<span class="mathjax-formula">$L^2$</span> stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods

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DOI:10.1051/m2an:2008018 www.esaim-m2an.org

L² STABILITY ANALYSIS OF THE CENTRAL DISCONTINUOUS GALERKIN METHOD AND A COMPARISON BETWEEN THE CENTRAL AND REGULAR

DISCONTINUOUS GALERKIN METHODS

Yingjie Liu

, Chi-Wang Shu

, Eitan Tadmor

and Mengping Zhang

Abstract. We prove stability and derive error estimates for the recently introduced central discon- tinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central discontinuous Galerkin method and the regular discon- tinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis.

Mathematics Subject Classification. 65M60.

Received April 17, 2007.

Published online May 27, 2008.

1. Introduction

We continue our study of the recently introduced central discontinuous Galerkin (DG) method [8]. In this paper we prove stability and derive error estimates of central DG approximation for linear hyperbolic equations.

We also compare the central DG method with the regular DG method in this context.

The central DG method is based on two existing techniques: the central scheme framework (e.g.[7,9]) and the DG framework (e.g.[2,4]). It uses overlapping cells and hence duplicative information, but avoids numerical fluxes (Riemann solvers) which is a distinct advantage of central schemes. The central DG scheme also avoids the excessive numerical dissipation for small time steps, common to some of the earlier central schemes, by a suitable choice of the numerical dissipation. Being a variant of the DG method, it shares many of its advantages,

Keywords and phrases. Central discontinuous Galerkin method, discontinuous Galerkin method, linear hyperbolic equation, stability, error estimate.

1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA.yingjie@math.gatech.edu Research supported in part by NSF grant DMS-0511815.

2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.shu@dam.brown.edu

Research supported in part by NSFC grant 10671190 while he was visiting the Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China. Additional support was provided by ARO grant W911NF-04-1-0291 and NSF grant DMS-0510345.

3 Department of Mathematics, Institute for Physical Science and Technology and Center of Scientific Computation and Math- ematical Modeling (CSCAMM), University of Maryland, College Park, MD 20742, USA.tadmor@cscamm.umd.edu

Research supported in part by NSF grant 07-07949 and ONR grant N00014-91-J-1076.

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 10671190.

Article published by EDP Sciences c EDP Sciences, SMAI 2008

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such as compact stencil, easy parallel implementation, etc. The central DG method performs well in numerical simulations of linear and nonlinear scalar and systems of conservation laws [8].

We proceed with a description of the regular DG method and the central DG method. For simplicity, we consider a scalar one dimensional conservation law

ut+f(u)x= 0, (x, t)∈[a, b]×[0, T] (1.1) with periodic or compactly supported boundary conditions. The DG and central DG methods can be defined for nonlinear, multi-dimensional and system cases, and with other boundary conditions as well, see [4,8].

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

2 = 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 degreekover the subintervals{I_j}with no continuity assumed across the subinterval boundaries. Likewise,W_h is the set of piecewise polynomials of degreek over the subintervals{I_j+¹

2} with no continuity assumed across the subinterval boundaries.

The regular DG method is defined using the spaceV_h only. The semi-discrete version is as follows. Find u_h(·, t)∈V_h such that for anyϕ_h∈V_h and for allj,

Ij

∂tuhϕhdx =

Ij

f(uh)∂xϕhdx−fˆ uh

x⁻_j+1 2, t

, uh

x⁺_j+1 2, t

ϕh

x⁻_j+1 2

+ ˆf

uh

x⁻_j−1 2, t

, uh

x⁺_j−1 2, t

ϕh

x⁺_j−1 2

(1.2) where ˆf(u⁻, u⁺) is a monotone numerical flux, namely it is increasing in the first argument and decreasing in the second argument, or symbolically ˆf(↑,↓); it is consistent with the physical flux ˆf(u, u) =f(u); and it is at least Lipschitz continuous with both arguments. For monotone fluxes suitable for the DG method, see, e.g.[2]. For systems the monotone numerical flux is replaced by a numerical flux obtained through an exact or approximate Riemann solver, see,e.g.[10].

