OPTIMAL ERROR ESTIMATES OF THE SEMIDISCRETE CENTRAL DISCONTINUOUS GALERKIN METHODS FOR LINEAR HYPERBOLIC EQUATIONS

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CENTRAL DISCONTINUOUS GALERKIN METHODS FOR LINEAR HYPERBOLIC EQUATIONS

YONG LIU^†, CHI-WANG SHU^‡, AND MENGPING ZHANG^§

Abstract. We analyze the central discontinuous Galerkin (DG) method for time-dependent linear conservation laws. In one dimension, optimal a prioriL²error estimates of orderk+ 1 are obtained for the semidiscrete scheme when piecewise polynomials of degree at mostk(k≥0) are used on overlapping uniform meshes. We then extend the analysis to multidimensions on uniform Carte- sian meshes when piecewise tensor product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results.

Key words. Optimal error estimate; central DG; superconvergence points AMS subject classifications.65M60, 65M12

1. Introduction. We study the central discontinuous Galerkin (DG) method for solving hyperbolic equations [14]. Even though central DG methods give optimal convergence rate in numerical tests, previous error estimates can only provide sub- optimal results [14]. In this paper we prove optimal error estimates of the central DG approximation based on tensor-product polynomials for solving linear hyperbolic conservation laws

ut+ Xd i=1

ciuxi= 0, (x, t)∈Ω×(0, T] (1.1a)

u(x,0) =u0(x), x∈Ω (1.1b)

where Ω is a bounded rectangular domain in R^d, and x = (x1, . . . , xd). Here ci

are constants and u0(x) is a given smooth function. We assume periodic boundary condition for simplicity, although this is not essential for the analysis, inflow-outflow boundary conditions can also be considered along the same lines.

The central scheme of Nessyahu and Tadmor [17] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries.

The excessive numerical dissipation for small time steps is considered by Kurganov and Tadmor [10] and is controlled by a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Liu [12] uses another coupling technique to avoid the excessive numerical dissipation for small time steps. The staggered meshes can be viewed as a collection of overlapping cells and the numerical solution is realized by its overlapping cell averages. Overlapping cells lend themselves to the development of the central DG method [13], following the series of work by Cockburn and Shu on DG methods [5, 7]. Besides the advantage of avoiding Riemann solvers, another advantage of central DG schemes is that they allow much larger time steps (proportional to O(h/k) where his the the spatial mesh size and k is the polynomial degree) for

†School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China. E-mail: [email protected]

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

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

§School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China. E-mail: [email protected]. Research supported by NSFC grant 11471305.

1

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linear stability than that for regular DG schemes (which are proportional toO(h/k²)), particularly for higher spatial orders [18]. Liu et al in [14] giveL² stability analysis and sub-optimal error estimates of the central DG method for linear hyperbolic conservation laws. They use the standardL² projection in the error estimate, resulting in a sub-optimalk-th order accuracy. The difficulty leading to this loss of optimality is the lack of a suitable projection, similar to the Gauss-Radau projection used for regular DG schemes, to eliminate the trouble-some cell boundary terms belonging to the approximation error in the error estimates. Later, central DG schemes have been extended to diffusion equations in [15], again with stability analysis and sub-optimal error estimates for the linear heat equation.

For smooth solutions of linear conservation laws, optimal a priori error estimates of orderk+1 for one-dimensional and some multidimensional cases [6, 22, 4, 11, 19, 20]

can be obtained for regular DG schemes when upwind fluxes are used. Similar optimal a priori error estimates can also be obtained when upwind-biased fluxes are used [16].

Xu and Shu [21] introduced a general approach for proving optimal error estimates by utilizing the local DG (LDG) scheme and its time derivatives with different test functions and fully making use of the so-called Gauss-Radau projections. We should also mention works on superconvergence of DG schemes [2, 1]. A key ingredient of all the optimal error estimates mentioned above is a suitable projection, to deal with the troublesome intercell terms in the approximation error. For upwind fluxes, this suitable projection is the Gauss-Radau projection; for the upwind-biased fluxes, a special global projection is introduced in [16] for this purpose. Unfortunately, up to now a suitable projection is elusive for central DG schemes, thus leading to only sub-optimal error estimates in [14]. The main contribution of the present paper is a careful con- struction and analysis of special projections, suitable for central DG schemes yielding optimal error estimates. In one dimension, we find the superconvergence points of the central DG scheme and successfully construct a proper local projectionP^∗_haccording to duality of overlapping cells. The existence and optimal approximation properties of this projection are proved by standard finite element techniques. Moreover, the pro- jectionP^∗_hcan eliminate the space-discrete terms involvingu−P^∗_huwhenu∈P^k+1(Ω), hereP^k+1(Ω) is the polynomial space of degree at most k+ 1 over each cell. This superconvergence property leads to the derivation of optimal convergence rate. The proof of optimal convergence results is valid for uniform meshes and for polynomials of arbitrary degreek≥0.

For multidimensional Cartesian meshes, we follow the same arguments as in the one-dimensional case to construct a suitable projectionP^∗_h and analyze its existence and approximation properties. We would like to remark that this new projection utilizesQ^k, the space of tensor-product polynomials of degree at mostkin each variable. The superconvergence result ofP^∗_h on Cartesian meshes helps to yield optimal convergence results.

