SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN METHODS BASED ON UPWIND-BIASED FLUXES FOR 1D LINEAR HYPERBOLIC EQUATIONS

DOI:10.1051/m2an/2016026 www.esaim-m2an.org

SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN METHODS BASED ON UPWIND-BIASED FLUXES FOR 1D LINEAR HYPERBOLIC EQUATIONS

Waixiang Cao

, Dongfang Li

, Yang Yang

and Zhimin Zhang

Abstract.In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A (2k+ 1)th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average is obtained under quasi-uniform meshes and some suitable initial discretization, when piecewise poly- nomials of degreekare used. Furthermore, surprisingly, we find that the derivative and function value approximation of the DG solution are superconvergent at a class of special points, with an orderk+ 1 andk+ 2, respectively. These superconvergent points can be regarded as the generalized Radau points.

All theoretical findings are confirmed by numerical experiments.

Mathematics Subject Classification. 65M15, 65M60, 65N30.

Received January 20, 2016. Revised March 24, 2016. Accepted April 21, 2016.

1. Introduction

In this paper, we study and analyze the discontinuous Galerkin (DG) method for the following one-dimensional linear hyperbolic conservation laws

u_t+u_x= 0, (x, t)∈[0,2π]×(0, T],

u(x,0) =u₀(x), x∈R, (1.1)

where u₀ is sufficiently smooth. We will consider both the periodic boundary condition u(0, t) =u(2π, t) and the Dirichlet boundary conditionu(0, t) =g(t).

Keywords and phrases.Discontinuous Galerkin methods, superconvergence, generalized Radau points, upwind-biased fluxes.

∗This work is supported in part by the China Postdoctoral Science Foundation 2015M570026, and the National Natural Science Foundation of China (NSFC) under Grants Nos. 91430216, 11471031, 11501026, and the US National Science Foundation (NSF) through grant DMS-1419040.

1 Beijing Computational Science Research Center, Zhongguancun Software Park II, No. 10 West Dongbeiwang Road, Haidian District, Beijing 100094, P.R. China.

2 School of Mathematics and Statistics, Huazhong University of Science and Technology, 1037 Luoyu Rd, Hongshan, Wuhan, Hubei 430074, P.R. China.

3 Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA.

[email protected]

4 Department of Mathematics, Wayne State University, 42 W. Warren Ave. Detroit, MI 48202, USA.

Article published by EDP Sciences c EDP Sciences, SMAI 2017

W. CAOET AL.

DG methods are a class of finite element methods using completely discontinuous piecewise polynomial space.

During the past several decades, it has gained more popularity in solving various differential equations (see, e.g., [9–14]). It is well known that to guarantee the stability of DG methods, a suitable choice of numerical fluxes is of great significance. Traditionally, purely upwind numerical fluxes are used for DG methods applied to the linear hyperbolic equation. However, when it comes to the complex systems or the nonlinear problems, purely upwind numerical fluxes may be difficult to construct. Therefore, study of the more general numerical fluxes, e.g., upwind-biased fluxes, becomes very necessary and significant. Very recently, Meng et al. in [17]

studied DG methods using upwind-biased numerical fluxes for linear conservation laws. They proved an optimal convergence orderk+ 1 for the semi-discrete DG schemes.

This paper is devoted to the study of superconvergence of DG methods using upwind-biased fluxes for the hyperbolic conservation laws (1.1). Note that many superconvergence studies of DG methods for hyperbolic equations are based on the purely upwind fluxes, see,e.g., [1–4,7,8,16,18–20]. The contribution of this paper is to establish the superconvergence theory for upwind-biased fluxes by providing a rigorous mathematical proof, and set a solid theoretical foundation on the fact that all upwind-biased fluxes share the same superconvergence properties with the purely upwind fluxes in designing DG schemes for linear hyperbolic equations. By doing so, we present a full picture for superconvergence properties of the DG method using upwind-biased fluxes, and thus extend the superconvergence results in [6] to more general cases. Furthermore, our current work is also part of an ongoing effort to develop superconvergence of DG methods using upwind-biased or Lax–Friedrichs fluxes for nonlinear hyperbolic equations.