The central DG method is defined on overlapping cells and uses both spaces Vh and Wh. We start with the description of the method for the fully discrete version, with forward Euler time discretization from tn to tn+1=tn+τ(here the time stepτcould change withn, but for simplicity of notations we will use a constantτ).

The scheme is defined by the following procedure: Finduⁿ⁺¹_h ∈Vh andvⁿ⁺¹_h ∈Wh, such that for anyϕh∈Vh

andψh∈Wh,

Ij

uⁿ⁺¹_h ϕhdx=θ

Ij

v_hⁿϕhdx+ (1−θ)

Ij

uⁿ_hϕhdx

+τ

Ij

f(v_hⁿ)∂_xϕ_hdx−f v_hⁿ

x_j+¹

2

ϕ_h x⁻_j+1

2

+f

vⁿ_h x_j−¹

2

ϕ_h x⁺_j−1

2

(1.3)

I_{j+ 1}

2

v_hⁿ⁺¹ψ_hdx=θ

I_{j+ 1}

2

uⁿ_hψ_hdx+ (1−θ)

I_{j+ 1}

2

vⁿ_hψ_hdx

+τ

⎛

⎝

I_{j+ 1}

2

f(uⁿ_h)∂xψhdx−f(uⁿ_h(xj+1))ψh(x⁻_j+1) +f(uⁿ_h(xj))ψh(x⁺_j)

⎞

⎠ (1.4)

whereθ= _τmax^τ andτmaxis an upper bound for the time step size due to the CFL restriction, that is,τmax=c h with a given constant CFL numbercdictated by stability. Computationally, the scheme (1.3)–(1.4) is used with

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the forward Euler replaced by a Runge-Kutta method of suitable temporal accuracy,e.g. the SSP Runge-Kutta methods in [5,11]. The time stepτ can be chosen arbitrarily subject to the stability restrictionτ ≤τmax, hence 0≤θ≤1. Ifθ= 1, the scheme is similar in spirit to the original central scheme [9].

We now take the limitτ →0 to obtain the semi-discrete version of the central DG scheme: Finduh(·, t)∈Vh

andvh(·, t)∈Wh, such that for any ϕh∈Vh andψh∈Wh,

Ij

∂_tu_hϕ_hdx = 1 τmax

Ij

(v_h−u_h)ϕ_hdx+

Ij

f(v_h)∂_xϕ_hdx

−f v_h

x_j+¹₂, t ϕ_h

x⁻_j+1 2

+f

v_h

x_j−¹₂, t ϕ_h

x⁺_j−1 2

(1.5)

I_{j+ 1}

2

∂_tv_hψ_hdx = 1 τmax

I_{j+ 1}

2

(u_h−v_h)ψ_hdx+

I_{j+ 1}

2

f(u_h)∂_xψ_hdx

−f(uh(xj+1, t))ψh(x⁻_j+1) +f(uh(xj, t))ψh(x⁺_j). (1.6) Notice that, unlike the regular DG scheme (1.2), the central DG scheme (1.5)–(1.6) does 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 first term on the right side of (1.5) or (1.6) is a numerical dissipation term. This will become clear when we discuss the stability of the scheme in Section 2.1. In all the DG schemes (1.2), (1.3)–(1.4) and (1.5)–(1.6), the initial condition is taken as theL² projection of the PDE initial condition into the relevant finite element space.

The organization of the paper is as follows. In Section 2, we first analyze the L² stability and give an a priori error estimate for the semi-discrete central DG scheme (1.5)–(1.6) for the linear hyperbolic equation, using similar techniques of stability and error analysis for standard DG methods [3,6]. We then perform a Fourier analysis for the semi-discrete central DG scheme (1.5)–(1.6) for the linear hyperbolic equation with uniform meshes for piecewise constant and linear elements, using the techniques in [13]. This analysis is more explicit and allows us to compare the errors quantitatively between the central and standard DG schemes, which is performed in Section 3. In Section 3 we also perform numerical experiments to verify such quantitative conclusions. We give a few concluding remarks in Section 4.

