SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN METHODS FOR 1-D LINEAR HYPERBOLIC EQUATIONS WITH DEGENERATE VARIABLE COEFFICIENTS

SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN METHODS FOR 1-D LINEAR HYPERBOLIC EQUATIONS WITH

DEGENERATE VARIABLE COEFFICIENTS

WAIXIANG CAO^∗, CHI-WANG SHU^†, AND ZHIMIN ZHANG^‡

Abstract. In this paper, we study the superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with de- generate variable coefficients. The study establishes superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order ofO(h^k+2), when piecewise polynomials of degreekare used. We then prove that the highest superconvergence rate of the DG solution itself isO(h^k+32) as the variable coefficient degenerates or achieves the value zero in the domain. As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell averages. All theoretical findings are confirmed by numerical experiments.

Key words. Discontinuous Galerkin methods, superconvergence, degenerate variable coeffi- cients, Radau points, upwind fluxes

AMS subject classifications. 65M15, 65M60, 65N30

1. Introduction. In this paper, we study and analyze the discontinuous Galerkin (DG) method for the following one-dimensional linear hyperbolic equation

ut+ (αu)x=g(x, t), (x, t)∈[0,2π]×(0, T],

u(x,0) =u0(x), x∈R, (1.1)

whereu0(x),α(x) andg(x, t) are all smooth functions. Without loss of generality, we only consider periodic boundary conditions. For simplicity, we assume that there are a finite number of zeros ofα(x) on the whole domain [0,2π].

The DG method was first introduced by Reed and Hill [19] for solving the neutron transport problem, and was later developed by Cockburn et al. [12, 14, 15, 16] for solving time dependent nonlinear equations. Since then, the DG method has been intensively studied and successfully applied to various problems in a wide range of applications. We refer to [13] and references cited therein for the development of DG methods.

Superconvergence of the DG method has been studied for many years. The first superconvergence result for the DG solution can be found in [1]. Later, in [2], Ad- jerid and Massey showed a superconvergence rate of k+ 2 for the DG solution at

∗Beijing Computational Science Research Center, Beijing 100193, China. E-mail: wx- [email protected]. Research supported in part by NSFC grant No.11501026, and the China Postdoctoral Science Foundation grant 2015M570026.

†Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:

[email protected]. Research supported in part by DOE grant DE-FG02-08ER25863 and NSF grant DMS-1418750.

‡Beijing Computational Science Research Center, Beijing 100193, China. E-mail:

[email protected]; and Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. E-mail: [email protected]. Research supported in part by NSFC grants No.11471031 and No.91430216, and NSF grant DMS-1419040.

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the downwind-biased Radau points for some ordinary differential equations, when piecewise polynomials of degreekare used. In 2007, Celiker and Cockburn proved a (2k+ 1)-th order superconvergence for the numerical traces of the DG and hybridiz- able DG solutions when applied to steady state problems [9]. Later, Xie and Zhang proved that the same superconvergence rate of 2k+ 1 holds for the DG approxima- tion at downwind points for singularly perturbed problems [20]. For time dependent problems, Cheng and Shu in [11] proved that the DG solution for 1-D hyperbolic and convection-diffusion equations is superconvergent towards a particular projection of the exact solution with an order ofk+³₂, and later this convergence rate was improved tok+ 2 in [21, 22] by Yang and Shu based on the duality arguments. Recently, Cao et al. revisited the 1-D and 2-D hyperbolic problems and proved a (2k+ 1)-th supercon- vergence rate for the DG solution at downwind points and for cell averages by using the idea of correction functions, see. e.g., [6, 8]. We also refer to [3, 4, 10, 17, 18, 24]

as an incomplete list of references on the superconvergence of DG methods. Note that almost all the superconvergence studies in the literature are based on problems with constant or non-degenerate (variable or nonlinear) coefficients. To the best of our knowledge, there has been no superconvergence result of the DG methods when they are applied to problems with general variable coefficients which may degenerate, i.e.

the coefficients either change signs or otherwise achieve the value zero in the domain.

This paper is concerned with the superconvergence properties of the DG method for the 1-D linear hyperbolic equation (1.1) withαbeing any smooth function which may generate or have zeros in the considered domain. As superconvergence has been extensively studied for the problem (1.1) with α = 1 (see, e.g., [8, 11, 21]), it is natural to ask whether the superconvergence phenomenon still exists and the same superconvergence results still hold for problems with variable coefficients. As we will see from the result of this paper, as well as the results in [18] for the more complicated nonlinear case, if the variable or nonlinear coefficients are bounded away from zero, we are able to obtain superconvergence results using the same methods as the con- stant coefficient case. However, the approach used in such analysis breaks down when the coefficient changes signs or otherwise achieves the value zero. In this work, we will prove that the superconvergence phenomenon of the DG methods for (1.1) still exists when the coefficients degenerate, and the superconvergence rate may depend upon specific properties of the variable coefficient functionα. The highest supercon- vergence rate for the DG solution at some special points is obtained, which is a half order higher than the optimal convergence rate. Our numerical examples demonstrate that this highest superconvergence rate for the DG solution approximation is sharp in many situations. As we may recall, these results are very different from the super- convergence results in [8] for the constant coefficient problems. Another important result we establish in this work is the superconvergence for the flux function of the DG methods. For the first time, we prove that the DG flux function is superconvergent towards a particular flux function of the exact solution with an order ofk+ 2. This superconvergence result can be viewed as the generalization of the result in [8, 21]

for the constant coefficient problems. As byproducts, superconvergence properties for the DG solution and the DG flux function at some special points (Radau points), as well as for cell averages, are also obtained.

