maximum-principle-satisfying and positivity …

MAXIMUM-PRINCIPLE-SATISFYING AND1

POSITIVITY-PRESERVING HIGH ORDER CENTRAL DG2

METHODS FOR HYPERBOLIC CONSERVATION LAWS∗3

MAOJUN LI† , FENGYAN LI‡ , ZHEN LI§ , AND LIWEI XU¶4

Abstract. Maximum principle or positivity-preserving property holds for many mathematical5models. When the models are approximated numerically, it is preferred that these important prop-6erties can be preserved by numerical discretizations for the robustness and the physical relevance of7the approximate solutions. In this paper, we investigate such discretizations of high order accuracy8within the central discontinuous Galerkin framework. More specifically, we design and analyze high9order maximum-principle-satisfying central discontinuous Galerkin methods for scaler conservation10laws, and high order positivity-preserving central discontinuous Galerkin for compressible Euler sys-11tems. The performance of the proposed methods will be demonstrated through a set of numerical12experiments.13

Key words. Hyperbolic conservation laws; Central discontinuous Galerkin method; Maximum14principle; Positivity preserving; High order accuracy15

1. Introduction. Maximum principle or positivity-preserving property holds16

for many mathematical models, which are often in the form of PDEs. Such property17

plays an important role for the theoretical understanding of the equations. When18

these models are approximated numerically, it is preferred that such properties can19

be preserved by numerical discretizations for the robustness and the physical relevance20

of the approximate solutions. In this paper, we are concerned with the preservation21

of these properties within high order central discontinuous Galerkin framework.22

One mathematical model we will consider is the scalar conservation law, given as23

(1.1) ut + F (u)x = 0, u(x, 0) = u0(x)24

in one dimension, and25

(1.2) ut + F (u)x +G(u)y = 0, u(x, y, 0) = u0(x, y)26

in two dimensions. Here the initial data u0 is a function with bounded variation.27

The unique entropy solution to (1.1), similarly to (1.2), satisfies a strict maximum28

principle [3]. That is, if29

m = minx

(u0(x)), M = maxx

(u0(x)),30

∗ML is partially supported by the Fundamental Research Funds for the Central Universities inChina (Project No. 106112015CDJXY100008) and the National Natural Science Foundation of China(Grant No. 11501062). FL is partially supported by the National Science Foundation (Grant No.DMS-1318409). ZL is partially supported by the National Natural Science Foundation of China(Grant No. 11371385). LX is partially supported by the National Natural Science Foundation ofChina (Grant No. 11371385), the start-up fund of Youth 1000 plan of China.

†College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China;Institute of Computing and Data Sciences, Chongqing University, Chongqing, 400044, P.R. China([email protected]).

‡Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA([email protected]).

§College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China([email protected]).

¶School of Mathematical Sciences, University of Electronic Science and Technology of China,Sichuan, 611731, R.R. China; Institute of Computing and Data Sciences, Chongqing University,Chongqing, 400044, P.R. China ([email protected], [email protected]).

1

This manuscript is for review purposes only.

2 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU

then u(x, t) ∈ [m,M ] for any x and t. This property is also desired for numerical31

schemes solving (1.1) and (1.2), for instance in some applications where u represents32

a volume ratio, and it should always be in the range of [0, 1].33

Many numerical schemes have been developed for solving the conservation laws34

(1.1) and (1.2), with examples including Runge-Kutta discontinuous Galerkin (DG)35

methods [2], weighted essentially non-oscillatory (WENO) finite difference or finite36

volume schemes [11, 5], and central schemes such as central DG methods [12]. How-37

ever, these methods, especially when they are of high order accuracy, do not in general38

satisfy the strict maximum principle. One important breakthrough was made in [15],39

where the authors proposed a very general framework to achieve maximum principle40

especially for high order methods. There a sufficient condition was first given to ensure41

that the cell averages of the numerical solution, which is from a high order finite vol-42

ume WENO method or a DG method with the Euler forward method in time, satisfy43

the maximum principle. A linear scaling limiter was then designed and applied to44

enforce this condition without destroying the local conservation and accuracy. For45

higher order temporal accuracy, strong stability preserving (SSP) time discretizations46

were used, and they can be expressed as a convex combination of the first order Euler47

forward method and therefore keep the maximum principle property.48

With the success in [15], the authors further generalized the techniques to com-49

pressible Euler equations in [16]. Here high order positivity-preserving DG methods50

were developed, which preserve the positivity of density and pressure. Such property51

is important for the well-posedness of the equations, and the violation of numerical52

schemes can lead to instability and the breakdown of the simulation. A simpler and53

more robust strategy was later proposed in [14]. In [1], two authors of the present54

paper extended the positivity-preserving techniques in [16, 14] to ideal MHD equa-55

tions, which, in the absence of the magnetic field, will become compressible Euler56

equations. In [1], high order DG methods and central DG methods were considered,57

and theoretical analysis was established in one-dimensional case; in two dimensions,58

the positivity-preserving property of the methods was mainly verified numerically.59

Our development with the central DG methods in [1], together with the missing anal-60

ysis in two dimensional case for ideal MHD equations, motivates us to examine both61

theoretically and numerically the central DG methods in this paper for solving s-62

calar conservation laws and compressible Euler equations while preserving maximum63

principle or positivity of density and pressure.64

Central DG methods are a family of high order numerical methods defined on65

overlapping meshes, which combine features of central schemes and DG methods, and66

were introduced originally for hyperbolic conservation laws [12] and then for diffusion67

equations [13]. By evolving two sets of numerical solutions without using any numer-68

ical flux at element interfaces as in Godunov schemes, central DG methods provide69

new opportunities to designing accurate and stable schemes such as for Hamilton-70

Jacobi equations [8], ideal MHD equations [6, 7], Camassa-Holm equation [9], and71

Green-Naghdi equations [10].72

In this paper, we first develop a maximum-principle-satisfying high order central73

DG method for the one- and two-dimensional scalar conservation laws in Section 2.74

A sufficient condition is proved for the cell averages of the numerical solution to be75

bounded in [m,M ] when the Euler forward method is used in time. This condition76

can be ensured through a linear scaling limiter as in [15] without destroying accuracy77

and conservation under a suitable CFL condition. In Section 3, we construct and78

analyze positivity-preserving central DG methods for compressible Euler equations in79

one- and two-dimensional spaces, employing the positivity-preserving limiting tech-80


