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J Sci Comput
DOI 10.1007/s10915-017-0587-4
TECHNICAL NOTE
An Improved Third-Order WENO-Z Scheme
Weizheng Xu1
· Weiguo Wu1
Received: 7 May 2017 / Revised: 19 September 2017 / Accepted: 19 October 2017
© Springer Science+Business Media, LLC 2017
Abstract In this paper, we develop an improved third-order WENO-Z scheme. Firstly, a
new reference smoothness indicator is derived by slightly modifying that of WENO-N3
scheme proposed by Wu and Zhang (Int. J. Numer. Meth. Fl. 78:162–187, 2015). Then a
new term is added to the weights of the developed scheme to further slightly increase the
weight of less-smooth stencil. Some numerical experiments are provided to demonstrate that
the improved scheme is stable and significantly outperforms the conventional third-order
WENO scheme of Jiang and Shu, while providing essentially non-oscillatory solutions near
strong discontinuities.
Keywords WENO schemes · Smoothness indicators · High-resolution · Hyperbolic
conservation laws
Mathematics Subject Classification 65M12 · 65M70 · 41A10
1 Introduction
The classical Weighted essentially non-oscillatory scheme (WENO-JS scheme) which is
first proposed in [13] and then improved by Jiang and Shu [9] has become a quite successful methodology for solving problems with strong discontinuities and complicated smooth
solutions in computational fluid dynamics. However, WENO-JS is too dissipative to solve
problems with lots of structures in the smooth part of the solution, such as direct numerical
simulation of turbulent flows. In [8], Henrick et al. find that this scheme fails to achieve the
maximum convergence order of the scheme at critical points where the first derivative van-
B
Weizheng Xu
[email protected]
Weiguo Wu
[email protected]
1
Key Laboratory of High Performance Ship Technology, Wuhan University of Technology,
Wuhan 430063, China
123
J Sci Comput
ishes but the third order derivative does not. In order to amend this drawback, an improved
fifth-order WENO scheme (called WENO-M) has been suggested by incorporating a mapping procedure to maintain the nonlinear weights of the convex combination of stencils
as near as possible to the linear weights of fifth order accuracy except for discontinuities.
Inspired by the study in [8], Borges et al. [2] designed another fifth-order WENO scheme
(called WENO-Z) by adding a higher order reference new smoothness indicator which is
obtained by a linear combination of the classical smoothness indicator of WENO-JS. The
new scheme can achieve superior results with almost the same computational effort of the
classical WENO method. Castro et al. [3] developed a general formula for the higher order
smoothness indicators and extended the WENO-Z scheme to all odd orders of accuracy.
Several approaches have been considered for the improvement of the WENO-Z scheme
in recent years. One of the effective approaches is through the design of the smoothness
indicators [4–7,19–21]. Aforementioned articles mainly focus on improving the order of
accuracy of the WENO-Z scheme at critical points. However, It should be mentioned that the
common characteristic of improved schemes (WENO-Z and WENO-M schemes) is that they
both assign larger weights to less-smooth stencils and while keep the ENO property. And
such a strategy indeed decreases the dissipation around the discontinuous region. Acker et al.
[1] present an improved fifth-order WENO-Z scheme (called WENO-Z+ scheme) through
adding a new term to the fifth-order WENO-Z weights to further increase the relevance of
less-smooth substencils. The improved scheme attains much better resolution at the smooth
parts of the solution, while keeping the same numerical stability of the original WENO-Z at
shocks and discontinuities.
Based on the idea that increasing the contribution of the less smooth substencils can
improve the performance of the conventional WENO-Z scheme, in the present work, we firstly
devised a new reference smoothness indicator for the third-order WENO-Z scheme through
slightly modifying that of the WENO-N3 scheme proposed in [20]. In fact, the purpose of
this modification is to slightly increase the weight of less-smooth stencil. Then a new term
was added to the weights of the developed scheme to further improve its performance.
