Central local discontinuous galerkin methods on overlapping cells for diffusion equations

DOI:10.1051/m2an/2011007 www.esaim-m2an.org

CENTRAL LOCAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS FOR DIFFUSION EQUATIONS

Yingjie Liu

, Chi-Wang Shu

, Eitan Tadmor

and Mengping Zhang

Abstract. In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made.

Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions for general polynomial degrees.

Mathematics Subject Classification. 65M60.

Received December 9, 2009. Revised January 6, 2011.

Published online June 13, 2011. .

1. Introduction

In this paper we present two versions of the central local discontinuous Galerkin (LDG) methods on overlap- ping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. We also compare the traditional LDG method on a single mesh and the two versions of the central LDG methods on overlapping cells in this context.

The LDG method is designed by suitably rewriting the diffusion partial differential equation (PDE) as a first-order system and then discretizing it by the DG method [3]. For a general review of DG methods we refer to [4]. In this paper we use overlapping cells and hence duplicative information, thereby avoiding numerical fluxes which is a distinct advantage of central schemes. This work is a continuation of our earlier work in [6,7]

in which we presented and analyzed central DG schemes for hyperbolic PDEs. We would like to point out that the discussion on the difficulties related to numerical fluxes for DG methods solving diffusion equations has

Keywords and phrases.Central discontinuous Galerkin method, local discontinuous Galerkin method, overlapping cells, diffusion equation, heat equation, stability, error estimate.

1 School of Mathematics, Georgia Institute of Technology, Atlanta, 30332-0160 GA, USA.yingjie@math.gatech.edu Research supported in part by NSF grant DMS-0810913.

2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.shu@dam.brown.edu Research supported in part by DOE grant DE-FG02-08ER25863 and NSF grant DMS-0809086.

3 Department of Mathematics, Institute for Physical Science and Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD 20742, USA.tadmor@cscamm.umd.edu Research supported in part by NSF grants DMS-1008397, FRG-0757227 and ONR grant N00014-09-10385.

4 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China.

mpzhang@ustc.edu.cn

Research supported in part by NSFC grant 11071234 and CAS grant KJCX1-YW-21.

Article published by EDP Sciences c EDP Sciences, SMAI 2011

attracted a lot of attention in the literature, for example we mention the recovery type DG method by van Leer and Nomura [8] (see also [9]), and the hybridizable discontinuous Galerkin (HDG) method of Cockburn et al.[5].

Our overall goal is to design (central) LDG methods which serve as effective solvers for convection diffusion equations, ranging from scalar equations of the form

u_t+f(u)_x= (c(u)u_x)_x, c(u)≥0, (1.1)

to general multidimensional systems of equations. The purely convective case was studied by us in [6,7]. The main focus of this paper is therefore to initiate the study of central LDG methods for the diffusive case, where we limit ourselves to a modest beginning with the linear heat equation

ut=uxx, (x, t)∈[a, b]×[0, T]. (1.2)

The formulation of the LDG and central LDG methods for the heat equation (1.2), can be extended in a straightforward manner to general nonlinear convection diffusion equations, see [3,6]. However, the stability and convergence analysis is more subtle: our stability analysis and error estimates are restricted to the heat equation with periodic boundary conditions and can be generalized to non-periodic boundary conditions, based on the finite element techniques. Stability analysis of the (central) LDG method for more general nonlinear diffusion is left for a future work.

The starting point of the LDG method is to rewrite the heat equation (1.2) as a first-order system

ut−rx= 0, r−ux= 0. (1.3)

Let{xj}be a partition of [a, b] withh_j+¹₂ =xj+1−xj and h= max_jh_j+¹₂. The mesh is regular, in the sense that max_jh_j+¹₂/min_jh_j+¹₂ is upper-bounded by a fixed constant during mesh refinements. Denote x_j+¹₂ =

12(x_j+1 +x_j), I_j = (x_j−¹

2, x_j+¹

2), and I_j+¹

2 = (x_j, x_j+1). V_h is the set of piecewise polynomials of degree k over the subintervals {I_j} with no continuity assumed across the subinterval boundaries. Likewise, W_h is the set of piecewise polynomials of degree k over the subintervals {I_j+¹

2} with no continuity assumed across the subinterval boundaries.