The contribution of this paper is to present and reveal the superconvergence be- havior of the DG methods for possibly degenerate variable coefficient problems. On the one hand, we uncover the superconvergence phenomenon of the DG methods for hyperbolic equations with variable coefficients, and prove that the superconvergence

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rate may depend upon the property of the variable coefficient function. On the other hand, we establish the superconvergence results for the flux function of the DG meth- ods, and extend the superconvergence results for constant coefficient problems to the general case. By doing so, we present a full picture for the superconvergence proper- ties and enrich the superconvergence theory of the DG method for linear hyperbolic equations in one dimension. Furthermore, our current work is also part of an ongoing effort to develop DG superconvergence results for degenerate nonlinear hyperbolic equations.

The key step of our superconvergence analysis is still based upon the idea of correction functions. That is, we construct special functions to correct the error between the DG solution and specific projections of the exact solution. However, the correction function here is not the same as that in [8] for constant coefficients, as the correction function in [8] fails to work in cells where the variable coefficient either degenerates to zero or is very close to zero. New techniques are developed in our analysis to guarantee the superconvergence results, especially the superconvergence results for the DG solution itself.

To end this introduction, we would like to emphasize that the superconvergence analysis for linear hyperbolic equations with degenerate variable coefficients is signif- icant and meaningful. First, this is a necessary step towards the analysis of nonlinear hyperbolic problemsut+f(u)x= 0, for which the wind directionf⁰(u) changes sign generically and the assumption made in Meng et al [18] for |f⁰(u)| ≥ δ >0 is quite artificial. Second, the analysis is much more difficult than the case without sonic points, i.e., points at whichf⁰(u) = 0, studied before in [8, 21]. The tools developed in this paper are expected to be useful in future analysis, e.g., for the analysis of non- linear problems. Third, superconvergence for hyperbolic problems with sonic points is important for adaptive computation for such problems, which arises in many appli- cations such as computational fluid dynamics and computational electro-magnetism.

The rest of the paper is organized as follows. In Section 2, we present DG schemes and discuss their stability for linear hyperbolic equations with variable coefficients.

Section 3 is dedicated to the superconvergence analysis of the flux function of the DG methods. Using special correction functions, we obtain (k+ 2)-th order superconver- gence rate for flux function approximation. In Section 4, we study the superconver- gence behavior of the DG approximation itself, and reveal a very important fact that the superconvergence phenomenon exists for general variable coefficient hyperbolic equations, and the superconvergence rate may depend upon specific properties of the variable coefficient function. In Section 5, we provide numerical examples to support our theoretical findings. Finally, concluding remarks and remarks for possible future work are presented in Section 6.

Throughout this paper, we adopt standard notations for Sobolev spaces such as W^m,p(D) on sub-domainD ⊂Ω equipped with the normk · km,p,D and semi-norm

|·|m,p,D. WhenD= Ω, we omit the indexD; and ifp= 2, we setW^m,p(D) =H^m(D), k · km,p,D=k · km,D, and| · |m,p,D=| · |m,D.

2. The DG methods.

2.1. Numerical schemes. Let Ω = [0,2π] and 0 =x1 2 < x3

2 < . . . < x_N+1

2 =

2π be N+ 1 distinct points on the interval Ω. For all positive integers r, we define

3

Z^r={1, . . . , r}and denote by τ_j= (x_j−1

2, x_j+1

2), x_j= 1 2(x_j−1

2 +x_j+1

2), j∈ZN

the cells and cell centers, respectively. Let hj =x_j+1

2 −x_j−1

2, hj = hj/2 and h= maxj hj. We assume that the mesh is regular, i.e., the ratio between the maximum and minimum mesh sizes shall stay bounded during mesh refinements.

Define

V_h={v_h: v_h|_τj ∈Pk(τ_j), j∈ZN}

to be the finite element space, wherePkdenotes the space of polynomials of degree at mostk. The DG scheme for (1.1) reads as: Findu_h∈V_h such that for anyv_h∈V_h,

(uht, vh)j−(αuh, vhx)j+αˆuhv⁻_h|_j+¹

2 −αˆuhv⁺_h|_j−¹

2 = (g, vh)j, (2.1) where (u, v)j = R

τ_juvdx, v_h⁻|_j+¹

2 and v⁺_h|_j+¹

2 denote the left and right limits of vh

at the point x_j+1

2, respectively, and ˆuh is the numerical flux, which is the discrete approximation to the trace ofuon the boundary of each interval. The choice of the numerical flux is of great importance to ensure the stability of the numerical scheme.