The outline of this paper is given as follows. We present, in Sect. 2, a brief description of
WENO reconstructions and the conventional WENO-JS3 and WENO-Z3 scheme. Section 3
introduces the constructions of the improved third-order WENO-Z scheme. Some numerical examples are provided in Sect. 4 to demonstrate advantages of the proposed WENO-Z
scheme. Finally, concluding remarks are given in Sect. 5.
2 WENO Schemes
The one-dimensional hyperbolic conservation law in the form
∂u
∂ f (u)
+
= 0,
(1)
∂t
∂x
can be approximated by the semi-discrete form:
du i
∂ f (u) =−
(2)
x=xi ,
dt
∂x
The conservation property of the spatial discretization is obtained by defining an implicit
function H (x) through the following equation
1 x+h/2
f (u) =
H (ξ )dξ,
(3)
h x−h/2
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J Sci Comput
where h denotes the uniform grid size. The spatial derivative in Eq. (2) is exactly represented as
a conservative finite difference formula at the cell boundaries. A conservative finite difference
formulation looks for numerical flux approximating H (xi±1/2 ) at the cell boundaries with
higher accuracy. In practice, the approximation to the spatial derivative ∂ f /∂ x at xi will take
the conservative form
f i+1/2 − f i−1/2
∂ f (u) ,
x=xi ≈
∂x
h
(4)
Then the following relation can be obtained
f i+1/2 − f i−1/2
du i
= L (u i ) = −
.
dt
h
(5)
2.1 WENO-JS3 Scheme
The third-order WENO scheme uses two candidate stencils. Thus the numerical flux is
approximated by a convex combination of the second-order fluxes over each stencil,
f i+1/2 = ω0 f 0,i+1/2 + ω1 f 1,i+1/2 ,
(6)
1
3
1
1
f 0,i+1/2 = − f i−1 + f i , f 1,i+1/2 = f i + f i+1 .
2
2
2
2
(7)
where
To achieve the high order accuracy in the smooth regions, the WENO approximation should
simulate the central upwind scheme of optimal order, while retaining the ENO property
near discontinuities. Therefore, the WENO methods design the weight functions, ωi , to be
nonlinear in order to change dynamically based on the smoothness of the numerical solutions.
The classical weight functions proposed by Jiang and Shu [9] are as follows:
ωk =
αk
dk
, αk =
, k = 0, 1,
1
(ε + βk )2
αs
(8)
s=0
where the parameter ε in Eq. (8) is the small positive number that is used to avoid the division
by zero. It is set to be 10−6 in the present work, as recommended in [9]. dk is the optimal
weight:
d0 =
1
2
, d1 = ,
3
3
(9)
and the βk is the smoothness indicator:
β0 = ( f i−1 − f i )2 , β1 = ( f i − f i+1 )2 .
(10)
2.2 WENO-Z3 Scheme
For the third-order WENO-Z scheme (WENO-Z3), Don and Borges [5] considered the reference smoothness indicator of the form
τ
, k = 0, 1,
(11)
αk = dk 1 +
βk + ε
τ Z = |β0 − β1 | ,
(12)
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J Sci Comput
The authors in [20] have previously conducted some derivations on the accuracy of the
WENO-Z3 scheme at the non-critical points. However, they use the following Eq. (13) to
calculate the nonlinear weights ωk .
dk 1 + βτk
τ
αk
=
d
1
+
(13)
=
ωk =
k
1
1
βk
τ
αs
ds 1 + βs
s=0
s=0
In fact, the Eq. (13) is equal to functions αk not the nonlinear weights ωk . Then, they concluded
that the WENO-Z3 scheme can not achieve the convergence order at the non-critical points
according to the sufficient condition as stated in [3,5,6,20,21].
ωk − d k = o h 2 .
(14)
Now, we will again present a short derivation to investigation on the accuracy of the WENOZ3 scheme at the non-critical points in the smooth flow fields with the way of Taylor
expansion.