The main idea in our superconvergence analysis is to design some special correction functions. Motivated by the successful applications of the correction idea to the DG methods for hyperbolic and parabolic equations (see, e.g. [4–6]), we will construct a special correction functionw, which vanishes or is of high order at some special points, to correct the error between the DG solution and some particular projection of the exact solution. To this end, the first key ingredient in our superconvergence analysis is to construct some suitable projection P u(for the periodic boundary condition) or ˜P u (for the Dirichlet boundary condition). Different from those in [4–6], the particular projection should be globally defined so as to eliminate the error on the inter-element boundary when the upwind-biased fluxes are concerned. The second key ingredient is to analyze the superconvergent approximation properties of the globally defined projection. However, the nonlocality of the projection makes the study of superconvergent approximation properties of the global projection more complicated than that in [4–6]. The latter is the Gauss–Radau projection, which is a local operator and is superconvergent at left and right Radau points. Based on the projectionP uor ˜P u, our final step is to construct a correction functionwto correct the error between DG solution and the globally defined projectionP uor ˜P u.

Thanks to the correction function, we establish the supercloseness between the DG solution and the specially constructed interpolation function u_I = P u−w or ˜P u−w. It is this supercloseness that gives us desired superconvergence results of the DG solution. To be more precise, we prove a (2k+ 1)th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average. As a by-product, we also prove that the DG solution is (k+2)th order superconvergent for the error to the particular globally defined projection P uor ˜P u. An unexpected discovery is that the derivative and function value approximations of the DG solution are superconvergent with an order k+ 1 and k+ 2 at a class of special points, which can be viewed as the generalized Radau points. As we may recall, all the results are similar to that for the purely upwind fluxes in [6]. In some special case, the superconvergence results established in this paper are reduced to that in [6].

Even though we study the problem in one space dimension only, the extension to the 2D case can be obtained follow the same analysis in [4] and this will be discussed in the future.

The rest of the paper is organized as follows. In Section 2, we present semi-discrete DG schemes using upwind- biased numerical fluxes for linear conservation laws under the periodic and Dirichlet boundary conditions.

Section 3 is devoted to the superconvergence analysis for the periodic boundary condition, where we first study the superconvergence properties of the global prejectionP u, and then design the correction function to correct the error between DG solution and the globally defined projectionP u, and finally prove the supercloseness of

the DG solution towards a particular interpolation function of the exact solution, which gives superconvergence results of the DG approximation. Extensions of the analysis to the Dirichlet boundary case are carried out in Section 4, and the same superconvergence results are obtained. In Section 5, we provide some numerical examples to support our theoretical findings. Finally, some concluding remarks are presented in Section 6.

2. DG schemes

LetΩ= [0,2π] and 0 =x1 2 < x3

2 < . . . < x_N+1

2 be N+ 1 distinct points on the intervalΩ. For all positive integersr, we defineZr={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. Leth_j=x_j+1

2 −x_j−1

2,h_j =h_j/2 andh= max

j h_j. We assume that the mesh is quasi-uniform,i.e., there exists a constantc such that

h≤ch_j, j∈ZN. Define

V_h={v: v|τj ∈Pk(τ_j), j∈ZN}

to be the finite element space, wherePk denotes the space of polynomials of degree at mostkwith coefficients as functions oft. The DG scheme for (1.1) reads as: findu_h∈V_h such that for anyv∈V_h

(u_ht, v)_j−(u_h, v_x)_j+ ˆu_h|_j+¹₂v⁻_j+1

2 −uˆ_h|_j−¹₂v_j−⁺ ₁

2 = 0, (2.1)

where (u, v)_j =

τjuvdx, v⁻

j+¹₂ and v⁺

j+¹₂ denote the left and right limits ofv at the point x_j+1

2, respectively, and ˆu_his the numerical flux. In this paper, instead of using the purely upwind flux, we adopt the upwind-biased flux. That is, we choose

uˆ_h

j+¹₂ =

θu⁻_h + (1−θ)u⁺_h

j+¹₂, j= 0, . . . , N (2.2)

for the periodic boundary condition, and

uˆ_h

j+¹₂ =

⎧⎪

⎪⎨

⎪⎪

⎩

(θu⁻_h +

1−θ)u⁺_h

j+¹₂ , j∈Z_N−1, (u_h)⁻1

2 =g(t), j= 0, (u_h)⁻_N+1

2, j=N

(2.3)

for the Dirichlet boundary condition. Here and in what follows, we takeθ > ¹₂· We denote

a_j(u, v) = (u_t, v)_j−(u, v_x)_j+ ˆuv⁻|_j+¹₂ −uvˆ ⁺|_j−¹₂ and

a(u, v) = N j=1

a_j(u, v), then (2.1) can be rewritten as

a_j(u_h, v) = 0, ∀v∈V_h, ∀j∈ZN.

