemma 4.2. For all t, τ ∈ R+,y∈ R,n,m∈ N we have

(a) (R1, R2)(y, τ) ∈ Kρ0 , 2 (b) i=1 TV(Ri(·,τ)) ≤ C0, (c) M (¯u(·,t+)) − R (·,t+) ≤C , i n i n 0 2 + + (d) i=1 Ri(·,tn ) − Ri(·,tm) ≤C0(|tn − tm|), 2 (e) i=1 Ri(·,t) − Ri(·,τ) ≤C0(|t − τ| +∆t). Proof. The relations (4.16) imply directly that (c)–(d) hold. For the proof of (e), 1 one has to use the time-L Lipschitz continuity of Ri; 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 Ri to Ri as ∆x ↓ 0. 1332 HAILIANG LIU AND GERALD WARNECKE

Since Ri(y, τ) is a weak solution (also an entropy solution due to its uniqueness) of the system (4.11) in (tn,tn+1), the Kruˇzkov-type entropy formulation is valid. This means that 2 2 tn+1 + |Ri(y, τ) − qi|[φτ + λiφy]dydτ + |Ri(y, tn ) − qi|φ(y, tn)dy tn i=1 i=1 2 − (4.17) − |Ri(y, tn+1) − qi|φ(y, tn+1)dy ≥ 0 i=1 ,q ) ∈ K and all nonnegative C∞-functions φ with compact support in ∀ values (q1 2 ρ0 R × [0,T]. On the other hand, from step (iii) above we observe that for any (q 1,q2) ∈ Kρ0 , i =1, 2, + + + + |Ri(y, t ) − qi| = (Ri(y, tn) − qi)+κ[Mi(¯u(y, t )) − Ri(y, t )] sgn Ri(y, t ) − qi n n n n + + + (4.18) ≤|Ri(y, tn) − qi| + κ Mi(¯u(y, tn )) − Ri(y, tn ) sgn Ri(y, tn ) − qi . + Inserting (4.18) into (4.17), summing over n from n =0toN−1, and using Ri(y, t0 )= − Ri(y, t0 )=Ri(y, t0), we obtain the following entropy inequality for the discrete solution: T 2 |Ri(y, τ) − qi|[φτ + λiφy]dydτ 0 i=1 N−1 2 − + |Ri(y, tn) − qi|−|Ri(y, tn ) − qi| φ(y, tn)dy n=0 i=1 2 2 − + |Ri(y, 0) − qi|φ(y, 0)dy − |Ri(y, tN ) − qi|φ(y, T)dy i=1 i=1 N 2 + + + (4.19) ≥− κ sgn(Ri(y, tn ) − qi) Mi(¯u(y, tn )) − Ri(y, tn ) φ(y, tn)dy. n=1 i=1 Inspired by the papers [30] and [25] of Tveito and others, we use the following Kruˇzkov-type inequality for our problem since the exact solution Ri(x, t) is the unique weak solution of (1.8):

T 2 |Ri(x, t) − qi|[ψt + λiψx]dxdt 0 i=1 2 2 + |Ri(x, 0) − qi|ψ(x, 0)dx − |Ri(x, T ) − qi|ψ(x, T )dx i=1 i=1 1 T 2 (4.20) ≥− sgn(R (x, t) − q ) M (u(x, t)) − R (x, t) ψ(x, t)dxdt i i i i 0 i=1 ,q ) ∈ Ω and all ψ ∈ C∞ with ψ ≥ 0. As in Kruˇzkov [11], any for any constants (q1 2 ρ0 0 weak solution to (1.8) satisﬁes (4.20). Equipped with the above variational inequalities, we return to estimate the error + bound of ||Ri(·,T) − Ri(·,tN )||1. CONVERGENCE RATES FOR RELAXATION SCHEMES 1333

Proof of Theorem 2.3. Our proof in this section is inspired by the one given by Schroll, Tveito, and Winther [25]. They show an L1-error bound for a model arising in chromatography. Their argument is in the spirit of the work by Kruˇzkov [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

1 x η (x)= η , δ δ δ where η is any nonnegative smooth function with support in [−1, 1], even, i.e., η(x)= η(−x), and unit mass. This molliﬁer therefore satisﬁes 1 η (x)dx =1, |x|η (x)dx ≤ δ, |η (x)|dx ≤ , δ δ δ δ

and supp(ηδ) ⊆ [−δ, δ]. Let T>0 be given. We proceed by selecting the constants (q1,q2) and the test functions φ, ψ. First taking qi = Ri(x, t),φ(y, τ)=ηδ(x−y)ηδ(t− τ) in (4.19) and integrating in x and t, we obtain using Ri := Ri(y, τ),Ri := Ri(x, t)

+ In the inequality (4.20) we set qi = Ri(y, tn ) and ψ(x, t):=ηδ(x−y)ηδ(t−τ), integrate over (y, τ) ∈ R×]tn−1,tn[, and sum n from 1 to N:

N 2 tn T + |Ri − Ri(y, tn )| ηδ(x − y)ηδ(t − τ) t 0 n=1 n−1 i=1 +λiηδ(x − y)ηδ(t − τ) dxdtdydτ N 2 tn + + Ri(x, 0) − Ri(y, tn ) ηδ(x − y)ηδ(−τ)dxdydt n=1 tn−1 i=1 N 2 tn + − Ri(x, T ) − Ri(y, tn ) ηδ(x − y)ηδ(T − τ)dxdydτ n=1 tn−1 i=1 N 2 1 tn T ≥− sgn R (x, t) − R (y, t+) M (u(x, t)) − R (x, t) i i n i i n=1 tn−1 0 i=1