The smoothness indicators of the candidate stencils in Eq. (10) can be expanded as a
Taylor series at xi
⎧
⎨ β0 = f 2 h 2 − f f h 3 + 1 f 2 + 1 f f h 4 − fi fi h 5 − fi fi h 5 + o h 6 ,
i
i i
4 i
3 i i
12
6
(15)
1 2 1 4
f i f i 5
f i f i 5
⎩
2
2
3
β1 = f i h + f i f i h + 4 f i + 3 f i f i h + 12 h + 6 h + o h 6 ,
At the non-critical points, substituting Eqs. (15) and (12) into Eq. (11) gives
3
2 f i2 2
2 f i
τZ
= d0 1 + h + 2 h + o h
α0 = d0 1 +
β0
fi
fi
3
2 f i2 2
2 f i
τZ
= d1 1 + h − 2 h + o h
α1 = d1 1 +
β1
fi
fi
(16)
(17)
Then, with the weighting procedure in Eq. (8), we can obtain the following relation
ω0 =
α0
1 8 f i2 2
= +
h + o h3
α0 + α1
3 9 f i2
(18)
ω1 =
α1
2 8 f i2 2
= −
h + o h3
2
α0 + α1
3 9 fi
(19)
Thus the numerical flux f i+1/2 in Eq. (5) is given by
1 1 2
1 3
1 (4) 4 f i3
f i+1/2 = f i + f i h +
h4 + o h5
fi h +
fi h +
fi −
2
2
12
12
144
9 fi
Similarly, the numerical flux f i−1/2 is given by
f i−1/2
1
1 2
1 3
= f i − f i h +
f h +
f h −
2
12 i
12 i
11 (4) 4 f i4
f +
144 i
9 f i2
h4 + o h5
(20)
(21)
Finally we get the following results
f i+1/2 − f i−1/2
h
123
= f i +
1 (4) 3
f i h + o h 4 = f i + o h 3
12
(22)
J Sci Comput
From above results, it is clear that the WENO-Z3 scheme can achieve 3rd-order accuracy at
the smooth flow field excluding the critical points.
2.3 Time Discretization
After the spatial derivative is discretized with one of the WENO schemes, an ordinary differential equation (ODE) system is obtained as follow
du
= L (u) ,
dt
(23)
where the spatial operator L(u) is represented in Eq. (5). In all the numerical simulations in
this paper, Shu’s 3rd order TVD time stepping method [9] is applied for the unsteady time
integrations of the ODE system. The third-order TVD Runge–Kutta method is written as:
⎧
n
n
(1)
⎪
⎨ u = u + t L (u ) ,
3
1
(24)
u (2) = 4 u n + 4 u (1) + 41 t L u (1) ,
⎪
⎩ n+1
1 n
1 (2)
2
(2)
= 3 u + 3 u + 3 t L u
.
u
3 Improved WENO Scheme
3.1 WENO-P3 Scheme
In reference [20], Wu and Zhao proposed the WENO-N3 scheme by introducing the new
reference smoothness indicator of the form
β0 + β1
τ N = (25)
− β3 ,
2
where β3 is the smoothness indicator of the whole stencil {xi−1 , xi , xi+1 } which is expressed
as [6]
β3 =
13
1
( f i−1 − 2 f i + f i+1 )2 + ( f i−1 − f i+1 )2 .
12
4
In the present work, a new smoothness indicator is derived in such a way
β0 + β1
1
2
τP = − ( f i−1 − f i+1 ) ,
2
4
(26)
(27)
For simplicity, we denote the WENO scheme with the new smoothness indicator as WENOP3 scheme.
3.1.1 Local Accuracy of the WENO-P3 Scheme
In order to investigate on the characteristics of the WENO-P3 scheme, firstly, we give a short
derivation to investigate on the accuracy of the improved WENO-P3 scheme at the critical
points.