In this paper, we take the upwind flux defined by ˆ

uh

_j+1

2

=

( uh(x⁺_j+1 2

), if α_j+1 2 ≤0, uh(x⁻_j+1

2

), if α_j+1

2 >0, j∈ZN (2.2) whereα_j+1

2 =α(x_j+1 2).

2.2. Stability. Define

H_h¹={v:v|τ_j ∈H¹(τj), j∈ZN}, and for allu, v∈H_h¹, let

aj(u, v) = (ut, v)j−(αu, vx)j+αˆuv⁻|_j+¹

2 −αˆuv⁺|_j−¹

2, a(u, v) =

N

X

j=1

aj(u, v), where ˆuis the numerical flux taken as (2.2). Apparently, (2.1) can be rewritten as

aj(uh, vh) = (g, vh)j, ∀vh∈Vh, ∀j∈ZN.

For any periodic boundary conditionv, a direct calculation from integration by parts yields

a(v, v) = (v_t, v) +1

2(α_xv, v) +

N

X

j=1

α_j+1

2({v} −v)ˆ _j+1 2[v]_j+1

2,

where (u, v) = PN

j=1(u, v)j and {v} = ^v++v₂ ⁻. Thanks to the special choice of the numerical flux, we have

N

X

j=1

α_j+1

2({v} −v)ˆ _j+1 2[v]_j+1

2 ≥0,

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which yields

(vt, v)≤a(v, v)−1

2(αxv, v). (2.3)

Especially, by choosingv=u_h in the above inequality, we get (u_ht, u_h)≤ −1

2(α_xu_h, u_h) + (g, u_h).

Then the stability of the numerical scheme is guaranteed by using the Gronwall in- equality.

To end this section, we would like to present another stability inequality, which will be used in our superconvergence analysis later. Let

A_j(u, v) = (α²_j+1 2

+α²_j−1 2

)a_j(u, v), A(u, v) =

N

X

j=1

A_j(u, v), u, v∈H_h¹.

Recalling the definition ofaj(·,·), we obtain from a direct calculation,

A(v, v) =

N

X

j=1

(α²_j+1 2

+α²_j−1 2

)((v_t, v)_j−(αv, v_x)_j+αˆvv⁻|_j+1

2 −αˆvv⁺|_j−1 2)

=

N

X

j=1

(α²_j+1

2

+α²_j−1 2

)((vt, v)j+1

2(αxv, v)j) + 2α³_j+1 2

({v} −ˆv)_j+1 2[v]_j+1

2

+I,

where I=

N

X

j=1

α_j−¹

2(α²_j+1

2 −α²_j−1 2)

v⁺ 2 −vˆ

j−¹₂

v⁺_j−1 2

+α_j+¹

2(α²_j+1

2 −α²_j−1 2)

v⁻ 2 −ˆv

j+¹₂

v_j+⁻ 1 2

! ,

Again, the special choice of the numerical flux in (2.2) ensures that

N

X

j=1

α³_j+1 2

({v} −v)ˆ _j+1 2[v]_j+1

2 ≥0.

Therefore, by defining kvkα=

N

X

j=1

(α²_j+1 2

+α²_j−1 2

) Z

τj

v²(x)dx, ∀v∈H_h¹, (2.4)

we have

1 2

d

dtkvk²_α≤A(v, v)−I−1 2

N

X

j=1

(α²_j+1 2

+α_j−² 1 2

)(αxv, v)j

≤ |A(v, v)|+|α|_1,∞

2 kvk²_α+|I|.

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Noticing that|ˆv_j−1

2| ≤ kvk_0,∞,τj+kvk_{0,∞,τj−1}, j∈Z^N, we get

|I|.h

N

X

j=1

(α_j+1

2 +α_j−1 2)(α_j−1

2(kvk_0,∞,τj +kvk_{0,∞,τj−1}) +α_j+1

2(kvk_0,∞,τj +kvk_0,∞,τj+1))kvk_0,∞,τj

.h

N

X

j=1

(α²_j+1 2

+α²_j−1 2

)kvk²_0,∞,τ

j.

Here and in the following,A.B denotes thatAcan be bounded byB multiplied by a constant independent of the mesh sizeh. Consequently,

1 2

d

dtkvk²_α.|A(v, v)|+kvk²_α+h

N

X

j=1

(α²_j+1 2

+α²_j−1 2

)kvk²_0,∞,τj, ∀v∈H_h¹. By the inverse inequality, we have for allv_h∈V_h

kvhk_0,∞,τj .h⁻¹²kvhk0,τ_j, j∈ZN. Then

d

dtkv_hk²_α.A(v_h, v_h) +kv_hk²_α, ∀v_h∈V_h. (2.5) 3. Superconvergence for the DG flux function approximation. This sec- tion is dedicated to the superconvergence analysis for the DG flux function approxima- tion. We shall prove that the flux function of the DG methodαu_his superconvergent with an order of k+ 2 towards a particular flux function of the exact solution. It is this supercloseness that gives us the superconvergence properties of the flux function at some special points as well as for the cell average.

We begin with some preliminaries.