W. CAOET AL.

A direct calculation from integration by parts yields a(v, v) = (v_t, v) +

θ−1

2 N

j=1

[v]²_j+1

2 (2.4)

for flux (2.2), and

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

θ−1 2

_N−1

j=1

[v]²_j+1

2 +1 2[v]²₁

2 +1 2

v⁻

N+¹₂v⁻

N+¹₂ −v⁻₁

2v⁻₁

2

(2.5)

for flux (2.3).

3. Analysis for the periodic boundary condition

Following the superconvergence technique in [5,6], our goal here is to construct a special interpolation function u_I of u such that the DG solution is superclose to the specially designed function u_I. By doing this, the superconvergence analysis of the DG solution is reduced to the study of the superconvergent approximation properties ofu_I. Therefore, in the rest of this section, we first design the special functionu_I, and then analyze its superconvergence behavior.

We begin with some preliminaries.

3.1. Priliminaries

For any integer m > 0, let W^m,p(D) be the standard Sobolev spaces on sub-domain D ⊂ Ω equipped with the norm · m,p,D and semi-norm | · |m,p,D. When D =Ω, we omit the index D; and if p= 2, we set W^m,p(D) =H^m(D), · m,p,D= · m,D, and| · |m,p,D=| · |m,D.

We denote by L_m(s), s∈[−1,1] the standard Legendre polynomial of degreemon the interval [−1,1], and (L_k+1−αL_k)(s) the generalized Radau polynomial of degreek+ 1 for some constant α. Noticing that when α= 1 andα=−1, (L_k+1−αL_k)(s) is reduced to the standard right and left Radau polynomials separately.

For any positiveα, we call (L_k+1−αL_k)(s) to be the generalized right Radau polynomial of degreek+ 1 and the zeros the generalized right Radau points of degreek+ 1. Moreover, we call the zeros ofD_s(L_k+1−αL_k)(s) to be the generalized derivative right Radau points of degreek. The following lemma shows the properties of the generalized Radau points and generalized derivative Radau points. For simplicity, we considerα >0 only, and the case forα <0 can be obtained following the same line.

Lemma 3.1. Let r₁ < r₂ < . . . < r_k+1 = 1 be the standard right Radau points on the interval [−1,1], i.e., r_i, i ∈ Z_k+1 are zeros of (L_k+1 −L_k)(s). Then for any positive constant α, the zeros of (L_k+1 −αL_k)(s) and D_s(L_k+1−αL_k)(s) are all simple, and there are at least k zeros of (L_k+1−αL_k)(s) and k−1 zeros of D_s(L_k+1−αL_k)(s) in [−1,1]. Moreover, there exists exactly one zero of(L_k+1−αL_k)(s)in each subinterval [r_i, r_i+1], i= 1, . . . , k. The position of the left one zero is dependent on the the choice ofα:

• If0< α≤1, the left one zero of(L_k+1−αL_k)(s)lies in the interval [−1, r1].

• Ifα >1, we extend the domain of Legendre polynomial from[−1,1]to the domain[−1,∞), then there exists one zero of(L_k+1−αL_k)(s) in the interval(1,∞).

Proof. We first prove that there are k zeros of (L_k+1−αL_k)(s) in [−1,1] for all constant α, and each of the interior subinterval [r_i, r_i+1], i= 1, . . . , k contains at least one zero of L_k+1−αL_k. To this end, it is sufficient to prove

(L_k+1−αL_k)(r_i)(L_k+1−αL_k)(r_i+1)≤0.

We denote byφ_i, i∈Z_k+1 the Lagrange basis polynomials corresponding to the pointsr_i, i∈Z_k+1, that is, φ_i(r_j) =δ_i,j.

Obviously, eachφ_i∈Pk and

φ_i(x) =c_ix^k+v(x), v∈P_k−1, wherec_i= 1/(Π_j=1ⁱ⁻¹(r_i−r_j)Π_j=i+1^k+1 (r_i−r_j)). A direct calculation leads to

c_ic_i+1<0.

On the other hand, noticing thatL_kφ_i∈P2k, we have, from Gauss–Radau quadrature and orthogonal properties of Legerdre polynomials,

L_k(r_i) = 1 w_i

₁

−1

L_k(x)φ_i(x)dx= c_i w_i

₁

−1

L_k(x)x^kdx.

Here w_i>0 is the weight of Gauss–Radau quadrature. Then L_k(r_i)L_k(r_i+1) = c_ic_i+1

w_iw_{i+1 1}

−1

L_k(x)x^kdx 2

<0.