2. Analysis of the central DG scheme

For the purpose of analysis we will consider the linear hyperbolic equation, namely (1.1) withf(u) =aufor a constanta. Our analysis can be easily generalized to multi-dimensional linear equations and hyperbolic linear systems. In particular, these results apply to hyperbolic equations with propagating discontinuities.

Without loss of generality we consider the following linear hyperbolic equation

ut+ux= 0, (x, t)∈[a, b]×[0, T]. (2.1) We study theL²stability of the central DG scheme (1.5)–(1.6) for the equation (2.1) in Section 2.1, and compare the result with that for the regular DG scheme in [6]. In Section 2.2 we provide anL² a priorierror estimate for smooth solutions, and compare the result with that for the regular DG scheme in [3]. In Section 2.3 we give an quantitative error estimate for the central DG scheme for polynomial degree up to 1 using Fourier analysis, similar to the technique used in [12,13].

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2.1. L² stability

Theorem 2.1. The numerical solutionuh andvh of the central DG scheme (1.5)–(1.6)for the equation (2.1) satisfies the followingL² stability condition

1 2

d dt

_b

a ((uh)²+ (vh)²)dx=− 1 τmax

_b

a (uh−vh)²dx≤0. (2.2) Proof. Taking the test functions ϕ_h = u_h and ψ_h = v_h in (1.5) and (1.6) respectively, summing up over j, observingf(u) =uand the periodic (or compactly supported) boundary condition, we have

1 2

d dt

_b

a ((uh)²+(vh)²)dx= 1 τ_max

_b

a

vhuh−(uh)²+uhvh−(vh)²

dx+

j Ij

vh∂xuhdx+

I_{j+ 1}

2

uh∂xvhdx

−vh

x_j+¹₂, t uh

x⁻_j+1 2, t

+vh

x_j−¹₂, t uh

x⁺_j−1 2, t

−uh(xj+1, t)vh x⁻_j+1, t

+uh(xj, t)vh x⁻_j, t

=− 1 τmax

_b

a (u_h−v_h)²dx+

j xj

x_j−1 2

∂_x(u_hv_h)dx+ _x

j+ 12

xj

∂_x(u_hv_h)dx

−v_h x_j+¹

2, t u_h

x⁻_j+1 2, t

+v_h x_j−¹

2, t u_h

x⁺_j−1 2, t

−u_h(x_j+1, t)v_h x⁻_j+1, t

+u_h(x_j, t)v_h x⁻_j, t

=− 1 τmax

_b

a (uh−vh)²dx ≤ 0.

Remark 2.1. The proof of Theorem 2.1 is similar to the proof of the cell entropy inequality for the regular DG method in [6]. However, we cannot prove a similarL² stability result for the central DG scheme when applied to the nonlinear scalar conservation law (1.1), even though the proof of Theorem 2.1 can be easily generalized to multi-dimensional central DG schemes for linear equations. This is in contrary to the cell entropy inequality for regular DG schemes, which holds for arbitrary nonlinear scalar conservation laws [6].

Remark 2.2. Theorem 2.1 indicates that the energy dissipation term is _τ¹

max

_b

a(u_h−v_h)²dx, that is, it is directly related to the difference of the two duplicative representationsuhandvhof the solution in overlapping cells. In contrast, for the regular DG method, the energy dissipation term is directly related to the jumps of the numerical solution at cell interfaces.

2.2. L² a priori error estimate

In this subsection we use the standard DG techniques [3] to obtain an a priori L² error estimate for the central DG scheme.

Theorem 2.2. The numerical solutionu_h andv_h of the central DG scheme (1.5)–(1.6)for the equation (2.1) with a smooth initial condition u(·,0)∈H^k+1 satisfies the followingL² error estimate

u−uh²+u−vh²≤Ch^2k (2.3)

where uis the exact solution of (2.1),k is the polynomial degree in the finite element spaces V_h and W_h, and the constantCdepends on the(k+ 1)-th order Sobolev norm of the initial condition ||u(·,0)||_H^k+1 as well as on the final timet, but is independent of the mesh sizeh.