MPS AND PP CDG METHODS FOR CONSERVATION LAWS 3

niques similar to those in [1, 14, 15]. The SSP high order time discretizations will81

keep the maximum principle and positivity-preserving property. Compared with DG82

methods, central DG methods do not depend on any numerical Riemann solvers to83

devise maximum-principle-satisfying or positivity-preserving methods. In Section 4,84

numerical experiments are presented, which are followed by some concluding remarks85

in Section 5.86

2. Maximum-principle-satisfying central DG method.87

2.1. One-dimensional case. In this section, we consider a one-dimensional s-88

calar conservation law in (1.1) and will develop a maximum-principle-preserving cen-89

tral DG method. Periodic boundary condition is assumed for the equation.90

The proposed method is based on the standard central DG method [12], which is91

formulated on two overlapping meshes and evolves two copies of numerical solutions.92

Let xjj be a uniform partition of the computational domain Ω = [xmin, xmax], and93

∆x be the mesh size. With xj+ 12= 1

2 (xj + xj+1), Ij = (xj− 12, xj+ 1

2) and Ij+ 1

2=94

(xj , xj+1), we define two discrete function spaces VCh and VD

h , associated with the95

primal mesh Ijj and the dual mesh Ij+ 12j , respectively,96

VCh = VC,k

h = v : v|Ij ∈ P k(Ij) , ∀ j ,97

VDh = VD,k

h = v : v|Ij+1

2

∈ P k(Ij+ 12) , ∀ j .98

Here P k(I) denotes the space of polynomials in I with degree at most k.99

To describe the central DG method, assume the numerical solutions are available100

at t = tn, denoted by un,⋆h ∈ V⋆

h, we want to update the numerical solutions un+1,⋆h ∈101

V⋆h at t = tn+1 = tn+∆tn. Here and below, ⋆ stands for C and D. Our focus for now102

will be on the first order forward Euler time discretization, and time discretizations103

with higher order accuracy will be discussed in Section 2.3.104

To obtain un+1,⋆h , we apply to (1.1) the central DG method of [12] in space105

and the forward Euler method in time. That is, to look for un+1,⋆h ∈ V⋆

h, such that106

∀v⋆ ∈ V⋆h with any j,107 ∫

Ij

un+1,Ch · vCdx =

∫Ij

(θnu

n,Dh + (1− θn)u

n,Ch

)· vCdx108

+ ∆tn

∫Ij

F (un,Dh ) · vCx dx109

− ∆tn

[F (un,D

h (xj+ 12)) · vC(x−

j+ 12

)110

− F (un,Dh (xj− 1

2)) · vC(x+

j− 12

)],(2.1)111

112 ∫Ij− 1

2

un+1,Dh · vDdx =

∫Ij− 1

2

(θnu

n,Ch + (1− θn)u

n,Dh

)· vDdx113

+ ∆tn

∫Ij− 1

2

F (un,Ch ) · vDx dx114

− ∆tn

[F (un,C

h (xj)) · vD(x−j )115

− F (un,Dh (xj−1)) · vD(x+

j−1)].(2.2)116



Here θn = ∆tn/τn ∈ [0, 1], with τn being the maximal time step allowed by the CFL117

restriction at tn. And w(x±) = limh→0± w(x+ h). In general, the numerical scheme118

(2.1)-(2.2) does not satisfy the maximum principle.119

2.1.1. Maximum-principle-satisfying central DG method. To construct120

a maximum-principle-satisfying central DG method, the most important step is to121

achieve such property for the cell averages of the computed solutions. To this end, let122

v⋆ = 1 in (2.1) and (2.2), respectively, we get the scheme satisfied by the cell averages,123

un+1,Ch,j =(1− θn)u

n,Ch,j +

θn∆x

∫Ij

un,Dh dx− λx

(F (un,D

h,j+ 12

)− F (un,D

h,j− 12

)),(2.3)124

un+1,D

h,j− 12

=(1− θn)un,D

h,j− 12

+θn∆x

∫Ij− 1

2

un,Ch dx− λx

(F (un,C

h,j )− F (un,Ch,j−1)

),(2.4)125

126

where λx = ∆tn∆x , un,⋆

h,i denotes the cell average of u⋆h on Ii at time tn, and un,⋆

h,i =127

u⋆h(xi, tn).128

Assume un,Ch,j , u

n,D

h,j− 12

∈ [m,M ], ∀j, we want to come up with a sufficient condition129

to ensure un+1,Ch,j , un+1,D

h,j− 12

∈ [m,M ], ∀j. Let L1,xj = x1,β

j , β = 1, 2, ..., N and L2,xj =130

x2,βj , β = 1, 2, ..., N be the Legendre Gauss-Lobatto quadrature points on [xj− 1

2, xj ]131

and [xj , xj+ 12], respectively, and ωβ , β = 1, 2, ..., N be the corresponding quadrature132

weights on the interval [− 12 ,

12 ]. This quadrature formula is exact for polynomials of133

degree up to 2N − 3. We choose N such that 2N − 3 ≥ k. Note that∑N

β=1 ωβ = 1,134

ω1 = ωN , and xj− 12= x1,1

j = x2,Nj−1, xj = x1,N

j = x2,1j , ∀j.135

Theorem 2.1. For the scheme (2.3)-(2.4), assume un,Ch,j , u

n,D

h,j− 12

∈ [m,M ], ∀j. If136

uCh (x⋄, tn), u

Dh (x⋄, tn) ∈ [m,M ] for all x⋄ ∈ Ll,x

j , ∀j and l = 1, 2, then un+1,Ch,j and137

un+1,D

h,j− 12

will belong to [m,M ], ∀j, under the CFL condition138

(2.5) λxax ≤ θn2ω1139

with ax = max(||F ′(uCh (·, tn))||∞, ||F ′(uD

h (·, tn))||∞).140

Proof. Using the Legendre Gauss-Lobatto quadrature rule, one has141

(2.6)1

∆x

∫Ij

un,Dh dx =

1

2

N∑β=1

ωβun,D1,β +

N∑β=1

ωβun,D2,β

142



where un,Dl,β = uD

h (xl,βj , tn). Now, substituting (2.6) into (2.3) yields143

un+1,Ch,j = (1− θn)u

n,Ch,j +

θn2

N∑β=1

ωβun,D1,β +

N∑β=1

ωβun,D2,β

144

− λx

[F (un,D

h,j+ 12

)− F (un,D

h,j− 12

)]

145

= (1− θn)un,Ch,j +

θn2

N∑β=2

ωβun,D1,β +

N−1∑β=1

ωβun,D2,β

146

+

(θn2ω1u

n,D1,1 + λxF (un,D

1,1 )

)+

(θn2ω1u

n,D

2,N− λxF (un,D

2,N)