3.1. Preliminaries. LetLmandLj,mbe the standard Legendre polynomials of degreemon the interval [−1,1] andτj, respectively. That is,

L_j,m(x) =L_m(s), s= (x−x_j)/h_j ∈[−1,1].

For any functionv∈H_h¹, we define the primal functionD_x⁻¹vof vby D_x⁻¹v

_τ

j = 1

¯h_j Z x

x_j−1

2

v(x)dx.

By the properties of Legendre polynomials and a scaling fromτj to [−1,1], we have D⁻¹_x Lj,m(x) =

Z s

−1

Lm(s)ds= 1

m(m+ 1)(s²−1)d

dsLm(s) (3.1)

= 1

2m+ 1(Lm+1−Lm−1)(s) = 1

2m+ 1(Lj,m+1−Lj,m−1)(x).(3.2) Given a function ψ, we denote byRhψ andP_h^±ψ the traditionalL² and Gauss- Radau projections ofψ, respectively. That is,

(R_hψ, v_h) = (ψ, v_h), ∀v_h∈Pk(τ_j),

6

and

(P_h⁺ψ, v_h)_j = (ψ, v_h)_j, ∀vh∈Pk−1(τ_j), P_h⁺ψ(x⁺

j−¹₂) =ψ(x⁺

j−¹₂), (3.3) (P_h⁻ψ, vh)j= (ψ, vh)j, ∀vh∈Pk−1(τj), P_h⁻ψ(x⁻_j+1

2

) =ψ(x⁻_j+1 2

). (3.4) In addition, we also define the Gauss-Lobatto projectionQhψ(fork≥2) as follows (Qhψ, vh) = (ψ, vh), ∀vh∈Pk−2(τj), Qhψ(x⁻_j+1

2

) =ψ(x⁻_j+1 2

), Qhψ(x⁺

j−¹₂) =ψ(x⁺

j−¹₂).

While fork= 1,Qhψis the interpolant ofψsatisfying Q_hψ(x⁻_j+1

2

) =ψ(x⁻_j+1 2

), Q_hψ(x⁺

j−¹₂) =ψ(x⁺

j−¹₂).

The standard approximation theory gives us

kψ−P_h⁺ψk0,p+kψ−P_h⁻ψk0,p+kψ−R_hψk0,p+kψ−Q_hψk0,p.h^k+1|ψ|k+1,p, p≥1.

(3.5) With the above four projections, we define a particular projectionPhψofψby

P_hψ=











R_hψ, if α_j+1

2 ≤0, α_j−1 2 >0 P_h⁺ψ, if α_j+1

2 ≤0, α_j−1 2 ≤0 P_h⁻ψ, if α_j+¹

2 >0, α_j−¹

2 >0 Q_hψ, if α_j+1

2 >0, α_j−1 2 ≤0

. (3.6)

In each element τ_j, j ∈ ZN, we denote by r_j,i, l_j,i, g_j,i, gl_j,i, i ∈ Zk+1 the right Radau, left Radau, Gauss, and Gauss-Lobatto points of degreek+1, respectively, That is,r_j,iandl_j,iare separately the zeros of the right Radau polynomialL_j,k+1−L_j,kand the left Radau polynomialL_j,k+1+L_j,k, andg_j,iare zeros of the Legendre polynomial L_j,k+1, and gl_j,i are zeros of the Lobatto polynomial L_j,k+1−L_j,k−1. Similarly, we denote by l_j,m^∗ , m ∈ Z^k the interior left Radau points (the roots of Lj,k+1+Lj,k

except the point x=x_j−1

2); r^∗_j,m, m∈Zk the interior right Radau points (the roots ofLj,k+1−Lj,kexcept the point x=x_j+1

2);g^∗_j,m, m∈Zk the Gauss points of degree k(i.e., the roots ofLj,k); andgl_j,m^∗ , m∈Zk thekinterior Gauss-Lobatto points (i.e., the roots ofLj,k+2−Lj,k except the two boundary pointsx=x_j+1

2, x_j−1 2).

Now we define a class of special pointsyj,i, zj,m, i∈Zk+1, m∈Zk in each element τj, j∈ZN as follows:

y_j,i=











g_j,i, if α_j+1

2 ≤0, α_j−1 2 >0 l_j,i, if α_j+1

2 ≤0, α_j−1 2 ≤0 rj,i, if α_j+1

2 >0, α_j−1 2 >0 glj,i, if α_j+1

2 >0, α_j−1 2 ≤0

, (3.7)

and

zj,m=











gl^∗_j,m, ifα_j+1

2 ≤0, α_j−1 2 >0 r^∗_j,m, ifα_j+1

2 ≤0, α_j−1 2 ≤0 l^∗_j,m, ifα_j+1

2 >0, α_j−1 2 >0 g_j,m^∗ , ifα_j+1

2 >0, α_j−1 2 ≤0

. (3.8)

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We have the following approximation properties of Phψ at the special points y_j,i, z_j,m,(j, i, m)∈ZN ×Zk+1×Zk.