The organization of the paper is as follows. In section 2, we first recall the central discontinuous Galerkin method for linear hyperbolic equations. Then, we construct a special projection, and study its existence, uniqueness and optimal approximation properties. This projection is used to prove optimal error estimate for the semidiscrete central DG scheme for the linear hyperbolic equation in one dimension in this section. We extend the analysis to multidimensions in section 3, in which optimal error estimates are proved, following the same lines of the one dimensional case. We provide numerical examples to show our theoretical results in section 4. In section 5 we give a few concluding remarks and perspectives for future work. Finally, in

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the appendix we provide proofs for some of the more technical results of the error estimates.

2. The central DG method in one dimension. In this section, we consider the one-dimensional scalar linear conversation law equation

(2.1)

(ut+ux= 0, x∈[a, b], t≥0 u(x,0) =u0(x), x∈[a, b]

with periodic boundary condition.

Let {xj} be a partition of [a, b] with h_j+¹

2 = xj+1 −xj and h = maxjh_j+¹

2. Denote x_j+¹

2 = ¹₂(xj+1 +xj), Ij = (x_j+¹

2, x_j−¹

2), and I_j+¹

2 = (xj, xj+1). Vh is the set of piecewise polynomials of degree at mostk over the subintervals{Ij} with no continuity assumed across the subinterval boundaries. Likewise, Wh is the set of piecewise polynomials of degree at most k over the subintervals I_j+¹

2 with no continuity assumed across the subinterval boundaries.

The central DG method is defined on overlapping cells and uses both spacesVh

and Wh. The semi-discrete version of the central DG scheme is defined as follows.

Finduh(·, t)∈Vhandvh(·, t)∈Wh, such that for anyϕh∈Vh andψh∈Wh, Z

Ij

∂tuhϕhdx = 1 τmax

Z

Ij

(vh−uh)ϕhdx+ Z

Ij

vh∂xϕhdx

−vh(x_j+¹

2, t)ϕh(x⁻_j+1 2

) +vh(x_j−¹

2, t)ϕh(x⁺_j₋1 2

) (2.2a)

Z

I_{j+ 1}

2

∂tvhψhdx = 1 τmax

Z

I_{j+ 1}

2

(uh−vh)ψhdx+ Z

I_{j+ 1}

2

uh∂xψhdx

−uh(xj+1, t)ψh(x⁻_j+1) +uh(xj, t)ψh(x⁺_j ) (2.2b)

whereτmax is an upper bound for the time step size due to the CFL restriction, that is,τmax=c hwith a given constant CFL numberc dictated by stability. In [14], the following stability result is proved for this scheme.

Theorem 2.1. (Liu et al [14]). The numerical solutionsuhandvhof the central DG scheme (2.2) for the equation (2.1) have the following L² stability property

(2.3) 1

2 d dt

Z b a

(u²_h+v_h²)dx=− 1 τmax

Z b a

(vh−uh)²dx≤0.

Remark 2.1. The sketch of the proof is taking the test functions ϕh =uh and ψh = vh in (2.2) respectively, summing up over j, and using the periodic boundary condition. The details are given in [14]. The proof of Theorem 2.1 is similar to the proof of the cell entropy inequality for the regular DG method in [9].

2.1. Optimal L² error estimate. In this subsection, we show the optimal a priori error estimate of the scheme (2.2). The main idea is to establish a special projection to facilitate the proof of the optimalL² error estimate. We first state our main result in the following theorem.

Theorem 2.2. The numerical solutions uh and vh of the central DG scheme (2.2) using uniform meshes for the equation (2.1) with a smooth initial condition u(·,0)∈H^k+2 satisfies the followingL² error estimate

(2.4) ku−uhk²+ku−vhk²≤Ch^2k+2

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whereuis the exact solution of (2.1),kis the polynomial degree in the finite element spacesVh andWh, and the constantC depends on the (k+ 2)-th order Sobolev norm of the initial conditionku(·,0)k_H^k+2 as well as on the final timet, but is independent of the mesh size h.

Let us first introduce a few notations. We define Bj(uh, vh;ϕh, ψh) =

Z

Ij

∂tuhϕhdx− 1 τmax

Z

Ij

(vh−uh)ϕhdx− Z

Ij

vh∂xϕhdx +vh(x_j+¹

2, t)ϕh(x⁻_j+1 2

)−vh(x_j−¹

2, t)ϕh(x⁺_j₋1 2

) +

Z

I_{j+ 1}

2

∂tvhψhdx− 1 τmax

Z

I_{j+ 1}

2

(uh−vh)ψhdx− Z

I_{j+ 1}

2

uh∂xψhdx +uh(xj+1, t)ψh(x⁻_j+1)−uh(xj, t)ψh(x⁺_j)

(2.5)

We also define

Bej(uh, vh;ϕh) = 1 τmax

Z

Ij

(vh−uh)ϕhdx+ Z

Ij

vh∂xϕhdx

−vh(x_j+¹

2, t)ϕh(x⁻_j+1 2

) +vh(x_j−¹

2, t)ϕh(x⁺_j−1 2

) (2.6)

