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Journal of Computational Physics 281 (2015) 334–351
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Journal of Computational Physics
www.elsevier.com/locate/jcp
High order parametrized maximum-principle-preserving and
positivity-preserving WENO schemes on unstructured meshes
Andrew J. Christlieb a,b, Yuan Liu a,∗, Qi Tang a, Zhengfu Xu c
a Department of Mathematics, Michigan State University, East Lansing, MI 48824, USAb Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USAc Department of Mathematical Science, Michigan Technological University, Houghton, MI 49931, USA
a r t i c l e i n f o a b s t r a c t
Article history:Received 11 April 2014Received in revised form 21 August 2014Accepted 11 October 2014Available online 24 October 2014
Keywords:Hyperbolic conservation lawsMaximum principle preservingPositivity preservingUnstructured meshesFinite volume schemesWENO schemesCompressible Euler system
In this paper, we generalize the maximum-principle-preserving (MPP) flux limiting technique developed by Xu (2013) [20] to a class of high order finite volume weighted essentially non-oscillatory (WENO) schemes for scalar conservation laws and the compress-ible Euler system on unstructured meshes in one and two dimensions. The key idea of this parameterized limiting technique is to limit the high order numerical flux with a first order flux which preserves the MPP or positivity-preserving (PP) property. The main purpose of this paper is to investigate the flux limiting approach with high order finite volume method on unstructured meshes which are often needed for solving some important problems on irregular domains. Truncation error analysis based on one-dimensional nonuniform meshes is presented to justify that the proposed MPP schemes can maintain third order accuracy in space and time. We also demonstrate through smooth test problems that the proposed third order MPP/PP WENO schemes coupled with a third order Runge–Kutta (RK) method attain the desired order of accuracy. Several test problems containing strong shocks and complex domain geometries are also presented to assess the performance of the schemes.
© 2014 Published by Elsevier Inc.
1. Introduction
In this paper, we are interested in the scalar conservation law:
ut + ∇ · F(u) = 0 (1.1)
and the compressible Euler system:
ξt + f (ξ)x + g(ξ)y = 0, (1.2)
with
ξ =⎛⎜⎝
ρmu
mv
E
⎞⎟⎠ , f (ξ) =
⎛⎜⎝
ρuρu2 + p
ρuvu(E + p)
⎞⎟⎠ , g(ξ) =
⎛⎜⎝
ρvρuv
ρv2 + pv(E + p)
⎞⎟⎠ ,
* Corresponding author.E-mail addresses: [email protected] (A.J. Christlieb), [email protected] (Y. Liu), [email protected] (Q. Tang), [email protected] (Z. Xu).
http://dx.doi.org/10.1016/j.jcp.2014.10.0290021-9991/© 2014 Published by Elsevier Inc.
A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351 335
and
mu = ρu, mv = ρv, E = p
γ − 1+ 1
2ρ(u2 + v2)
where ρ is the density, (u, v)T is the velocity, mu and mv are the momentums, p is the pressure, E is the total energy and γ is the specific heat ratio.
Many numerical methods have been developed for solving (1.1) and (1.2) over the recent decades, such as the discon-tinuous Galerkin (DG) method [2], the finite volume/finite difference essentially non-oscillatory (ENO) schemes [6,10], and finite volume/finite difference WENO schemes [10,12]. Among various methods, WENO schemes are shown to be very robust and efficient especially when solutions may contain discontinuities, sharp gradient regions and other complicated solution structures. Moreover, finite volume WENO schemes have more flexibility in terms of mesh structure compared to finite difference WENO schemes. In particular, finite volume WENO schemes have been applied to unstructured meshes with ar-bitrary partition for complex domain geometries [3,7,13,29], while finite difference WENO schemes can only be applied to a uniform or smoothly varying mapped grid. In this paper, we focus our discussion on finite volume WENO schemes.
The solution to the scalar conservation law (1.1) has MPP property such that, if the initial condition is bounded um ≤u(x, t0) ≤ uM , then um ≤ u(x, t) ≤ uM for all the future times t > t0. Similarly, the solutions to the compressible Euler system have a PP property such that both densities and pressures must maintain positive in every situation. However, the existing high order schemes for solving (1.1) and (1.2) do not necessarily retain the MPP/PP property in the numerical solutions. In particular, if the solution to (1.2) contains low density or pressure, high order schemes might produce negative density or pressure, leading to an ill-posed problem that typically causes failure of the numerical algorithm. The situation was restively stagnant until the recent work by Zhang and Shu [23,24]. In this work, arbitrary high order finite volume WENO schemes and DG methods are developed to preserve MPP and PP properties, by limiting the reconstructed polynomials around cell averages. Following a similar idea, Zhang and Shu extended their schemes to two-dimensional unstructured meshes [25,27]. They further demonstrated that the schemes were able to the maintain designed order of accuracy under a CFL constraint. Afterwards, Hu et al. [9] also developed a flux cut-off limiter applied to finite difference WENO schemes which maintains the PP property for compressible Euler system. More recently, a parametrized MPP flux limiter is developed to maintain the MPP property for scalar hyperbolic conservation laws [11,20]. The main advantage of this new parametrized limiter is that the designed order of accuracy of the base WENO/DG schemes are maintained without excessively restricting the CFL. Later in [19], Xiong et al. improved the CFL constraint and reduced computational cost by applying the parametrized MPP flux limiter to the final RK stage only. It was also proven in [19] that the parametrized MPP flux limiter can maintain up to third order accuracy in space and time for one-dimensional nonlinear scalar conservation laws on uniform meshes. This limiter is also extended to high order PP finite difference WENO schemes for the compressible Euler system in [18] and high order MPP finite volume WENO methods for scalar convection-dominated problems on uniform meshes [21].
In this paper, we will generalize the parametrized flux limiter to finite volume WENO schemes on unstructured meshes, and perform numerical experiments for both scalar equations and the Euler system of compressible gas dynamics. To ac-company the numerical results, error analysis on one-dimensional nonuniform meshes is provided. There are two main types of WENO reconstruction in the literature. In the first type of reconstruction, the order of WENO schemes is not higher than the degree of the reconstructed polynomials on each small stencil. i.e., the nonlinear WENO weights are only designed for the purpose of stability, see [3–5,15,16] for details. The second type consists of WENO schemes whose order of accuracy is higher than the degree of the reconstructed polynomials on each small stencil, see, for example, [7,28,29]. Compared with the first type of WENO schemes, the second type of WENO schemes are more difficult to construct but have a more compact stencil of the same accuracy. Based on those two different approaches, a hybrid approach was also proposed in [13]to deal with distorted local mesh geometries. In the implementation of this work, we will use the second type of WENO reconstruction and the hybrid approach. We refer them as WENO-C and WENO-H respectively in the following sections.