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Proof. Let us first introduce the standard notation B_j(u_h, v_h;ϕ_h, ψ_h) =

Ij

∂_tu_hϕ_hdx− 1 τmax

Ij

(v_h−u_h)ϕ_hdx−

Ij

v_h∂_xϕ_hdx +vh(x_j+¹₂, t)ϕh(x⁻_j+1

2)−vh(x_j−¹₂, t)ϕh(x⁺_j−1

2) (2.4)

+

I_{j+ 1}

2

∂tvhψhdx− 1 τ_max

I_{j+ 1}

2

(uh−vh)ψhdx−

I_{j+ 1}

2

uh∂xψhdx +u_h(x_j+1, t)ψ_h(x⁻_j+1)−u_h(x_j, t)ψ_h(x⁺_j).

Clearly, we have:

B_j(u_h, v_h;ϕ_h, ψ_h) = 0 (2.5)

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

Bj(u, u;ϕh, ψh) = 0 (2.6)

for allj and allϕ_h∈V_h andψ_h∈W_h. Subtracting (2.5) from (2.6), we obtain the error equation

Bj(u−uh, u−vh;ϕh, ψh) = 0 (2.7)

for allj and allϕ_h∈V_h andψ_h∈W_h.

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) (2.8)

and

I_{j+ 1}

2

(Qw(x)−w(x))ψh(x)dx= 0 ∀ψh∈P^k(I_j+¹₂) (2.9) 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+¹

2 respectively. Standard approximation theory [1] implies, for a smooth functionw,

(P w(x)−w(x))+h^1/2(P w(x)−w(x))Γ≤Ch^k+1 (2.10) and

(Qw(x)−w(x))+h^1/2(Qw(x)−w(x))_Γ≤Ch^k+1 (2.11) where Γ denotes the set of boundary points of all elements Ij or I_j+1/2 respectively, the norm || · ||Γ is the standardL² norm, and the positive constantC, here and below, solely depending onw(x) and its derivatives, is independent ofh.

We also recall that [1], for anyw_h ∈V_h or w_h ∈W_h, there exists a positive constant C independent of w_h andh, such that

∂xwh ≤Ch⁻¹wh; whΓ≤Ch^−1/2wh (2.12) where Γ is the set of boundary points of all elementsI_j orI_j+1/2.

We now take:

ϕ_h=P u−u_h, ψ_h=Qu−v_h (2.13)

in the error equation (2.7), and denote

ϕ^e=P u−u, ψ^e=Qu−u (2.14)

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to obtain

Bj(ϕh, ψh;ϕh, ψh) =Bj(ϕ^e, ψ^e;ϕh, ψh). (2.15) For the left-hand side of (2.15), we use Theorem 2.1 to conclude

j

Bj(ϕh, ψh;ϕh, ψh) =1 2

d dt

_b

a

((ϕh)²+ (ψh)²)dx+ 1 τmax

_b

a

(ϕh−ψh)²dx. (2.16)

We then write the right-hand side of (2.15) as a sum of three terms

B_j(ϕ^e, ψ^e;ϕ_h, ψ_h) =B_j¹+B_j²+B_j³ (2.17) where

B_j¹= 1 τ_max

Ij

(ϕ^e−ψ^e)ϕhdx+ 1 τ_max

I_{j+ 1}

2

(ψ^e−ϕ^e)ψhdx+

Ij

∂tϕ^eϕhdx+

I_{j+ 1}

2

∂tψ^eψhdx

B_j²=−

Ij

ψ^e∂xϕhdx−

I_{j+ 1}

2

ϕ^e∂xψhdx B_j³=ψ^e(x_j+¹₂, t)ϕh(x⁻_j+1

2, t)−ψ^e(x_j−¹₂, t)ϕh(x⁺_j−1

2, t) +ϕ^e(xj+1, t)ψh(x⁻_j+1, t)−ϕ^e(xj, t)ψh(x⁺_j, t) and we will estimate each term separately.