The traditional LDG method is defined by formally using the DG method in equations (1.3). The semi- discrete version is as follows. Find u_h(·, t)∈V_hand r_h(·, t)∈V_h, such that for anyϕ_h ∈V_h andψ_h∈V_h,

Ij

∂tuhϕhdx=−

Ij

rh∂xϕhdx+ ˆr_j+¹₂ϕh

x⁻_j+1 2

−ˆr_j−¹₂ϕh

x⁺_j−1 2

(1.4)

Ij

rhψhdx=−

Ij

uh∂xψhdx+ ˆu_j+¹₂ψh

x⁻_j+1 2

−uˆ_j−¹₂ψh

x⁺_j−1 2

. (1.5)

Here ϕh(x^±_j+1

2) refers to the right and left limits, respectively, of the possibly discontinuous function ϕh(x) at the cell interface x_j+¹

2. A good choice for the numerical fluxes (the hat terms), both in accuracy and in compactness of the eventual stencil after the auxiliary variabler_his eliminated, is

uˆ_j+¹₂ =uh

x⁺_j+1 2, t

, rˆ_j+¹₂ =rh

x⁻_j+1 2, t

(1.6) i.e. we alternatively take the left and right limits for the fluxes inuandr(we could of course also take the pair uh(x⁻_j+1

2, t) andrh(x⁺_j+1

2, t) as the fluxes).

The first version of central LDG method. On overlapping cells uses both the spacesV_h andW_h. The first semi-discrete version has the following form. Finduh(·, t), rh(·, t)∈Vh andvh(·, t), sh(·, t)∈Wh, such that

for anyϕ_h,ϕ¯_h∈V_h andψ_h,ψ¯_h∈W_h, we have

Ij

∂tuhϕhdx=−

Ij

sh∂xϕhdx+sh

x_j+¹₂, t ϕh

x⁻_j+1 2

−sh

x_j−¹₂, t ϕh

x⁺_j−1 2

(1.7)

Ij

r_hϕ¯_hdx=−

Ij

v_h∂_xϕ¯_hdx+v_h x_j+¹

2, t ϕ¯_h

x⁻_j+1 2

−v_h x_j−¹

2, t ϕ¯_h

x⁺_j−1 2

; (1.8)

I_{j+ 1}

2

∂_tv_hψ_hdx=−

I_{j+ 1}

2

r_h∂_xψ_hdx+r_h(x_j+1, t)ψ_h x⁻_j+1

−r_h(x_j, t) (1.9)

ψ_h(x⁺_j)

I_{j+ 1}

2

s_hψ¯_hdx=−

I_{j+ 1}

2

u_h∂_xψ¯_hdx+u_h(x_j+1, t) ¯ψ_h x⁻_j+1

−u_h(x_j, t) ¯ψ_h x⁺_j

. (1.10)

The second version of the central LDG method. Has a similar form to the first one: the only difference from (1.7)–(1.10) is the addition of an extra term involving the difference between the two duplicative solutions on different cells. This term serves as an additional numerical dissipation and we will see later that the addition of this term allows us to recover optimal convergence rate for the piecewise linear case, as well as providing smaller errors for non-smooth initial conditions. The semi-discrete version is as follows. Find uh(·, t), rh(·, t)∈Vh andvh(·, t), sh(·, t)∈Wh, such that for anyϕh,ϕ¯h∈Vh andψh,ψ¯h∈Wh, we have

Ij

∂_tu_hϕ_hdx= 1 τ_max

Ij

(v_h−u_h)ϕ_hdx−

Ij

s_h∂_xϕ_hdx +sh

x_j+¹₂, t ϕh

x⁻_j+1 2

−sh

x_j−¹₂, t ϕh

x⁺_j−1 2

(1.11)

Ij

rhϕ¯hdx=−

Ij

vh∂xϕ¯hdx +vh

x_j+¹₂, t ϕ¯h

x⁻_j+1 2

−vh

x_j−¹₂, t ϕ¯h

x⁺_j−1 2

; (1.12)

I_{j+ 1}

2

∂tvhψhdx= 1 τmax

I_{j+ 1}

2

(uh−vh)ψhdx−

I_{j+ 1}

2

rh∂xψhdx +r_h(x_j+1, t)ψ_h

x⁻_j+1

−r_h(x_j, t)ψ_h(x⁺_j) (1.13)

I_{j+ 1}

2

shψ¯hdx=−

I_{j+ 1}

2

uh∂xψ¯hdx +u_h(x_j+1, t) ¯ψ_h

x⁻_j+1

−u_h(x_j, t) ¯ψ_h(x⁺_j) (1.14)

whereτmax is an upper bound of the time step size due to the CFL restriction, that is,τmax=c h²with a given constant CFL numbercdictated by stability.