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J Sci Comput
At the first-order critical point ( f i = 0, f i = 0, f i = 0), substituting Eqs. (15) and
(27) into Eq. (11) gives
2
τp
2 f i
α0 = d0 1 +
= d0 2 +
h
+
o
h
(28)
β0
3 f i
2
τp
2 f i
= d1 2 −
h
+
o
h
(29)
α1 = d1 1 +
β1
3 f i
Then, with the weighting procedure in Eq. (8), we can obtain the following relation
ω0 =
α0
1
α1
2
4 f i
4 f i
= +
h + o h 2 , ω1 =
= −
h + o h2
α0 + α1
3 27 f i
α0 + α1
3 27 f i
(30)
Thus the numerical flux f i+1/2 in Eq. (5) is given by
1 1
1 2
2 3
1 (4) 4
f i+1/2 = f i + f i h +
fi h +
fi −
fi h +
fi h + o h 5
2
12
12
27
144
(31)
Similarly, the numerical flux f i−1/2 is given by
f i−1/2 = f i −
1 13 2
fi h +
fi h + o h 3
2
92
(32)
Finally we get the following results
f i+1/2 − f i−1/2
h
= f i −
55 f h + o h 2 = f i + o (h)
276 i
(33)
Thus, the WENO-P3 methods only achieve 1rd-order accuracy at critical points.
3.1.2 Weights of Less-Smooth Substencils of the WENO-P3 Scheme
In this subsection, we give the comparison of the weights of less-smooth substencils between
the WENO-P3 scheme and WENO-N3 scheme. After some symbolic derivations, the following relation can be obtained
β0 + β1
10
τ N = (34)
− β3 =
( f i − 2 f i−1 + f i+1 )2
2
12
β0 + β1
3
1
− ( f i−1 − f i+1 )2 =
(35)
τ P = ( f i − 2 f i−1 + f i+1 )2
2
4
12
Proposition Suppose SC and S D are two substencils of the same stencil, such
the
that
ωD
solution is smoother at SC than at S D (meaning that βC < β D ). Then ωC
>
τP
ωD
, (τ N > τ P ).
ωC
τN
Proof Firstly we have the following relation disregarding ε
ωD
αD
αD
ωD
dD 1 +
−
=
−
=
ωC τ P
ωC τ N
αC τ P
αC τ N
dC 1 +
123
τP
βD
τP
βC
−
1+
1+
τN
βD
τN
βC
(36)
J Sci Comput
Since β D
>
βC , we have
a = β D , b = βC ⇒ 0 < b < a
Therefore
1+
1+
τP
a
τP
b
−
1+
1+
τN
a
τN
b
=
(a − b)
(τ N − τ P )
>0
ab
1 + τbP 1 + τbN
(37)
From Eqs. (36) and (37), we can obtain the conclusion
ωD
ωC
τP
ωD
>
ωC
τN
, (τ N > τ P )
(38)
From above conclusion, it is obvious that WENO-P3 scheme is expected to be less dissipative giving better resolution results than the WENO-N3 scheme. And, this conclusion is in
agreement with the following simulation results.
3.2 WENO-P+3 Scheme
It is demonstrated in [1] that, at relatively coarse grids, increasing the weights of less-smooth
stencils is the most relevant cause than the order of accuracy at critical points for better
resolution of waves. Acker et al. proposed the fifth-order WENO-Z+ scheme based upon
this idea, which shows substantial improvement on the numerical resolution of problems
containing shocks and high gradients. Here, we apply this new technique to WENO-P3
scheme to further increase the weights of less smooth substentils, whereas, the function αk
is defined by
βk + ε
τP
+λ
, k = 0, 1,
(39)
αk = dk 1 +
βk + ε
τP + ε
β0 + β1
1
− ( f i−1 − f i+1 )2 , ε = 10−40 .
(40)
τ P = 2
4
where λ is a parameter for fine-tuning the size of the increment of the weight of less smooth
stencils. The value of λ was empirically determined to be λ = h 1/6 as the one yielding the
best results in terms of stability and resolution power for the standard tests suite, as we shall
see in Sects. 4.1.5, 4.2.3, 4.2.4, and 4.2.5.