SinceL_k+1(r_i) =L_k(r_i), i∈Z_k+1, we have

(L_k+1−αL_k)(r_i)(L_k+1−αL_k)(r_i+1) = (1−α)²L_k(r_i)L_k(r_i+1)≤0. (3.1) Therefore, for anyα, there are at least one zero ofL_k+1−αL_k in each subinterval [r_i, r_i+1], i= 1, . . . , k.

When α ≤ 1, we shall prove that there exists at least one zero of L_k+1−αL_k in the subinterval [r₀, r₁].

Actually, by (3.1),

(L_k+1−αL_k)(r₁)(L_k+1−αL_k)(r_k+1)(−1)^k >0.

Note that

(L_k+1−αL_k)(r₀)(L_k+1−αL_k)(r_k+1) = (1−α²)(−1)^k+1, we have

(L_k+1−αL_k)(r₀)(L_k+1−αL_k)(r₁)≤0, which implies [−1, r₁] contains at least one zero ofL_k+1−αL_k.

Now we consider the caseα >1. Noticing that

(L_k+1−αL_k)(1) = 1−α <0,

On the other hand, by the properties of Legendre polynomials, there exists a pointr_k+2∈(1,∞) such that (L_k+1−αL_k)(r_k+2)≥0.

Then there is at least one zero of L_k+1−αL_k in (1,∞). Since L_k+1−αL_k ∈Pk+1, each subinterval contains exactly one zero of (L_k+1−αL_k)(s).

By Rolle’s theorem, there existkzeros ofD_s(L_k+1−αL_k)(s). Moreover, the zeros ofD_s(L_k+1−αL_k)(s) are simple, and there exists one zero ofD_s(L_k+1−αL_k)(s) between two consecutive zeros of (L_k+1−αL_k)(s). The

proof is complete.

W. CAOET AL.

Let L_j,m be the standard Legendre polynomial of degree m on the interval [x_j−1 2, x_j+1

2]. For smooth func- tionu, we suppose uhas the following Legendre expansion in each element τ_j

u= ∞ m=0

u_j,mL_j,m, u_j,m= 2m+ 1 h_j

τj

u(x)L_j,m(x)dx. (3.2)

Denoting ˆu(s) = u(x), s= (x−x_j)/h_j ∈[−1,1], we have from the integration by parts and the property of Legendre polynomials

u_j,m=

2m+ 1 2

1 (2m)!

₁

−1u(s)Dˆ ^m_s (s²−1)^mds

=

2m+ 1 2

(−1)ⁱ (2m)!

₁

−1

Dⁱ_su(s)Dˆ ^m−i_s (s²−1)^mds, i≤m. (3.3)

Noticing thatD_sⁱuˆ=¯h_ji

Dⁱ_xu, then

|u_j,m|h^i−1pui+1,p,τj, 0≤i≤m. (3.4)

Here and in the following,AB denotes thatAcan be bounded by B multiplied by a constant independent of the mesh sizeh. The above estimate will be frequently used in our later superconvergence analysis.

We define a special global projectionP u∈V_h on functionuas

(P u, v)_j = (u, v)_j, ∀v∈Pk−1(τ_j), (3.5)

P u = (θP u⁻+ (1−θ)P u⁺)|_j+¹₂ = ˆu_j+1

2, j∈ZN. (3.6)

It has been shown in [17] that the projectionP u is well defined and there holds

u−P u0,τj+h¹²u−P u0,∞,τj h^k+32uk+1,∞. (3.7) For any functionv, we define an integral projectionD⁻¹ by

D⁻¹v(x) = 1 h_j

_x

x_j−1 2

v(x)dx= _s

−1

v(s)ds, x∈τ_j. (3.8)

wheres= (x−x_j)/h_j∈[−1,1].

To end this section, we would like to introduce a class of functions F_i(x), i∈Zk, which will be used in the construction of the interpolation function. For alli, j∈Zk, we define

F₀(x)

τj =L_j,k(x), F_i=P D⁻¹F_i−1. (3.9) Note that these functionsF_i, i∈Zk are different from those in [6], and the latter ones are locally defined.

Lemma 3.2. Suppose A is an N ×N circulant matrix with the first row (θ,(−1)^k(1−θ),0, . . . ,0) and the last row ((−1)^k(1−θ),0,0, . . . , θ), where ¹₂ < θ ≤ 1. Then A is non-singular. Moreover, For any vectors X = (x₁, . . . , x_N)^T, b= (b₁, . . . , b_N)^T satisfying AX=b, there holds

|x_j| max

1≤l≤N|b_l|, ∀j≤N.