·ηδ(x − y)ηδ(t − τ)dxdtdydτ. 1334 HAILIANG LIU AND GERALD WARNECKE

Adding this inequality to (4.21) and suitably grouping the terms, we obtain an inequality that we write in the shorthand form

4 (4.22) L(δ) ≤ Ei(δ). i=1

The individual expressions are

Here Ri = Ri(y, τ),Ri = Ri(x, t). For these 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 , and δ such that 2 + (i) |L(δ) − i=1 Ri(·,T) − Ri(·,tN ) 1|≤C(δ +∆t), 2 (ii) |E1(δ) − i=1 Ri(·, 0) − Ri(·, 0) 1|≤C(δ +∆t), ∆t (iii) |E2(δ)|≤C δ , ∆t (iv) E3(δ) ≤ C δ , ∆t (v) |E4(δ)|≤C δ . Equipped with these estimates we continue the proof of Theorem 2.3. Using L(δ) ≥ 0, (4.22), and Lemma 4.3 we ﬁnd for a suitably large constant C>0 CONVERGENCE RATES FOR RELAXATION SCHEMES 1335

2 2 + + Ri(·,T) − Ri(·,tN ) 1 ≤|L(δ) − Ri(·,T) − Ri(·,tN ) 1| + L(δ) i=1 i=1 4 ≤ C(δ +∆t)+ Ei(δ) i=1 2 ∆t (4.23) ≤ R (·, 0) − R (·, 0) + C δ +∆t + . i i 1 δ i=1 Using an obvious bound like (2.11) for the initial error, we have 2 2 (4.24) Ri(·, 0) − Ri(·, 0) 1 ≤ TV(Ri(·, 0))∆x ≤ C∆x. i=1 i=1 √ Inserting this into (4.23) and picking δ = ∆t to minimize the right-hand side of the relation (4.23), we have 2 √ + Ri(·,T) − Ri(·,tN ) ≤C(∆x +∆t + ∆t). i=1 √ Using the CFL condition ∆t ≤ ∆x/ a we have 2 √ + (4.25) Ri(·,T) − Ri(·,tN ) ≤C ∆x. i=1 Returning to the original variables (u, v) and noting the fact that we have 2 2 N N + u∆ = Ri,∆ = Ri(y, tN ), i=1 i=1 we obtain √ 2 √ N N + u(·,T) − u∆ + v(·,T) − v∆ ≤(1 + a) Ri(·,T) − Ri(·,tN ) ≤C ∆x. i=1 This completes the proof of Theorem 2.3. Finally, we return to 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, Tveito, and Winther [25]. We next estimate the term E3(δ). Let us analogously as above use the notations

Zi(x, t):=Mi(u(x, t)) − Ri(x, t), Zi(y, τ):=Mi(¯u(y, τ)) − Ri(y, τ).

Then E3(δ) can be rewritten as

T N t 2 1 n E (δ)= sgn(R (x, t) − R (y, t+)) Z (x, t) − Z (y, t+) 3 i i n i i n 0 n=1 tn−1 i=1 ·ηδ(t − tn)ηδ(x − y)dydτdxdt N 2 1 T tn + sgn(R (x, t) − R (y, t+))Z (x, t) i i n i 0 n=1 tn−1 i=1 (4.26) · [ηδ(t − τ) − ηδ(t − tn)] ηδ(x − y)dydτdxdt. 1336 HAILIANG LIU AND GERALD WARNECKE

The ﬁrst term in E3(δ) is nonpositive due to the relations (3.3) and (3.6): 2 2 2 ¯ sgn(Ri(x, t) − Ri)[Zi − Zi] ≤ |Mi(u) − Mi(¯u)|− |Ri − Ri| i=1 i=1 i=1 2 = |u − u¯|− |Ri − Ri|≤0. i=1 Therefore

N 2 1 T tn E (δ) ≤ Z (·,t) |η (t − τ) − η (t − t )|dτdt. 3 i 1 δ δ n 0 n=1 tn−1 i=1 2 − t − t Due to Lemma 3.2 the sum i=1 Zi(·,t) 1 is bounded by C[ω( )e + (1 − e )]. The integral of this sum over [0,T] is clearly bounded by C . Noting that

t−τ ηδ(t − τ) − ηδ(t − tn)= ηδ(s)ds, t−tn we ﬁnd the estimates ∆t T T 2 ∆t E (δ) ≤ C Z (·,t) |η (s − t)|dsdt ≤ C , 3 i 1 δ δ 0 0 i=1 which completes the proof. − t Remark 4.4. The above proof uses the exponential rate e without resorting to the restriction ω( )= on the initial error. To see this, note that from

2 − t − t Zi(·,t) 1 ≤ C[ω( )e + (1 − e )], i=1 we can easily deduce the bound 1 T 2 Z (·,t) dτ ≤ C. i 1 0 i=1

Here, we do not use any speciﬁc choice of ω( ). We mention that the authors in [25] obtained the same error bound in the discretization step by assuming that ω( )= .

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