)147

=: H(un,Ch,j , u

n,D1,1 , · · · , un,D

1,N, un,D

2,1 , · · · , un,D

2,N).(2.7)148

We here used ω1 = ωN , un,D1,1 = un,D

h,j− 12

and un,D

2,N= un,D

h,j+ 12

.149

Under the CFL condition (2.5), one can easily verify that H is monotonically150

non-decreasing with respect to each input, more specifically,151∂un+1,C

h,j

∂un,Ch,j

= 1− θn ≥ 0,152

∂un+1,Ch,j

∂un,D1,β

= θn2 ωβ ≥ 0, β = 2, 3, ..., N ,153

∂un+1,Ch,j

∂un,D2,β

= θn2 ωβ ≥ 0, β = 1, 2, ..., N − 1,154

∂un+1,Ch,j

∂un,D1,1

= θn2 ω1 + λxF

′(un,D1,1 ) ≥ θn

2 ω1 − λxax ≥ 0,155

∂un+1,Ch,j

∂un,D

2,N

= θn2 ω1 − λxF

′(un,D

2,N) ≥ θn

2 ω1 − λxax ≥ 0.156

This implies157

m = H(m, · · · ,m) ≤ H(un,Ch,j , u

n,D1,1 , · · · , un,D

1,N, un,D

2,1 , · · · , un,D

2,N) = un+1,C

h,j158

≤ H(M, · · · ,M) = M,(2.8)159

hence un+1,Ch,j ∈ [m,M ], ∀j. The first and last equalities in (2.8) are related to the160

consistency of the scheme, and can be verified directly. Similarly, one can show161

un+1,D

h,j− 12

∈ [m,M ], ∀j.162

Next, we will give a maximum-principle-satisfying limiter which modifies the cen-163

tral DG solutions uCh and uD

h at time tn, with their cell averages in [m,M ], into uCh164

and uDh such that the modified solutions will satisfy the sufficient condition in Theo-165

rem 1, while maintaining accuracy and local conservation (see [15] for the analysis on166

accuracy). This limiter is almost the same as the one for DG methods [15], as long167

as it is applied to uCh and uD

h separately.168

LetK denote a mesh element Ij on the primal mesh or Ij− 12on the dual mesh, and

let LK represent the set of relevant quadrature points in K, namely LK = L1,xj ∪ L2,x

j

when K = Ij , or LK = L2,xj−1 ∪ L1,x

j when K = Ij− 12. At time tn, the numerical

solution on K is denoted as uK . Following [15], the maximum-principle-satisfyinglimiter is given as follows. On each mesh element K, we modify the solution uK into

uK = αK(uK − uK) + uK , αK = min

1, | M − uK

MK − uK|, | m− uK

mK − uK|



withMk = maxx∈LK(uK) andmk = minx∈LK

(uK). uK is a convex combination of the169

original central DG solution uK and its average, hence does not change the average of170

uK . This implies the local conservation property. It can be seen that the number of171

total quadrature points involved in the maximum-principle-satisfying limiter for the172

central DG method is twice (resp. four times) of that of the standard DG method in173

one dimension (resp. two dimensions).174

2.2. Two-dimensional case. In this section, we will develop a maximum-175

principle-preserving central DG method to solve the two-dimensional scalar conserva-176

tion law (1.2). Periodic boundary condition is assumed.177

Let T Ch = Cij , ∀i, j and T D

h = Dij , ∀i, j define two overlapping meshes for178

the computational domain Ω = [xmin, xmax]× [ymin, ymax], with Cij = [xi− 12, xi+ 1

2]×179

[yj− 12, yj− 1

2] and Dij = [xi−1, xi] × [yj−1, yj ], where xii and yjj are uniform180

partitions of [xmin, xmax] and [ymin, ymax], respectively, and xi+ 12= 1

2 (xi + xi+1),181

yj+ 12= 1

2 (yj + yj+1). The mesh size is ∆x in x direction and ∆y in y direction.182

Associated with each mesh, we define the following discrete spaces,183

WCh = WC,k

h = v : v|Cij ∈ P k(Cij) , ∀ i, j ,184

WDh = WD,k

h = v : v|Dij ∈ P k(Dij) , ∀ i, j .185

To describe the central DG method, assume the numerical solutions are available186

at t = tn, denoted by un,⋆h ∈ W⋆

h, and we want to update the numerical solutions187

un+1,⋆h ∈ W⋆

h at t = tn+1 = tn + ∆tn. Our focus will be on the first order forward188

Euler time discretization for now. For the brevity of the presentation, we only describe189

the procedure to update un+1,Ch , as the one for un+1,D

h is similar.190

To obtain un+1,Ch , we apply to (1.2) the central DG method of [12] in space and191

the forward Euler method in time. That is, to look for un+1,Ch ∈ WC,k

h such that ∀192

v ∈ WC,kh with any i and j,193 ∫

Cij

un+1,Ch vdxdy =

∫Cij

(θnu

n,Dh + (1− θn)u

n,Ch

)· vdxdy194

+ ∆tn

∫Cij

[F (un,D

h ) · vx +G(un,Dh ) · vy

]dxdy195

− ∆tn

∫ yj+1

2

yj− 1

2

[F (un,D

h (xi+ 12, y)) · v(x−

i+ 12

, y)196

− F (un,Dh (xi− 1

2, y)) · v(x+

i− 12

, y)]dy197

− ∆tn

∫ xi+1

2

xi− 1

2

[G(un,D

h (x, yj+ 12)) · v(x, y−

j+ 12

)198

− G(un,Dh (x, yj− 1

2)) · v(x, y+

j− 12

)]dx .(2.9)199


restriction at tn. Similar as in one dimension, the numerical scheme (2.9) in general201

does not satisfy the maximum principle.202

2.2.1. Maximum-principle-satisfying central DG method. In this section,203

we extend the maximum-principle-satisfying central DG method in Section 2.1.1 to204

solve the two-dimensional conservation law. We consider the scheme satisfied by the205



cell averages of the central DG solution with the forward Euler time discretization,206

which can be obtained by taking the test function v = 1 in (2.9),207

un+1,Ch,ij = (1− θn)u

n,Ch,ij +

θn∆x∆y

∫Cij

un,Dh dxdy208

− ∆tn∆x∆y

∫ yj+1

2

yj− 1

2

[F (un,D

h (xi+ 12, y))− F (un,D

h (xi− 12, y))