Lemma 3.1. Letψ∈W^k+2,∞, and Phψ∈Vh be defined by (3.6). The following approximation properties hold true.

• The numerical flux is exact, i.e., Pˆhψ

_j+1 2

=ψ_j+1

2, ∀j ∈ZN. (3.9)

Here the numerical flux is defined by (2.2)with uh replaced by Phψ.

• The function value of Phψ is k+ 2-th order superconvergent at the special pointsyj,i,(j, i)∈ZN ×Zk+1 defined by (3.7), i.e.,

|(ψ−Phψ)(yj,i)|.h^k+2|ψ|_k+2,∞,τj. (3.10)

• The derivative value ofPhψis superconvergent at the special pointszj,m,(j, i)∈ ZN×Zk, with a convergence ratek+ 1, wherez_j,m is defined by (3.8). That is,

|∂x(ψ−Phψ)(zj,m)|.h^k+1|ψ|_k+2,∞,τj. (3.11) Proof. For anyj∈ZN, by (2.2) and (3.6), there holds forα_j+1

2 >0 Pˆhψ

_j+1 2

=Phψ(x⁻_j+1 2

) =P_h⁻ψ(x⁻_j+1 2

) or Qhψ(x⁻_j+1 2

), and forα_j+1

2 ≤0 Pˆhψ

_j+1 2

=Phψ(x⁺_j+1 2

) =P_h⁺ψ(x⁺_j+1 2

) or Qhψ(x⁺_j+1 2

).

In both cases, we have, from the properties ofP_h^±ψ andQhψ Pˆhψ

_j+1 2

=ψ(x_j+1 2).

Now we show (3.10)-(3.11). Using the result in [23] (see Theorem 2.1) and a scaling from [−1,1] toτj, we have

(ψ−Qhψ) _τ

j =

∞

X

p=k

bj,p

2p+ 1(Lj,p+1−L_j,p−1), ∂x(ψ−Qhψ) _τ

j = 1

¯hj

∞

X

p=k

bj,p

2p+ 1Lj,p

with

bj,p= 2^pp!

(2p)!(¯hj)^p+1∂_x^p+1u(ξj), ξj∈τj. Then

|(ψ−Qhψ)(glj,i)|.h^k+2|ψ|k+2,∞,τj, |∂x(ψ−Qhψ)(g_j,m^∗ )|.h^k+1|ψ|k+2,∞,τj. On the other hand, it has been proved in [7, 8]

|(ψ−P_h⁻ψ)(rj,i)|+|(ψ−P_h⁺ψ)(lj,i)|.h^k+2|ψ|k+2,∞,τj,

|∂_x(ψ−P_h⁻ψ)(l^∗_j,m)|+|∂_x(ψ−P_h⁺ψ)(r^∗_j,m)|.h^k+1|ψ|_k+2,∞,τj.

8

Therefore, we only need to analyze the superconvergent approximation properties of theL² projectionRhψ. To this end, we suppose in each elementτj,

ψ _τ

j =

∞

X

p=0

aj,pLj,p, aj,p= 2p+ 1 h_j

Z

τj

ψLj,pdx. (3.12)

By the definition ofRhψ, we easily obtain (ψ−R_hψ)

_τ

j =

∞

X

p=k+1

a_j,pL_j,p, ∂_x(ψ−R_hψ) _τ

j =

∞

X

p=k+1

a_j,p∂_xL_j,p.

By (3.1)-(3.2)

∂_xL_j,k+1(gl^∗_j,m) = c_k hj

(L_j,k+2−L_j,k)(gl^∗_j,m) = 0, m∈Zk, wherec_k is a constant dependent onk. Then

(ψ−Rhψ)(gj,i) =

∞

X

p=k+2

aj,pLj,p(gj,i), ∂x(ψ−Rhψ)(gl_j,m^∗ ) =

∞

X

p=k+2

aj,p∂xLj,p(gl^∗_j,m).

By denotingψ(s, t) =ψ(x, t), s= (x−xj)/hj∈[−1,1], we have, from the integration by parts and the properties of Legendre polynomials,

a_j,m=

2m+ 1 2

1 (2m)!

Z 1

−1

ψd^m

ds(s²−1)^mds

=

2m+ 1 2

(−1)ⁱ (2m)!

Z 1

−1

∂_sⁱψd^m−i

ds (s²−1)^mds, i≤m.

Noticing that∂ⁱ_sψ= ¯h_jⁱ

∂_xⁱψ, we have

|aj,m|.h^i−1p|ψ|i,p,τ_j, 1≤i≤m. (3.13) Consequently,

|(ψ−R_hψ)(g_j,i)|+h_j

∂_x(ψ−R_hψ)(gl^∗_j,m) .

∞

X

m=k+2

|a_j,m|.h^k+2|ψ|_k+2,∞,τj.

This finishes the proof of (3.10)-(3.11).

