SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN METHODS FOR 2-D HYPERBOLIC EQUATIONS∗

SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN METHODS FOR 2-D HYPERBOLIC EQUATIONS^∗

WAIXIANG CAO ^†, CHI-WANG SHU^‡, YANG YANG ^§, AND ZHIMIN ZHANG ^{† ¶}

Abstract. This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for 2-D linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the (2k+ 1)- th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degreekare used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of (k+ 1)-th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of (k+ 2)-th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp.

Key words. Discontinuous Galerkin method, superconvergence, hyperbolic equations, Radau points, cell averages, initialand boundarydiscretizations

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

1. Introduction. In this paper, we present and analyze the discontinuous Galerkin (DG) method for the two-dimensional linear hyperbolic conservation laws

ut+ux+uy= 0, (x, y, t)∈[0,2π]×[0,2π]×(0, T],

u(x, y,0) =u0(x, y), (1.1)

whereu0is sufficiently smooth. We will consider both the periodic boundary condition u(0, y, t) =u(2π, y, t), u(x,0, t) =u(x,2π, t),

and the Dirichlet boundary condition

u(0, y, t) =g0(y, t), u(x,0, t) =g1(x, t).

This paper is the fourth in a series ([10, 11, 13]) devoted to the study of su- perconvergence phenomena of the DG method for time-dependent partial differential equations. Superconvergence phenomena of finite element methods (or continuous Galerkin methods) were discussed as early as 1967 by Zienkiewicz and Cheung [38].

Since then the superconvergence behavior had been studied intensively. For an in- complete list of references, we refer to [7, 8, 15, 16, 21, 24, 25, 26, 27, 28, 29, 37] for finite element methods (FEM), [9, 12, 14, 20, 31] for finite volume methods (FVM), and [4, 5, 6, 18, 19, 23, 30, 32, 36] for DG methods.

In [10] and [13], we considered the discontinuous and local discontinuous Galerkin method for 1D hyperbolic conservation laws and parabolic equations when upwind fluxes (for hyperbolic conservation laws) and alternating fluxes (for parabolic equa- tions) were used. For piecewise polynomials of degree k, these methods were shown

∗This work is supported in part by the National Natural Science Foundation of China under grants 11471031 and 91430216, the US National Science Foundation through grants DMS-1115530, DMS- 1419040 and DMS-1418750, and the US Department of Energy through grant DE-FG02-08ER25863.

†Beijing Computational Science Research Center, Beijing,100094, China.

‡Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.

§Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA.

¶Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.

1

to be superconvergent for the numerical traces at all nodes of the mesh, with a con- vergence rate of (2k+ 1)-th order (in an average sense). A pointwise (k+ 2)-th order superconvergence rate for the function value approximation and (k+1)-th order super- convergence rate for the derivative approximation at the (left or right) Radau points were also proved. Later, we provided in [11] a proof of (2k+ 1)-th order supercon- vergence rate for the cell averages and the pointwise error at nodes (downwind points for hyperbolic equations and numerical traces at nodes for parabolic equations).

In this paper, we continue our study of DG method applied to the 2-D linear hyperbolic conservation laws (1.1). To the best of our knowledge, there was not any global superconvergence result for these problems in the literature. Some previous works are based on local error estimates [1, 2, 3, 4, 5, 6], where the numerical fluxes are given as a projection of the boundary condition. Our contribution of this paper is to establish the superconvergence theory of DG methods for liner hyperbolic equations in two space dimensions. To be more precise, we shall provide a rigorous mathematical proof of the (2k+ 1)-th order superconvergence rate of the DG approximation at the downwind points and for the cell averages. We also prove the DG solution is superconvergent with a rate of (k+ 2)-th order at the right Radau points (function value approximation) and a rate of (k+ 1)-th order at the interior left Radau points (gradient approximation). As the reader may recall, these rates are the same as the counterparts in the 1D case. In other words, all superconvergence results in 1D can be extended to 2D. However, the analysis is by no means a trivial extension. Moreover, there are some new phenomena in the 2D situation, which are not shared by the 1D case.

A key step of our superconvergence analysis is the construction of a correction function. We have successfully applied the correction function idea to the DG method for 1D hyperbolic and parabolic equations (see, e.g. [10, 13]). However, when it comes to the 2D case, the procedure of constructing the correction function is more sophisticated. More special cares are needed. Especially for the Dirichlet boundary conditions, to achieve our desired superconvergence rate, e.g. (2k+ 1)-th order, both the (numerical) initial and boundary conditions need to be adjusted. This is very different from the 1D case, where only the initial condition needs to be adjusted. Our approach here is to construct a correction function w to reduce the error between the DG solution uh and the truncated Radau expansion P_h⁻u of the exact solution, such that the errors uh−P_h⁻u+w at the initial time and at the boundary are both convergent with order 2k+ 1. By doing so, we prove that the DG solution uh is superclose (with order 2k+ 1) toP_h⁻u−w, which yields the superconvergence properties for the cell averages and at downwind points and Radau points.