Bb_j+¹

2(uh, vh;ψh) = 1 τmax

Z

I_{j+ 1}

2

(uh−vh)ψhdx+ Z

I_{j+ 1}

2

uh∂xψhdx

−uh(xj+1, t)ψh(x⁻_j+1) +uh(xj, t)ψh(x⁺_j) (2.7)

Thus

Bj(uh, vh;ϕh, ψh) = Z

Ij

∂tuhϕhdx+ Z

I_{j+ 1}

2

∂tvhψhdx

−Bej(uh, vh;ϕh)−Bb_j+¹

2(uh, vh;ψh) (2.8)

Clearly, we have:

(2.9) Bj(uh, vh;ϕh, ψh) = 0

for all j and allϕh ∈ Vh and ψh ∈Wh. It is also clear that the exact solutionuof the PDE (2.1) satisfies

(2.10) Bj(u, u;ϕh, ψh) = 0

for allj and allϕh∈Vh andψh ∈Wh. Subtracting (2.9) from (2.10), we obtain the error equation

(2.11) Bj(u−uh, u−vh;ϕh, ψh) = 0 for allj and allϕh∈Vh andψh∈Wh.

We now defineP^∗_handQ^∗_has the following projections intoVhandWhrespectively.

That is, for eachj, (2.12a)

Z

Ij

P^∗_hω(x)dx= Z

Ij

ω(x)dx

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(2.12b) Pfh(P^∗_hω;ϕh)j =Pfh(ω;ϕh)j ∀ϕh∈P^k(Ij) wherePfh(ω;ϕh)j is defined as follows

Pfh(ω;ϕh)j = 1 τmax



Z xj

x_j₋1 2

ω(x+h

2)ϕh(x)dx+ Z x_{j+ 1}

2

xj

ω(x−h

2)ϕh(x)dx

− Z x_{j+ 1}

2

x_j₋1 2

ω(x)ϕh(x)dx



+ Z xj

x_j₋1 2

ω(x+h

2)(ϕh(x))xdx +

Z x_{j+ 1}

2

xj

ω(x−h

2)(ϕh(x))xdx−ω(xj)

ϕh(x⁻_j+1 2

)−ϕh(x⁺_j−1 2

) and, similarly,

(2.13a)

Z

I_{j+ 1}

2

Q^∗_hω(x)dx= Z

I_{j+ 1}

2

ω(x)dx

(2.13b) Qfh(Q^∗_hω;ψh)_j+¹

2 =Qfh(ω;ψh)_j+¹

2 ∀ψh(x)∈P^k(I_j+¹

2) whereQfh(ω;ψh)_j+¹

2 is defined as follows Qfh(ω;ψh)_j+¹

2 = 1

τmax



 Z x_{j+ 1}

2

xj

ω(x+h

2)ψh(x)dx+ Z xj+1

x_{j+ 1}

2

ω(x−h

2)ψh(x)dx

− Z xj+1

xj

ω(x)ψh(x)dx

! +

Z x_{j+ 1}

2

xj

ω(x+h

2)(ψh(x))xdx +

Z xj+1

x_{j+ 1}

2

ω(x−h

2)(ψh(x))xdx−ω(x_j+¹

2) ψh(x⁻_j+1)−ψh(x⁺_j) Here P^k(Ij) andP^k(I_j+¹

2) denote the spaces of polynomials of degree up tok in the cellIj and the cellI_j+¹

2 respectively. Next, we prove the projectionsP^∗_h andQ^∗_h are well defined. Note the projectionsP^∗_handQ^∗_hare local projections, so we only consider the projections defined on the reference interval [−1,1]. In this case,h= 2, τmax= 2c.

Without loss of generality we will only considerP^∗_h.

Remark 2.2. The equation (2.12a) is required by the conservation laws. Note that Pfh(ω, ϕh)j = 0,∀ω when ϕh = 1, so (2.12b) alone misses one condition which is provided by (2.12a). The projection is a local one on cell Ij. Therefore we only need to prove the existence as well as uniqueness and boundedness in the L^∞-norm.

Classical approximation results then imply optimal approximation of the projections.

Lemma 2.1. The projection P^∗_h defined by (2.12) on the interval [−1,1] exists and is unique for any smooth function ω, and the projection is bounded in the L^∞ norm,i.e.

(2.14) kP^∗_hωk∞≤C(k)kωk∞

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whereC(k)is a constant that only depends on kbut is independent of ω.

Proof. The proof of this lemma is provided in the Appendix; see section A.1.

Since the projections P^∗_h and Q^∗_h are k-th degree polynomial preserving local projections, standard approximation theory [3] implies, for a smooth functionω, (2.15) kP^∗_hω(x)−ω(x)k+h¹²kP^∗_hω(x)−ω(x)kΓ≤Ch^k+1

and

(2.16) kQ^∗_hω(x)−ω(x)k+h¹²kQ^∗_hω(x)−ω(x)kΓ≤Ch^k+1 where Γ denotes the set of boundary points of all elements Ij or I_j+¹

2 respectively, the normk · kΓis the standardL²norm, and the positive constantC, here and below, solely depending onωand its derivatives, is independent of h.