We end this subsection with the optimal error estimate forkuh−Phuk0. Lemma 3.2. Let u(·, t)∈H^k+2,∀t∈[0, T]be the solution of (1.1), anduhbe the solution of (2.1)with the initial solution u_h(x,0) =P_hu₀. Then

k(uh−Phu)(·, t)k0.h^k+1 sup

τ∈[0,t]

ku(·, τ)kk+2. (3.14)

Proof. In each element τj, j ∈ Z^N, if Phu|τ_j = Rhu, P_h^±u, we have, from the orthogonality ofRhu, P_h^±u, and (3.9)

a_j(u−P_hu, v_h) = (u_t−P_hu_t, v_h)_j+ ((α−α_j)(u−P_hu), ∂_xv_h)_j,

9

whereαj =α(xj) is a constant. IfPhu=Qhu, which indicatesα_j+1 2α_j−1

2 ≤0, then there exists at least one pointξj∈τj such thatα(ξj) = 0, and thus,

aj(u−Phu, vh) = (ut−Qhut, vh)j−((α−α(ξj))(u−Qhu), ∂xvh).

Combining the above two equations, we have for allv_h∈V_h andj∈ZN, aj(u−Phu, vh).kut−Phutk0,τ_jkvhk0,τ_j +hku−Phuk0,τ_j|vh|1,τ_j

.h^k+1(kukk+1,τ_j +kutkk+1,τ_j)kvhk0,τ_j, (3.15) where in the last step, we have used (3.5) and the inverse inequality |vh|1,τ_j . h⁻¹_j kvhk0,τj for allv_h∈V_h.

Now we obtain, by choosing v = uh −Phu in (2.3), and using (3.9) and the orthogonalitya(u−uh, vh) = 0, vh∈Vh,

d

dtkuh−Phuk²₀.a(u−Phu, uh−Phu) +kuh−Phuk²₀

=

N

X

j=1

aj(u−Phu, uh−Phu) +kuh−Phuk²₀.

Then (3.14) follows from (3.15), the Gronwall inequality and the fact that u_t =

−(αu)x.

3.2. Analysis. In this subsection, we will study the superconvergence properties of the DG flux function αuh towards the particular flux functionαPhuof the exact solution in theL² norm. In light of (3.14) and the fact that

kα(uh−Phu)k0.kuh−Phukα+hkuh−Phuk0.h^k+2 sup

t∈[0,T]

ku(·, t)kk+2+kuh−Phukα, (3.16) wherek · kα is defined by (2.4), our superconvergence goal is reduced to the estimate of the errorkuh−Phukα.

As indicated by (2.5), the errork(uh−Phu)(·, t)kαfor any timet >0 is dependent on the initial error and the term A(uh−Phu, uh−Phu), or equivalently the term A(u−Phu, uh−Phu) due to the orthogonalityA(u−uh, vh) = 0, vh∈Vh. However, as a direct consequence of (3.15),

A(u−Phu, uh−Phu) =

N

X

j=1

(α²_j+1 2

+α²_j−1 2

)aj(u−Phu, uh−Phu) .h^k+1kukk+2kuh−Phukα.

Then the standard error estimate only gives us the optimal convergence rate, i.e., k(uh−Phu)(·, t)kα.h^k+1 sup

τ∈[0,t]

ku(·, τ)kk+2,

which is far from our superconvergence goal. Therefore, to improve the convergence rate, some extra term or function is needed to correct the errorA(u−Phu, vh), vh∈Vh

based on the projectionPhudefined in (3.6). In other words, we need to construct a correction functionw∈Vh such thatuI =Phu−wand

A(u−u_I, v_h) =A(u−P_hu, v_h) +A(w, v_h).h^k+1+lkv_hk_α, ∀v_h∈V_h

10

for somel >0. By doing so, we obtain the superconvergence result forαuh−αuI (or αuh−αPhu). Our analysis is along this line. Consequently, the rest of this subsection is dedicated to the construction of the correction functionw.

For functionsu, v∈H_h¹, we define b(u, v) =

N

X

j=1

bj(u, v), bj(u, v) = (ut, v)j−(αu, vx)j. Apparently, we have from (3.9) and the definition ofaj(·,·),

bj(u−Phu, vh) =aj(u−Phu, vh), ∀vh∈Vh.

On the other hand, in light of the definition ofPhuin (3.6), the whole domain Ω can be divided into four parts, that is

Ω =

4

[

i=1

Ωi, where

Ω1={τj:α_j−1

2 >0, α_j+1

2 >0}, Ω2={τj :α_j−1

2 ≤0, α_j+1 2 ≤0}, Ω3={τj:α_j−1

2 ≤0, α_j+1

2 >0}, Ω4={τj :α_j−1

2 >0, α_j+1

2 ≤0}. (3.17) Then

A(u−P_hu, v_h) = X

τ_j∈Ω1

˜ α²_j+1

2

b_j(u−P_h⁻u, v_h) + X

τ_j∈Ω2

˜ α²_j+1

2

b_j(u−P_h⁺u, v_h)

+ X

τ_j∈Ω3

˜ α²_j+1

2

b_j(u−Q_hu, v_h) + X

τ_j∈Ω4

˜ α²_j+1

2

b_j(u−R_hu, v_h), (3.18)

where ˜α²_j+1 2

=α²_j+1 2

+α²_j−1 2

.