To end this introduction, we would like to point out that in principle it is straight- forward to generalize the methodology we adopt in this paper to linear transport equations in higher dimensions. However, it requires very tedious and lengthy ar- guments to carry on the argument for 3D, 4D, etc.... in a mathematically rigorous way. On the other hand, one of our on-going project is to extend the investigation to nonlinear cases, where we require locally sufficiently smooth solution and nonlinear flux function, which will rule out possible shockwave regions.

The rest of the paper is organized as follows. In section 2, we present DG schemes for two-dimensional linear hyperbolic conservation laws. Section 3 is the most techni- cal part, where we design a special correction function toreducethe error between the DG solution and the truncated Radau expansion of the exact solution. Section 4 is the main body of the paper, where the superconvergence results are proved with suit-

2

able initial and boundary discretizations. Some numerical examples are presented in section 5 to support our theoretical findings. Finally, we provide concluding remarks 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,Dand the 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. NotationA.B implies thatA can be bounded byB multiplied by a constant independent of the mesh sizeh.

2. DG schemes. Let 0 = x¹

2 < x³

2 < · · · < x_m+¹

2 = 2π and 0 = y¹

2 < y³

2 <

· · ·< y_n+¹

2 = 2π. For any positive integerr, we defineZ_r={1,2, . . . , r}, and denote byTh the rectangular partition of Ω. That is

Th={τi,j = [x_i−¹

2, x_i+¹

2]×[y_j−¹

2, y_j+¹

2] : (i, j)∈Z_m×Z_n}.

For any τ ∈ Th, we denote by h^x_τ, h^y_τ the lengths of x- and y-directional edges of τ, respectively. h is the maximal length of all edges, and hmin = minτ(h^x_τ, h^y_τ). We assume that the mesh Th is quasi-uniform in the sense that there exist constants c1, c2>0 such that

h≤c1h^x_τ, h≤c2h^y_τ ∀τ∈ Th. Define the finite element space

Vh={v: v|τ ∈Q_k(x, y) =P_k(x)×P_k(y), τ ∈ Th},

where P_k denotes the space of polynomials of degree at mostk with coefficients as functions oft. The DG solution for (1.1) is to finduh∈Vh such that

aτ(uh, v) = 0 ∀τ∈ Th, v∈Vh, (2.1) where

aτ(uh, v) = Z

τ

(uhtv−uh(vx+vy))dxdy+ Z

∂τ

ˆ

uhvds, (2.2)

and for anyτ =τi,j∈ Th,(i, j)∈Z_m×Z_n, Z

∂τi,j

ˆ uhvds=

Z y_{j+ 1}

2

y_j−1 2

uˆh(x_i+¹

2, y)v(x⁻_i+1 2

, y)−uˆh(x_i−¹

2, y)v(x⁺_i−1 2

, y) dy

+ Z x_{i+ 1}

2

x_i−1 2

uˆh(x, y_j+¹

2)v(x, y_j+⁻ 1 2

)−ˆuh(x, y_j−¹

2)v(x, y_j⁺₋1 2

) dx.

(2.3)

Herev(x⁻_i−1

2,·), v(x⁺_i−1

2,·) denote the left and right limits ofvacrossx_i−¹

2, respectively, and ˆuhis the numerical flux. In this paper, we consider the upwind flux

ˆ

uh=u⁻_h.

To complete the DG scheme, we still need to define the numerical flux at the boundary (∂Ω)⁻, where

(∂Ω)⁻={(x, y)∈∂Ω :n0·n(x, y)≤0}

3

with n0 = (1,1) and n the outward normal unit vector at the boundary of a given domain. For the periodic boundary condition, the numerical flux at the boundary (∂Ω)⁻ is taken as

ˆ uh(x¹

2, y) =uh(x⁻1 2

, y) =uh(x⁻_m+1 2

, y) = ˆuh(x_m+¹

2, y), ˆ

uh(x, y¹

2) =uh(x, y⁻1 2

) =uh(x, y_n+⁻ 1 2

) = ˆuh(x, y_n+¹

2).

While for the Dirichlet boundary condition, the numerical flux at (∂Ω)⁻ is some- what sophisticated. For the purpose of our superconvergence proof later, we take the numerical flux at (∂Ω)⁻ as

uh(x⁻1 2

, y) = (P_h⁻u−w)(x⁻1 2

, y), uh(x, y⁻1 2

) = (P_h⁻u−w)(x, y⁻1 2

). (2.4) Here P_h⁻uand w (defined in section 2) denote the truncated Radau expansion ofu and the specially constructed correction function, respectively.