We also recall that [3], for any ωh ∈ Vh or ωh ∈ Wh, there exists a positive constantC independent ofωh andh, such that

(2.17) k∂xωhk ≤Ch⁻¹kωhk; kωhkΓ≤Ch^−1/2kωhk where Γ is the set of boundary points of all elementsIj orI_j+¹

2.

Besides the standard approximation results (2.15) and (2.16), the special projec- tionsP^∗_h andQ^∗_h also have the following superconvergence result.

Proposition 2.1. Assume that uis a (k+ 1)-th degree polynomial function in P^k+1([a, b]). For a uniform partition on the interval [a, b], set uI =P^∗_hu ∈ Vh and vI =Q^∗_hu∈Wh where P^∗_h andQ^∗_h are defined by (2.12) and (2.13). Then we have

Bej(uI, vI;ϕh) =Bej(u, u;ϕh) ∀ϕh∈P^k(Ij) (2.18a)

Bb_j+¹

2(uI, vI;ψh) =Bb_j+¹

2(u, u;ψh) ∀ψh∈P^k(I_j+¹

2) (2.18b)

whereBe andBb are defined by (2.6) and (2.7).

Proof. The proof of this proposition is provided in the Appendix; see section A.2.

We now take:

(2.19) ϕh=P^∗_hu−uh, ψh=Q^∗_hu−uh

in the error equation (2.11), and define

(2.20) ϕ^e=P^∗_hu−u, ψ^e=Q^∗_hu−u to obtain

(2.21) Bj(ϕh, ψh;ϕh, ψh) =Bj(ϕ^e, ψ^e;ϕh, ψh).

For the left-hand side of (2.21), we use Theorem 2.1 to conclude

(2.22) X

j

Bj(ϕh, ψh;ϕh, ψh) = 1 2

d dt

Z b a

(ϕ²_h+ψ²_h)dx+ 1 τmax

Z b a

(ϕh−ψh)²dx We then write the right-hand side of (2.21) as a sum of three terms

(2.23) Bj(ϕ^e, ψ^e;ϕh, ψh) =B_j¹+B_j²+B_j³

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where

B_j¹=−Bej(ϕ^e, ψ^e;ϕh) B_j²=−Bb_j+¹

2(ϕ^e, ψ^e;ψh) B³_j =

Z

Ij

∂tϕ^eϕhdx+ Z

I_{j+ 1}

2

∂tψ^eψhdx and we will estimate each term separately.

By using the simple inequality

(2.24) µν ≤1

2(µ²+ν²)

and the special projection properties (2.15)-(2.16) for∂tϕ^eand∂tψ^e, we have:

(2.25) X

j

B_j³≤ Z b

a

(ϕh)²dx+ Z b

a

(ψh)²dx+Ch^2k+2. ForB_j¹, letcuIj

be the Taylor expansion polynomial of orderk+1 ofuover the interval Ij, i.e. ucIj

=Pk+1 i=0 1

i!u⁽ⁱ⁾(xj)(x−xj)ⁱ, x∈(xj−1, xj+1). Letru denote the residual term i.e. r_u^j =u−cuIj

. Recalling the Bramble-Hilbert lemma in [3], we have (2.26) kr^j_ukL^∞(Ij)≤Ch^k+³²|u|H^k+2(Ij)

Then we rewriteϕ^e andψ^e

ϕ^e=P^∗_hu−u=P^∗_h(cuIj

+r_u^j)−ucIj

−ru

=P^∗_hcuIj

−cuIj

+P^∗_hr^j_u−r^j_u (2.27)

ψ^e=Q^∗_hu−u=Q^∗_h(cuIj

+r^j_u)−cuIj

−r^j_u

=Q^∗_hcuIj

−cuIj

+Q^∗_hr^j_u−r^j_u (2.28)

Thus, using Proposition 2.1 we have B_j¹=−Bej(ϕ^e, ψ^e;ϕh)

=−Bej(P^∗_hcuIj

−ucIj

+P^∗_hr^j_u−r^j_u,Q^∗_hcuIj

−cuIj

+Q^∗_hr^j_u−r_u^j;ϕh)

=−Bej(P^∗_hcuIj

−ucIj

,Q^∗_hcuIj

−cuIj

;ϕh)−Bej(P^∗_hr^j_u−r^j_u,Q^∗_hr_u^j −r_u^j;ϕh)

=−Bej(P^∗_hr_u^j−r^j_u,Q^∗_hr_u^j −r_u^j;ϕh) (2.29)

Therefore, by using the simple inequality (2.24), the special projection property (2.15) and (2.16), the property (2.26) forru, and the inequality in (2.17) forϕh, we have:

(2.30) X

j

B_j¹≤Ch^2k+2|u|²_Hk+2([a,b])+C Z b

a

(ϕh)²dx Similarly, forB²_j we have

(2.31) X

j

B_j²≤Ch^2k+2|u|²_Hk+2([a,b])+C Z b

a

(ψh)²dx

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Combining (2.25), (2.30) and (2.31) with (2.22), we obtain from (2.21) 1

2 d dt

Z b a

(ϕh)²+ (ψh)² dx≤C

Z b a

(ϕh)²+ (ψh)²

dx+Ch^2k+2|u|²_Hk+2([a,b]). This, together with the approximation results (2.15) and (2.16), implies the desired error estimate (2.4).