]dy209

− ∆tn∆x∆y

∫ xi+1

2

xi− 1

2

[G(un,D

h (x, yj+ 12))−G(un,D

h (x, yj− 12))]dx.(2.10)210

Here un,Ch,ij denotes the cell average of uC

h on Cij at time tn.211

Assume un,Ch,ij , u

n,Dh,ij ∈ [m,M ], ∀i, j, we want to come up with a sufficient con-212

dition to ensure un+1,Ch,ij , un+1,D

h,ij ∈ [m,M ], ∀i, j. Let L1,xi = x1,β

i , β = 1, 2, ..., N213

and L2,xi = x2,β

i , β = 1, 2, ..., N be the Legendre Gauss-Lobatto quadrature points214

on [xi− 12, xi] and [xi, xi+ 1

2], respectively, L1,y

j = y1,βj , β = 1, 2, ..., N and L2,yj =215

y2,βj , β = 1, 2, ..., N be the Legendre Gauss-Lobatto quadrature points on [yj− 12, yj ]216

and [yj , yj+ 12], respectively. The corresponding quadrature weights ωβ , β = 1, 2, ..., N217

are on the interval [−12 ,

12 ] and N is chosen such that 2N − 3 ≥ k. In addition, let218

L1,xi = x1,α

i , α = 1, 2, ..., N and L2,xi = x2,α

i , α = 1, 2, ..., N denote the Gaus-219

sian quadrature points on [xi− 12, xi] and [xi, xi+ 1

2], respectively, L1,y

j = y1,αj , α =220

1, 2, ..., N and L2,yj = y2,αj , α = 1, 2, ..., N be the Gaussian quadrature points221

on [yj− 12, yj ] and [yj , yj+ 1

2], respectively. The corresponding quadrature weights222

ωα, α = 1, 2, ..., N are on the interval [−12 ,

12 ] and N is chosen such that the Gaus-223

sian quadrature is exact for polynomials of degree up to 2k + 1. Define Ll,mi,j =224

(Ll,xi ⊗ Lm,y

j ) ∪ (Ll,xi ⊗ Lm,y

j ) with l,m = 1, 2. We further approximate the boundary225

integrals in (2.10) using the Gaussian quadrature rule we just described. For the226

resulting scheme that is still referred to as (2.10), the following theorem holds.227

Theorem 2.2. For the scheme (2.10) and its counter part for un+1,Dh,ij , assume228

un,Ch,ij , u

n,Dh,ij ∈ [m,M ], ∀i, j. If uC

h (x⋄, y⋄, tn), uDh (x⋄, y⋄, tn) ∈ [m,M ], ∀(x⋄, y⋄) ∈229

Ll,mi,j , ∀i, j and l,m = 1, 2, then un+1,C

h,ij and un+1,Dh,ij will belong to [m,M ], ∀i, j, under230

the CFL condition231

(2.11) λxax + λyay ≤ θn2ω1.232

where λx = ∆tn∆x , λy = ∆tn

∆y , ax = max(||F ′(uCh (·, ·, tn))||∞, ||F ′(uD

h (·, ·, tn))||∞), and233

ay = max(||G′(uCh (·, ·, tn))||∞, ||G′(uD

h (·, ·, tn))||∞).234

The proof is essentially a combination of that for Theorem 2.1, and some (now235

standard) techniques in [16] for two dimensions. It is omitted here.236

Next, we will give a maximum-principle-satisfying limiter which modifies the cen-237

tral DG solution uCh and uD

h into uCh and uD

h such that the modified solutions will238

satisfy the sufficient condition given in Theorem 2. The limiter is in the same form as239

that in one dimension, as long as the notation K and LK are reinterpreted as follows:240

On the primal mesh, K denotes a mesh element Cij and LK represents the set of241

relevant quadrature points in K, namely LK = ∪2l,m=1L

l,mi,j . On the dual mesh, K242

denotes a mesh element Dij and LK represents the set of relevant quadrature points243



in K, namely LK = L1,1i,j ∪ L1,2

i,j−1 ∪ L2,1i−1,j ∪ L2,2

i−1,j−1. At time tn, the numerical244

solution on K is denoted by uK .245

2.3. High order time discretizations. To achieve better accuracy in time,246

strong stability preserving (SSP) high order time discretizations will be used [4]. Such247

discretizations can be written as a convex combination of the forward Euler method,248

and therefore the proposed schemes with a high order SSP time discretization are still249

maximum-principle-satisfying.250

3. Positivity-preserving central DG method for compressible Euler e-251

quations.252

3.1. One-dimensional case. In this section, we consider the following com-253

pressible Euler equations in one dimension,254

(3.1) Ut + F (U)x = 0 ,255

with U = (ρ, ρu, ρE)⊤ and F (U) =(ρu, ρu2 + p, ρEu+ pu

)⊤, where ρ is the density,256

u is the velocity, E = 12u

2 + e is the total energy with e being the internal energy,257

and p is the pressure. For the closure of the system (3.1), the following equation of258

state is used,259

(3.2) p = (γ − 1)ρe ,260

where γ > 1 is the specific heat ratio (γ = 1.4 for air). For this nonlinear hyperbolic261

system, we want to design a positivity-preserving central DG method which preserves262

the positivity of density and pressure.263

As in Section 2.1, we introduce two discrete function spaces associated with two264

overlapping meshes Ijj and Ij+ 12j ,265

VCh = VC,k

h = v : v|Ij ∈ [P k(Ij)]3 , ∀ j ,266

VDh = VD,k

h = v : v|Ij+1

2

∈ [P k(Ij+ 12)]3 , ∀ j ,267

where [P k(I)]3 = v = (v1, v2, v3)⊤ : vi ∈ P k(I), i = 1, 2, 3 is the vector version268

of P k(I). Assume two copies of the numerical solutions are available at t = tn,269

denoted by Un,⋆h = (ρn,⋆h , (ρu)n,⋆h , (ρE)n,⋆h )⊤ ∈ V⋆

h, and we want to find the solutions270

at t = tn+1 = tn + ∆tn using central DG methods. For brevity of the presentation,271

we only describe the procedure to update Un+1,Ch = (ρn,Ch , (ρu)n,Ch , (ρE)n,Ch )⊤.272

To obtain Un+1,Ch , we apply to (3.1) the central DG method of [12] in space and273

the forward Euler method in time. That is, to look for Un+1,Ch ∈ VC

h such that for274

any V ∈ VCh with any j,275 ∫

Ij

Un+1,Ch · V dx =

∫Ij

(θnU

n,Dh + (1− θn)U

n,Ch

)· V dx276

+ ∆tn

∫Ij

F (Un,Dh ) · Vxdx277

− ∆tn

[F (Un,D

h (xj+ 12)) · V (x−

j+ 12

)278

− F (Un,Dh (xj− 1

2)) · V (x+

j− 12

)].(3.3)279




restriction at tn.281

For problems with low density or pressure, the numerical scheme (3.3) may pro-282

duce negative values for density or pressure. This can result in numerical instability283

and the breakdown of the algorithm.284

3.1.1. Positivity-preserving central DG method. We start with an admis-285

sible set286

(3.4) H = U = (ρ, ρu, ρE)⊤ : ρ > 0, p(U) = (γ − 1)(ρE − 1

2ρu2) > 0,287

which defines a convex set [15]. For the standard central DG method in (3.3), we288

want to propose a positivity-preserving limiter, with which the cell averages of the289

numerical solution on each mesh belong to the admissible set H at each time tn. To290

achieve this, we consider the first order central DG method using the forward Euler291

time discretization, given as292

Un+1,Cj = (1− θn)U

n,Cj + θn

Un,D

j− 12

+ Un,D

j+ 12

2− λx

(F (Un,D

j+ 12

)− F (Un,D

j− 12

)),293

(3.5)294

where λx = ∆tn∆x , Un,C

j denotes the cell average of UCh on Ij at time tn, and Un,D

j+ 12

295

denotes the cell average of UDh on Ij+ 1

2at time tn.296

Lemma 3.1. The first order central DG method in (3.5) and its counter part for297