Remark 2.1. The special choice of the numerical flux at the boundary (∂Ω)⁻ for the Dirichlet boundary condition is to guarantee that the boundary errors of DG approximation are small enough to be compatible with superconvergence error estimate, especially the (2k+ 1)-th superconvergence error at the downwind points. This choice is very different from the traditional one, which is usually taken as theL² projection, or the truncated Radau expansion, or the Radau interpolation function, of the exact solution u. As we shall demonstrate in the numerical experiments, the numerical fluxes at the boundary have a significant influence on the superconvergence at the downwind points.

By denoting

a(u, v) = X

τ∈T_h

aτ(u, v), we obtain from a direct calculation

a(v, v) = (vt, v) +1 2



 Z 2π

0 m

X

i=1

[v]²_i−¹

2(y)dy+ Z 2π

0 n

X

j=1

[v]²_j−¹

2(x)dx+ Z

∂Ω

[v²]ds



, (2.5) where

[v]_i−¹

2(y) =v(x⁺_i−1 2

, y)−v(x⁻_i−1 2

, y), [v]_j−¹

2(x) =v(x, y⁺_j−1 2

)−v(x, y_j⁻₋1 2

) denote the jump ofvacross the points (x_i−¹

2, y) and (x, y_j−¹

2), respectively, and Z

∂Ω

[v²]ds= Z 2π

0

v²(x⁻_m+1 2

, y)−v²(x⁻1 2

, y) dy+

Z 2π 0

v²(x, y_n+⁻ 1 2

)−v²(x, y⁻1 2

) dx.

Ifv satisfies the periodic boundary condition v(x⁻_m+1

2

, y) =v(x⁻1 2

, y), v(x, y⁻_n+1 2

) =v(x, y⁻1 2

), (2.6)

or

v(x⁻1 2

, y) = 0, v(x, y⁻1 2

) = 0, (2.7)

then

1 2

d

dtkvk²₀= (vt, v)≤a(v, v). (2.8)

4

3. Correction function. To study the superconvergence properties of the DG solution, we first construct a specially constructed interpolation function uI of u such thatuI is superclose to the DG solutionuh. Then by using this super-closeness betweenuIanduh, we prove the superconvergence of the DG solution at some special points as well as for the cell averages.

In light of (2.8), by choosing v = uh−uI and using the orthogonal property a(u−uh, v) = 0, v∈Vh, we obtain

1 2

d

dtkuh−uIk²₀≤a(u−uI, uh−uI).

Consequently, for all t > 0, the errorkuh−uIk0(t) depends on two terms: a(u− uI, uh−uI) and the initial errorkuh−uIk0(0). Since we can control the initial error by taking a special initial discretization, the superconvergence analysis ofuh−uI is reduced to the estimate ofa(u−uI, uh−uI).Therefore, our next goal is to construct a special interpolation functionuI such that

a(u−uI, v) ∀v∈Vh

is of high order.

We begin the construction ofuI with the truncated Radau expansionP_h⁻u∈Vh

ofu. In each elementτi,j,(i, j)∈Z_m×Z_n, supposeu(x, y), for (x, y)∈τi,j, has the following Radau expansion

u(x, y) =

∞

X

p=0

∞

X

q=0

up,q(Li,p−Li,p−1)(x)(Lj,q−Lj,q−1)(y), (3.1)

where Li,p(x), Lj,p(y) denote the Legendre polynomial of degree p on the interval τ_i^x = [x_i−¹

2, x_i+¹

2] andτ_j^y = [y_j−¹

2, y_j+¹

2], respectively withLi,−1(x) =Lj,−1(y) = 0, and

up,q =u(x⁻_i+1 2

, y_j+⁻ 1 2

) +

p−1

X

l=0 q−1

X

r=0

(2l+ 1)(2r+ 1) h^x_ih^y_j

Z

τi,j

u(x, y)Li,l(x)Lj,r(y)dxdy

−

p−1

X

l=0

2l+ 1 h^x_i

Z x_{i+ 1}

2

x_i−1 2

u(x, y_j+⁻ 1 2

)Li,l(x)dx−

q−1

X

r=0

2r+ 1 h^y_j

Z y_{j+ 1}

2

y_j−1 2

u(x⁻_i+1 2

, y)Lj,r(y)dy

withh^x_i =x_i+¹

2 −x_i−¹

2, h^y_j =y_j+¹

2 −y_j−¹

2. Then the Radau truncation P_h⁻uof uis defined as

P_h⁻u(x, y) =

k

X

p=0 k

X

q=0

up,q(Li,p−Li,p−1)(x)(Lj,q−Lj,q−1)(y).