Remark 2.3. Notice that, the first term on the right side of (2.2) is a numer- ical dissipation term. This is directly related to the difference of the two duplicative representationsuh andvh of the solution in overlapping cells. It is important for the uniqueness as well as existence of the special projectionsP^∗_h defined in (2.12) andQ^∗_h defined in (2.13). In fact, if we eliminate this term, the stability of central scheme is still preserved, however the optimal convergence accuracy is lost and the special projections do not exist for somek.

Remark 2.4. The special projections P^∗_h defined in (2.12) and Q^∗_h defined in (2.13) are related to the superconvergence points of the central DG method for one- dimensional linear scalar hyperbolic equation. In fact, we find that the zeros of the polynomialx^k+1−P^∗_h(x^k+1)are the superconvergence points of the central DG scheme.

We will display the numerical results in section 4. This is the reason that the special projection can help us to obtain the optimal a prior error estimates.

3. The central DG method in multidimensions. In this section, we consider the semidiscrete central DG method for multidimensional linear conservation laws.

Without loss of generality, we describe our central DG scheme and prove the optimal a priori error estimates in two dimensions (d = 2); all the arguments we present in our analysis depend on the tensor product structure of the mesh and finite element space and can be easily extended to the more general casesd >2. Hence, from now on, we shall restrict ourselves to the following two-dimensional problem

(3.1)

(ut+ux+uy= 0, (x, y, t)∈Ω×(0, T] u(x, y,0) =u0(x, y), (x, y)∈Ω again with periodic boundary conditions.

We recall the 2D formulation of the central DG scheme in [13]. Let {CI = [x_i−¹

2, x_i+¹

2]×[y_j−¹

2, y_j+¹

2]}, I = (i, j), be a partition of Ω into uniform square cells, depicted by the solid lines in Figure 3.1, and tagged by their cell centroid at thex_I :=

I∆x. LetVh:={v∈L²(Ω) :v|CI ∈Q^k(CI)∀I}, whereQ^k(CI) denotes the space of tensor-product polynomials of degrees at most k in each variable defined onCI; no continuity is assumed across cell boundaries. Let{D_I+¹

2 = [xi, xi+1]×[yj, yj+1]} be the dual mesh which consists of a ∆x/2 shift of theCI’s, depicted by the dashed lines in Figure 3.1. Let x_I+¹

2 = (I+¹₂)∆xbe the cell centroid of the cell D_I₊¹

2 and let Wh:={v∈L²(Ω) :v|D_{I+ 1}

2

∈Q^k(D_I₊¹

2)∀I}, whereQ^k(D_I+¹

2) denotes the space of tensor-product polynomials of degrees at mostkin each variable defined on{D_I+¹

2};

again, no continuity is assumed across the cell boundary.

3.1. The central DG method and optimal error estimate.

3.1.1. The central DG scheme. The semidiscrete central DG approximations uh∈Vh andvh ∈Wh of (3.1) are defined such that for all admissible test functions ϕhandψh and allI’s,

d dt

Z

CI

uhϕhdxdy= 1 τmax

Z

CI

(vh−uh)ϕhdxdy

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Fig. 3.1. 2D overlapping cells formed by collapsing the staggered dual cells on two adjacent time levels to one time level.

+ Z

CI

vh(∂xϕh+∂yϕh)dxdy

− Z y_{j+ 1}

2

y_j₋1 2

vh(x_i+¹

2, y)ϕh(x⁻_i+1 2

, y)−vh(x_i−¹

2, y)ϕh(x⁺_i₋1 2

, y) dy

− Z x_{i+ 1}

2

x_i₋1 2

vh(x, y_j+¹

2)ϕh(x, y⁻_j+1 2

)−vh(x, y_j−¹

2)ϕh(x, y_j−⁺ 1 2

) dx

= :B(ue h, vh;ϕh)i,j ∀ϕh∈Vh

(3.2)

d dt

Z

D_{I+ 1}

2

vhψhdxdy= 1 τmax



Z

D_{I+ 1}

2

(uh−vh)ψhdxdy



 +

Z

D_{I+ 1}

2

uh(∂xψh+∂yψh)dxdy

− Z yj+1

yj

uh(xi+1, y)ψh(x⁻_i+1, y)−uh(xi, y)ψh(x⁺_i , y) dy

− Z xi+1

xi

uh(x, yj+1)ψh(x, y_j+1⁻ )−uh(x, yj)ψh(x, y⁺_j) dx

= :B(vb h, uh;ψh)_i+¹

2,j+¹₂ ∀ψh∈Wh

(3.3)

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where τmax is the max step size, determined byτmax =(CFL factor)×h/(maximum characteristic speed), in which the CFL factor should be less than 1/2. For the initial condition, we simply takeuh(0) =P_hu0 andvh(0) =Q_hu0, whereP_h andQ_hare the L² projections intoVhand Wh, respectively, and we have

ku0−P_hu0kL²(CI)≤Ch^k+1ku0kH^k+1(CI)

ku0−Q_hu0kL²(D_{I+ 1}

2

)≤Ch^k+1ku0kH^k+1(D_{I+ 1}

2

).

TheL²-stability for the central DG scheme (3.2)-(3.3) is proved in [14].