Both versions of the central LDG methods on overlapping cells do not need a numerical flux to define the interface values of the solution, since the evaluation of the solution at the interface is in the middle of the staggered mesh, hence in the continuous region of the solution. The initial condition is taken as the L² projection of the PDE initial condition into the relevant finite element space.

The organization of the paper is as follows. In Section 2 we analyze the stability and give an error estimate for the first version of the central LDG scheme (1.7)–(1.10) on overlapping cells. We will see that only a sub-optimal O(h^k) error estimate can be proved for piecewise polynomials of degreek. This sub-optimal convergence rate

is also verified by numerical experiments to be sharp for k = 1. In Section 3 we analyze the stability and give an error estimate for the second version of the central LDG scheme (1.11)–(1.14) on overlapping cells. We still can only get the sub-optimalO(h^k) error estimate using the finite element technique, however a Fourier analysis indicates the optimalO(h^k+1) convergence rate fork= 0 andk= 1 and this optimal rate is confirmed numerically together with the cases with higher k in Section 4. A comparison among the traditional LDG scheme and the two versions of central LDG schemes on overlapping cells is also given in Section 4. Concluding remarks are provided in Section 5.

2. Analysis of the first version of the central LDG scheme on overlapping cells

We study the L² stability of the LDG scheme on overlapping cells (1.7)–(1.10) for the equation (1.2) in Section 2.1. In Section 2.2 we provide an L² a priori error estimate for smooth solutions. In Section 2.3 we give a quantitative error estimate for this central LDG scheme for the polynomial degree up to 1 using Fourier analysis, similar to the technique used in [7,10,11]. As indicated before, we assume periodic boundary conditions.

2.1. L

stability

Theorem 2.1. The numerical solutionuh,rh,vh andshof the LDG scheme(1.7)–(1.10)for the equation(1.2) satisfies the following L² stability condition

1 2

d dt

_b

a

(u_h)²+ (v_h)² dx+

_b

a

r²_h+s²_h

dx= 0. (2.1)

Proof. Taking the test functionsϕh=uhand ¯ψh=sh in (1.7) and (1.10) respectively, summing up overj, and observing the periodic boundary condition, we have

1 2

d dt

_b

a u²_hdx+ _b

a (s_h)²dx

= −

j xj

x_j−1 2

∂_x(u_hs_h)dx+ _x

j+ 12

xj

∂_x(u_hs_h)dx−s_h x_j+¹

2, t u_h

x⁻_j+1 2, t +s_h

x_j−¹₂, t u_h

x⁺_j−1 2, t

−u_h(x_j+1, t)s_h x⁻_j+1, t

+u_h(x_j, t)s_h x⁺_j , t

= 0.

Similarly, taking the test functions ¯ϕh=rh andψh=vhin (1.8) and (1.9) respectively, we have 1

2 d dt

_b

a v²_hdx+ _b

a r²_hdx= 0.

Summing up the two equalities proves the theorem.

Remark 2.1. The proof of Theorem 2.1 is similar to the proof of the cell entropy inequality for the traditional LDG method in [3]. However, we cannot prove a similar L² stability result for the central LDG scheme on overlapping cells when applied to the nonlinear diffusion equation (1.1), even though the scheme itself can be easily defined for such nonlinear problems. The proof of Theorem 2.1 can be easily generalized to multi- dimensional central LDG schemes on overlapping cells for the linear diffusion equation.

2.2. L

a priori error estimate

In this subsection we use the standard DG techniques [3] to obtain ana prioriL²error estimates for the first version of the central LDG scheme on overlapping cells given in (1.7)–(1.10).

Theorem 2.2. The numerical solutionu_h,r_h,v_h ands_hof the central LDG scheme on overlapping cells(1.7)–

(1.10) for the equation (1.2) with a smooth initial condition u(·,0) ∈ H^k+2 satisfies the following L² error estimate

u−u_h²+u−v_h²+|||u_x−r_h|||²+|||u_x−s_h|||²≤Ch^2k (2.2) where uis the exact solution of (1.2), k is the polynomial degree in the finite element spaces V_h andW_h, and the constantCdepends on the(k+ 2)-th order Sobolev norm of the initial condition ||u(·,0)||_H^k+2 as well as on the final timet, but is independent of the mesh sizeh. The unmarked norm · here and below is the standard L² norm. The norm ||| · |||is defined as

|||r(·, t)|||²= _t

0 r(·, τ)²dτ.