A direct calculation yields

u−P_h⁻u=E^xu+E^yu−E^xE^yu, (3.2)

5

where

E^xu(x, y) =

∞

X

p=k+1

∞

X

q=0

up,q(Li,p−Li,p−1)(x)(Lj,q−Lj,q−1)(y), (3.3)

E^yu(x, y) =

∞

X

p=0

∞

X

q=k+1

up,q(Li,p−Li,p−1)(x)(Lj,q−Lj,q−1)(y), (3.4)

E^xE^yu(x, y) =

∞

X

p=k+1

∞

X

q=k+1

up,q(Li,p−Li,p−1)(x)(Lj,q−Lj,q−1)(y). (3.5) The property of Legendre polynomials gives

E^xu(x⁻_i+1 2

, y) = 0,

Z x_{i+ 1}

2

x_i−1 2

E^xuvxdx= 0, v∈P_k(x), (3.6) E^yu(x, y_j+⁻ 1

2

) = 0,

Z y_{j+ 1}

2

y_j−1 2

E^yuvydy= 0, v∈P_k(y), (3.7) E^xE^yu(x⁻_i+1

2

, y) =E^xE^yu(x, y_j+⁻ 1 2

) = 0, Z

τi,j

E^xE^yu∇ ·vdxdy= 0, v∈Vh.(3.8) If we choose uI =P_h⁻u, then a straightforward analysis using the standard ap- proximation theory yields

|a(u−P_h⁻u, v)|.h^k+1, v∈Vh,

which is far from our need for superconvergence. To achieve our superconvergence goal, we need a properly designed functionwto correct the error betweenuandP_h⁻u such that

a(u−P_h⁻u+w, v).h^k+1+l ∀v∈Vh

for somel >0. Once the correction functionwis designed, by lettinguI =P_h⁻u−w, we finish the construction of the interpolation functionuI.

To construct the correction functionw, we first study the terma(u−P_h⁻u, v), v∈ Vh. By the decomposition ofu−P_h⁻uin (3.2), we have

a(u−P_h⁻u, v) =a(E^xu, v) +a(E^yu, v)−a(E^xE^yu, v). (3.9) 3.1. Correction function for a(E^xu, v). We begin with some preliminaries.

First, we define, for anyv(s)∈L¹[−1,1], a special Gauss-Radau projectionP⁻ by P⁻v(1) =v(1),

Z 1

−1

(P⁻v−v)(s)ϕ(s)ds= 0 ∀ϕ∈P_k−1, and an integral operatorD⁻¹by

D⁻¹v(s) = Z s

−1

v(s^′)ds^′. Define

F1(s) =P⁻D⁻¹Lk(s), Fi(s) =P⁻D⁻¹Fi−1(s), 2≤i≤k, s∈[−1,1], (3.10)

6

where Lk(s) denotes the Legendre polynomial of degreek on [−1,1]. It is proved in [13] thatFi(s) has the following representation

Fi(s) =

k

X

p=k−i+1

bp(Lp−Lp−1)(s) (3.11) with b^′_ps being bounded constants. By the properties of Legendre polynomials, we obtain

Fi(1) = 0, |Fi|.1, Fi⊥Pk−i−1, i∈Z_k. (3.12) Second, we define a special operator Q^x_h along the x-direction as follows: for any smooth functionv,Q^x_hv(x,·)|τ_i^x ∈P_k(τ_i^x), τ_i^x= [x_i−¹

2, x_i+¹

2] and Q^x_hv(x⁺_i−1

2

,·) =v(x⁺_i−1 2

,·), Q^x_hv(x⁻_i+1 2

,·) =v(x⁻_i+1 2

,·), (3.13) Z x_{i+ 1}

2

x_i

−1 2

(v−Q^x_hv)(x,·)ϕdx= 0 ∀ϕ∈P_k−2(τ_i^x), k≥2. (3.14) Note that in the casek= 1,Q^x_hvonly satisfies the condition (3.13). It is easy to show the existence and uniqueness ofQ^x_hv. Moreover, we have the following error estimate (c.f., [14, 17])

kv−Q^x_hvkp,∞,τ_i^x .(h^x_i)^l+1−pkvkl+1,∞,τ_i^x, 1≤p≤l. (3.15) Similarly, we can define the special operatorQ^y_halong the y-direction.

For all τ = τi,j ∈ Th, recalling the definition of aτ(·,·) in (2.2)-(2.3), we have, from (3.6) and the integration by parts,

aτ(E^xu, v) = Z

τ

((E^xu)t+ (E^xu)y)vdxdy+ Z x_{i+ 1}

2

x_i−1 2

[E^xu]_j−¹

2(x)v(x, y_j⁺₋1 2

)dx.

Since the functionE^xu(x, y) is continuous abouty, we have [E^xu]_j−¹

2(x) =E^xu(x, y_j⁺₋1 2

)−E^xu(x, y⁻_j−1 2

) = 0, which yields

aτ(E^xu, v) = Z

τ

((E^xu)t+ (E^xu)y)vdxdy= Z

τ

E^x(ut+uy)vdxdy.