Un+1,D

j+ 12

is positivity-preserving under the CFL condition298

(3.6) λxax ≤ 1

2θn.299

That is, if Un,Cj , Un,D

j− 12

∈ H, ∀j, then Un+1,Cj , Un+1,D

j− 12

∈ H, ∀j. Here ax = max(∥|uCh |+300

cC∥∞, ∥|uDh |+ cD∥∞), with c⋆ =

√γp⋆h/ρ

⋆h being the sound speed.301

Proof. Equation (3.5) can be written as302

Un+1,Cj = (1− θn)U

n,Cj +

θn2

(Un,D

j− 12

+2λx

θnF (Un,D

j− 12

)

)303

+θn2

(Un,D

j+ 12

− 2λx

θnF (Un,D

j+ 12

)

).(3.7)304

Notice that Un+1,Cj is a convex combination of three vectors, Un,C

j , Un,D

j− 12

+305

2λx

θnF (Un,D

j− 12

), and Un,D

j+ 12

− 2λx

θnF (Un,D

j+ 12

). Since Un,Cj , Un,D

j− 12

and Un,D

j+ 12

are in the set H306

which is convex, we only need to show Un,D

j− 12

+ 2λx

θnF (Un,D

j− 12

) and Un,D

j+ 12

− 2λx

θnF (Un,D

j+ 12

)307

are both in H.308

For simplicity, we drop the subscripts and superscripts, and prove that if U ∈ H,309

then U ± 2λx

θnF (U) ∈ H under the CFL condition (3.6). For density, we have310

ρ

(U ± 2λx

θnF (U)

)= ρ± 2λx

θnρu = ρ

(1± 2λx

θnu

)311

> ρ

(1− 2λx

θnax

)≥ 0.312



For pressure,313

p

(U ± 2λx

θnF (U)

)= p

((1± 2λx

θnu

)ρ,

(1± 2λx

θnu

)ρu± 2λx

θnp,314 (

1± 2λx

θnu

)ρE ± 2λx

θnpu

)315

=

(1± 2λx

θnu

)(1− p(γ − 1)

2ρ(θn/2λx ± u)2

)p.316

Given 1± 2λx

θnu > 0 and p > 0, one can see317

p

(U ± 2λx

θnF (U)

)> 0 ⇔ p(γ − 1)

2ρ(θn/2λx ± u)2< 1318

⇔ γp

ρ<

2γ

γ − 1

(θn2λx

± u

)2

.319

The fact γ > 1 gives 2γγ−1 > 1. In addition, the CFL condition (3.6) implies | θn

2λx±u| ≥320

ax −u ≥ c =√

γpρ , thus p

(U ± 2λx

θnF (U)

)> 0. Similarly, we can prove Un+1,D

j− 12

∈ H,321

∀j.322

Lemma 3.1 is for the first order central DG method. We are now ready to discuss323

central DG methods of general accuracy. We consider the scheme satisfied by the cell324

averages of the central DG solution with the forward Euler method in time, given as325

Un+1,Ch,j = (1− θn)U

n,Ch,j +

θn∆x

∫Ij

Un,Dh dx326

− λx

[F (Un,D

h,j+ 12

)− F (Un,D

h,j− 12

)].(3.8)327

Here Un,Ch,j denotes the cell average of UC

h on Ij at time tn, and Un,D

h,j+ 12

denotes the328

cell average of UDh Ij+ 1

2at time tn.329

Theorem 3.2. For the scheme (3.8) and its counter part for Un+1,Dh,j , assume330

Un+1,Ch,j , Un+1,D

h,j− 12

∈ H, ∀j. If UCh (x⋄, tn), U

Dh (x⋄, tn) ∈ H, ∀x⋄ ∈ Ll,x

j , ∀j and l = 1, 2,331

then Un+1,Ch,j and Un+1,D

h,j− 12

will belong to H, ∀j, under the CFL condition332

(3.9) λxax ≤ θn2ω1.333

Here ax is defined in Lemma 3.1.334

Proof. Using the Legendre Gauss-Lobatto quadrature rule, one has335

(3.10)1

∆x

∫Ij

Un,Dh dx =

1

2

N∑β=1

ωβUn,D1,β +

N∑β=1

ωβUn,D2,β

.336



Here Un,Dl,β = Un,D

h (xl,βj ). Now, substituting (3.10) into (3.8) yields337

Un+1,Ch,j = (1− θn)U

n,Ch,j +

θn2

N∑β=1

ωβUn,D1,β +

N∑β=1

ωβUn,D2,β

338

− λx

[F (Un,D

h,j+ 12

)− F (Un,D

h,j− 12

)]

(3.11)339

= (1− θn)Un,Ch,j +

θn2

N∑β=2

ωβUn,D1,β +

N−1∑β=1

ωβUn,D2,β

+ θnω1Uj340

where341

Uj =Un,D

h,j− 12

+Un,D

h,j+12

2 − λx

θnω1

(F (Un,D

h,j+ 12

)− F (Un,D

h,j− 12

)).342

We here used ω1 = ωN , Un,D1,1 = Un,D

h,j− 12

, and Un,D

2,N= Un,D

h,j+ 12

. Based on Lemma343

1, we know Uj ∈ H as long as λx

θnω1ax ≤ 1

2 , or equivalently λxax ≤ 12θnω1. Note344

that Un+1,Ch,j is a convex combination of Un,C

h,j , Un,D1,β with β = 2, 3, ..., N , Un,D

2,β with345

β = 1, 2, ..., N − 1, and Uj , which all belong to the convex admissible set H, therefore346

Un+1,Ch,j ∈ H, ∀j. Similarly, one can show Un+1,D

h,− 12

∈ H, ∀j.347

Next, we will give a positivity-preserving limiter which modifies the central DG348

solution UCh and UD

h at time tn, with the cell averages in the set H, into UCh and349

UDh , such that the modified solutions will satisfy the sufficient condition in Theorem350

3, while maintaining accuracy and local conservation (see [16]).351

Let K denote a mesh element, with the numerical solution on K being UK =352

(ρ, ρu, ρE), and let LK represent the set of relevant quadrature points in K. Following353

[14, 1], the positivity-preserving limiter is given as follows.354

(a) First, we enforce the positivity of density. In each element K, we modify the355

density ρ into ρ,356

ρ = αK(ρ− ρ) + ρ , αK = minx∈LK

1, |(ρ− ε)/(ρ− ρ)| .357

Here ε is a small number such that ρ ≥ ε for all K. In our simulation,358

ε = 10−13 is taken. We now define UK = (ρ, ρu, ρE).359

(b.) Second, we enforce the positivity of pressure. For each x ∈ LK , we define360

δx =

1, if p(UK) > 0

p(UK)

p(UK)−p(UK), otherwise.