The rest of the paper is organized as follows. In Section 2, we will briefly review finite volume schemes for scalar hyperbolic conservation laws and the parametrized MPP flux limiter. In Section 3, we will provide error analysis on one-dimensional nonuniform meshes to show that the MPP schemes can preserve high order accuracy without any sacrifice on CFL constraints. In Section 4, we will present high order MPP and PP finite volume WENO schemes on two-dimensional unstructured meshes for scalar conservation laws and compressible Euler system. Numerical examples of the third order finite volume WENO schemes for smooth problems and some problems containing strong shocks and complex geometries will be shown in Section 5. The conclusion is given in Section 6.
2. One-dimensional finite volume MPP flux limiter for scalar equations
In this section, we first present the general framework of an MPP limiter applied to a finite volume scheme on a nonuni-form mesh.
We consider a one-dimensional scalar conservation law:{ut + f (u)x = 0, x ∈ [0,1];u(x,0) = u (x),
(2.1)
0
336 A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351
with suitable boundary conditions. We divide the spatial domain [0, 1] into N nonuniform cells:
0 = x 12
< x 32
< · · · < xN+ 12
= 1, (2.2)
and denote
I j = [x j− 12, x j+ 1
2], x j = 1
2(x j+ 1
2+ x j− 1
2),
�x j = x j+ 12
− x j− 12, �x = max
j�x j .
Let u j(t) be the cell averaged solution on the cell I j . Finite volume schemes solve Eq. (2.1) in a conservative form:
d
dtu j(t) + 1
�x j
(H
(u−
j+ 12, u+
j+ 12
) − H(u−
j− 12, u+
j− 12
)) = 0, (2.3)
where u±j+ 1
2are high order approximations to the point values u(x j+ 1
2, tn) from the left and right hand sides respectively,
and H is a numerical flux.The semi-discrete equation (2.3) can be further discretized in time using a high order time integrator. For instance, if a
third order total variation diminishing (TVD) RK method is adopted, we will get
u(1)j = un
j + �tL(un
j
),
u(2)j = un
j + 1
4�t
(L(un
j
) + L(u(1)
j
)),
un+1j = un
j + 1
6�t
(L(un
j
) + 4L(u(2)
j
) + L(u(1)
j
)), (2.4)
where u(k)j and un
j denote the numerical solutions at kth RK stage and at t = tn respectively, and
L(un
j
) = − 1
�x j
(Hn
j+ 12
− Hnj− 1
2
)with
Hnj+ 1
2= H
(u−
j+ 12, u+
j+ 12
).
Then the final stage of RK discretization (2.4) can be rewritten as
un+1j = un
j − λ j(
Hrkj+ 1
2− Hrk
j− 12
), (2.5)
where
λ j = �t
�x j, Hrk
j+ 12
:= 1
6
(Hn
j+ 12
+ 4H (2)
j+ 12
+ H (1)
j+ 12
).
Define um = minx u(x, 0) and uM = maxx u(x, 0). To achieve the MPP property, i.e.,
um ≤ unj − λ j
(Hrk
j+ 12
− Hrkj− 1
2
) ≤ uM , (2.6)
Xu [20] proposed modifying the numerical fluxes Hrkj+ 1
2as
Hrkj+ 1
2= θ j+ 1
2
(Hrk
j+ 12
− h j+ 12
) + h j+ 12. (2.7)
Here h j+ 12
is chosen as a first order monotone flux, which has been shown to be MPP. θ j+ 12
is a limiting parameter to be determined. We are looking for a pair (Λ− 1
2 ,I j, Λ+ 1
2 ,I j) such that the inequality (2.6) holds for any (θ j− 1
2, θ j+ 1
2) ∈
[0, Λ− 12 ,I j
] × [0, Λ+ 12 ,I j
]. Let
Γ Mj = uM − u j + λ j(h j+ 1
2− h j− 1
2), (2.8)
Γ mj = um − u j + λ j(h j+ 1
2− h j− 1
2), (2.9)
and it is easy to see Γ Mj ≥ 0 and Γ m
j ≤ 0. With this notation, the decoupling of inequality (2.6) is carried out as follows,
λ jθ j− 12
(Hrk
j− 12
− h j− 12
) − λ jθ j+ 12
(Hrk
j+ 12
− h j+ 12
) − Γ Mj ≤ 0, (2.10)
λ jθ j− 1
(Hrk
1 − h j− 1
) − λ jθ j+ 1
(Hrk
1 − h j+ 1
) − Γ mj ≥ 0. (2.11)
2 j− 2 2 2 j+ 2 2
A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351 337
As an abbreviation, we denote F j+ 12
= Hrkj+ 1
2− h j+ 1
2. The parameter (θ j− 1
2, θ j+ 1
2) ∈ [0, Λ− 1
2 ,I j] × [0, Λ+ 1
2 ,I j] is determined
by a case-by-case discussion according to the sign of F j− 12
and F j+ 12
. For the inequality regarding the maximum part (2.10), we have the following four cases:
• Case (1): If F j− 12
≤ 0 and F j+ 12
≥ 0, then(ΛM
− 12 ,I j
,ΛM+ 1
2 ,I j
) = (1,1).
• Case (2): If F j− 12
≤ 0 and F j+ 12
< 0, then
(ΛM
− 12 ,I j
,ΛM+ 1
2 ,I j
) =(
1,min
(1,
Γ Mj
−λ j F j+ 12
)).
• Case (3): If F j− 12
> 0 and F j+ 12
≥ 0, then
(ΛM
− 12 ,I j
,ΛM+ 1
2 ,I j
) =(
min
(1,
Γ Mj
λ j F j− 12
),1
).
• Case (4): If F j− 12
> 0 and F j+ 12
< 0,
– If the inequality (2.10) is satisfied with (θ j− 12, θ j+ 1
2) = (1, 1) then(
ΛM− 1
2 ,I j,ΛM
+ 12 ,I j
) = (1,1).
– Otherwise, we choose
(ΛM
− 12 ,I j
,ΛM+ 1
2 ,I j
) =(
Γ Mj
λ j F j− 12
− λ j F j+ 12
,Γ M
j
λ j F j− 12
− λ j F j+ 12
).
The discussion of the inequality regarding the minimum part (2.11) can be carried out similarly and we can obtain the bounds for (Λm
− 12 ,I j
, Λm+ 1
2 ,I j).
Finally, the locally defined limiting parameter is given as
Λ j+ 12
= min(ΛM
+ 12 ,I j
,ΛM− 1
2 ,I j+1,Λm
+ 12 ,I j
,Λm− 1
2 ,I j+1
).
3. Accuracy assessment
In this section, we will show that the flux limiter does not affect the order of accuracy in either space or time when it is applied to a third order finite volume schemes on one-dimensional nonuniform meshes. The result will serve as the foundation for our generalization of this parametrized flux limiter to the two-dimensional unstructured meshes in Section 4.