Here we omit the proof since the similar argument can be found in [17] (see, Lem. 2.6).

Now we study the properties of functionsF_i, i∈Zk.

Lemma 3.3. SupposeF_i, i∈Zk are functions defined by (3.9). Then there hold F_i(x_j+1

2) = 0, ∀j∈Z_N, (3.10)

and in each elementτ_j, j∈ZN

F_i

τj = k m=k−i

bⁱ_j,mL_j,m, (3.11)

where the coefficientsbⁱ_j,m are some bounded constants independent of the mesh size h_j.

Proof. We will show (3.10)–(3.11) by induction. First, by the properties of Legendre polynomials, D⁻¹F₀

x⁻_j+1

2

= _x

j+ 12

x_j−1 2

L_j,kdx= 0 =D⁻¹F₀ x⁺_j+1

2

.

Then D⁻¹F₀

x_j+1 2

= 0.

In light of (3.6), we have F₁

x_j+1 2

=P D⁻¹F₀ x_j+1

2

=D⁻¹F₀ x_j+1

2

= 0, ∀j∈ZN. (3.12)

On the other hand, noticing thatF_i∈V_h, we have the following representation in each element τ_j F_i

τj = k m=0

bⁱ_j,mL_j,m, i∈Zk.

Recalling the definition of the global projectionP, we have

(F₁, v)_j= (D⁻¹F₀, v)_j = (D⁻¹L_j,k, v)_j, ∀v ∈Pk−1. Since

D⁻¹L_j,m= 1

2m+ 1(L_j,m+1−L_j,m−1), ∀m≥1, (3.13)

we have, by choosingv=L_j,m, m≤k−1,

b¹_j,m= 0, ∀m≤k−2, b¹_j,k−1=− 1

2k+ 1, ∀j∈ZN. Denoting

X = (b¹_1,k, . . . , b¹_N,k)^T, b=−^k−1

m=0

(θb¹_1,m+ (−1)^m(1−θ)b¹_2,m, . . . , θb¹_N,m+ (−1)^m(1−θ)b¹_1,m)^T,

then (3.12) can by rewritten as a linear systemAX=b, whereAis the same as in Lemma3.2. By the conclusion of Lemma3.2, we easily obtain

|b¹_j,k| max

1≤l≤N|b¹_l,k−1|1.

W. CAOET AL.

Consequently, both (3.10) and (3.11) are valid fori= 1. Suppose (3.11) is valid for alli≤k−1, we now prove that it also holds fori+ 1. Since

(F_i+1, v)_j = (D⁻¹F_i, v)_j= k r=k−i

bⁱ_j,r(D⁻¹L_j,r, v)_j, v∈P_k−1,

we derive, by choosingv=L_j,m, m≤k−1 and using (3.13), bⁱ⁺¹_j,m= 0, m≤k−i−2, bⁱ⁺¹_j,m= bⁱ_j,m−1

2m−1 − bⁱ_j,m+1

2m+ 3, k−i−1≤m≤k−1.

To estimate bⁱ⁺¹_j,k, we first obtain, from the fact that F_i⊥P_k−i−1, i ≤ k−1 and the orthogonal property of Legendre polynomials,

D⁻¹F_i x⁻_j+1

2

=D⁻¹F_i x⁺_j−1

2

= 0, (3.14)

which yields

F_i+1 x_j+1

2

=P D⁻¹F_i x_j+1

2

=D⁻¹F_i x_j+1

2

= 0, ∀j∈ZN.

Then by the same argument as what we did fori= 1, we obtain

|bⁱ⁺¹_j,k| max

1≤l≤N k−1

m=0

bⁱ⁺¹_j,m1.

Consequently, (3.10) and (3.11) are also valid fori+ 1. This finishes our proof.

3.2. Construction of a special interpolation function

To construct the special interpolation functionu_I, we first design a correction functionw, which is used to correct the error between the DG solution and P u. Note that the standard analysis only leads to the optimal error estimate (see [17])

u_h−P u0h^k+1.

Here we use the correction idea to achieve our goal,i.e. we design a correction functionwsuch that u_h−P u+w₀=u_h−u_I0h^k+l+1

for somel >0.

Since (u−P u)⊥P_k−1, we have

(u−P u)

τj = ˜u_j,kL_j,k+ ∞ m=k+1

u_j,mL_j,m, (3.15)

where ˜u_j,k= ^2k+1_h

j (u−P u, L_j,k)_j and u_j,m is the same as in (3.2).