3.2.1 Local Accuracy of the WENO-P+3 Scheme
Firstly, we give a short derivation to investigate on the accuracy of the improved WENO-P+3
scheme at the critical points.
At the first-order critical point ( f i = 0, f i = 0, f i = 0), substituting Eqs. (15) and
(40) into Eq. (39) gives
τp
2 f i
β0
α0 = d0 1 +
= d0 2 + λ +
+λ
h
+
o
(41)
(λh)
β0
τp
3 f i
τp
β1
2 f i
+λ
h
+
o
= d1 2 + λ −
α1 = d1 1 +
(42)
(λh)
β1
τp
3 f i
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J Sci Comput
Then, with the weighting procedure in Eq. (8), we can obtain the following relation
ω0 =
α0
1
α1
2
4 f i
4 f i
= +
h
+
o
,
ω
=
=
h + o (λh)
−
(λh)
1
α0 + α1
3 27 f i
α0 + α1
3 27 f i
(43)
Finally we get the following results
f i+1/2 − f i−1/2
h
= f i −
55 f i h + o h 2 = f i + o (h)
276
(44)
Thus, the WENO-P+3 methods only achieve 1rd-order accuracy at critical points. From
Eqs. (33) to (44), we can see that the WENO-P+3 almost share the same accuracy with the
WENO-P3 scheme. It is also indicated that the parameter λ almost has no influence on the
accuracy of the WENO-P+3 scheme.
3.2.2 Weights of Less-Smooth Substencils of the WENO-P3 Scheme
Similarly, we still give a short derivation to verify that the WENO-P+3 scheme puts a relatively
larger weight on less-smooth substencils than WENO-P3 scheme.
Proposition Suppose SC and S D are two substencils of the same stencil,
such that the
solution is smoother at SC than at S D (meaning that βC < β D ). Then ωωCD
>
W E N O−P+3
ωD
.
ωC
W E N O−P3
Proof Firstly we have the following relation disregarding the parameter
ωD
ωD
−
ωC W E N O−P+3
ωC W E N O−P3
βD
τP
1+
αD
αD
dD 1 + βD + λ τP
=
−
=
−
β
τ
C
P
αC W E N O−P+3
αC W E N O−P3
dC 1 +
1+
βC + λ τ P
τP
βD
τP
βC
(45)
Since β D
>
βC , we have
A=
τP
τP
,B =
⇒B> A>0
βD
βC
Therefore
1+ A+
1+ B+
λ
A
λ
B
−
1+ A
(B + A + 1)
>0
= λ (B − A)
1+ B
AB 1 + B + Bλ (1 + B)
Considering Eqs. (45) and (46), we can obtain the conclusion
ωD
ωD
>
ωC W E N O−P+3
ωC W E N O−P3
(46)
(47)
From above conclusion, it can be expected that WENO-P+3 scheme will give better resolution
results than the WENO-P3 scheme. And, this conclusion is in agreement with the following
simulation results as well.
123
J Sci Comput
4 Numerical Results
In this section, we consider several benchmark problems to illustrate the improvement produced by the present method.
4.1 Scalar Test Problems
In order to observe the shock capturing ability of the proposed WENO-P and WENO-P+3
schemes, we test them for one dimensional scalar advection equations with five initial data.
The results of the proposed scheme are compared to those of WENO-JS3, WENO-N3 and the
classical WENO-JS5 scheme [9]. Let us first consider the one-dimensional linear advection
equation controlled by
∂u
∂t
+ ∂∂ux = 0, −1 < x < 1
u (x, 0) = u 0 (x) , periodic
(48)
4.1.1 Critical Point Problem 1
In this case, the initial condition in [8] is chosen as follow
sin (π x)
u 0 (x) = sin π x −
π
(49)
which contains two first-order critical points at which f = 0 and f = 0. The L 1 and
L ∞ errors along with the numerical order of accuracy are provided in Tables 1 and 2 for
WENO-JS3, WENO-Z3, WENO-N3 and WENO-P3 schemes.