Theorem 3.1. Consider using third order finite volume spatial discretization and third order RK time stepping to solve the one-dimensional conservation law (2.1) on a nonuniform mesh, assume the global error satisfies
enj = ∣∣un
j − u(
I j, tn)∣∣ = O(�x3 + �t3), ∀n, j, (3.1)
where u(I j, tn) is defined as the averaged cell value of the exact solution u(x, t) at the cell I j and time tn.Then, after applying the parameterized MPP flux limiter, the modified numerical flux Hrk satisfies,∣∣Hrk
j+ 12
− Hrkj+ 1
2
∣∣ = O(�x3 + �t3), ∀ j (3.2)
under the CFL constraint
λmaxu
∣∣ f ′(u)∣∣ ≤ 1 (3.3)
where
λ j = �t
�x j, λ = max
jλ j �x = max
j�x j .
and h 1 in (2.7) is taken as the local Lax–Friedrichs flux.
j+ 2
338 A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351
Proof. In this proof, we only consider the limiter applied to the maximum value part with the understanding that it is similar for the minimum value part. Throughout the proof, we use u j for un
j and u(x) for the exact solution u(x, tn) without ambiguity. We assume the nonuniform mesh satisfies, �x j = r j�x, where r j ∈ [m, 1], m is a positive lower boundary of the mesh partition ratio r j .Case (1): The conclusion holds since Hrk
j+ 12
= Hrkj+ 1
2.
Case (4): In this case, F j− 12
> 0 and F j+ 12
< 0. Due to the construction of the flux limiter, we only need to show (3.2) holds
when θ j+ 12
is taken as Γ M
jλ j F
j− 12−λ j F
j+ 12
. Thus we have,
Hrkj+ 1
2− Hrk
j+ 12
= (θ j+ 12
− 1)F j+ 12
=Γ M
j − (λ j F j− 12
− λ j F j+ 12)
λ j F j− 12
− λ j F j+ 12
F j+ 12. (3.4)
Noticing that F j− 12
> 0 and F j+ 12
< 0, we can see it is sufficient to show∣∣Γ Mj − (λ j F j− 1
2− λ j F j+ 1
2)∣∣ = O
(�x3 + �t3). (3.5)
Equivalently, we only need to show,∣∣uM − (u j − λ j
(Hrk
j+ 12
− Hrkj− 1
2
))∣∣ = O(�x3 + �t3). (3.6)
Because u(I j, tn+1) − (u j − λ(Hrkj+ 1
2− Hrk
j− 12)) = O (�x3 + �t3) by the assumption of accuracy and u(I j, tn+1) ≤ uM ≤
(u j − λ(Hrkj+ 1
2− Hrk
j− 12)), it is easy to see (3.6) holds.
Case (2): Similarly to Case (4), we only need to show,∣∣Hrkj+ 1
2− Hrk
j+ 12
∣∣ = O(�x3 + �t3), (3.7)
when θ j+ 12
= Γ Mj
−λ j Fj+ 1
2
< 1. We get,
Hrkj+ 1
2− Hrk
j+ 12
= (θ j+ 12
− 1)F j+ 12
=uM − (u j − λ j(Hrk
j+ 12
− h j− 12))
−λ j. (3.8)
This is equivalent to show∣∣uM − (u j − λ j
(Hrk
j+ 12
− h j− 12
))∣∣ = O(�x3 + �t3), (3.9)
when
uM − (u j − λ j
(Hrk
j+ 12
− h j− 12
))< 0. (3.10)
We first consider the case where u reaches its maximum at xM ∈ [x j− 12, x j+ 1
2], which implies u′
M = 0 and u′′M ≤ 0. The
Taylor expansion about xM can be written as
u(x) = uM + u′′(xM)
2! (x − xM)2 + u′′′(xM)
3! (x − xM)3 + · · · (3.11)
Therefore, we have
u j = 1
�x j
xj+ 1
2∫x
j− 12
u(x)dx = uM + u′′M
12
(6(x j − xM)2 + 1
2�x2
j
)+ O
(�x3
j
), (3.12)
and
u j−1 = 1
�x j−1
xj− 1
2∫x
j− 32
u(x)dx
= uM + u′′M
(6(x j − xM)2 − 6(x j − xM)(�x j−1 + �x j) + 2�x2
j−1 + 3�x j−1�x j + 3�x2
j
)
12 2
A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351 339
+ O((xm − x j− 3
2)3)
= uM + u′′M
12
(6(x j − xM)2 − 6(r + 1)(x j − xM)�x j +
(2r2 + 3r + 3
2
)�x2
j
)+ O
((�x j + �x j−1)
3), (3.13)
where we use a local factor r = �x j−1/�x j = r j−1/r j for simplicity, and r ∈ [m, 1/m] is a positive number depending on the mesh partition. Note r is still dependent on the index j.
Given the expansions (3.12) and (3.13) and the fact that u′M = 0, the first-order local Lax–Friedrichs flux h j− 1
2=
12 ( f (u j) + f (u j−1) − α j− 1
2(u j − u j−1)) can be expanded as follows,
h j− 12
= f (uM) + f ′(uM)u′′
M
12
(6(x j − xM)2 − 3(r + 1)(x j − xM)�x j +
(r2 + 3
2r + 1
)�x2
j
)
− α j− 12
u′′M
12
(3(r + 1)(x j − xM)�x j −
(r2 + 3
2r + 1
2
)�x2
j
)+ O
((�x j + �x j−1)
3). (3.14)
Here α j− 12
= maxu∈[A,B] | f ′(u)| with A = min(u j−1, u j) and B = max(u j−1, u j).