Consequently,

a(E^xu, v) = Z

Ω

E^x(ut+uy)vdxdy ∀v∈Vh. (3.16) In light of (3.3),E^x(ut+uy) has the following representation in each elementτi,j,(i, j)∈ Z_m×Z_n,

E^x(ut+uy) = (∂t+∂y)E^xu=

∞

X

p=k+1

(∂t+∂y)¯ui,p(y, t)(Li,p−Li,p−1)(x),

7

where

¯

ui,p(y, t) =u(x⁻_i+1 2

, y, t)−

p−1

X

l=0

2l+ 1 h^x_i

Z x_{i+ 1}

2

x_i−1 2

u(x, y, t)Li,l(x)dx. (3.17)

Then

aτi,j(E^xu, v) = Z

τi,j

(∂t+∂y)E^xuvdxdy=− Z

τi,j

(∂t+∂y)¯ui,k+1Li,kvdxdy. (3.18) Now we are ready to construct the correction function corresponding to the term a(E^xu, v). We define, for any positive integerlwith 1≤l≤k,

w^l1|τi,j :=

l

X

p=1

w1,p, w1,p= (¯h^x_i)^p(Q^y_hGp)(y, t)Fp(s), (3.19)

where ¯h^x_i =h^x_i/2 = (x_i+¹

2 −x_i−¹

2)/2,s= (2x−x_i+¹

2 −x_i−¹

2)/h^x_i and Gp= (−1)^p−1(∂t+∂y)^pu¯i,k+1, p≥1.

Note thatui,k+1 = 0 whenu(x,·)∈P_k(τ_i^x), we obtain from Bramble-Hilbert lemma and (3.17)

kGpk0,∞,τi,j =k(∂t+∂y)^pu¯i,k+1k0,∞,τi,j.h^k+1kukk+1+p,∞,τi,j. (3.20) Theorem 3.1. Let u∈W^k+l+2,∞(Ω),1≤l≤kbe the solution of (1.1), andw^l₁ be the correction function defined by (3.19). Then

w^l1(x⁻_i+1 2

, y, t) = 0, kw1,pk0,∞,τi,j .h^k+1+pkukk+1+p,∞,τi,j. (3.21) Moreover, there holds for all v∈Vh

a(w^l1, v) +a(E^xu, v)

.h^k+l+1kukk+l+2,∞kvk0,1. (3.22) Proof. From the definition of (3.19), the first equation of (3.21) is a direct conse- quence of (3.12). On the other hand, the standard approximation theory and (3.20) give

kQ^y_hGpk0,∞,τi,j .kGpk0,∞,τi,j .h^k+1kukk+1+p,∞,τi,j. Then the second inequality of (3.21) follows.

Now we consider (3.22). For allτ =τi,j, by (2.2)-(2.3), the fact thatw1^l(x⁻_i+1 2

,·) = 0 and the integration by parts, we derive

aτ(w^l1, v) = Z

τ

(w^l1)tv−w^l1vx+ (w^l1)yv dxdy+

Z x_{i+ 1}

2

x_i

−1 2

[w1^l]_j−¹

2(x, t)v(x, y⁺_j−1 2

)dx

= Z

τ

(∂t+∂y)w^l1v−w^l1vx

dxdy =

l

X

p=1

Z

τ

(∂t+∂y)w1,pv−w1,pvx

dxdy.

8

Here in the second step, we have used the fact thatGp(y, t), forp≥1, is a continuous function abouty, which yields

[w^l₁]_j−¹

2(x, t) =w^l₁(x, y⁺_j−1 2

, t)−w^l₁(x, y⁻_j−1 2

, t) = 0.

Noticing that∂tQ^y_hGp=Q^y_h∂tGp,1≤p≤l and

∂yQ^y_hGp=∂yGp+O(h^l−p)k∂_y^l+1−pGpk0,∞,τi,j

=Q^y_h∂yGp+O(h^l−p)k∂_y^l+1−pGpk0,∞,τi,j, we have

(∂t+∂y)Q^y_hGp=Q^y_h(∂t+∂y)Gp+O(h^l−p)k∂_y^l+1−pGpk0,∞,τi,j

=−Q^y_hGp+1+O(h^k+1+l−p)kukk+l+2,∞,τi,j. Then for all 1≤p < l≤k,

Z

τi,j

v(∂t+∂y)w1,pdxdy= (¯h^x_i)^p Z

τi,j

v(∂t+∂y)Q^y_hGpFpdxdy

=−(¯h^x_i)^p Z

τi,j

vQ^y_hGp+1Fpdxdy+O(h^k+1+l)kukk+l+2,∞,τi,jkvk0,1,τi,j

= (¯h^x_i)^p+1 Z

τi,j

vxQ^y_hGp+1D⁻¹Fpdxdy+O(h^k+1+l)kukk+l+2,∞,τi,jkvk0,1,τi,j

= Z

τi,j

w1,p+1vxdxdy+O(h^k+1+l)kukk+l+2,∞,τi,jkvk0,1,τi,j, where in the third step, we have used the integration by parts and the fact that

D⁻¹Fp(1) =D⁻¹Fp(−1) = 0, 1≤p < l≤k.