By using the simple inequality

αβ≤ 1

2(α²+β²), (2.18)

theL² projection property (2.10)–(2.11) forϕ^e,ψ^e, ∂tϕ^eand∂tψ^e, and the fact thatτmax=O(h), we have:

j

B_j¹≤ _b

a (ϕ_h)²dx+ _b

a (ψ_h)²dx+Ch^2k. (2.19)

Likewise, by using the simple inequality (2.18), theL²projection property (2.10) forϕ^eandψ^e, and the first inequality in (2.12) forϕh andψh, we have:

j

B_j²≤ _b

a (ϕh)²dx+ _b

a (ψh)²dx+Ch^2k. (2.20)

Finally, by using the simple inequality (2.18), theL²projection property (2.10) forϕ^eandψ^e, and the second inequality in (2.12) forϕ_h andψ_h, we have:

j

B_j³≤ _b

a

(ϕh)²dx+ _b

a

(ψh)²dx+Ch^2k. (2.21)

Combining (2.19), (2.20) and (2.21) with with (2.16), we obtain from (2.15) 1

2 d dt

_b

a ((ϕ_h)²+ (ψ_h)²)dx≤C _b

a ((ϕ_h)²+ (ψ_h)²)dx+Ch^2k. This, together with the approximation result (2.10), implies the desired error estimate (2.3).

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Remark 2.3. The error estimate of Theorem 2.2 is sub-optimal. Numerically we observe the optimal (k+ 1)-th order accuracy, see [8] and also the numerical results in next section. In contrast, the error estimate for the regular DG method in such one dimensional and also in multi-dimensional tensor product cases is optimal [3].

The major technical difficulty leading to this loss of optimality in the proof is theB_j²term. For the regular DG method this term is zero for both the regular L² projection and for a special projection which is orthogonal to polynomials of one degree lower and can render the boundary terms inB³_j also to vanish. However, for the central DG method, since it involvesψ^e and ∂_xϕ_h and they are defined by polynomials in different cells, it is impossible to make the B_j² term vanish no matter how the projection is chosen, although a special projection like that used for regular DG methods can renderB_j³to be zero. A better technique is needed to prove the opti- mal (k+1)-th order accuracy observed numerically, however we have been unable to find such a technique so far.

Remark 2.4. The error estimate of Theorem 2.2 can be easily generalized to one dimensional linear hyperbolic systems, multi-dimensional scalar linear hyperbolic equations, and multi-dimensional symmetric linear systems.

2.3. A quantitative error estimate via Fourier analysis

In this subsection we perform a Fourier analysis for the semi-discrete central DG scheme (1.5)–(1.6) for the linear hyperbolic equation with uniform meshes for piecewise constant and linear elements, using the techniques in [13]. This analysis is more explicit and allows us to compare the errors quantitatively between the central and standard DG schemes.

For this purpose we rewrite the scheme (1.5)–(1.6) for the linear equation (2.1) as a finite difference scheme on a uniform mesh. Towards this goal we choose the degrees of freedom for the k-th degree polynomial inside the cellsIj andI_j+¹₂ respectively as the point values of the solution, denoted by

u_j+ ^2i−k

2(k+1)(t), i= 0, ..., k and

v_j+^2i+1−k

2(k+1)(t), i= 0, ..., k at thek+1 equally spaced points

j+ 2i−k 2(k+ 1)

h, i= 0, ..., k

and

j+2i+ 1−k 2(k+ 1)

h, i= 0, ..., k.

The schemes written in terms of these degrees of freedom become finite difference schemes on a globally uniform mesh (with a mesh sizeh/(k+ 1)), however they are not standard finite difference schemes because each point in the group ofk+1 points belonging to the cellI_j orI_j+¹₂ obeys a different form of the finite difference scheme.