Proof. Let us first introduce the shorthand notation Bj(uh, sh;ϕh,ψ¯h)

=

Ij

∂_tu_hϕ_hdx+

Ij

s_h∂_xϕ_hdx−s_h x_j+¹

2, t ϕ_h

x⁻_j+1 2

+s_h

x_j−¹

2, t ϕ_h

x⁺_j−1 2

(2.3) +

I_{j+ 1}

2

s_hψ¯_hdx+

I_{j+ 1}

2

u_h∂_xψ¯_hdx−u_h(x_j+1, t) ¯ψ_h x⁻_j+1

+u_h(x_j, t) ¯ψ_h x⁺_j

.

By the definition of the scheme, we have:

Bj(uh, sh;ϕh,ψ¯h) = 0 (2.4)

for all j and allϕ_h ∈V_h and ¯ψ_h∈W_h. It can also be easily verified that the exact solutionsuandr=u_x of the PDE (1.2) satisfy

B_j(u, r;ϕ_h,ψ¯_h) = 0 (2.5)

for allj and allϕh∈Vh and ¯ψh∈Wh. This is usually referred to as the consistency of the scheme. Subtract- ing (2.4) from (2.5), we obtain the error equation which represents the Galerkin orthogonality

Bj(u−uh, r−sh;ϕh,ψ¯h) = 0 (2.6) for allj and allϕh∈Vh and ¯ψh∈Wh.

We now defineP andQas the standardL² projection intoVh andWh respectively. That is, for eachj,

Ij

(P w(x)−w(x))ϕh(x)dx= 0 ∀ϕh∈P^k(Ij) (2.7)

and

I_{j+ 1}

2

(Qw(x)−w(x))ψ_h(x)dx= 0 ∀ψ_h∈P^k(I_j+¹

2) (2.8)

where P^k(I_j) andP^k(I_j+¹

2) denote the spaces of polynomials of the degree up tok in the cell I_j and the cell I_j+¹₂ respectively. Standard approximation theory [2] implies, for a smooth functionw,

(P w(x)−w(x))+h^1/2(P w(x)−w(x))Γj ≤Ch^k+1 (2.9)

and

(Qw(x)−w(x))+h^1/2(Qw(x)−w(x))Γ_{j+ 1}

2 ≤Ch^k+1 (2.10)

where Γ_jand Γ_j+1

2 denote the set of boundary points of all elementsIjandI_j+1/2respectively, and the positive constantC, here and below, solely depending onw(x) and its firstk+ 2 derivatives, is independent ofh.

We also recall that [2], for anywh ∈Vh or wh ∈Wh, there exists a positive constant C independent of wh

andh, such that

∂xwh ≤Ch⁻¹wh; whΓ≤Ch^−1/2wh (2.11) where Γ = Γ_j or Γ_j+1

2. We now take:

ϕh=P u−uh, ψ¯h=Qr−sh (2.12)

in the error equation (2.6), and denote

ϕ^e=P u−u, ψ^e=Qr−r (2.13)

to obtain

Bj(ϕh,ψ¯h;ϕh,ψ¯h) =Bj(ϕ^e, ψ^e;ϕh,ψ¯h). (2.14) For the left-hand side of (2.14), we use a similar proof as that for Theorem 2.1 to conclude

j

B_j(ϕ_h,ψ¯_h;ϕ_h,ψ¯_h) = 1 2

d dt

_b

a (ϕ_h)²dx+ _b

a ( ¯ψ_h)²dx. (2.15) We then write the right-hand side of (2.14) as a sum of three terms

B_j(ϕ^e, ψ^e;ϕ_h,ψ¯_h) =B_j¹+B_j²+B_j³ (2.16) where

B¹_j =

Ij

∂tϕ^eϕhdx+

I_{j+ 1}

2

ψ^eψ¯hdx

B²_j =

Ij

ψ^e∂xϕhdx+

I_{j+ 1}

2

ϕ^e∂xψ¯hdx B_j³=−ψ^e

x_j+¹

2, t ϕ_h

x⁻_j+1 2, t

+ψ^e x_j−¹

2, t ϕ_h

x⁺_j−1 2, t

−ϕ^e(x_j+1, t) ¯ψ_h x⁻_j+1, t

+ϕ^e(x_j, t) ¯ψ_h x⁺_j, t and we will estimate each term separately.