Summing over allp, p= 1, . . . , lgives aτ(w1^l, v) =

l

X

p=1

Z

τ

(∂t+∂y)w1,pv−w1,pvx

dxdy

= Z

τ

(∂t+∂y)w1,lv−w1,1vx

dxdy+O(h^k+1+l)kukk+l+2,∞,τkvk0,1,τ. On the other hand, we have from (3.18)

aτ(E^xu, v) =− Z

τ

Li,kG1vdxdy= ¯h^x_i Z

τ

D⁻¹Li,kG1vxdxdy

= Z

τ

w1,1vxdxdy+O(h^k+1+l)kukk+l+2,∞,τkvk0,1,τ. Then

aτ(w^l1+E^xu, v) = Z

τ

(∂t+∂y)w1,lvdxdy+O(h^k+1+l)kukk+l+2,∞,τkvk0,1,τ. (3.23) Substituting the second inequality of (3.21) into (3.23) and summing up all elements τ∈ Th, we obtain the desired result (3.22) directly.

9

3.2. Correction function for a(E^yu, v). Since E^yuis a continuous function aboutx, andE^yu(x, y⁻_j+1

2

) = 0, by the same arguments as we did for a(E^xu, v), we obtain

a(E^yu, v) = Z

Ω

E^y(ut+ux)vdxdy ∀v∈Vh. (3.24) By (3.4), we have, in each elementτi,j,(i, j)∈Z_m×Z_n,

E^y(ut+ux) = (∂t+∂x)E^yu=

∞

X

q=k+1

(∂t+∂x)˜uj,p(x, t)(Lj,q−Lj,q−1)(y),

where

˜

uj,p(x, t) =u(x, y⁻_j+1 2

, t)−

p−1

X

l=0

2l+ 1 h^y_j

Z y_{j+ 1}

2

y_j−1 2

u(x, y, t)Lj,l(y)dy. (3.25)

Then a direct calculation from (3.24) yields aτi,j(E^yu, v) =

Z

τi,j

(∂t+∂x)E^yuvdxdy=− Z

τi,j

(∂t+∂x)˜uj,k+1Lj,kvdxdy.

The construction of the correction functionw2^l,1≤l≤k fora(E^yu, v) is similar tow^l1, which is defined as:

w^l2|τi,j :=

l

X

p=1

w2,p, w2,p= (¯h^y_j)^p(Q^x_hG˜p)(x, t)Fp(s). (3.26) Here ¯h^y_j =h^y_j/2 = (y_j+¹

2 −y_j−¹

2)/2,s= (2y−y_j+¹

2 −y_j−¹

2)/h^y_j and G˜p= (−1)^p−1(∂t+∂x)^pu˜i,k+1, p≥1.

Following the same line as in Theorem 3.1, we obtain w^l₂(x, y⁻_j+1

2

, t) = 0, kw2,pk0,∞,τi,j .h^k+p+1kukk+p+1,∞,τi,j, (3.27) and

a(w^l2, v) +a(E^yu, v)

.h^k+l+1kukk+l+2,∞kvk0,1. (3.28) 3.3. Estimates. For any given l, where 1 ≤ l ≤ k, we now define the final correction functionw^l as

w^l=w₁^l +w₂^l. (3.29)

Here w_i^l, i=Z2 are defined in (3.19) and (3.26), respectively. We have the following estimates for the correction functionw^l.

Theorem 3.2. Let u∈W^k+l+2,∞(Ω),1≤l≤k be the solution of (1.1), andw^l be the correction function defined by (3.29),(3.19), and (3.26). Then

w^l(x⁻_i+1 2

, y⁻_j+1 2

, t) = 0, kw^lk0,∞.h^k+2kukk+l+1,∞. (3.30)

10

Moreover, there holds for all v∈Vh

a(w^l, v) +a(u−P_h⁻u, v)

.h^k+l+1kukk+l+2,∞kvk0,1. (3.31) Proof. First, (3.30) is a direct consequence of (3.29), (3.21) and (3.27). By (3.9), (3.22) and (3.28),

a(w^l, v) +a(u−P_h⁻u, v)

.h^k+l+1kukk+l+2,∞kvk0,1+|a(E^xE^yu, v)|. (3.32) Now we estimatea(E^xE^yu, v). A direct calculation from (2.3) and (3.8) yields

Z

∂τ

E^xE^yuvds= 0, and thus

a(E^xE^yu, v) = Z

Ω

(E^xE^yut)vdxdy, v∈Vh. (3.33) Sinceu=E^xuwhenu|τ_i^x∈P_k(τ_i^x), by Bramble-Hilbert lemma, we obtain

kE^xuk0,∞,τ_i^x.(h^x_i)^r Z x_{i+ 1}

2

x_i−1 2

∂_x^r+1u

dx, 1≤r≤k.