To be more specific, we first concentrate on the simplest piecewise constant k = 0 case. For this case, we choose the degrees of freedom as the point values at theN uniformly spaced points

u_j(t), j= 0, ..., N−1 or

v_j+¹

2(t), j= 0, ..., N−1.

The solution inside the cellIj orI_j+¹₂ is then represented by uh(x, t) =uj(t)ϕ⁰_h(x) or

vh(x, t) =v_j+¹₂(t)ψ_h⁰(x)

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whereϕ⁰_h(x) is the constant function which equals 1 insideI_j, and similarlyψ⁰_h(x) is the constant function which equals 1 insideI_j+¹

2. With this representation, taking the test functionsϕh as ϕ⁰_h, andψh as ψ_h⁰, respectively, we obtain easily the finite difference schemes corresponding to the central DG scheme:

u_j = − 1 τ_maxuj+

1 2τ_max+ 1

h

v_j−¹

2 + 1

2τ_max −1 h

v_j+¹

2

v_j+ 1

2 = − 1

τ_maxv_j+¹₂ + 1

2τ_max +1 h

uj+

1 2τ_max −1

h

uj+1 (2.22)

forj= 0, ..., N−1. Hereu andv denote the time derivatives ofuandv. The scheme can be rewritten into a more compact form

u_j v_j+ 1

2

=A

u_j−1 v_j−¹₂

+B

u_j v_j+¹₂

+C

u_j+1 v_j+³₂

(2.23) with

A =

0 _2τ¹

max+_h¹

0 0

, B =

−_τmax¹ _2τmax¹ −_h¹

2τmax1 +_h¹ −_τmax¹

, C =

0 0

2τmax1 −¹_h 0

. (2.24)

We now perform the following standard Fourier analysis for the finite difference scheme (2.23)–(2.24). This analysis depends heavily on the assumption of uniform mesh sizes and periodic boundary conditions. We make

an ansatz of the form

u_j(t) v_j+¹

2(t)

=

uˆ_m(t) ˆv_m(t)

e^imxj (2.25)

and substitute this into the scheme (2.23)–(2.24) to find the evolution equation for the coefficient vector as uˆ_m(t)

ˆv_m (t)

=G(m, h)

uˆm(t) vˆm(t)

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

G(m, h) = Ae^−imh+ B +Ce^imh (2.27)

with the matricesA,B andC defined by (2.24). The two eigenvalues of the amplification matrixG(m, h) are λ1=− 1

τmax −αe^imh2 , λ2=− 1

τmax +αe^imh2 (2.28)

whereα= (_2τ¹

max+_h¹)e^−imh+_2τ¹

max−_h¹. The general solution of the ODE (2.26) is given by ˆum(t)

vˆm(t)

= a1e^λ1tV1 +a2e^λ2tV2, (2.29) where the eigenvaluesλ1 andλ2are given by (2.28), andV1 andV2are the corresponding eigenvectors given by

V1 =

−e^−iξ2 1

, V2 =

e^−iξ2 1

, (2.30)

withξ=mh. For accuracy we look at the low modes, in particular atm= 1. To fit the given initial condition uj(0) = e^ixj, v_j+¹₂(0) = e^{ixj+ 12} (2.31)

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whose imaginary part is the initial condition

uj(0) = sin(xj), v_j+¹₂(0) = sin(x_j+¹₂), (2.32) we require, att= 0,

uˆ1(0) vˆ₁(0)

= 1

e^ih2

,

hence we obtain the coefficients a1 anda2 in (2.29) as

a1 = 0, a2 = e^ih2. (2.33)

We remark that the usual way of taking initial conditions in a finite element method isvia anL² projection, not by a point value collocation (2.32), however we have verified that this does not affect the final results in the analysis in this paper. We thus have the explicit solutions of the scheme (2.23)–(2.24) with the initial condition (2.31), for example

u_j(t) =a₂e^{ixj+λ2t−ih2} (2.34)

with the eigenvalue λ₂ given by (2.28) with m= 1 and the coefficienta₂ given by (2.33). By a simple Taylor expansion, we obtain the imaginary part ofu_j(t) to be

Im{uj(t)}= sin(xj−t)− t

8csin(xj−t)

h+O(h²) (2.35)

wherec=^τmax_h =O(1) is the maximum CFL number. This is clearly consistent with the exact solution to first order accuracy. We can similarly check the first order accuracy ofIm{v_j+¹

2(t)}.