By the definition of theL²projection (2.7)–(2.8), we have

j

B_j¹= 0. (2.17)

By using the simple inequality

αβ≤ 1 2

α²+β²

, (2.18)

theL²projection error property (2.9)–(2.10) forϕ^eandψ^e, and the first inequality in (2.11) forϕhand ¯ψh, we have:

j

|B_j²| ≤ 1 4

_b

a (ϕh)²dx+ _b

a ( ¯ψh)²dx

+Ch^2k. (2.19)

Finally, by using the simple inequality (2.18), the L² projection error property (2.9)–(2.10) for ϕ^e and ψ^e, and the second inequality in (2.11) forϕhand ¯ψh, we have:

j

|B_j³| ≤ 1 4

_b

a (ϕh)²dx+ _b

a ( ¯ψh)²dx

+Ch^2k. (2.20)

Summing up (2.17), (2.19) and (2.20) and combining with (2.15), we obtain from (2.14) d

dt _b

a (ϕ_h)²dx+ _b

a ( ¯ψ_h)²dx≤C _b

a (ϕ_h)²dx+Ch^2k. or

d dt

_b

a (P u−uh)²dx+ _b

a (Qr−sh)²dx≤C _b

a (P u−uh)²dx+Ch^2k. (2.21) Similarly we have:

d dt

_b

a (Qu−v_h)²dx+ _b

a (P r−r_h)²dx≤C _b

a (Qu−v_h)²dx+Ch^2k. (2.22) This, together with an application of the Gronwall’s inequality [1] and the approximation result (2.9), (2.10) for the projection errors and the errors of the initial condition, implies the desired error estimate (2.2).

Remark 2.2. The error estimate of Theorem 2.2 is sub-optimal. The loss of accuracy mainly comes from the estimate onB_j². Because it involves the products of piecewise polynomials on different meshes, the standard trick of introducing a projection to make it zero does not apply. The boundary termB_j³can be brought to zero by special projections, but this will not help in improving the global error estimate. We will see from numerical experiments later that this suboptimal error estimate is actually sharp for at leastk= 1.

Remark 2.3. The error estimate of Theorem 2.2 can be easily generalized to multi-dimensional scalar linear diffusion equations.

2.3. A quantitative error estimate via Fourier analysis

In this subsection we perform a Fourier analysis for the semi-discrete central LDG scheme on overlapping cells (1.7)–(1.10) solving the heat equation (1.2) with uniform meshes for piecewise constant and piecewise linear elements, using the techniques in [10,11], to obtain quantitative error estimates. We also use numerical experiments to verify these error estimates.

For the simplest piecewise constantk= 0 case, we obtain the finite difference schemes corresponding to the central LDG scheme on overlapping cells (1.7)–(1.10) as follows:

u_j = 1 h

s_j+¹

2 −s_j−¹

2

s_j+¹

2 = 1

h(u_j+1−u_j) (2.23)

for j = 0, . . . , N −1. Here u denotes the time derivatives of u, and wα denotes the value of the numerical solutionwat the pointxα, withw=uorsandα=jor j+¹₂. Eliminatings, we get:

u_j = 1

h²(uj+1−2uj+uj−1). (2.24)

This is apparently the standard second order centered finite difference scheme for solving the heat equation (1.2).

Nevertheless, in order to demonstrate our procedure we still perform the following standard Fourier analysis.

Note that this analysis depends heavily on the assumption of uniform mesh sizes and periodic boundary condi- tions. We make an ansatz of the form

uj(t) = ˆum(t)e^imxj (2.25)

and substitute this into the scheme (2.24) to find the evolution equation for the coefficient as uˆ_m(t) = 2

h²(cos(mh)−1)ˆum(t). (2.26)

The general solution of the ODE (2.26) is given by

uˆ_m(t) =a₀e^h22(cos(mh)−1)t. (2.27) For accuracy we look at the low modes, in particular atm= 1. To fit the given initial condition

uj(0) = e^ixj (2.28)

whose imaginary part is the initial condition

uj(0) = sin(xj) (2.29)

we require, att= 0,

uˆ₁(0) = 1, hence we obtain the coefficients a0 in (2.27) as

a₀= 1. (2.30)

We remark that the usual way of taking initial conditions in a finite element method isviaanL²projection, not by a point value collocation (2.29), however this does not affect the final results in the analysis in this paper.

We thus have the explicit solutions of the scheme (2.24) with the initial condition (2.28), for example

u_j(t) =a₀e^ixj+h22(cos(mh)−1)t (2.31) withm= 1 and the coefficienta₀given by (2.30). By a simple Taylor expansion, we obtain the imaginary part ofu_j(t) to be

Im{uj(t)}= e^−tsin(xj) + 1

12h²te^−tsin(xj) +O(h³). (2.32) This is clearly consistent, and it is second order accurate at the cell center xj. This second order accuracy is however a superconvergence phenomenon for uniform meshes at the cell center. We can do a similar analysis for Im{v_j+¹₂(t)}.