361

With this, the newly limited numerical solution is given as362

UK = βK(UK − UK) + UK , βK = minx∈LK

(δx).363

3.2. Two-dimensional case. In this section, we will extend the positivity-364

preserving central DG method in Section 3.1 to the two-dimensional compressible365

Euler equations,366

(3.12) Ut + F (U)x +G(U)y = 0,367

where368

U = (ρ, ρu, ρv, ρE)⊤,369



370

F (U) =(ρu, ρu2 + p, ρuv, ρEu+ pu

)⊤,371

372

G(U) =(ρv, ρuv, ρv2 + p, ρEv + pv

)⊤.373

Here ρ denotes the density, u and v are the velocity components in the x and y374

direction, respectively, p = (γ − 1)ρe is the pressure with e being the internal energy,375

and E = e+ 12 (u

2 + v2) is the total energy.376

Following the same notation in Section 2.2 for overlapping meshes Cijij and377

Dijij , we define two discrete spaces,378

WCh = WC,k

h = v : v|Cij ∈ [P k(Cij)]4 , ∀ i, j ,379

WDh = WD,k

h = v : v|Dij∈ [P k(Dij)]

4 , ∀ i, j.380

Assume two copies of numerical solutions are available at t = tn, denoted by Un,⋆h =381

(ρn,⋆h , (ρu)n,⋆h , (ρv)n,⋆h , (ρE)n,⋆h )⊤ ∈ W⋆h, we want to find the solutions at t = tn+1 =382

tn +∆tn. Again, only the procedure to update Un+1,Ch is presented.383

To obtain Un+1,Ch , we apply to (3.12) the central DG method of [12] in space and384

the forward Euler method in time. That is, to look for Un+1,Ch ∈ WC

h such that for385

any V ∈ WCh with any i and j,386 ∫

Cij

Un+1,Ch · V dxdy =

∫Cij

(θnU

n,Dh + (1− θn)U

n,Ch

)· V dxdy387

+ ∆tn

∫Cij

[F (Un,D

h ) · Vx +G(Un,Dh ) · Vy

]dxdy388

− ∆tn

∫ yj+1

2

yj− 1

2

[F (Un,D

h (xi+ 12, y)) · V (x−

i+ 12

, y)389

− F (Un,Dh (xi− 1

2, y)) · V (x+

i− 12

, y)]dy390

− ∆tn

∫ xi+1

2

xi− 1

2

[G(Un,D

h (x, yj+ 12)) · V (x, y−

j+ 12

)391

− G(Un,Dh (x, yj− 1

2)) · V (x, y+

j− 12

)]dx .(3.13)392

We define an admissible set393

H =

U = (ρ, ρu, ρv, ρE)⊤ : ρ > 0, p(U) = (γ − 1)

(ρE − 1

2ρ(u2 + v2)

)> 0

,394

which is known to be a convex set [15]. Similar to the one-dimensional case, we395

consider the scheme satisfied by the cell averages of the central DG solution with the396

forward Euler method in time, given as397

Un+1,Ch,ij = (1− θn)U

n,Ch,ij +

θn∆x∆y

∫Cij

Un,Dh dxdy398

− ∆tn∆x∆y

∫ yj+1

2

yj− 1

2

[F (Un,D

h (xi+ 12, y))− F (Un,D

h (xi− 12, y))

]dy399

− ∆tn∆x∆y

∫ xi+1

2

xi− 1

2

[G(Un,D

h (x, yj+ 12))−G(Un,D

h (x, yj− 12))]dx ,(3.14)400

401



where Un,Ch,ij denotes the cell average of the central DG solution UC

h on Cij at time402

tn. We further approximate the boundary integrals in (3.14) using the Gaussian403

quadrature rule given in Section 2.2.1. For the resulting scheme that is still referred404

to as (3.14), the following theorem holds.405

Theorem 3.3. For the scheme (3.14) and its counter part for Un+1,Dh,ij , assume406

Un,Ch,ij , U

n,Dh,ij ∈ H, ∀i, j. If UC

h (x⋄, y⋄, tn), UDh (x⋄, y⋄, tn) ∈ H, ∀(x⋄, y⋄) ∈ Ll,m

i,j , ∀i, j407

and l,m = 1, 2, then Un+1,Ch,ij and Un+1,D

h,ij will belong to H, ∀i, j, under the CFL408

condition409

(3.15) λxax + λyay ≤ θn2ω1.410

Here λx = ∆tn∆x , λy = ∆tn

∆y , ax = max(∥|uCh (·, ·, tn)| + cC(·, ·, tn)∥∞, ∥|uD

h (·, ·, tn)| +411

cD(·, ·, tn)∥∞), and ay = max(∥|vCh (·, ·, tn)|+cC(·, ·, tn)∥∞, ∥|vDh (·, ·, tn)|+cD(·, ·, tn)∥∞),412

with c⋆ =√γp⋆h/ρ

⋆h being the sound speed.413

To enforce the sufficient condition in Theorem 3.3, we will apply a two-dimensional414

version of the positivity-preserving limiter in Section 3.1, and the detail will be omit-415

ted.416

Similar to the discussions in Section 2.3, for higher order temporal accuracy, high417

order SSP time discretizations will be used [4]. With their intrinsic structure of being418

a convex combination of the Euler forward method, and the admissible set H be-419

ing convex, SSP temporal discretizations will keep the positivity-preserving property420

of the overall methods. When the solutions contains non-smooth structures such as421

strong shocks, nonlinear limiters are often needed to further improve the numerical422

stability. In our numerical experiments, a total variation bounded (TVB) corrected423

minmod slope limiter [2] is applied for some examples. When the nonlinear lim-424

iter is used, it is implemented in local characteristic fields and is applied before the425

maximum-principle-satisfying limiter.426

4. Numerical examples. In this section, numerical experiments are present-427

ed to demonstrate the performance of the proposed methods. We first examine in428

Section 4.1 the maximum-principle-satisfying central DG methods applied to scalar429

conservation laws. In Section 4.2, we test positivity-preserving central DG methods to430

solve compressible Euler equations, the parameter γ is taken to be 1.4 for all tests. In431

addition, a third order TVD Runge-Kutta method is used as the time discretization432