Proposition 3.1. [14]. The numerical solutions uh andvh of the semidiscrete central DG method (3.2)-(3.3) for the equation (3.1) has the following L² stability property

(3.4) kuh(T)k²_L2(Ω)+kvh(T)k²_L2(Ω)≤ kuh(0)k²_L2(Ω)+kvh(0)k²_L2(Ω)

3.2. A priori error estimates. Let us state a priori error estimate for the two-dimensional case, whose proof will be given in the next subsection.

Theorem 3.1. The numerical solutions uh and vh of the central DG scheme (3.2)-(3.3) using uniform meshes for the equation (3.1) with a smooth initial condition u(·,0)∈H^k+2 satisfies the followingL² error estimate

(3.5) ku(T)−uh(T)k²_L²_(Ω)+ku(T)−vh(T)k²_L²_(Ω)≤Ch^2k+2

whereu is the exact solution of (3.1),k is the degree of the piecewise tensor product polynomials in the finite element spaces Vh andWh, and the constant C depends on the (k+ 2)-th order Sobolev norm of the initial condition ku(·,0)k_H^k+2 as well as on the final timeT, but is independent of the mesh size h.

3.3. Proof of the error estimates. In this subsection we prove Theorem 3.1 stated in the previous subsection. To do that, we proceed as follows. First, in subsection 3.3.1 we establish the existence as well as uniqueness of suitably defined special local projectionsP^∗_handQ^∗_h. In addition, the optimal approximation properties ofP^∗_h andQ^∗_h are derived. We prove a few propositions and superconvergence results of the special projections in subsection 3.3.2. Finally, we complete the proof Theorem 3.1 in subsection 3.3.3.

3.3.1. The special projections P^∗_h and Q^∗_h. Prior to giving the definition of the special projectionsP^∗_h andQ^∗_h, we would like to recall the projectionsP^∗_handQ^∗_h introduced in (2.12)-(2.13) for the one dimensional case. We note that we right shift by

h

2 when integrating on the subinterval [x_i−¹

2, xi] and left shift by ^h₂ when integrating on the subinterval [xi, x_i+¹

2]. We now continue to apply this technique in the two dimensional case, forxandyvariables respectively. We define the projectionP^∗_hfrom u∈L^∞(CI) intoP^∗_hu∈Q^k(CI) overCI satisfying the following two equations.

(3.6)

Z

CI

P^∗_hudxdy= Z

CI

udxdy

(3.7) Pfh(P^∗_hu, ϕh)i,j=Pfh(u, ϕh)i,j, ∀ϕh∈Q^k(CI)

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wherePfh(u, ϕh)i,j is defined as follows:

Pfh(u, ϕh)i,j= 1 τmax



 Z yj

y_j−1 2

Z xi

x_i−1 2

u(x+h 2, y+h

2)ϕhdxdy +

Z yj

y_j₋1 2

Z x_{i+ 1}

2

xi

u(x−h 2, y+h

2)ϕhdxdy +

Z y_{j+ 1}

2

yj

Z xi

x_i−1 2

u(x+h 2, y−h

2)ϕhdxdy +

Z y_{j+ 1}

2

yj

Z x_{i+ 1}

2

xi

u(x−h 2, y−h

2)ϕhdxdy

− Z y_{j+ 1}

2

y_j−1 2

Z x_{i+ 1}

2

x_i−1 2

u(x, y)ϕhdxdy



 +

Z yj

y_j₋1 2

Z xi

x_i₋1 2

u(x+h 2, y+h

2) (∂xϕh+∂yϕh)dxdy +

Z yj

y_j₋1 2

Z x_{i+ 1}

2

xi

u(x−h 2, y+h

Z y_{j+ 1}

2

yj

Z xi

x_i−1 2

u(x+h 2, y−h

Z y_{j+ 1}

2

yj

Z x_{i+ 1}

2

xi

u(x−h 2, y−h

2) (∂xϕh+∂yϕh)dxdy

− Z xi

x_i₋1 2

u(x+h 2, yj)

ϕh(x, y_j+⁻ 1 2

)−ϕh(x, y⁺_j−1 2

) dx

− Z x_{i+ 1}

2

xi

u(x−h 2, yj)

ϕh(x, y⁻_j+1 2

)−ϕh(x, y⁺_j−1 2

) dx

− Z yj

y_j₋1 2

u(xi, y+h 2)

ϕh(x⁻_i+1 2

, y)−ϕh(x⁺_i−1 2

, y) dy

− Z y_{j+ 1}

2

yj

u(xi, y−h 2)

ϕh(x⁻_i+1 2

, y)−ϕh(x⁺_i₋1 2

, y) dy (3.8)

Similarly, we can define the projection Q^∗_h from u ∈ L^∞(D_I+¹

2) into Q^k(D_I+¹

2) over D_I+¹

2. The equation (3.6) is required by the conservation laws. Note that Pfh(u, ϕh)i,j = 0,∀u whenϕh = 1, so (3.7) alone misses one condition which is provided by (3.6).

Existence and optimal approximate property of the projectionP^∗_hare established in the following lemma.