3.2.1. The local correction function for bj(u−P_h⁻u, vh) . Noticing that (u−P_h⁻u)⊥P0, we have

D_x⁻¹(u−P_h⁻u)(x⁻_j+1 2

) = 1 h¯j

Z x_{j+ 1}

2

x_j−1

2

(u−P_h⁻u)(x)dx= 0 =D⁻¹_x (u−P_h⁻u)(x⁺_j−1 2

).

Then a direct calculation from integration by parts yields bj(u−P_h⁻u, vh) = (ut−P_h⁻ut, vh)j−(α(u−P_h⁻u), ∂xvh)j

=−¯hj(D_x⁻¹(ut−P_h⁻ut), vhx)j−((α−α¯j)(u−P_h⁻u), ∂xvh)j, vh∈(3.19)Vh, where ¯αj is a constant satisfying

|α¯j|= max

x∈τj

|α(x)|. (3.20)

With ¯α_j, we now define a special function w₁∈V_h as follows.

¯

αj(w1, vh)j=−¯hj(D⁻¹_x (ut−P_h⁻ut), vh)j−((α−α¯j)(u−P_h⁻u), vh)j, vh∈Pk−1, w1(x⁻_j+1

2

) = 0, ∀j ∈ZN. (3.21)

11

Lemma 3.3. Let w1∈Vh be defined by (3.21). Then

bj(u−P_h⁻u+w1, vh) = (∂tw1, vh)j−((α−α¯j)w, ∂xvh)j,∀vh∈Vh, (3.22) whereα¯j is defined in (3.20). Moreover, ifu(·, t)∈W^k+3,∞, t∈[0, T], then

kw1(·, t)k0,τ_j +k∂tw1(·, t)k0,τ_j .h^k+52

k+3

X

l=0

k∂_x^lαk_0,∞,τj

kαk_0,∞,τj ku(·, t)k_k+3,∞. (3.23)

Proof. First, (3.22) follows directly from (3.19) and (3.21). To estimatew1, we supposew1 has the following Legendre expansion in eachτj

w1

_τ

j =

k

X

m=0

cmLj,m(x).

By choosingvh=Lj,m, m≤k−1 in (3.21), we get

¯ αjhj

2m+ 1cm=−¯hj(D_x⁻¹(ut−P_h⁻ut), Lj,m)j−((α−α¯j)(u−P_h⁻u), Lj,m)j. Noticing that

kD⁻¹_x (ut−P_h⁻ut)k_0,∞,τj ≤ kut−P_h⁻utk_0,∞,τj .h^k+1k∂_x^k+1utk_0,∞,τj .h^k+1k∂_x^k+2(αu)k_0,∞,τj, we have for allm≤k−1,

|cm|.h^k+2

¯

α_j k∂_x^k+2(αu)k_0,∞,τj +k∂xαk_0,∞,τjk∂_x^k+1uk_0,∞,τj .h^k+2

¯ αj

k+2

X

l=0

k∂_x^lαk_0,∞,τjkuk_k+2,∞.

Sincew1(x⁻_j+1 2

, t) = 0, then

|c_k|=|

k−1

X

m=0

c_m|. h^k+2

¯ α_j

k+2

X

l=0

k∂_x^lαk_0,∞,τjkuk_k+2,∞.

Consequently,

kw₁(·, t)k²_0,τ

j .h

k

X

m=0

c²_m.h^2k+5

k+2

X

l=0

k∂_x^lαk²_0,∞,τj

¯

α²_j ku(·, t)k²_k+2,∞.

Similarly, taking time derivative on both sides of (3.21), the identity still holds. Then we follow the same argument to derive

k∂tw1(·, t)k²_0,τj .h^2k+5

k+3

X

l=0

k∂_x^lαk²_0,∞,τj

¯

α²_j ku(·, t)k²_k+3,∞.

Then (3.23) follows. The proof is complete.

12

3.2.2. The local correction function forbj(u−P_h⁺u, vh). SinceP_h⁺ushares almost the same properties with P_h⁻u, the correction function for the term bj(u− P_h⁺u, vh) is similar to that forbj(u−P_h⁻u, vh), which is defined as follows. In each elementτj, letw2∈Vh satisfy

¯

α_j(w₂, v_h)_j=−¯h_j(D⁻¹_x (u_t−P_h⁺u_t), v_h)_j−((α−α¯_j)(u−P_h⁺u), v_h)_j, v_h∈Pk−1(τ_j), w2(x⁺_j−1

2

) = 0, j∈ZN. (3.24)

Here ¯αj is given in (3.20). Following the same argument as in Lemma 3.3, we get b_j(u−P_h⁺u+w₂, v_h) = (∂_tw₂, v_h)_j−((α−α¯_j)w₂, ∂_xv_h)_j, (3.25) kw2(·, t)k0,τ_j+k∂tw2(·, t)k0,τ_j .h^k+52

k+3

X

l=0

k∂^l_xαk_0,∞,τj kαk0,∞,τj

ku(·, t)k_k+3,∞. (3.26)

3.2.3. The global correction function forb(u−P_hu, v_h). Now we are ready to define the global correction function on the whole domain. Let w ∈ V_h be the correction function such that

w=











0, α_j+¹

2 ≤0, α_j−¹

2 >0, w₂, α_j+1

2 ≤0, α_j−1 2 ≤0, w₁, α_j+1

2 >0, α_j−1 2 >0, 0 α_j+1

2 >0, α_j−1 2 ≤0,

(3.27)

wherew₁, w₂ are separately defined in (3.21) and (3.24).