When a third order finite volume scheme is coupled with third order RK method, the numerical flux generally satisfies,
Hrkj+ 1
2= 1
�t
tn+1∫tn
f(u(x j+ 1
2, t)
)dt + O
(�x3
j + �t3). (3.15)
Using the 3-point Gauss–Lobatto quadrature for (3.15), we have
Hrkj+ 1
2= 1
6f(u(x j+ 1
2, tn + �t
)) + 2
3f
(u
(x j+ 1
2, tn + �t
2
))+ 1
6f(u(x j+ 1
2, tn)) + O
(�x3
j + �t3). (3.16)
Following the characteristics, the flux Hrkj+ 1
2can be approximated by the solution u(x, t) at t = tn ,
Hrkj+ 1
2= 1
6f(u(x j+ 1
2− λ j,1�x j, tn)) + 2
3f(u(x j+ 1
2− λ j,2�x j, tn)) + 1
6f(u(x j+ 1
2, tn))
+ O(�x3
j + �t3), (3.17)
where
λ j,1 = λ j f ′(u(x j+ 1
2− λ j,1�x j, tn)), (3.18)
λ j,2 = λ j
2f ′(u
(x j+ 1
2− λ j,2�x j, tn))
. (3.19)
If we denote λ0 = λ j f ′(uM) and still use the fact u′M = 0, the expansions of (3.18) and (3.19) at x = xM lead to
λ j,1 = λ0 + O(�x2
j
), (3.20)
λ j,2 = λ0
2+ O
(�x2
j
). (3.21)
Substituting (3.20) and (3.21) into (3.17) and performing a Taylor expansion at x = xM results in
Hrkj+ 1
2= f (uM) + f ′(uM)
u′′M
12
(6(x j − xM)2 + (6 − 6λ0)(x j − xM)�x j +
(2λ2
0 − 3λ0 + 3
2
)�x2
j
)+ O
(�x3
j + �t3). (3.22)
If we use the expansions in (3.12), (3.14) and (3.22) and denote z = (x j − xM)/�x j , it is easy to get
u j − λ j(
Hrkj+ 1
2− h j− 1
2
) = uM + u′′M
12�x2
j g(z, λ0) + O((�x j + �x j−1)
3 + �t3)= uM + u′′
M
12�x2
j g(z, λ0) + O((r j + r j−1)
3�x3 + �t3) (3.23)
with
g(z, λ0) = 6z2 + λ0(6λ0 − 9 − 3r)z − λ0
(2λ2
0 − 3λ0 − r2 − 3
2r + 1
2
)+ 1
2
− λα
(3(r + 1)z − r2 − 3
2r − 1
2
), (3.24)
where we denote λα = λ jα 1 . We will discuss g(z, λ0) based on the following two cases:
j− 2
340 A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351
• If f ′(uM) ≥ 0, we have λ0 = λ j f ′(uM) ∈ [0, 1] since λ j maxu | f ′(u)| ≤ 1, and we can write g(z, λ0) as
g(z, λ0) = g1(z, λ0) +(
3(r + 1)z − r2 − 3
2r − 1
2
)(λ0 − λα), (3.25)
with
g1(z, λ0) = 6z2 + 6λ0(λ0 − 2 − r)z − λ0(2λ2
0 − 3λ0) + λ0
(2r2 + 3r
) + 1
2. (3.26)
One can numerically check that g1(z, λ0) ≥ 0 when λ0 ∈ [0, 1] and r ∈ [m, 1/m]. Notice that |λα − λ0| = O (�x), we will have g(z, λ0) = g1(z, λ0) + O (�x) with g1(z, λ0) ≥ 0. Since u′′
M ≤ 0, from (3.23) we obtain∣∣uM − (u j − λ j
(Hrk
j+ 12
− h j− 12
))∣∣ = O(r2
j �x3) + O((r j + r j−1)
3�x3 + �t3)= O
((r2
j + r2j−1
)�x3 + �t3). (3.27)
This is equivalent to (3.9) since r j and r j−1 is bounded.• If f ′(uM) < 0, we have λ0 = λ j f ′(uM) ∈ [−1, 0] and |λα + λ0| = O (�x). Hence, we can rewrite g(z, λ0) as
g(z, λ0) = g2(z, λ0) −(
3(r + 1)z − r2 − 3
2r − 1
2
)(λ0 + λα) (3.28)
with
g2(z, λ0) = 6z2 + 6λ0(λ0 − 1)z − λ0(2λ2
0 − 3λ0 + 1) + 1
2(3.29)
Similarly, one can numerically check that g2(z, λ0) ≥ 0 when λ0 ∈ [−1, 0]. This implies that g(z, λ0) = g2(z, λ0) + O (�x)with g2(z, λ0) ≥ 0. Since u′′
M ≤ 0, from (3.23), we can also obtain (3.9).
Thus, we finish the proof of (3.9) when xM ∈ I j .For the case when xM /∈ I j , if there is a local maximum point inside the cell, the above analysis still holds. We then
consider the case when u(x) reaches its local maximum over I j at x j− 12
, which implies that u′j− 1
2< 0. By performing a
similar Taylor expansion at x j− 12
and denoting λ0 = λ j f ′(u j− 12), we will have
u j − λ j(
Hrkj+ 1
2− h j− 1
2
) = u j− 12
+ �x ju′j− 1
2s1 + �x2
j
(u′
j− 12
)2s2 + �x2
j
2u′′
j− 12
s3 + O(�x3
j + �x3j−1 + �t3), (3.30)
where
s1 = λ20
2− r + 3
4λ0 + 1
2− r + 1
4λα, (3.31)
s2 = −λ j f ′′(u j− 12)
(1
2λ2
0 − λ0 + 1
2− r2 + 1
16
), (3.32)
s3 = 1
6g
(1
2, λ0
)= −1
3λ3
0 + λ20 + r2 − 5
6λ0 + r2 − 1
6λα + 1
3. (3.33)
Here s3 is computed simply by g(z, λ0) because the term involving �x2j u′′
j− 12
has the same formulae as the situation when xm ∈ I j and we just need to take xm = x j− 1
2here. Hence, we see immediately s3 ≥ 0 because of the previous discussion of
g(z, λ0) when xm ∈ I j . We will also discuss the expansion (3.30) using two scenarios:
• If f ′(u j− 12) ≥ 0, similarly as the previous discussion, we have λα − λ0 = O (�x) and λ0 ∈ [0, 1]. The expansion (3.30)
can be rewritten as,
u j − λ j(
Hrkj+ 1
2− h j− 1
2
) = u(x j− 12
− √s3�x j) + �x ju
′j− 1
2
(1
2
(λ2
0 − (r + 2)λ0 + 1) + √
s3
)
+ �x2j u′
j− 12
s4 + O((
r3j + r3
j−1
)�x3 + �t3), (3.34)
where s4 = u′j− 1
2s2 − (1 + r)(λα −λ0)/(4�x j) = O (1). One can numerically check 1
2 (λ20 − (r + 2)λ0 + 1) +√
s3 ≥ 0 when λ0 ∈ [0, 1] and r ∈ [m, 1/m].
A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351 341
• If f ′(u j− 12) < 0, similarly, we have λα + λ0 = O (�x) and λ0 ∈ [−1, 0]. The expansion (3.30) now becomes,
u j − λ j(
Hrkj+ 1
2− h j− 1
2
) = u(x j− 12
− √s3�x j) + �x ju
′j− 1
2
(1
2
(λ2
0 − λ0 + 1) + √
s3
)+ �x2
j u′j− 1
2s4 + O
((r3
j + r3j−1
)�x3 + �t3), (3.35)
where s4 = u′j− 1
2s2 − (r + 1)(λα + λ0)/(4�x j) = O (1). Obviously, 1
2 (λ20 − λ0 + 1) + √
s3 ≥ 0.