We construct the correction functionwas follows. For any positivel, where 1≤l≤k, we define w(x, t) =w^l(x, t) =

l i=1

w_i(x, t), w_i(x, t)|τj = (¯h_j)ⁱ∂_tⁱu˜_j,k(t)F_i(x). (3.16)

Now we are ready to construct our special interpolation functionu_I. We define, for alll,1≤l≤k,

u_I =u^l_I =P u−w^l. (3.17)

Theorem 3.4. For any given l, where 1 ≤ l ≤ k, suppose u ∈ W^k+l+2,∞, and w^l, u^l_I are defined in (3.16) and (3.17), respectively. Then

wˆ_i(x_j+1

2) = 0, j∈ZN, w_i0,∞h^k+i+1uk+i+1,∞, 1≤i≤l, (3.18) and

a(u−u^l_I, v)h^k+l+1u_k+l+2,∞v₀, ∀v∈V_h. (3.19)

Proof. As we can see, the first identity of (3.18) is a direct consequence of (3.10) and (3.16). By (3.7),

∂_tⁱu˜_j,k= 2k+ 1 h_j

τj

(∂_tⁱu−P(∂ⁱ_tu))L_j,kdxh^k+1∂_tⁱuk+1,∞.

Then the second inequality of (3.18) follows from (3.11), (3.16) and the fact∂_tⁱu_t= (−1)ⁱ∂_xⁱu.

Recalling the definitions ofa(·,·) andP u, we obtain

a_j(u−P u, v) = ((u−P u)_t, v)_j =∂_tu˜_j,k(L_j,k, v)_j

=−¯h_j∂_tu˜_j,k(D⁻¹L_j,k, v_x)_j = (w₁, v_x)_j, ∀v∈V_h. Due to the special construction ofw_i in (3.16),

(∂_tw_i, v)_j−(w_i+1, v_x)_j =∂_tⁱ⁺¹u˜_j,k(¯h_j)ⁱ

(F_i, v)_j−¯h_j(F_i+1, v_x)_j

=∂_tⁱ⁺¹u˜_j,k

(¯h_j)ⁱ⁺¹(D⁻¹F_i, v_x)_j−(F_i+1, v_x)_j

= 0, where in the second step, we have used the integration by parts and (3.14). Then

a_j(w^l, v) = ((w^l)_t, v)_j−(w^l, v_x)_j = (∂_tw_l, v)_j−(w₁, v_x)_j. Consequently,

a(u−u_I, v) =a(u−P u, v) +a(w^l, v) = (∂_tw_l, v), ∀v∈V_h. (3.20) Then the desired result (3.19) follows from the factu_t=−u_xand (3.18).

3.3. Approximation properties of the interpolation function u

The result of (3.18) indicates thatw is of high order as the mesh size hdecreases. Consequently, the term P uin the formula ofu_I is dominant. Then the analysis of the superconvergent approximation properties ofu_I is reduced to that ofP u. However, as we may recall,P uis a global projection, which makes the analysis more complicated. To overcome this difficulty, we first introduce a local projectionP_hu∈V_h ofu, and then show the supercolseness betweenP uandP_hu. Therefore, to achieve our ultimate goal, all we need to do is to analyze the approximation properties of the local projectionP_hu, which is an easier task.

For any coefficientsθ_j, j∈ZN, where θ_j= ¹₂, we define a local projection P_hu∈V_h as

(P_hu, v)_j = (u, v)_j, ∀v∈Pk−1(τ_j), (3.21)

θ_jP_hu x⁻

j+¹₂

+ (1−θ_j)P_hu x⁺

j−¹₂

=θ_ju⁻

j+¹₂ + (1−θ_j)u⁺

j−¹₂, j∈ZN. (3.22) Obviously,P_huis well defined whenθ_j= ¹₂. Moreover, we have the following approximation properties ofP_hu.