4.1.2 Critical Point Problem 2
In this case, the initial condition [6] is as follow
u 0 (x) = sin3 (π x)
Table 1 A comparison study of
L 1 (error and order) for linear
advection equation with initial
condition (49) at t = 2
N
WENO-JS3
(50)
WENO-Z3
L 1 (error)
L 1 (order)
L 1 (error)
L 1 (order)
25
1.2429E−1
–
5.7276E−2
–
50
4.6050E−2
1.4324
1.5116E−2
1.9219
100
1.3010E−2
1.8236
3.4645E−3
2.1254
200
1.5132E−3
3.1039
8.3721E−4
2.0490
400
1.4618E−4
3.3718
1.6823E−4
2.3152
N
WENO-N3
WENO-P3
L 1 (error)
L 1 (order)
L 1 (error)
L 1 (order)
25
4.8254E−2
–
3.2863E−2
–
50
1.1315E−2
2.0924
6.7032E−3
2.2935
100
2.5549E−3
2.1469
1.3325E−3
2.3307
200
4.9809E−4
2.3588
2.2705E−4
2.5531
400
1.0067E−4
2.3068
4.1760E−5
2.4428
123
J Sci Comput
Table 2 A comparison study of
L ∞ (error and order) for linear
advection equation with initial
condition (49) at t = 2
N
25
WENO-JS3
L ∞ (order)
L ∞ (error)
L ∞ (order)
2.6012E−1
–
1.4836E−1
-
50
1.1014E−1
1.2398
5.6752E−2
1.3864
100
4.6113E−2
1.2561
2.0758E−2
1.4510
200
1.0219E−2
2.1739
7.4819E−3
1.4722
400
1.6899E−3
2.5962
2.5623E−3
1.5460
N
WENO-N3
L ∞ (error)
Table 3 A comparison study of
L 1 (error and order) for linear
advection equation with initial
condition (50) at t = 2
WENO-Z3
L ∞ (error)
WENO-P3
L ∞ (order)
L ∞ (error)
L ∞ (order)
25
1.2411E−1
–
9.7985E−2
–
50
4.7462E−2
1.3868
3.1544E−2
1.6352
100
1.6716E−2
1.5055
9.5830E−3
1.7188
200
5.7271E−3
1.5454
2.7884E−3
1.7810
400
1.9697E−3
1.5398
8.4490E−4
1.7226
N
WENO-JS3
WENO-Z3
L 1 (error)
L 1 (order)
L 1 (error)
L 1 (order)
25
1.8499E−1
–
1.5221E−1
–
50
9.1142E−2
1.0213
4.1257E−2
1.8834
100
2.9843E−2
1.6107
9.1524E−3
2.1724
200
7.3790E−3
2.0159
2.0387E−3
2.1665
400
7.0804E−4
3.3815
4.1500E−4
2.2965
N
WENO-N3
WENO-P3
L 1 (error)
L 1 (order)
L 1 (error)
L 1 (order)
25
1.4067E−1
–
1.2275E−1
–
50
3.2229E−2
2.1259
2.3281E−2
2.3985
100
6.7468E−3
2.2561
4.2600E−3
2.4502
200
1.4009E−3
2.2678
8.0490E−4
2.4040
400
2.7477E−4
2.3501
1.4483E−4
2.4744
which contains a first-order critical point at which f = 0, f = 0, but f = 0. Tables 3
and 4 contain the L 1 and L ∞ errors along with the numerical order of accuracy for WENOJS3, WENO-Z3, WENO-N3 and WENO-P3 schemes. From Tables 1, 2, 3, and 4, it is clear
that the WENO-P3 scheme almost share the same accuracy with the WENO-Z3 and WENON3 scheme, which is in agreement with the discussions in Sects. 3.1.1 and 3.2.1.