In the above two scenarios, the expansions result in similar formulas to those given in the second part of the proof of Theorem 3.2 in [19], and satisfy the exact same relations as those in that proof. Using the same argument as the proof of [19], it can be shown one of the relations u(x j− 1
2− √
s3�x j) + �x2j u′
j− 12
s4 ≤ uM and u′j− 1
2= O (�x j) holds. When the
condition (3.10) is used, the relation u(x j− 12
− √s3�x j) + �x2
j u′j− 1
2s4 ≤ uM or u′
j− 12
= O (�x j) indicates,
∣∣uM − (u j − λ j
(Hrk
j+ 12
− h j− 12
))∣∣ = O((
r3j + r3
j−1
)�x3 + �t3). (3.36)
This is also equivalent to (3.9). Hence, we finish the proof for the case when u(x) reaches its local maximum at x j− 12
.
The proof for the case when u(x) reaches its local maximum at x j+ 12
is similar. Combining the above discussion, we complete the discussions of Case (2).Case (3): This case can be proved similarly to Case (2). �4. Generalization of flux limiter on triangular mesh
4.1. Finite volume WENO schemes on triangular meshes
In our generalization, we consider a two dimensional conservation law (1.1) solved by finite volume WENO schemes with the flux tensor F = ( f , g)T . The computational control volumes for our schemes are triangles.
Taking the triangle j as our control volume, we formulate the semi-discrete finite volume scheme for (1.1) as
du j(t)
dt+ 1
| j|∫
∂ j
F · n dS = 0, (4.1)
where the cell average u j(t) = 1| j |
∫ j
u dxdy, and n is the outward unit normal of the triangle boundary ∂ j .
In (4.1), the line integral is discretized by a q-point Gaussian quadrature formula,∫∂ j
F · n dS ≈3∑
k=1
S j,k
q∑i=1
wiF(u(G(k)
i , t)) · nk, (4.2)
where S j,k is the length of the kth side of ∂ j , G(k)j and w j are the Gaussian quadrature points and weights respectively,
and F(u(G(k)i , t)) ·nk is approximated by a numerical flux H
G(k)i
. For example, we choose the Lax–Friedrichs flux in this paper, which is given by
HG(k)
i= 1
2
[(F(u−(
G(k)i , t
)) + F(u+(
G(k)i , t
))) · nk − α(u+(
G(k)i , t
) − u−(G(k)
i , t))]
, (4.3)
where α is taken as an upper bound for the magnitude of the eigenvalues of the Jacobian in the nk direction, and u− and u+ are the approximations of the solution u(x, t) inside the triangle and outside the triangle (located at the neighboring triangle) at the Gaussian point.
In this paper, the third order finite volume WENO schemes are used to test the parametrized flux limiter. Hence, it is enough to use the two-point Gaussian quadrature (i.e., q = 2) for approximation of the flux integral. For a line with endpoints P1 and P2, the two-point Gaussian quadrature points are G1 = c P1 + (1 − c)P2, G2 = c P2 + (1 − c)P1, where c = 1
2 +√
36 ; and the corresponding Gaussian quadrature weights are w1 = w2 = 1
2 . And the values of u− and u+ are obtained through WENO reconstruction procedure.
4.2. High order MPP schemes on triangular meshes
In this subsection, we will present our finite volume MPP scheme on the triangular mesh for two dimensional (2D) scalar conservation law (1.1). The semi-discrete scheme (4.1) in this case can be rewritten as
342 A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351
du j(t)
dt+ 1
| j|3∑
k=1
S j,k H j,k = 0, (4.4)
where the numerical flux H j,k is a high order flux but not MPP in general. The semi-discrete form (4.4) can be further discretized by a general time integrator. For simplicity, we only present our limiters by using the forward Euler discretiza-tion and the extension to a high order RK method is straightforward if the technique in Section 2 is used. Under this discretization, the scheme becomes
un+1j = un
j −3∑
k=1
λ j,k H j,k, (4.5)
where λ j,k = �t| j | S j,k . To preserve the MPP property, the modified numerical flux H j,k takes the form of
H j,k = θ j,k(H j,k − h j,k) + h j,k (4.6)
where h j,k is chosen as the first order monotone flux. It is known that the numerical flux h j,k satisfies the MPP property under the CFL constraint,
α�t
| j|3∑
k=1
S j,k ≤ 1. (4.7)
Thus, we need to find out θ j,k ∈ [0, Λ j,k] such that um ≤ un+1j ≤ uM , i.e.,
um ≤ unj −
3∑k=1
λ j,k H j,k ≤ uM , (4.8)
which implies
um ≤ unj −
3∑k=1
λ j,k(θ j,k(H j,k − h j,k) + h j,k
) ≤ uM . (4.9)
To find the upper bound Λ j,k such that the inequalities (4.9) hold for θ j,k ∈ [0, Λ j,k], we need to discuss two inequalities separately. The right inequality of (4.9) can be rewritten as
3∑k=1
θ j,k(−λ j,k(H j,k − h j,k)
) ≤ Γ Mj , (4.10)
with
Γ Mj = uM −
(un
j −3∑
k=1
λ j,kh j,k
)≥ 0. (4.11)
Denoting F j,k = −λ j,k(H j,k − h j,k), we carry out the following discussion based on the sign of F j,k ,
• If F j,k ≤ 0 for k = 1, 2, 3, then(ΛM
j,1,ΛMj,2,Λ
Mj,3
) = (1,1,1).
• If F j,1 ≤ 0, F j,2 ≤ 0, and F j,3 > 0, then
(ΛM
j,1,ΛMj,2,Λ
Mj,3
) =(
1,1,min
(1,
Γ Mj
F j,3
)).
• If F j,1 ≤ 0, F j,2 > 0, and F j,3 > 0, then
(ΛM
j,1,ΛMj,2,Λ
Mj,3
) =(
1,min
(1,
Γ Mj
F j,2 + F j,3
),min
(1,
Γ Mj
F j,2 + F j,3
)).
• If F j,1 > 0, F j,2 > 0, and F j,3 > 0, then
(ΛM
j,1,ΛMj,2,Λ
Mj,3
) =(
min
(1,
Γ Mj
F j,1 + F j,3 + F j,2
),min
(1,
Γ Mj
F j,1 + F j,3 + F j,2
),min
(1,
Γ Mj
F j,1 + F j,3 + F j,2
)).
A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351 343
• For the other similar cases, there is a similar upper bound (ΛMj,1, Λ
Mj,2, Λ
Mj,3). We omit the details here.