Similarly, there holds

kE^yuk_0,∞,τ_j^y .(h^y_j)^r Z y_{j+ 1}

2

y_j−1 2

∂_y^r+1u

dy, 1≤r≤k.

Then

kE^xE^yuk0,∞,τi,j .h^k Z x_{i+ 1}

2

x_i−1 2

∂_x^k+1E^yu

dx=h^k Z x_{i+ 1}

2

x_i−1 2

E^y(∂_x^k+1u)

dx (3.34) .h^k+l−1

Z

τi,j

∂_y^l∂_x^k+1u

dxdy.h^k+l+1kukk+l+1,∞,τi,j,(3.35) which yields

|a(E^xE^yu, v)|= Z

Ω

(E^xE^yut)vdxdy

.h^k+l+1kukk+l+2,∞kvk0,1. (3.36) Plugging (3.36) into (3.32) yields the desired result (3.31).

Remark 3.3. From (3.36), the term a(E^xE^yu, v)is of high order, which means that a correction function for a(E^xE^yu, v)is not necessary.

4. Superconvergence. In this section, we shall study superconvergence proper- ties of the DG solution, including superconvergence for the cell averages and at some special points: downwind points and left and right Radau points.

We begin with the analysis of the super-closeness between the interpolation func- tionu^l_I =P_h⁻u−w^land the DG solutionuh.

11

Theorem 4.1. Let u ∈ W^k+l+2,∞(Ω),1 ≤ l ≤ k and uh ∈ Vh be solution of (1.1)and (2.1), respectively. Supposeu^l_I =P_h⁻u−w^l∈Vh withw^l defined by (3.29), (3.19), and (3.26). Then for both periodic and Dirichlet boundary conditions,

ku^l_I−uhk0(t).ku^l_I −uhk0(0) +th^k+l+1kukk+l+2,∞, ∀t >0. (4.1)

Proof. For the periodic boundary condition, we have from (3.12) and (3.19), w^l₁(x⁻_m+1

2

, y, t) =w^l₁(x⁻1 2

, y, t) = 0 ∀y, t.

Since u satisfies the periodic boundary condition, then P_h⁻u satisfies the periodic boundary condition (2.6). Moreover, by (3.17),

¯

ui,k+1(y_n+⁻ 1 2

, t) = ¯ui,k+1(y⁻1 2

, t) ∀t≥0, which yields

Q^y_hGp(y⁻_n+1 2

, t) =Gp(y_n+⁻ 1 2

, t) =Gp(y⁻1 2

, t) =Q^y_hGp(y⁻1 2

, t).

Then

w^l1(x, y_n+⁻ 1

2, t) =w1^l(x, y⁻1

2, t) ∀x, t.

Consequently,w^l1satisfies the periodic boundary condition (2.6). Similar result holds true for the correction functionw^l2defined by (3.26). SinceP_h⁻u, uhandw^lall satisfy (2.6), then (2.6) is valid forv=u^l_I−uh. For the Dirichlet boundary condition, due to the special choice of the numerical fluxes at the boundary, it is easy to see that (2.7) holds true forv =u^l_I−uh. Therefore, (2.8) is valid for both periodic and Dirichlet boundary conditions withv=u^l_I−uh. Then

ku^l_I−uhk0

d

dtku^l_I−uhk0≤

a(uh−u^l_I, u^l_I−uh)

=

a(u−u^l_I, u^l_I−uh)

.h^k+l+1kukk+l+2,∞ku^l_I−uhk0, which yields

d

dtku^l_I−uhk0.h^k+l+1kukk+l+2,∞. Then (4.1) follows.

Remark 4.2. To guarantee the superconvergence rate of (k+l+ 1)-th order for ku^l_I −uhk0, we know from Theorem 4.1 that the initial error should reach the same convergence rate. Namely,

kuh(·,0)−u^l_I(·,0)k0.h^k+l+1kukk+l+2,∞. (4.2) To obtain (4.2), a natural way of initial discretization is to choose

uh(x, y,0) =u^l_I(x, y,0). (4.3)

12

Remark 4.3. For the Dirichlet boundary condition, if we choose ˆ

uh(x, y) =Phu(x, y), (x, y)∈(∂Ω)⁻

instead of the choice of the numerical flux (2.4), where Phu = Rhu, P_h⁻u, Ihu with Rhu and Ihu denoting the L² projection and the Radau interpolation function of u respectively, then the standard approximation theory yields

Z

∂Ω

(u^l_I−uh)²ds

.kP_h⁻u−Phuk²_0,∞+kw^lk²_0,∞.h^2p,

wherep=k+ 1for Phu=Rhu, and p=k+ 2for Phu=P_h⁻u, Ihu. This means that the boundary error of u^l_I−uh can not be ignored. Therefore, we have from (2.5),

ku^l_I−uhk0

d

dtku^l_I−uhk0≤

a(uh−u^l_I, u^l_I−uh) +

Z

∂Ω

(u^l_I−uh)²ds .h^k+l+1kukk+l+2,∞ku^l_I−uhk0+h^2p.