[4], with the time step dynamically determined by433

∆tn =Ccflax

∆x

(1D case), ∆tn =Ccfl

ax

∆x +ay

∆y

(2D case).434

Although θn can be chosen from [0, 1], we take θn = 1 in all simulations for better435

computational efficiency. According to the theoretical results, we take N = 2 for436

k = 1 with ω1 = 1/2, and N = 3 for k = 2 with ω1 = 1/6. Together with θn = 1, the437

requirement on the CFL number becomes Ccfl ≤ 1/4 for k = 1 and Ccfl ≤ 1/12 for438

k = 2. In this paper, we use Ccfl = 1/4 for P 1 approximation and Ccfl = 1/12 for439

P 2 approximation in our numerical experiments. The numerical results shown in all440

figures are based on P 2 approximation unless otherwise stated.441

4.1. Scalar conservation laws. In this section, we examine the numerical per-442

formance of the maximum-principle-satisfying central DG methods for scalar conser-443

vation laws.444



4.1.1. The 1D linear equation. We consider the linear advection equation in445

one-dimensional space446

(4.1) ut + ux = 0447

with a smooth initial condition u(x, 0) = sin(2πx) and periodic boundary condition.448

The computational domain is [0, 1]. In Table 1, we present L1, L2 and L∞ errors449

and orders of accuracy for u at t = 0.1. Even though the errors in L1 norm show450

optimal (k + 1)-st order accuracy, the accuracy degeneracy can be seen especially in451

L∞ errors. This was also observed in the numerical solutions of the finite volume and452

the DG methods in [15] with the same Runge-Kutta temporal discretization, and was453

attributed in [15] to the less accurate approximations from the inner stages of such454

multi-stage time discretizations.455

For the linear equation (4.1), we now consider a discontinuous initial data456

u(x, 0) =

1, −1 < x ≤ 0 ,

−1, 0 < x ≤ 1457

with periodic boundary condition. The computational domain is taken as [−1, 1]458

with 160 uniform elements. In Figure 1, we compare the results from the central DG459

methods with and without using maximum-principle-satisfying limiter. It is observed460

that the numerical solution with the maximum-principle-satisfying limiter maintains461

maximum principle and has much more satisfactory resolution around the discontinu-462

ity, while overshoots and undershoots are observed near the discontinuity when such463

limiter is not applied.464

It turns out the predicted bound for CFL number is not sharp. For instance,465

using the discontinuous initial data and P 2 approximation, we tested the proposed466

method with Ccfl = 1/12, 0.1, 0.105, 0.108, 0.109, 0.11, 0.15, 0.2. The numerical467

results show that the method can maintain maximum principle when Ccfl ≤ 0.108,468

yet it cannot when Ccfl ≥ 0.109. We did not test the CFL number when it is between469

0.108 and 0.109.470

Table 1The L1, L2 and L∞ errors and orders of accuracy of u at t = 0.1 for the 1D linear equation.

Mesh L1 error Order L2 error Order L∞ error OrderP 1

20 2.81E-03 — 4.12E-03 — 1.79E-02 —40 6.77E-04 2.05 9.70E-04 2.09 4.95E-03 1.8580 1.63E-04 2.05 2.30E-04 2.07 1.27E-03 1.96160 3.93E-05 2.05 5.53E-05 2.06 3.17E-04 2.00320 9.65E-06 2.03 1.34E-05 2.04 8.23E-05 1.95

P 2

20 7.36E-05 — 9.48E-05 — 2.18E-04 —40 1.08E-05 2.76 1.55E-05 2.61 6.45E-05 1.7580 1.53E-06 2.82 2.81E-06 2.46 1.78E-05 1.86160 2.15E-07 2.83 5.32E-07 2.40 4.71E-06 1.91320 2.94E-08 2.87 1.02E-07 2.39 1.22E-06 1.95



−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u

Fig. 1. Numerical results for the 1D linear equation with discontinuous initial data. 160uniform elements. t = 100. Solid line: exact solution; circles: numerical solution. Left: withlimiter; right: without limiter.

−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

x

u

Fig. 2. Numerical results at t = 0.4 for the scalar conservation law with the nonconvex Buckley-Leverett flux. Solid line: 2560 uniform elements; circles: 160 uniform elements.

4.1.2. The Buckley-Leverett equation. In this test, we solve the scalar con-471

servation law (1.1) with the nonconvex Buckley-Leverett flux472

(4.2) F (u) =4u2

4u2 + (1− u)2473

on a computational domain [−1, 1]. The initial condition is given by474

u(x, 0) =

1, −0.5 < x ≤ 0 ,0, otherwise ,

475

and outflow boundary condition is used. The numerical solutions on 160 and 2560476

uniform elements at t = 0.4 are plotted in Figure 2. The convergence to the entropy477

solutions is observed for this non-standard hyperbolic equation, and the numerical478

solutions stay in the range of [0, 1] and compare well with that in [15].479

4.1.3. A traffic flow model. In this test, we solve the scalar conservation law480

(1.1) with the flux481

F (u) =

−0.4u2 + 100u, 0 ≤ u ≤ 50 ,−0.1u2 + 15u+ 3500, 50 ≤ u ≤ 100 ,−0.024u2 − 5.2u+ 4760, 100 ≤ u ≤ 350 ,

482

which provides a traffic flow model. Here u ∈ [0, 350] denotes the density of the483

vehicles on a homogeneous highway, F (u) is the traffic flow flux function.484



0 5 10 15 200

50

100

150

200

250

300

350

x

u

0 5 10 15 200

50

100

150

200

250

300

350

x

u

Fig. 3. Numerical results at t = 0.5 (left) and 1.5 (right) for the traffic flow model. Solid line:exact solution; circles: numerical solution.

In the computation, the initial and boundary conditions are taken from [15].485

The computational domain is [0, 20] with 800 uniform elements. Figure 3 shows the486

numerical solutions at t = 0.5 and 1.5, as well as the exact solutions. The computed487

solutions stay in the range of [0, 350] and compare well with the exact ones.488

4.1.4. The 2D Burgers’ equation. We consider the two-dimensional Burgers’489

equation490

(4.3) ut +

(u2

2

)x

+

(u2

2

)y

= 0491

with a smooth initial condition u(x, 0) = 0.5 + sin(π(x + y)) and periodic boundary492

condition. The computational domain is [−1, 1] × [−1, 1]. In Table 2, we report the493

L1, L2 and L∞ errors and orders of accuracy for u at t = 0.05 when the solution is494

still smooth. Numerical solutions at t = 0.23 and t = 0.6 along the diagonal of the495

domain, namely y = x, are plotted in Figure 4 and compare quite well with the exact496

solutions.497

Table 2The L1, L2 and L∞ errors and orders of accuracy of u at t = 0.05 for the 2D Burgers’ equation.