We now repeat this analysis for the piecewise linear k = 1 case. The solution inside the cell I_j or I_j+¹

2 is then represented by

u_h(x, t) =u_j−¹

4(t)ϕ¹_h(ξ) +u_j+¹

4(t)ϕ²_h(ξ) or

v_h(x, t) =v_j+¹₄(t)ψ¹_h(ξ) +v_j+³₄ψ_h²(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 functionsϕ_h andψ_has ϕ¹_h,ϕ²_h andψ¹_h, ψ_h² respectively, and inverting the small 4×4 mass matrix by hand, we obtain easily the finite difference scheme corresponding to the central DG scheme as

⎛

⎜⎜

⎜⎝ u_j−1

u_j+41

v_j+ 14

v_j+ 43 4

⎞

⎟⎟

⎟⎠=A

⎛

⎜⎜

⎜⎝ u_j−⁵₄ u_j−³₄ v_j−³₄ v_j−¹₄

⎞

⎟⎟

⎟⎠+B

⎛

⎜⎜

⎜⎝ u_j−¹₄ u_j+¹₄ v_j+¹₄ v_j+³₄

⎞

⎟⎟

⎟⎠+C

⎛

⎜⎜

⎜⎝ u_j+³₄ u_j+⁵₄ v_j+⁵₄ v_j+⁷₄

⎞

⎟⎟

⎟⎠ (2.36)

Rapide Not Highlight Paper

with

A =

⎛

⎜⎜

⎜⎝

0 0 _16τ¹

max+_4h⁵ _16τ¹³

max−_4h¹ 0 0 _16τ⁻¹

max−_4h¹ _16τ³_max+_4h⁵

0 0 0 0

0 0 0 0

⎞

⎟⎟

⎟⎠,

B =

⎛

⎜⎜

⎜⎜

⎝

τ−1max 0 _16τ³

max −_4h⁵ _16τ⁻¹_max+_4h¹

0 _τ⁻¹

max

16τ13max +_4h¹ _16τ¹

max−_4h⁵

16τ1max +_4h⁵ _16τ¹³

max −_4h¹ _τ⁻¹_max 0

16τ−1max −_4h¹ _16τ³_max +_4h⁵ 0 _τ⁻¹

max

⎞

⎟⎟

⎟⎟

⎠, (2.37)

C =

⎛

⎜⎜

⎝

0 0 0 0

0 0 0 0

16τ3max −_4h⁵ _16τ⁻¹_max +_4h¹ 0 0

16τ13max +_4h¹ _16τ¹

max −_4h⁵ 0 0

⎞

⎟⎟

⎠.

We make an ansatz of the form

⎛

⎜⎜

⎜⎝

u_j−¹₄(t) u_j+¹

4(t) v_j+¹

4(t) v_j+³₄(t)

⎞

⎟⎟

⎟⎠=

⎛

⎜⎜

⎜⎝

uˆ_m,−¹₄(t) uˆ_m,¹

4(t) vˆ_m,¹

4(t) vˆ_m,³₄(t)

⎞

⎟⎟

⎟⎠e^imxj (2.38)

and substitute this into the scheme (2.36)–(2.37) to find the evolution equation for the coefficient vector as

⎛

⎜⎜

⎜⎝ uˆ_m,−1

4(t) uˆ_m,+1

4(t) vˆ_m, 1

4(t) vˆ_m, 3

4(t)

⎞

⎟⎟

⎟⎠=G(k, h)