We now compute thek= 0 central LDG solutions on overlapping cells (1.7)–(1.10) withu(x,0) = sin(x) as the initial condition and with periodic boundary conditions, up tot= 4π, to verify the quantitative comparison above. We use forward Euler time discretization and take a small time step Δt= 0.01h²to reduce the effect from the time discretization. In order to be consistent with the error analysis above, the errors are computed for u_h at the pointsx_j. The discrete L² and L^∞ errors (measuring the errors at cell centers x_j only, namely 1

N

_N

j=1|u(xj, t)−uj(t)|²for theL²error and max_j|u(xj, t)−uj(t)|for theL^∞error) and order of accuracy of the central LDG method on overlapping cells are listed in Table 1. We also list the predicted errors by the analysis, namely the leading terms in the Taylor expansions in (2.32) in the table. We can see that the predicted errors and the actual errors are very close, validating the quantitative analysis in (2.32). The error at cell centers is apparently second order accurate. When measuring as finite element solutions (using 40 uniformly spaced sampling points per cell to compute the L² error

1 2π

_2π

0 |u(x, t)−u_h(x, t)|²dx and the

Table 1. DiscreteL² andL^∞errors foru_h, measured at the center of the cells, and orders of accuracy of the central LDG method (1.7)–(1.10) fork= 0.

Numerical solutions Predicted by analysis h L² error Order L^∞ error Order L² error Order L^∞error Order

2π/20 2.68E-07 – 3.78E-07 – 2.55E-07 – 3.60E-07 –

2π/40 6.45E-08 2.05 9.12E-08 2.05 6.37E-08 2.00 9.01E-01 2.00 2π/80 1.60E-08 2.01 2.26E-08 2.01 1.59E-08 2.00 2.25E-08 2.00 2π/160 3.99E-09 2.00 5.64E-09 2.00 3.98E-09 2.00 5.63E-09 2.00 2π/320 9.96E-10 2.00 1.41E-09 2.00 9.96E-10 2.00 1.41E-09 2.00 2π/640 2.49E-10 2.00 3.52E-10 2.00 2.49E-10 2.00 3.52E-10 2.00 Table 2. L² andL^∞ errors, measured as finite element solutions, and orders of accuracy of the central LDG method (1.7)–(1.10) fork= 0.

uh

h L² error Order L^∞error Order 2π/20 2.01E-06 – 6.89E-07 – 2π/40 7.08E-07 1.51 2.92E-07 1.24 2π/80 3.17E-07 1.16 1.39E-07 1.07 2π/160 1.54E-07 1.04 6.88E-08 1.02 2π/320 7.67E-08 1.01 3.43E-08 1.00 2π/640 3.83E-08 1.00 1.71E-08 1.00

L^∞ error max_x|u(x, t)−uh(x, t)|), the L² and L^∞ errors and order of accuracy of the central LDG scheme (1.7)–(1.10) are listed in Table2. Apparently, these errors are first order.

We now repeat this analysis for the piecewise linear k = 1 case. The solution inside the cell Ij or I_j+¹₂ is then represented by

u_h(x, t) =u_j−¹

4(t)ϕ¹_h(ξ) +u_j+¹

4(t)ϕ²_h(ξ) and

sh(x, t) =s_j+¹₄(t) ¯ψ¹_h(ξ) +s_j+³₄ψ¯²_h(ξ)

where ϕ¹_h(ξ) = −ξ+ ¹₂, ϕ²_h(ξ) = ξ+¹₂, ¯ψ¹_h(ξ) = −ξ+ ³₂ and ¯ψ_h²(ξ) = ξ−¹₂, with ξ = ^2(x−x_h ^j). With this representation, taking the test functionsϕh and ¯ψh asϕ¹_h,ϕ²_h and ¯ψ_h¹, ¯ψ²_hrespectively, eliminating the variable s, and inverting the small 2×2 mass matrix by hand, we obtain easily the finite difference scheme corresponding to the central LDG scheme (1.7) and (1.10) as

u_j−1

u_j+41 4

=A

u_j−⁵₄ u_j−³₄

+B

u_j−¹₄ u_j+¹₄

+C

u_j+³₄ u_j+⁵₄

. (2.33)

with

A = 1 h²

₁₃

8 −5

−5 8 8 13

8

, B = 1

h² ₋₁₃

4 5

5 4 4 −13

4

, C = 1

h² ₁₃

8 −5

−5 8 8 13

8

(2.34) forj= 0, . . . , N−1. Hereu denotes the time derivatives ofu. We again perform the following standard Fourier analysis for the finite difference scheme (2.33). We make an ansatz of the form

u_j−¹₄(t) u_j+¹₄(t)