Similarly, the left inequality of (4.9) can be rewritten as
3∑k=1
θ j,k(−λ j,k(H j,k − h j,k)
) ≥ Γ mj , (4.12)
with
Γ mj = um −
(un
j −3∑
k=1
λ j,kh j,k
)≤ 0. (4.13)
Similar to the maximum case, the discussion is also carried out case-by-case,
• If Fk ≥ 0 for k = 1, 2, 3, then(Λm
j,1,Λmj,2,Λ
mj,3
) = (1,1,1).
• If F j,1 ≥ 0, F j,2 ≥ 0, and F j,3 < 0, then
(Λm
j,1,Λmj,2,Λ
mj,3
) =(
1,1,min
(1,
Γ mj
F j,3
)).
• If F j,1 ≥ 0, F j,2 < 0, and F j,3 < 0, then
(Λm
j,1,Λmj,2,Λ
mj,3
) =(
1,min
(1,
Γ mj
F j,2 + F j,3
),min
(1,
Γ mj
F j,2 + F j,3
)).
• If F j,1 < 0, F j,2 < 0, and F j,3 < 0, then
(Λm
j,1,Λmj,2,Λ
mj,3
) =(
min
(1,
Γ mj
F j,1 + F j,3 + F j,2
),min
(1,
Γ mj
F j,1 + F j,3 + F j,2
),min
(1,
Γ mj
F j,1 + F j,3 + F j,2
)).
• For the other cases, there is a similar upper bound (Λmj,1, Λ
mj,2, Λ
mj,3). We omit the details here.
Finally, this whole procedure will result in four upper bounds on each edge of the triangles. To be conservative on each edge, θ j,k can be simply taken as the minimum of the calculated bounds, i.e.
θ j,k = min(ΛM
j,k,Λmj,k,Λ
Mj,k
,Λmj,k
), (4.14)
where j is the immediate neighbor cell of j sharing the edge k and k denote the marker of the edge k on the neighbor cell j.
4.3. High order PP schemes for compressible Euler system on triangular meshes
In this subsection, we will present our finite volume PP schemes for the compressible Euler system (1.2) on triangular meshes when the WENO reconstructions [7,13] are used.
For the compressible Euler system, we still present the PP limiter by using a WENO scheme coupled with a forward Euler method for simplicity. In this case, the scheme has the form of
ξn+1j = ξn
j −3∑
k=1
λ j,k H j,k, (4.15)
where the numerical flux H j,k is a high order flux but can possibly result in a negative density or pressure. To achieve the positivity of density and pressure, we need to find a modification of the numerical flux,
H j,k = θ j,k(H j,k − h j,k) + h j,k (4.16)
such that,{ρn+1
j > 0,
pn+1 > 0.(4.17)
j
344 A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351
Here the numerical flux h j,k is typically a first order flux that guarantees the positivity of density and pressure. In simula-tions, positivity preservation is implemented using{
ρn+1j ≥ ερ,
pn+1j ≥ εp,
(4.18)
where the small positive number is defined by
ερ = minj
(ρn+1
j ,10−13),εp = min
j
(pn+1
j ,10−13).Here ρn+1
j and pn+1j are density and pressure obtained using the first order PP flux h j,k . The process needs two main steps
to satisfy (4.18). In the following steps, we denote the flux component of the density as f ρ for H , f ρ for H and f ρ for h.
1. Find the limiting parameter θ j,k to preserve the positivity of the density
ρn+1j = ρn
j −3∑
k=1
λ j,k f ρj,k ≥ ερ.
The limiting parameters θ j,k needs to satisfy,
Γ j −3∑
k=1
λ j,k(θ j,k
(f ρ
j,k − f ρj,k
)) ≥ ερ
where Γ j = ρnj − ∑3
k=1 λ j,k f ρj,k ≥ ερ . Equivalently, the inequality can be rewritten as
3∑k=1
θ j,k(−λ j,k
(f ρ
j,k − f ρj,k
)) ≥ ερ − Γ j,
where ερ − Γ j ≤ 0. This is exactly the same as the form used to determine the upper bound of θ j,k for the minimum case of the scalar conservation law in Section 4.2. Following the same procedure discussed in that case, we can find a set where the calculated density is positive (i.e., ρn+1
j ≥ ερ ):
Sρ = {(θ j,1, θ j,2, θ j,3) : (θ j,1, θ j,2, θ j,3) ∈ [
0,Λρj,1
] × [0,Λ
ρj,2
] × [0,Λ
ρj,3
]}.
2. Find the limiting parameters θ j,k within the region Sρ to guarantee the positivity of the pressure. In other words, we seek a sufficient condition such that the pressure pn+1
j obtained using the modified flux satisfies,
pn+1j (θ j,1, θ j,2, θ j,3) = (γ − 1)
(En+1
j − 1
2
((mu)n+1j )2 + ((mv)n+1
j )2
ρn+1j
)≥ εp .
Because the pressure p is a concave function of (ρ, mu, mv , E), the set
S p = {(θ j,1, θ j,2, θ j,3) ∈ Sρ : pn+1
j (θ j,1, θ j,2, θ j,3) ≥ εp}
is convex. Thanks to this property, we only need to discuss the vertices of the set Sρ to find this set.Let the eight vertices of Sρ be denoted by
Ak1,k2,k3 = (k1Λ
ρj,1,k2Λ
ρj,2,k3Λ
ρj,3
),
with k j equal 0 or 1. We propose the following strategy to find the upper bounds of the limiting parameters θ j,k to maintain positive density and pressure.(a) For (k1, k2, k3) �= (0, 0, 0), if p(Ak1,k2,k3 ) ≥ εp , let Bk1,k2,k3 = Ak1,k2,k3 ; otherwise find r such that p(r Ak1,k2,k3 ) ≥ εp
and let Bk1,k2,k3 = r Ak1,k2,k3 . The resulting seven Bk1,k2,k3 with the origin (0, 0, 0) form a convex polyhedron S p in Sρ .
(b) We need to find a rectangular cuboid inside this convex polyhedron, which can be simply determined as,
Rρ,p = [0,Λ j,1] × [0,Λ j,2] × [0,Λ j,3],where
Λ j,1 = mink2=0,1k3=0,1
(B1,k2,k3
1
), Λ j,2 = min
k1=0,1k3=0,1
(Bk1,1,k3
2
), Λ j,3 = min
k1=0,1k2=0,1
(Bk1,k2,1
3
).
A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351 345
Fig. 5.1. Sample mesh used for linear equation and Burgers’ equation.
Table 5.1Accuracy for 2D linear equation without parametrized flux limiter. Computational meshes are refined versions of the mesh in Fig. 5.1. T = 2.0, CFL = 0.6.