We have the following properties of the correction functionw.

Lemma 3.4. Let u(·, t) ∈ W^k+3,∞,∀t ∈ [0, T] be the solution of (1.1), and w∈Vh be the correction function defined by (3.27). Then

ˆ w(x_j+¹

2, t) = 0, j ∈ZN, kw(·, t)kα+k∂tw(·, t)kα.h^k+2ku(·, t)kk+3,∞, (3.28) and

A(u−Phu+w, vh).h^k+2kuk_k+3,∞kvhkα, ∀vh∈Vh. (3.29)

Proof. By the definition of w in (3.27), (3.21) and (3.24), we can easily obtain ˆ

w(x_j+¹

2, t) = 0, j ∈ ZN. The second inequality of (3.28) follows from (3.23) and (3.26).

LetuI =Phu−w. Recall the definition ofA(·,·), we immediately get A(w, v) =

N

X

j=1

˜ α²_j+1

2

((wt, v)j−(αw, vx)j) =

2

X

i=1

X

τ_j∈Ω_i

˜ α_j+1

2bj(wi, v), (3.30) which yields, together with (3.18), (3.22), and (3.25),

A(u−uI, v_h) = X

τ_j∈Ω3∪Ω4

˜ α²_j+1

2

b_j(u−Phu, v_h)+ X

τ_j∈Ω1∪Ω2

˜ α²_j+1

2

(w_t, v_h)_j−((α−¯α_j)w, ∂_xv_h)_j ,

13

where ˜α²_j+1 2

and ¯αjis the same as in (3.18) and (3.20) respectively, and Ωi, i∈Z⁴ are defined by (3.17).

For anyτ_j ∈Ω₃∪Ω₄, noticing that α_j+1 2α_j−1

2 ≤0, then there exists at least one zero ofαonτ_j. Then

kαk0,∞,Ω₃∪Ω₄.hkαk1,∞,Ω₃∪Ω₄ .h, and thus

˜ α²_j+1

2

=α²_j+1 2

+α²_j−1

2 .h², τj∈Ω3∪Ω4.

Since α is smooth, there are a finite number of zeros ofα on the whole domain Ω, which indicates the number of elements in Ω3∪Ω4 is upper bounded independent of h. Then by (3.15) and the Cauchy-Schwartz inequality,

X

τ_j∈Ω₃∪Ω₄

˜ α²_j+1

2

bj(u−Phu, vh)

=

X

τ_j∈Ω₃∪Ω₄

˜ α²_j+1

2

aj(u−Phu, vh)

.h^k+1



 X

τ_j∈Ω₃∪Ω₄

˜ α²_j+1

2

kuk²_k+2,τj





1 2

kvhkα.h^k+2kukk+2kvhkα.

Plugging the above inequality into the formula ofA(u−u_I, v_h), we get for allv_h∈V_h

|A(u−u_I, v_h)|.h^k+2kukk+2kvhkα+ (kwtkα+kwkα)(kvhkα+hk∂xv_hkα) .h^k+2kuk_k+3,∞kvhkα,

where in the last step, we have used (3.28) and the inverse inequality.

Remark 3.5. Since the number of elements in the domain Ω₃∪Ω₄ is upper bounded independent of h, the terms P

τj⊂Ω3b_j(u−Q_hu, v_h) and P

τj⊂Ω4b_j(u− R_hu, v_h) in the formula (3.18) are of high order, which means the correction func- tion is not necessary in the case P_hu=Q_hu or R_hu. This is why we take w|_τj = 0 in those elementsτ_j∈Ω₃∪Ω₄.

3.3. The superconvergence result. Now we are ready to present the super- convergence properties of the DG flux function approximation.

Theorem 3.6. Let u(·, t) ∈ W^k+3,∞,∀t ∈ [0, T] be the solution of (1.1), and u_h be the solution of (2.1) with the initial solution u_h(x,0) =P_hu₀, where P_hu₀ is defined by (3.6). Then

ef =





N

X

j=1

Z

τ_j

α²(uh−Phu)²(x, t)dx





1 2

.h^k+2 sup

τ∈[0,t]

ku(·, τ)kk+3,∞. (3.31)

Proof. Let uI = Phu−w. By choosing vh = uh−uI in (2.5) and using the estimates in (3.28)-(3.29), we have

k(u_h−u_I)(·, t)k²_α.k(u_h−u_I)(·,0)k²_α+ Z t

0

A(u−u_I, u_h−u_I)dt .kw(·,0)k²_α+th^k+2 sup

τ∈[0,t]

ku(·, τ)k_k+3,∞k(u_h−u_I)(·, τ)k_α.

14