Lemma 3.1. Assume u is sufficiently smooth. Then, there exists a unique P^∗_hu ∈ Vh satisfying (3.6) and (3.7). Moreover, there holds the following approxi- mating property

(3.9) ku−P^∗_hukL²(C_I)≤Ch^k+1kuk_H^k+1_(C_I₎

where C=C(k) is independent of the elementCI and the mesh sizeh.

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Proof. The proof of this lemma is provided in the Appendix; see section A.3.

3.3.2. Properties of the projection P^∗_h. To obtain the optimalL² error estimate, we need the following lemmas.

Lemma 3.2. Assume that u=x^k+1 or y^k+1, let uI =P^∗_huand vI =Q^∗_hu, then

∀(x, y)∈CI we have following relations:

x^k+1−uI(x, y) =(x+h

2)^k+1−vI(x+h 2, y+h

2)

=(x+h

2)^k+1−vI(x+h 2, y−h

2)

=(x−h

2)^k+1−vI(x−h 2, y+h

2)

=(x−h

2)^k+1−vI(x−h 2, y−h

2) (3.10)

or

y^k+1−uI(x, y) =(y+h

2)^k+1−vI(x+h 2, y+h

2)

=(y+h

2)^k+1−vI(x−h 2, y+h

2)

=(y−h

2)^k+1−vI(x+h 2, y−h

2)

=(y−h

2)^k+1−vI(x−h 2, y−h

2) (3.11)

Proof. The details of the proof of this lemma are provided in Appendix; see section A.4.

Again, the projectionsP^∗_handQ^∗_h satisfy the following superconvergence result.

Lemma 3.3. Assume that u=x^k+1 or y^k+1, let uI=P^∗_huandvI =Q^∗_hu, then (3.12) B(ue I, vI;ϕh)i,j=B(u, u;e ϕh)i,j

(3.13) B(vb I, uI;ψ)_i+¹

2,j+¹₂ =B(u, u;b ψh)_i+¹

2,j+¹₂

whereBe andBb are defined by (3.2) and (3.3).

Proof. We will only give the details of the proof for one case, namelyB(ue I, vI;ϕ)i,j = B(u, u;e ϕh)i,jis true whenu=x^k+1, as the other cases can be handled similarly. We can use lemma 3.2 to replacevI byuI. After simplification, we have

B(ue I, vI;ϕh)i,j =Pfh(uI−x^k+1, ϕh)i,j+B(xe ^k+1, x^k+1;ϕh)i,j

=B(u, u;e ϕh)i,j

(3.14)

where we have used the definition of the special projectionP^∗_hin (3.7).

Next, we will use these lemmas to prove our finial result, Theorem 3.1.

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3.3.3. Proof of Theorem 3.1. We take

ϕh=P^∗_hu−uh; ψh=Q^∗_hu−vh

(3.15)

ϕ^e=P^∗_hu−u; ψ^e=Q^∗_hu−u (3.16)

The sum of the (3.2) and (3.3) gives Bi,j(uh, vh;ϕh, ψh) =

Z

CI

∂tuhϕhdxdy+ Z

D_{I+ 1}

2

∂tvhψhdxdy

−B(ue h, vh;ϕh)i,j−B(vb h, uh;ψh)_i+¹

2,j+¹₂

=0 (3.17)

The exact solution of the PDE (3.1) also satisfies the above scheme, hence we have (3.18) Bi,j(ϕh, ψh;ϕh, ψh) =Bi,j(ϕ^e, ψ^e;ϕh, ψh)

For the left-hand side of (3.18), we can use the stability results in [14] to obtain (3.19) X

i,j

Bi,j(ϕh, ψh;ϕh, ψh) =1 2

d dt

Z

Ω

ϕ²_h+ψ²_hdxdy+ 1 τmax

Z

Ω

(ϕh−ψh)²dxdy From Lemma 3.3, we know that on an arbitrary element CI, we have the following results

(3.20)

B(Pe ^∗_hu,Q^∗_hu;ϕh)i,j=B(u, u;e ϕh)i,j ∀u∈P^k+1([xi−1, xi+1]×[yj−1, yj+1]) ∀ϕh∈Q^k(CI) Next, on each elementCI, we consider the Taylor expansion ofuaround (xi, yj),

u=T u+Ru where

T u=

k+1X

l=0

Xl m=0

1 (l−m)!m!

∂^lu(xi, yj)

∂x^l−m∂y^m(x−xi)^l−m(y−yj)^m,

Ru= (k+ 2)

k+2X

m=0

(x−xi)^k+2⁻^m(y−yj)^m (k+ 2−m)!m!

Z 1 0

(1−s)^k+1∂^k+2u(x^s_i, y_j^s)

∂x^k+2−m∂y^mds with x^s_i = xi+s(x−xi), y_j^s = yj+s(y−yj). Clearly, T u ∈ P^k+1([xi−1, xi+1]× [yj−1, yj+1]). Note that the operator P^∗_h is linear and thus P^∗_hu = P^∗_hT u+P^∗_hRu.