Thus, the choice of numerical fluxes at the boundary has an influence on the super- convergence rate for the Dirichlet boundary condition.

4.1. Superconvergence for the cell averages. We have the following super- convergence results for the cell averages.

Theorem 4.4. Let u ∈ W^2k+2,∞(Ω) be the solution of (1.1), and uh be the solution of (2.1)with the initial valueuh(·,0)chosen such that (4.2)holds withl=k.

Then for both the periodic and Dirichlet boundary conditions,

eu,c= 1 nm

X

τ∈T_h

1

|τ|

Z

τ

(u−uh)2

dxdy

!¹2

.(1 +t)h^2k+1kuk2k+2,∞. (4.4)

Proof. Let eh = u−uh and uI = u^k_I = P_h⁻u−w^k. By the special initial discretization, we have from (4.1),

kuI−uhk0(t).(1 +t)h^2k+1kuk2k+2,∞. (4.5) On the other hand, the orthogonal property of (3.12) gives

Z

τ

w^kdxdy = Z

τ

w^k1+w^k2

dxdy= Z

τ

(w1,k+w2,k)dxdy, τ ∈ Th. Then

Z

τ

ehdxdy= Z

τ

u−P_h⁻u+w^k dxdy+

Z

τ

(uI−uh)dxdy

= Z

τ

(w1,k+w2,k)dxdy+ Z

τ

(uI −uh)dxdy.

In light of the estimates in (3.21) and (3.27), we derive 1

|τ|

Z

τ

ehdxdy²

.h^2k+1kuk2k+1,∞,τ+|τ|⁻¹kuI−uhk²_0,τ.

13

SinceTh is quasi-uniform, we have 1

nm|τ|⁻¹.1.

Then

eu,c.kuI−uhk0+h^2k+1kuk2k+1,∞.(1 +t)h^2k+1kuk2k+2,∞. The proof is completed.

4.2. Superconvergence at the downwind points. We are now ready to present our superconvergence result of the DG solution at the downwind points.

Theorem 4.5. Suppose all the conditions of Theorem 4.4 hold. Then for both the periodic and Dirichlet boundary conditions,

eu,d=



 1 mn

m

X

i=1 n

X

j=1

u−uh

2

x⁻_i+1 2

, y⁻_j+1 2

, t





1 2

.(1 +t)h^2k+1kuk2k+2,∞. (4.6)

Proof. For any fixedt, uI −uh ∈ Vh in eachτi,j,(i, j) ∈ Z_m×Z_n. Then the inverse inequality holds and thus,

(uI−uh)(x⁻_i+1 2

, y⁻_j+1 2

, t)

≤ kuI−uhk0,∞,τi,j(t) .h⁻¹kuI−uhk0,τi,j(t).

Then



 1 mn

m

X

i=1 n

X

j=1

uI−uh² x⁻_j+1

2

, y_j+⁻ 1 2

, t





1 2

.



 h⁻² mn

m

X

i=1 n

X

j=1

kuI−uhk²_0,τi,j(t)





1 2

.kuI−uhk0(t).(1 +t)h^2k+1kuk2k+2,∞. By (3.30) and the fact thatu(x⁻_i+1

2

, y_j+⁻ 1 2

, t) =P_h⁻u(x⁻_i+1 2

, y⁻_j+1 2

, t), we have uI(x⁻_i+1

2

, y_j+⁻ 1 2

, t) =u(x⁻_i+1 2

, y_j+⁻ 1 2

, t) ∀(i, j)∈Z_m×Z_n. Then the desired result (4.6) follows.

4.3. Superconvergence at the Radau points. Let R^l_p, R^r_p, p ∈ Z_k be the k interior left and right Radau points in the interval [−1,1], respectively. Namely, R_p^l, p∈Z_k are zeros ofLk+1+Lk except the points=−1, andR^r_p, p∈Z_k are zeros ofLk+1−Lk except the points= 1. Then for allτ=τi,j ∈ Th,(i, j)∈Z_m×Z_n,

R^l_τ={P :P = (R^l,x_τ,p, R^l,y_τ,q), p, q∈Z_k}, R^r_τ ={Q:Q= (R_τ,p^r,x, R^r,y_τ,q), p, q∈Z_k}, constitutek² interior left and right Radau points inτ, respectively. Here

R_τ,p^l,x = 1 2(x_i−¹

2 +x_i+¹

2 +h^x_iR^l_p), R^l,y_τ,p= 1 2(y_j−¹

2 +y_j+¹

2 +h^y_jR_p^l), and R^r,x_τ,p, R^r,y_τ,p are defined similarly. We denote by R^r = S

τ∈T_hR^r_τ the set of right Radau points on the whole domain, and

E_x^l ={(x, y) :x=R^l,x_τ,p, y∈[c, d], p∈Z_k, τ ∈ Th}

14