Nee L1 error Order L∞ error Order (u)min
2816 6.98E−02 1.69 3.35E−01 0.60 −8.68E−0311 264 1.05E−02 2.73 5.83E−02 2.52 −1.34E−0245 056 1.08E−03 3.28 4.80E−03 3.60 −2.22E−03
180 224 1.36E−04 2.99 5.03E−04 3.25 −2.87E−04720 896 1.71E−05 2.99 6.33E−05 2.99 −3.63E−05
Finally, this whole procedure will result in two upper bounds on each edge of the triangles. To be conservative on each edge, θ j,k can be simply taken as the minimum of all the calculated bounds.
5. Numerical examples
In this section, we test the proposed finite volume MPP and PP WENO schemes on two-dimensional unstructured meshes. A third order TVD RK method serves as the time integrator. For all the examples of the compressible Euler system, γ is taken as 1.4. We remark that if there are negative density and pressure in the intermediate stages of the RK method, we take the speed of the sound as c =
√γ |p|
|ρ| .
Example 5.1. Linear equation:
ut + ux + u y = 0, (5.1)
with the initial condition
u(x, y,0) = sin4(
π
2(x + y)
), (5.2)
and periodic boundary conditions on each side of the domain (x, y) ∈ [−2, 2] × [−2, 2].The problem is solved by the third order WENO-H scheme to test the performance of the parametrized MPP flux lim-
iter. The domain is discretized into a triangular mesh such as Fig. 5.1. The convergence study is based on the refinement of meshes by cutting each triangle into four smaller similar ones. The numerical results at T = 2.0 without and with a parametrized flux limiter are reported in Tables 5.1 and 5.2 respectively. We use Nee to denote the number of cells in the domain in the tables. From Tables 5.1 and 5.2, we can observe third order accuracy in L1 and L∞ errors, and the scheme with a parametrized flux limiter clearly maintains MPP property.
Example 5.2. Nonlinear Burgers’ equation:
ut +(
u2
2
)+
(u2
2
)= 0, (5.3)
x y
346 A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351
Table 5.2Accuracy for 2D linear equation with parametrized flux limiter. Computational meshes are refined versions of the mesh in Fig. 5.1. T = 2.0, CFL = 0.6.
Nee L1 error Order L∞ error Order (u)min
2816 6.99E−02 1.69 3.46E−01 0.55 8.32E−1311 264 9.14E−03 2.94 5.83E−02 2.57 1.03E−1645 056 1.07E−03 3.09 4.80E−03 3.60 1.48E−16
180 224 1.37E−04 2.97 5.03E−04 3.25 1.30E−18720 896 1.71E−05 3.00 7.13E−05 2.82 8.55E−20
Table 5.3Accuracy for 2D Burgers’ equation without parametrized flux limiter. Computational meshes are refined versions of the mesh in Fig. 5.1. T = 0.5/π2, CFL = 0.6.
Nee L1 error Order L∞ error Order (u)min
2816 9.19E−03 2.20 1.08E−01 1.34 −2.26E−0311 264 9.27E−04 3.31 2.05E−02 2.40 −6.83E−0445 056 8.12E−05 3.51 1.21E−03 4.08 −8.63E−05
180 224 1.01E−05 3.01 1.86E−04 2.70 −1.11E−05720 896 1.27E−06 2.99 2.97E−05 2.65 −1.36E−06
Table 5.4Accuracy for 2D Burgers’ equation with parametrized flux limiter. Computational meshes are refined versions of the mesh in Fig. 5.1. T = 0.5/π2, CFL = 0.6.
Nee L1 error Order L∞ error Order (u)min
2816 9.12E−03 2.21 1.08E−01 1.34 6.73E−1711 264 9.24E−04 3.30 2.05E−02 2.40 1.08E−2145 056 8.11E−05 3.51 1.21E−03 4.08 8.85E−22
180 224 1.01E−05 3.01 1.86E−04 2.70 2.81E−19720 896 1.27E−06 2.99 2.97E−05 2.65 4.89E−19
with the initial condition
u0(x, y) = sin4(
π
2(x + y)
), (5.4)
and periodic boundary conditions on each side of the domain (x, y) ∈ [−2, 2] × [−2, 2].The problem is solved by the third order WENO-H scheme. The errors of the scheme without and with the flux limiter
are shown in Tables 5.3 and 5.4 for T = 0.5/π2, when the solution is still smooth. It is easy to observe third order accuracy in both cases, and the scheme with flux limiter preserves the MPP property. It is also important to note the errors with or without the flux limiter are comparable, indicating the limiter does not affect the accuracy of the base WENO scheme.
Example 5.3 (2D vortex evolution problem [7]). In this example, we solve the compressible Euler equations (1.2) with the following setup. The mean flow is ρ = 1, p = 1, and (u, v) = (1, 1). An isentropic vortex is added to the mean flow, through perturbations in (u, v) and the temperature T = p/ρ , with no perturbation in the entropy S = p/ργ :
(δu, δv) = ε
2πe0.5(1−r2)(− y, x)
δT = − (γ − 1)ε2
8γπ2e1−r2
, δS = 0,
where (x, y) = (x − 5, y − 5), r2 = x2 + y2. We set the vortex strength ε = 10.0828 such that the lowest density and lowest pressure of the exact solution are 7.8 × 10−15 and 1.7 × 10−20. The computational domain is taken as [0, 10] ×[0, 10] and is partitioned by triangles. Periodic boundary conditions are used on each side. The exact solution of this problem is smooth, hence it is often used as a benchmark problem for testing accuracy of numerical schemes for solving Euler systems. We test the flux limiter with a third order WENO-H scheme on a triangular mesh and the numerical results are reported in Table 5.5 for T = 1.0. Again, we observe good third order accuracy with positive density and pressure.
Example 5.4 (Double mach reflection problem [17] on distorted meshes). We solve the Euler system (1.2) in the computational domain [0, 3.2] × [0, 1]. A reflecting wall lies at the bottom side of the domain starting from x = 1
6 . Initially a right moving Mach 10 shock is located at x = 1
6 , y = 0, making a 60◦ angle with the x axis. The reflective boundary condition is used at the wall. The exact postshock condition is imposed for the rest of the bottom side (i.e., the part from x = 0 to x = 1 ). At
6
A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351 347
Table 5.5Accuracy for 2D compressible Euler system with parametrized flux limiter for mean flow problem. CFL = 0.6, T = 1.0.