=

uˆ_m,−¹₄(t) uˆ_m,¹₄(t)

e^imxj (2.35)

and substitute this into the scheme (2.33) to find the evolution equation for the coefficient vector as uˆ_m,−1

4(t) uˆ_m,+1

4(t)

=G(m, h)

uˆ_m,−¹₄(t) uˆ_m,+¹₄(t)

(2.36)

where the amplification matrixG(m, h) is given by

G(m, h) = Ae^−imh+ B +Ce^imh (2.37)

with the matricesA,B andC defined by (2.34). The eigenvalues of the amplification matrixG(m, h) are λ1=−2 + e^−imh+ e^imh, λ2=−9

4e^−imh(−1 + e^imh)². (2.38) The general solution of the ODE (2.33) is given by

uˆ_m,−¹₄(t) uˆ_m,+¹

4(t)

= a1e^λ1tV1 +a2e^λ2tV2, (2.39)

where the eigenvaluesλ1andλ2are given by (2.38), andV1andV2are the corresponding eigenvectors given by V₁ =

1 1

, V₂ = −1

1

. (2.40)

For accuracy we look at the low modes, in particular atm= 1. To fit the given initial condition u_j−¹

4(0) = e^ixj−14, u_j+¹

4(0) = e^{ixj+ 14} (2.41)

whose imaginary part is our initial condition for (1.2), we require, att= 0, uˆ_m,−¹₄(0)

uˆ_m,+¹

4(0)

=

e^i−h4 e^ih4

.

This gives us the coefficientsa1anda2 in the solution (2.39). We thus have the explicit solution of the scheme (2.33), (2.34) with the initial condition (2.41), for example

u_j−¹

4 =a₁e^{ixj+λ1t−ih4}V₁+a₂e^{ixj+λ2t−ih4}V₂ (2.42) with the eigenvalues λ1 and λ2 given by (2.38) and the eigenvectors V1 and V2 given by (2.40) with m = 1, and the coefficientsa1 anda2obtained by fitting the initial condition. Through a simple Taylor expansion, we obtain the imaginary part ofu_j−¹₄(t) to be

Im{u_j−¹₄(t)}= e^−tsin(x_j−¹₄) +h 4e^−t

e^−5t4 + 1

cos(x_j−¹₄) +O(h²). (2.43) The result for Im{u_j+¹₄(t)} is similar. This analysis indicates a sub-optimal convergence rate of first order for piecewise linear polynomials.

Table 3. Discrete L² and L^∞ errors for u_h, measured at the points x_j−¹

4, and orders of accuracy of the central LDG method (1.7)–(1.10) fork= 1.

Numerical solutions Predicted by analysis h L² error Order L^∞ error Order L² error Order L^∞error Order

2π/20 3.30E-07 – 4.66E-07 – 1.94E-07 – 2.73E-07 –

2π/40 1.16E-07 1.50 1.64E-07 1.50 9.68E-08 1.00 1.37E-07 1.00 2π/80 5.10E-08 1.19 7.21E-08 1.19 4.48E-08 1.00 6.84E-08 1.00 2π/160 2.45E-08 1.06 3.47E-08 1.06 2.42E-08 1.00 3.42E-08 1.00 2π/320 1.22E-08 1.01 1.72E-08 1.01 1.21E-08 1.00 1.71E-08 1.00 2π/640 6.06E-09 1.00 8.57E-09 1.00 6.05E-09 1.00 8.56E-09 1.00

Table 4. L² andL^∞ errors, measured as finite element solutions, and orders of accuracy of the central LDG method (1.7)–(1.10) fork= 1.

uh

h L² error Order L^∞error Order 2π/20 1.97E-06 – 6.82E-07 – 2π/40 7.02E-07 1.49 2.91E-07 1.23 2π/80 3.16E-07 1.15 1.39E-07 1.07 2π/160 1.54E-07 1.04 6.87E-08 1.02 2π/320 7.67E-08 1.01 3.43E-08 1.00 2π/640 3.83E-08 1.00 1.71E-08 1.00

In principle this analysis can be performed for higher order polynomials in the central LDG scheme on overlapping cells (1.7)–(1.10), however the algebra becomes prohibitively complicated.