⎛

⎜⎜

⎜⎝

uˆ_m,−¹₄(t) uˆ_m,+¹₄(t) vˆ_m,¹₄(t) vˆ_m,³₄(t)

⎞

⎟⎟

⎟⎠ (2.39)

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

G(m, h) = Ae^−imh+ B +Ce^imh (2.40)

with the matricesA,B andC defined by (2.37). The eigenvalues of the amplification matrixG(m, h) are

λ1 = 1 τ_max

−1 +

√2

8 e^−imh√

−α1−α2

λ2 = 1 τmax

−1−

√2

8 e^−imh√

−α1−α2

λ3 = 1 τmax

−1 +

√2

8 e^−imh√

−α1+α2

λ₄ = 1 τmax

−1−

√2

8 e^−imh√

−α₁+α₂

Rapide Note Highlight Paper

where

α1 = (1−4c−52c²)e^imh−2(11−52c²)e^2imh+ (1 + 4c−52c²)e^3imh, α2 = e^imh(1 + e^imh+ 10c(1−e^imh))

−3−12c+ 4c² (2.41)

+ 2(21−4c²)e^imh+ 4(−3 + 12c+ 4c²)e^2imh]^1/2 withc=^τmax_h being the maximum CFL number.

The general solution of the ODE (2.39) is given by

⎛

⎜⎜

⎜⎝

ˆu_m,−¹₄(t) uˆ_m,+¹₄(t) ˆv_m,¹₄(t) ˆv_m,³

4(t)

⎞

⎟⎟

⎟⎠ = a1e^λ1tV1 +a2e^λ2tV2+a3e^λ3tV3 +a4e^λ4tV4, (2.42)

where the eigenvalues λ1, λ2, λ3 and λ4 are given by (2.41), and V1, V2, V3 and V4 are the corresponding eigenvectors given by

V1 =

⎛

⎜⎜

⎜⎜

⎝

√2e^−imh√

−α1−α2(α7+2α2) α5(α6−α2)

√−2(α1+α2)(1+e^imh−10c(−1+e^imh)) α6−α2

α4α2+α3

α5(α6−α2)

1

⎞

⎟⎟

⎟⎟

⎠,

V2 =

⎛

⎜⎜

⎜⎜

⎝

−^{√2e−imh√}_α5^−α_(α6¹_−α^−α₂²₎^(α7+2α2)

−√

−2(α1+α2)(1+e^imh−10c(−1+e^imh)) α6−α2

α4α2+α3

α5(α6−α2)

1

⎞

⎟⎟

⎟⎟

⎠,

V3 =

⎛

⎜⎜

⎜⎜

⎝

√2e^−imh√

−α1+α2(−α7+2α2) α5(α6+α2)

√2(−α1+α2)(1+e^imh−10c(−1+e^imh)) α6+α2

−α4α2+α3

α5(α6+α2)

1

⎞

⎟⎟

⎟⎟

⎠,

V4 =

⎛

⎜⎜

⎜⎜

⎝

−^{√2e−imh√}_α^−α_5(α¹₆^+α_+α²₂^(−α₎ ^7+2α2)

−√

2(−α1+α2)(1+e^imh−10c(−1+e^imh)) α6+α2

−α4α2+α3

α5(α6+α2)

1

⎞

⎟⎟

⎟⎟

⎠, (2.43)

where c= ^τmax_h is still the maximum CFL number, α1 andα2 are given by (2.41), and the remaining αs are given by

α₃ = −(7 + 104c+ 332c²−80c³)e^imh+ (77 + 896c+ 532c²−240c³)e^2imh + (79−856c−68c²+ 240c³)e^3imh−(5−64c+ 132c²+ 80c³)e^4imh,

α4 = −1−20c+ (−3 + 20c)e^imh, (2.44)

α5 = −1−4c+ (13 + 4c)e^imh,

α6 = (1 + 10c)²e^imh+ (2−200c²)e^2imh+ (1−10c)²e^3imh, α7 = (1 + 10c)e^imh−20c²e^2imh−(1−10c)²e^3imh.