20× 20 3.02E-02 — 2.33E-02 — 7.98E-02 —40× 40 6.87E-03 2.13 5.37E-03 2.12 1.58E-02 2.3480× 80 1.56E-03 2.14 1.22E-03 2.14 3.41E-03 2.21160× 160 3.66E-04 2.09 2.87E-04 2.09 8.30E-04 2.04320× 320 8.86E-05 2.05 6.92E-05 2.05 2.08E-04 2.00

P 2


4.2. Compressible Euler equations. In this section, we examine the numer-498

ical performance of the positivity-preserving central DG methods for compressible499



−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

1

1.5

r

u

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

1

1.5

r

u

Fig. 4. Numerical results along the diagonal (y = x, r =√

x2 + y2) for the 2D Burgersequation. 80 uniform elements. Solid line: exact solution; circles: numerical solution. Left: t =0.23; right: t = 0.6.

Euler equations. Except for example 4.2.1 where shocks are not present, the TVB500

minmod slope limiter is applied to the local characteristic fields before the positivity-501

preserving limiter is used. This slope limiter involves a parameter M , which is tuned502

for each example.503

4.2.1. Accuracy test. We start with a two-dimensional low density example to504

demonstrate the accuracy of the proposed methods. The initial condition is given by505

ρ(x, y, 0) = 1 + 0.99 sin(2π(x+ y)), u(x, y, 0) = v(x, y, 0) = 1, p(x, y, 0) = 1,506

with periodic boundary condition. The computational domain is [0, 1] × [0, 1]. In507

Table 3, we report L1, L2 and L∞ errors and orders of accuracy for ρ at t = 0.1. We508

want to mention that the positivity-preserving limiter does work on coarser meshes.509

For this example, the numerical solutions show optimal (k+1)-st order accuracy with510

k = 1, 2, and no accuracy degeneracy is observed.511

Table 3The L1, L2 and L∞ errors and orders of accuracy of ρ at t = 0.1 for the 2D Euler equations.



P 2




−10 −5 0 5 1010

−3

10−2

10−1

100

x

De

nsity

−10 −5 0 5 100

2

4

6

8x 10

4

x

Ve

locity

−10 −5 0 5 1010

0

105

1010

x

Pre

ssu

re

Fig. 5. Numerical results at t = 0.0001 for the shock tube problem. Solid line: exact solution;circles: numerical solutions on 800 uniform meshes.

4.2.2. Leblanc shock tube problem. In this test, we consider a one-dimensional512

Leblanc shock tube problem, with the initial condition given by513

(ρ, u, p) =

(2, 0, 109), x ≤ 0 ,(0.001, 0, 1), x > 0

514

on the computational domain [−10, 10]. Note that the discontinuity in pressure is515

fairly large. The limiter parameter M is set as 1010, 10, 1010 for variables ρ, ρu, ρE,516

respectively. In Figure 5, we plot the numerical density, velocity, and pressure at517

t = 0.0001 on 800 unform meshes. The results are quite stable, and they overall518

match well with the exact solution.519

4.2.3. The Sedov blast waves. Here, we test the Sedov point-blast wave in520

one-dimensional space and in two-dimensional space, respectively. The Sedov point-521

blast wave is a typical low density problem involving shocks.522

For the initial condition in one-dimensional space, the density is 1, the velocity is523

0, and the total energy is 10−12 everywhere except that in the center cell the energy524

is set as 3200000∆x . The computational domain is [−2, 2], and the numerical solutions at525

t = 0.001 are shown in Figure 6. For the initial condition in two-dimensional space,526

the density is 1, the velocity is 0, and the total energy is 10−12 everywhere except that527

in the lower left corner cell the energy is set as 0.244816∆x∆y . The computational domain is528

[0, 1.1]× [0, 1.1] with 160×160 uniform elements. The numerical boundary treatment529

follows that in [16]. The numerical solutions at t = 1 are shown in Figure 7.530

In both cases, shocks are well captured, and the results are comparable to the531

computed solutions by the positivity-preserving DG methods, see [16].532

4.2.4. Shock diffraction problem. In this test, we consider a shock diffraction533

problem where shock passing a backward facing corner is modeled [2, 16]. Standard534

numerical methods often result in negative density and/or negative pressure, and it535

is important to utilize positivity-preserving methods to simulate this example. The536



−2 −1 0 1 20

1

2

3

4

5

6

x

de

nsi

ty

−2 −1 0 1 2−1000

−500

0

500

1000

x

velo

city

−2 −1 0 1 20

2

4

6

8x 10

5

x

pre

ssu

re

Fig. 6. Numerical results at t = 0.001 for the Sedov blast wave in one-dimensional space. Solidline: results on 3200 elements; circles: results on 800 elements.

0 0.5 1 1.50

1

2

3

4

5

6

x

dens

ity

00.2

0.40.6

0.81 0

0.5

10

0.05

0.1

0.15

0.2

0.25

y

x

pres

sure

Fig. 7. Numerical results at t = 1 for the Sedov blast wave in two-dimensional space. Left:numerical result (circles) on 160 elements compared with the exact solution (solid line); right: pres-sure.

initial condition is given by537

(ρ, u, v, p) =

(7.04113, 4.07795, 0, 30.05945), x ≤ 0.5 ,(1.4, 0, 0, 1), x > 0.5 ,

538

on an L-shape computational domain [0, 1]× [6, 11] ∪ [1, 13]× [0, 11] . We use inflow539

boundary condition at x = 0, y ∈ [6, 11], reflective boundary condition at x ∈540

[0, 1], y = 6 and x = 1, y ∈ [0, 6], and outflow boundary condition elsewhere. The541

mesh size is set as 1/16 in both x and y directions. The limiter parameter M is equal542

to 100. In Figure 8, the computed density and pressure at t = 2.3 are plotted, and543

no negative pressure or density is encountered during the simulation.544



Fig. 8. Numerical results at t = 2.3 for shock diffraction problem. Left: density; right: pressure.

5. Conclusions. In this paper, we have developed and analyzed high order545

maximum-principle-satisfying central DG methods for scalar conservation laws as well546

as high order positivity-preserving central DG methods for compressible Euler equa-547

tions in one- and two-dimensional spaces on Cartesian meshes. Based on the standard548

central DG methods, the proposed methods call a maximum-principle-satisfying or549

positivity-preserving limiter at each discrete time and also at each inner stage if multi-550

stage SSP time discretizations are used. Such limiters are easy to implement, and they551

do not increase significantly the computational cost.552

Though the mathematical analysis for these limiters requires a CFL number s-553

maller than that used in standard central DG methods, in practice, one can adopt a554

dynamical strategy, by starting with the usual CFL number (for central DG method-555

s), until the computed solution fails to belong to [m,M ] or H. When this happens,556

the simulation returns to the previous discrete time and a smaller CFL number hence557

time step is taken based on the theorems.558

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