Nee L1 error Order L∞ error Order (u)min (p)min
3042 5.82E−03 1.41 1.34E−01 0.75 0.1334 0.172212 482 1.82E−03 1.68 4.36E−02 1.62 4.37E−05 1.00E−1350 562 1.11E−04 4.04 5.86E−03 2.90 1.46E−05 1.00E−13
203 522 9.59E−06 3.53 7.87E−04 2.90 2.17E−06 1.00E−13816 642 1.10E−06 3.12 8.17E−05 3.27 1.35E−06 1.00E−13
Fig. 5.2. Convergence study of double mach reflection. Density contours with 30 equally spaced contour lines from 1.5 to 21.5. CFL = 0.6. (a) 73 074 elements. (b) 294 630 elements. (c) 1 178 060 elements.
the top boundary, the flow values are set to describe the exact motion of the Mach 10 shock. The final time is T = 0.2. The mesh we used is the same as the one in [13], which is obtained by randomly perturbing the uniform computational meshes with equilateral triangles within 30% of every interior node. It was observed in [13] that the high order finite volume WENO-H schemes have severe CFL constraints due to poor mesh quality and the code blows up with the same setup. We test the parametrized flux limiter by WENO-H scheme with a much larger CFL number of 0.6 and present the numerical solution in Fig. 5.2, which is consistent with the result in [7]. It indicates the importance of maintaining PP in numerical solutions to the Euler system and our proposed approach can deal with the situation when the computational domain is partitioned into a poor quality mesh.
Example 5.5 (Shock diffraction problem). On a domain of {[0, 1] ×[6, 11]} ∪{[1, 13] ×[0, 11]}, a shock of Mach = 5.09, initially located at x = 0.5 and 6 ≤ y ≤ 11, moves into undisturbed air with a density of 1.4 and pressure of 1. The boundary conditions are inflow at {x = 0, 6 ≤ y ≤ 11}, outflow at {x = 13, 0 ≤ y ≤ 11}, {1 ≤ x ≤ 13, y = 0} and {0 ≤ x ≤ 13, y = 11}and reflective at walls {0 ≤ x ≤ 1, y = 6} and {x = 1, 0 ≤ y ≤ 6}. The computational domain is partitioned by triangular meshes. We test the PP flux limiter with WENO-C scheme. The density and pressure at T = 2.3 are presented in Fig. 5.3, which is consistent with the results obtained by finite difference WENO schemes in [26].
348 A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351
Fig. 5.3. Left: density, 20 equally spaced contour lines from ρ = 0.07 to ρ = 7.07; Right: pressure, 40 equally spaced contour lines from p = 0.01 to p = 31. CFL = 0.6.
Fig. 5.4. Left: density, 20 equally spaced contour lines from ρ = 0.5 to ρ = 7.5; Right: pressure, 40 equally spaced contour lines from p = 2 to p = 115. CFL = 0.6.
The second example is to study a Mach 10 shock diffracting at a 120◦ convex corner. We also partitioned the computa-tional domain with triangular meshes. The initial condition is a shock of Mach = 10, initially located at {x = 3.4, 6 ≤ y ≤ 11}, moving into undisturbed air with a density of 1.4 and pressure of 1. The contour plots of density and pressure at T = 0.9are given in Fig. 5.4.
Example 5.6 (NACA0012 airfoil problem). In this example, we consider inviscid Euler transonic flow past a single NACA0012 airfoil configuration. The computational domain is [−15, 15] × [−15, 15]. We consider two different setups for this test: the first case has Mach number M∞ = 0.8, angle of attack α = 1.25◦ and the second case has M∞ = 4.0, angle of attack α = 25◦ . The mesh used in the simulation is shown in Fig. 5.5.
We solve this problem using the third order WENO-C scheme. The contours of Mach number and the pressure distribu-tion around airfoil of the first case are presented in Fig. 5.6, where we can clearly observe a strong shock on the upper edge of the airfoil and a weak shock on the lower edge of the airfoil. This result also shows good agreement with the results in Refs. [1,8]. We remark here that there is no need of PP limiter for the case M∞ = 0.8.
However, with an increase in the Mach number and attack angle, a low pressure may appear in numerical solutions. This phenomenon is demonstrated by the second case M∞ = 4.0. Although it is not a standard test case, it is initialized in such a way as to make the WENO-C scheme without PP limiter blow up in finite time because of the presence of a negative pressure. For this case, the WENO-C scheme with the proposed PP limiter can successfully solve the problem. The result in this case is presented in Fig. 5.7.
A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351 349
Fig. 5.5. NACA0012 airfoil mesh.
Fig. 5.6. NACA0012 airfoil. Mach number is 0.8 with attack angle 1.25◦ . Left: 30 equally spaced contour lines for Mach number from 0.17 to 1.33; Right: Pressure distribution around airfoil. CFL = 0.8. Here CP is the pressure coefficient and C is the chord length.
Example 5.7 (Double ellipse problem). In this example, we consider supersonic flow around a double ellipse configuration. The double ellipse configuration is defined by
(x
0.06
)2
+(
y
0.015
)2
= 1,
(x
0.035
)2
+(
y
0.025
)2
= 1 (5.5)
with (x, y) ∈ [−0.1, 0.016] ×[−0.05, 0.1]. The mesh used in the real simulation is a refined version of that shown in Fig. 5.8. The initial condition is a freestream with Mach number of 8.15 at an angle of attack of 30◦ . A reflective boundary condition is imposed at the surface of the double ellipse configuration. Inflow and outflow boundary conditions are applied at the farfield boundaries. The simulation result solved by WENO-C scheme with PP limiter is reported in Fig. 5.8 which shows significant shock resolution. This result is comparable with the results in Refs. [14,22]. We also remark that the third order WENO-C code blows up in finite time if the PP limiter is not applied. The last two examples also show our numerical approach can deal with different complicated geometries.
350 A.J. Christlieb et al. / Journal of Computational Physics 281 (2015) 334–351
Fig. 5.7. NACA0012 airfoil. Mach number is 4.0 with attack angle 25◦ . Left: 30 equally spaced contour lines for Mach number from 0.5 to 3.5; Right: Pressure distribution around airfoil. CFL = 0.8.
Fig. 5.8. Double ellipse. Mach number is 8.15 with attack angle 30◦ . Left: Sample mesh; Right: 30 equally spaced contour lines for Mach number from 0.5 to 3.5. CFL = 0.8.
6. Conclusion
We generalized the MPP flux limiter developed in [20] to the high order finite volume WENO methods on two-dimensional unstructured meshes, solving scalar conservation laws as well as providing PP flux limiters for compressible Euler equations. We proved that the parametrized limiter preserves high order accuracy in one-dimensional case on a nonuniform mesh, without any CFL constraints. Extensive numerical tests in two dimensions were performed to demon-strate that the high order schemes with a parameterized limiter preserve the MPP/PP property while attaining high order accuracy. Similar approach can also be generalized to any flux based schemes for three-dimensional problems.
Acknowledgements
AJC is supported by AFOSR grants FA9550-11-1-0281, FA9550-12-1-0343 and FA9550-12-1-0455, NSF grant DMS-1115709, and MSU Foundation grant SPG-RG100059. ZX is supported by NSF grant DMS-1316662. YL and QT would like to thank Pro-fessor Guanghui Hu in University of Macau for helpful discussions.
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