We now compute the central LDG scheme (1.7)–(1.10) fork= 1 on overlapping cells to (1.2) withu(x,0) = sin(x) as the initial condition and with periodic boundary conditions, up tot= 4π, to verify the quantitative analysis above. We use a second order Runge-Kutta method and take a small time step Δt= 0.01h²to reduce the effect from the time discretization. In order to be consistent with the error analysis above, the errors are computed foru_h at the pointsx_j−¹

4. The discreteL² andL^∞errors and order of accuracy of the central LDG method (1.7)–(1.10) on overlapping cells are listed in Table3. We also list the predicted errors by the analysis, namely the leading terms in the Taylor expansions in (2.43) in the tables. We can see that the predicted errors and the actual errors are very close, validating our quantitative analysis in (2.43). It is only first order accurate for the LDG method on overlapping cells. When measuring as finite element solutions (using 40 uniformly spaced sampling points per cell), the L² and L^∞ errors and order of accuracy of the central LDG method (1.7)–(1.10) are listed in Table4. Again, the errors are only first order.

According to the analysis and numerical experiments in this section and in Section 4, it appears that the central LDG method (1.7)–(1.10) on overlapping cells is suboptimal of orderkfork= 1 but it seems to achieve the optimal (k+ 1)-th order fork >1.

3. Analysis of the second version of central LDG scheme on overlapping cells

In this section, we perform a similar analysis for the second version of the central LDG scheme on overlapping cells, given by (1.11)–(1.14), for solving the heat equation (1.2). As before, we study the L² stability of the central LDG scheme (1.11)–(1.14) in Section 3.1. In Section 3.2 we provide a L² a priori error estimate for

smooth solutions. In Section 3.3 we give a quantitative error estimate for this scheme for polynomial degree up to 1 using Fourier analysis, and provide numerical evidence to verify this analysis.

3.1. L

stability

Theorem 3.1. The numerical solution u_h, r_h,v_h and s_h of the central LDG scheme (1.11)–(1.14) for equa- tion (1.2) satisfies the following L² stability condition

1 2

d dt

_b

a

(u_h)²+ (v_h)² dx+

_b

a

r²_h+s²_h

dx≤0. (3.1)

Proof. Taking the test functionsϕh=uh,ψh=vh, ¯ϕh=rhand ¯ψh=shin (1.11)–(1.14) respectively, summing up overj, and observing the periodic boundary condition, we have

1 2

d dt

_b

a u²_hdx+1 2

d dt

_b

a v_h²dx+ _b

a r²_hdx+ _b

a s²_hdx

=

j

1 τ_max

⎛

⎝ _x

j+ 12

x_j−1 2

(vh−uh)uhdx+ _xj+1

xj

(uh−vh)vhdx

⎞

⎠

−

⎛

⎝ _x

j+ 12

x_j−1 2

∂x(uh)shdx+ _xj+1

xj

∂x(sh)uhdx

⎞

⎠−

⎛

⎝ _x

j+ 12

x_j−1 2

∂x(rh)vhdx+ _xj+1

xj

∂x(vh)rhdx

⎞

⎠ +sh

x_j+¹₂, t uh

x⁻_j+1 2, t

−sh

x_j−¹₂, t uh

x⁺_j−1 2, t

+rh(xj+1, t)vh x⁻_j+1, t

−rh(xj, t)vh x⁺_j, t

+vh

x_j+¹₂, t rh

x⁻_j+1 2, t

−vh

x_j−¹₂, t rh

x⁺_j−1 2, t + uh(xj+1, t)sh

x⁻_j+1, t

−uh(xj, t)sh x⁺_j, t

=− 1 τmax

_b

a (u_h−v_h)²dx

≤0.

Remark 3.1. Comparing with theL² stability of the first version of the central LDG scheme, we notice the additional numerical dissipation _b

a(uh−vh)²dx, which is related to the difference of the two duplicative sets of information in the computational domain. The second version of the central LDG scheme couples these two sets of duplicative solutions through the extra numerical diffusion term.

3.2. L

a priori error estimate

In this subsection we obtain ana prioriL²error estimate for the central LDG scheme (1.11)–(1.14).

Theorem 3.2. The numerical solution uh, rh, vh and sh of the central LDG scheme (1.11)–(1.14) for the equation (1.2)with a smooth initial conditionu(·,0)∈H^k+2 satisfies the following L² error estimate

u−u_h²+u−v_h²+|||u_x−r_h|||²+|||u_x−s_h|||²≤Ch^2k (3.2) whereu, the constantC, and the norms are the same as those in Theorem2.2.