W. CAOET AL.

Lemma 3.5. Suppose u ∈ W^k+2,∞ and P_hu is the special projection of u defined in (3.21) and (3.22) with θ_j =¹₂, j∈Z_N. Then

|(u−P_hu)(R^r_j,l)|h^k+2u_k+2,∞, |∂_x(u−P_hu)(R^d_j,m)|h^k+1u_k+2,∞. (3.23) HereR^r_j,l∈τ_j, l≤Zk+1andR^d_j,m∈τ_j, m≤Zk are zeros ofL_j,k+1−α_jL_j,kand∂_x(L_j,k+1−α_jL_j,k)respectively, where

α_j = 2θ_j−1, ifkis even, α_j = 1/(2θ_j−1), ifkis odd. (3.24) Proof. In light of (3.21), we obtain

P_hu

τj =

k−1

m=0

u_j,mL_j,m+ ¯u_j,kL_j,k,

whereu_j,m is the same as in (3.2) and ¯u_j,k is a constant to be determined. To obtain ¯u_j,k, we have from (3.22) (θ_j+ (−1)^k(1−θ_j))¯u_j,k =θ_j

∞ m=k

u_j,m+ (1−θ_j) ∞ m=k

(−1)^mu_j,m

= (θ_j+ (−1)^k(1−θ_j))u_j,k+ ∞ m=k+1

(θ_j+ (−1)^m(1−θ_j))u_j,m, which yields

(u−P_hu)

τj = (L_j,k+1−α_jL_j,k)u_j,k+1+ ∞ m=k+2

(L_j,m−α_j,mL_j,k)u_j,m. (3.25) Here α_j,m= (θ_j+ (−1)^m(1−θ_j))/(θ_j+ (−1)^k(1−θ_j)). Then the desired result (3.23) follows from (3.4).

Remark 3.6. The above lemma indicates that the function and derivative value approximations ofP_hu are superconvergent at the zeros of L_j,k+1−α_jL_j,k and ∂_x(L_j,k+1−α_jL_j,k), respectively. These results can be regarded as the generalization of the superconvergence results of the standard Gauss–Radau projection P_h⁻u provided in [6]. In fact, ifα_j= 1 orθ_j = 1, thenP_huis reduced to the standard Gauss–Radau projectionP_h⁻u.

Remark 3.7. We would like to point out that the number of superconvergence points of the local projection P_huin each interval τ_j may depend upon the choice ofθ_j orα_j, j ∈ZN. As indicated from Lemma3.1, there arek+ 1 superconvergence points in each elementτ_jfor the function value approximation ofP_huwhen|α_j| ≤1;

andksuperconvergence points when|α_j|>1. Similar results hold for the derivative approximation.

When choosing special θ_j, j ∈ZN in (3.21)–(3.22), we can obtain the following superconvergence result for P_hu−P u.

Lemma 3.8. Let u∈W^k+2,∞ anduhave the Legendre polynomial (3.2)in each element τ_j. Suppose P uand P_huare the projections of udefined in (3.5)–(3.6)and (3.21)–(3.22)withθ andθ_j satisfying

θ(h_j)^k+1α_j+ (−1)^k(1−θ)(h_j+1)^k+1α_j+1=θ(h_j)^k+1−(−1)^k(1−θ)(h_j+1)^k+1, (3.26) whereα_j is given by (3.24). Then

P_hu−P u0,∞h^k+2uk+2,∞. (3.27)

Consequently,

|(u−P u)(R^r_j,l)|h^k+2uk+2,∞, |∂_x(u−P u)(R^d_j,l)|h^k+1uk+2,∞, (3.28) whereR^r_j,l, R^d_j,l are the same as in (3.23).

Proof. Since (3.28) is a direct result of (3.23) and (3.27), we only prove (3.27) in the following.

By (3.5) and (3.21), we have (P u−P_hu)⊥P_k−1, which gives (P u−P_hu)

τj = ¯u_jL_j,k. Here ¯u_j is a constant to be determined. By (3.6) and (3.22),

θ(P u−P_hu) x⁻_j+1

2

+ (1−θ)(P u−P_hu) x⁺_j+1

2

=b_j, where

b_j=θ(u−P_hu) x⁻_j+1

2

+ (1−θ)(u−P_hu) x⁺_j+1

2

. (3.29)

Then

θu¯_j+ (1−θ)(−1)^ku¯_j+1=b_j. (3.30) By denoting

X= (¯u₁, . . . ,u¯_N)^T, b= (b₁, . . . , b_N)^T,

(3.30) can be rewritten as a linear system AX = b, whereA is the same as in Lemma 3.2. By the result of Lemma 3.2, we derive

|u¯_j| max

1≤l≤N|b_l|. We now estimateb_l. Substituting (3.25) into (3.29), we obtain

b_j =θ(1−α_j)u_j,k+1−(−1)^k(1−θ)(1 +α_j+1)u_j+1,k+1 +

∞ m=k+2

θ(1−α_j,m)u_j,m+ (1−θ)((−1)^m−α_j,m(−1)^k)u_j+1,m ,

whereα_j,m, m≥k+ 2 is the same as in (3.25). In light of (3.3) and the mean value theory of integrals, we have u_j,k+1=

2k+ 3 2

(−1)^k+1 (2k+ 2)!

