maximum-principle-satisfying and positivity …
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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
This manuscript is for review purposes only.
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
This manuscript is for review purposes only.
4 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
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
This manuscript is for review purposes only.
MPS AND PP CDG METHODS FOR CONSERVATION LAWS 5
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|
This manuscript is for review purposes only.
6 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
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
Here θn = ∆tn/τn ∈ [0, 1], with τn being the maximal time step allowed by the CFL200
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
This manuscript is for review purposes only.
MPS AND PP CDG METHODS FOR CONSERVATION LAWS 7
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
This manuscript is for review purposes only.
8 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
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
This manuscript is for review purposes only.
MPS AND PP CDG METHODS FOR CONSERVATION LAWS 9
Here θn = ∆tn/τn ∈ [0, 1], with τn being the maximal time step allowed by the CFL280
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
This manuscript is for review purposes only.
10 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
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
This manuscript is for review purposes only.
MPS AND PP CDG METHODS FOR CONSERVATION LAWS 11
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
This manuscript is for review purposes only.
12 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
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
This manuscript is for review purposes only.
MPS AND PP CDG METHODS FOR CONSERVATION LAWS 13
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
This manuscript is for review purposes only.
14 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
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
This manuscript is for review purposes only.
MPS AND PP CDG METHODS FOR CONSERVATION LAWS 15
−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
This manuscript is for review purposes only.
16 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
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.
Mesh L1 error Order L2 error Order L∞ error OrderP 1
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
20× 20 1.53E-03 — 1.44E-03 — 8.96E-03 —40× 40 2.11E-04 2.86 1.89E-04 2.93 1.17E-03 2.9380× 80 2.48E-05 3.09 2.33E-05 3.02 1.50E-04 2.97160× 160 3.08E-06 3.01 2.97E-06 2.97 1.89E-05 2.99320× 320 3.90E-07 2.98 3.95E-06 2.91 2.36E-05 3.00
4.2. Compressible Euler equations. In this section, we examine the numer-498
ical performance of the positivity-preserving central DG methods for compressible499
This manuscript is for review purposes only.
MPS AND PP CDG METHODS FOR CONSERVATION LAWS 17
−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.
Mesh L1 error Order L2 error Order L∞ error OrderP 1
8× 8 7.97E-02 — 9.20E-02 — 1.70E-01 —16× 16 1.55E-02 2.36 1.95E-02 2.24 4.63E-02 1.8832× 32 3.18E-03 2.29 3.82E-03 2.35 8.73E-03 2.4164× 64 6.28E-04 2.34 7.85E-04 2.28 2.56E-03 1.77128× 128 1.35E-04 2.22 1.75E-04 2.16 7.10E-04 1.85
P 2
8× 8 3.62E-02 — 4.42E-02 — 9.12E-02 —16× 16 4.69E-04 6.27 6.88E-04 6.01 4.01E-03 4.5132× 32 5.52E-05 3.09 8.47E-05 3.02 5.07E-04 2.9864× 64 6.82E-06 3.02 1.06E-05 2.99 6.35E-05 3.00128× 128 8.51E-07 3.00 1.32E-06 3.00 7.94E-06 3.00
This manuscript is for review purposes only.
18 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
−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
This manuscript is for review purposes only.
MPS AND PP CDG METHODS FOR CONSERVATION LAWS 19
−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
This manuscript is for review purposes only.
20 MAOJUN LI, FENGYAN LI, ZHEN LI AND LIWEI XU
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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