From (3.20) we then get

B(ϕe ^e, ψ^e;ϕh)i,j =B(Pe ^∗_hT u−T u,Q^∗_hT u−T u;ϕh)i,j+B(Pe ^∗_hRu−Ru,Q^∗_hRu−Ru, ϕh)i,j

=B(Pe ^∗_hRu−Ru,Q^∗_hRu−Ru, ϕh)i,j

(3.21)

Again recalling the Bramble-Hilbert lemma in [3], we have (3.22) kRukL^∞(CI)≤Ch^k+1|u|_H^k+2_(C_I₎

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Next using the same derivation as in the one dimensional case, we get (3.23) X

i,j

|B1(P^∗_hRu−Ru,Q^∗_hRu−Ru, ϕh)i,j| ≤Ch^k+2kuk²_Hk+2(Ω)+kϕhk²_L²_(Ω) Similarly, we get

(3.24) X

i,j

|B2(Q^∗_hRu−Ru,P^∗_hRu−Ru, ψh)i,j| ≤Ch^k+2kuk²_Hk+2(Ω)+kψhk²_L2(Ω)

We can easily use the approximation properties of the projections P^∗_h, Q^∗_h to show that

(3.25) X

i,j

Z

CI

∂tϕ^eϕhdxdy≤Ch^2k+2kuk²_Hk+1(Ω)+kϕhk²_L²_(Ω)

(3.26) X

i,j

Z

D_{I+ 1}

2

∂tψ^eψhdxdy≤Ch^2k+2kuk²_Hk+1(Ω)+kψhk²_L²_(Ω) From (3.23)-(3.26) and (3.19) we have

(3.27) kϕhk²_L²_(Ω)+kψhk²_L²_(Ω)≤Ch^2k+2kuk²_Hk+2(Ω)

Together with the approximation properties of the projection (3.9) we have finished the proof of Theorem 3.1.

4. Numerical examples. In this section, we present numerical examples to verify our theoretical findings. In our numerical experiments, we measure the maximum errors at the zeros of the polynomialx^k+1−P^∗_h(x^k+1) in each cell, and theL¹, L^∞ andL² errors respectively. They are defined by

Esuper= max

j∈ZN,xij∈Gj

|(u−uh)(xij, T)|

(4.1)

E1= Z 2π

0

|(u−uh)(x, T)|dx (4.2)

E2= Z 2π

0

(u−uh)²(x, T)dx ¹2

(4.3)

E∞= max

x |(u−uh)(x, T)|

(4.4)

whereGj is a set of zeros of the polynomialx^k+1−P^∗_h(x^k+1) in cellIj.

Example 4.1. We consider the following equation with periodic boundary condi- tion:

ut+ux= 0, (x, t)∈[0,2π]×(0,2π) u(x,0) = sin(x)

u(0, t) =u(2π, t) (4.5)

The exact solution to this problem is

(4.6) u(x, t) = sin(x−t).

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The problem is solved by the central DG scheme (2.2) with k = 1,2,3, respectively. Uniform meshes are used in our experiments, which are constructed by equally dividing the interval, [0,2π], into N subintervals with N = 10,20,40,80,160. The seventh order strong-stability preserving Runge-Kutta method [8] with the time step

∆t = 0.05h,h= ^2π_N is used to reduce the time discretization error. Table 4.1 shows that the order of convergence of the error achieves the expected (k+ 1)-th order of accuracy. Also the superconvergence phenomenon is found. According to this numerical example, we find that the central DG scheme for the liner hyperbolic equation has (k+ 2)-th order superconvergence accuracy at the zeros of the special polynomial x^k+1−P^∗_h(x^k+1) in each element.

Table 4.1

Errors and the corresponding convergence rates. T= 1,τmax=_2k+1¹ h

k= 1

N E_super Rate E₁ Rate E₂ Rate E_∞ Rate

10 5.63E-03 − 1.20E-02 − 1.34E-02 − 1.85E-02 −

20 7.29E-04 2.95 2.85E-03 2.07 3.15E-03 2.09 4.43E-03 2.07 40 9.06E-05 3.01 6.85E-04 2.06 7.59E-04 2.06 1.07E-03 2.05 80 1.13E-05 3.00 1.67E-04 2.03 1.86E-04 2.03 2.63E-04 2.03 160 1.41E-06 3.00 4.14E-05 2.01 4.59E-05 2.01 6.49E-05 2.01

k= 2

10 1.12E-04 − 1.35E-04 − 1.52E-04 − 2.09E-04 −

k= 3

10 7.94E-06 − 1.77E-05 − 1.93E-05 − 2.73E-05 −

Example 4.2. We consider the following equation with periodic boundary condi- tion:

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

u(0, y, t) =u(2π, y, t) u(x,0, t) =u(x,2π, t) (4.7)

The exact solution to this problem is

(4.8) u(x, y, t) = 2 + sin(x+y−2t)

We test this example using Q^k polynomials with 0≤k ≤2 on a uniform mesh with N×N cells. Again the seventh order strong-stability preserving Runge-Kutta method [8] with time step ∆t= 0.05h,h=^2π_N is used to reduce the time discretization errors. The results in Table 4.2 show that the order of convergence of the error, ku−uhkL²(Ω), achieves the expected (k+ 1)-th order of accuracy. We have also observed the superconvergence phenomenon ofEsuper.

5. Concluding remarks. In this paper, optimal L² error estimates to central DG methods applied to linear conservation laws are proved. Our analysis is carried out both in one dimension and in multidimensions for uniform Cartesian meshes and