₁

−1

D^k+1_s u(s)(sˆ ²−1)^k+1ds

=c_k(h_j)^k+1D^k+1_x u(ζ_j) ₁

−1

(s²−1)^k+1ds= ¯c_k(h_j)^k+1D_x^k+1u(ζ_j),

where ζ_j ∈ τ_j is some point and c_k =_2k+3

2

₍₋₁₎^k+1

(2k+2)!. Plugging the above identity into the formula ofb_j and using the identity (3.26), we get

b_j = ¯c_kθ(1−α_j)h^k+1_j

D^k+1_x u(ζ_j)−D_x^k+1u(ζ_j+1) +

∞ m=k+2

θ(1−α_j,m)u_j,m+ (1−θ)((−1)^m−α_j,m(−1)^k)u_j+1,m ,

By (3.4) and the fact that|D^k+1_x u(ζ_j)−D_x^k+1u(ζ_j+1)| ≤huk+2,∞,Ωj, whereΩ_j=τ_j∪τ_j+1 withτ_N+1 =τ₁, we obtain

P_hu−P u0,∞,τj |¯u_j|h^k+2,∞uk+2,∞,Ωj.

The proof is complete.

As a direct consequence of Lemmas3.5and3.8, we have the following superconvergence result of the specially designed functionu_I.

W. CAOET AL.

Corollary 3.9. Suppose u ∈ W^k+2,∞ and u_I is the special interpolation function defined by (3.17), (3.16).

There hold

(u−uˆ_I)(x_j+1

2) = 0, j∈ZN, (3.31)

and

|(u−u_I)(R^r_j,l)|h^k+2uk+2,∞, |∂_x(u−u_I)(R^d_j,m)|h^k+1uk+2,∞. (3.32) Here R^r_j,l, l≤Zk+1 andR^d_j,m, m≤Zk are the same as in Lemma 3.5.

Remark 3.10. As we may see from Lemmas 3.5 and 3.8, the superconvergence points ofu_I or P_hudepend on the lengths of all the cells. In practice, to compute the interior superconvergence points, we need to find α_js by solving (3.26). Then the interior superconvergence points in cell τ_j are the zeros of the polynomial L_j,k+1−α_jL_j,k.

3.4. Superconvergence of the DG approximation

We begin with the supercloseness between u_I and the DG solutionu_h. Choosing v =u_h−u^l_I in (2.4) and using (3.19), we get

1 2

d

dtu_h−u^l_I²₀≤a(u_h−u^l_I, u_h−u^l_I) =a(u−u^l_I, u_h−u^l_I) h^k+l+1u_k+l+2,∞u_h−u^l_I0.

By the Gronwall inequality,

(u^l_I−u_h)(·, t)(u^l_I−u_h)(·,0)+th^k+l+1 sup

τ∈[0,t]u(·, τ)k+l+2,∞. (3.33) The above inequality indicates that the initial dicretization should be carefully chosen to ensure the superclose- ness betweenu_handu^l_I. To obtain the desired superconvergence ratek+l+ 1 foru_h−u_I0, the initial value u_h(·,0) should satisfy

(u^l_I −u_h)(·,0)h^k+l+1 sup

τ∈[0,t]u(·, τ)_k+l+1,∞. (3.34)

As a direct consequence of (3.33) and (3.18), we have the following superconvergence ofu_htowards the global projectionP u.

Corollary 3.11. Let u∈W^k+3,∞ andu_h be the solutions of (1.1) and (2.1), respectively. Suppose the initial discretization is taken asu_h(·,0) =P u₀(·,0). Then

u_h−P u0th^k+2 sup

τ∈[0,t]u(·, τ)k+3,∞. (3.35)

3.4.1. Superconvergence at the numerical flux and for the cell-average

We denote by e_u,f and e_u,c the errors ofu−u_h at the numerical flux and for the cell-average, respectively.

That is,

e_u,f =

⎛

⎝1 N

N j=1

(u−uˆ_h) x_j+1

2, t₂⎞

⎠

12

, e_u,c=

⎛

⎝1 N

N j=1

1 h_j

τj

(u−u_h)(x)dx₂⎞

⎠

12

.

We have the following superconvergence results.