4.1.3 Linear Discontinuity Problem
A solution with an initial discontinuity [2] is tested
− sin (π x) − 21 x 3 ,
u 0 (x) =
− sin (π x) − 21 x 3 + 1,
123
−1 < x ≤ 0
0<x ≤1
(51)
J Sci Comput
Table 4 A comparison study of
L ∞ (error and order) for linear
advection equation with initial
condition (50) at t = 2
N
25
WENO-JS3
WENO-Z3
L ∞ (error)
L ∞ (order)
L ∞ (error)
L ∞ (order)
3.8696E−1
–
3.1669E−1
–
50
2.2025E−1
0.8130
1.2435E−1
1.3487
100
9.7458E−2
1.1763
4.8963E−2
1.3446
200
3.6946E−2
1.3994
1.7450E−2
1.4885
400
7.1187E−3
2.3757
6.1192E−3
1.5118
N
WENO-N3
L ∞ (error)
WENO-P3
L ∞ (order)
L ∞ (error)
L ∞ (order)
25
2.9180E−1
–
2.5016E−1
–
50
1.0164E−1
1.5215
7.1777E−2
1.8013
100
4.0100E−2
1.3418
2.8144E−2
1.3507
200
1.3554E−2
1.5649
8.5412E−3
1.7203
400
4.5504E−3
1.5747
2.6072E−3
1.7119
1.2
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-P+3
WENO-JS5
1.0
0.8
u
0.6
0.4
0.2
0.0
-0.2
-1.0
-0.5
0.0
0.5
1.0
x
Fig. 1 Numerical results with the initial condition (51), t = 6
which is a piecewise sine function with a jump discontinuity at x = 0. We solve it up to t = 6
to see the behaviors of the WENO schemes at the jump discontinuity. Numerical solutions for
the WENO-JS3, WENO-N3, WENO-P3, WENO-P+3 and WENO-JS5 schemes with grid
number N = 200 at the final time are shown in Fig. 1. It is clear that, near the discontinuity,
the WENO-P+3 scheme obtains more sharper solution than the WENO-JS3, WENO-N3
schemes. WENO-P3 scheme is slightly better than WENO-N3 scheme due to slightly larger
weight assignment to the discontinuous stencils as shown in Fig. 2b and c. What is more,
WENO-P+3 does not perform better than the WENO-JS5 due to the accuracy loss at the
critical points as demonstrated in Sect. 3.1.1.
123
J Sci Comput
0
0
10
10
-1
-1
10
10
-2
-2
10
10
-3
-3
10
10
-4
-4
10
10
ω0
ω1
-5
10
ω0
ω1
-5
10
d0
d0
d1
-6
10
-7
10
d1
-6
10
-7
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
10
-0.04
-0.03
-0.02
-0.01
x
(a)
0.02
0.03
0.04
(b)
0
10
10
-1
-1
10
-2
10
-3
10
10
-2
10
-3
10
-4
-4
10
10
ω0
ω1
-5
10
ω0
ω1
-5
10
d0
-6
d0
-6
d1
10
d1
10
-7
-7
-0.04
0.01
x
0
10
0.00
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
10
-0.04
-0.03
-0.02
-0.01
0.00
x
x
(c)
(d)
0.01
0.02
0.03
0.04
Fig. 2 a WENO-JS3. b WENO-N3. c WENO-P3. d WENO-P+3. The distribution of the ideal weights dk
and the weights ωk for the schemes at the first step of the numerical solution of the nonlinear discontinuity
problem
Figure 2 shows the weights ωk for the WENO-JS3, WENO-N3, WENO-P3 and WENOP+3 schemes at the first step of the numerical solution of the wave equation with the initial
condition (51). The ideal weights dk are also plotted as lines and the vertical axis is in log10
scale. And from this, one can observe that WENO-N3, WENO-P3 and WENO-P+3 assign
larger weights for the discontinuous stencils as compared with WENO-JS3. The WENO-P3
scheme assigns larger weights for the discontinuous stencils as compared with WENO-N3
scheme. The WENO-P+3 scheme further assigns larger weights for the discontinuous stencils
as compared with WENO-P3 due to be added term. From above analysis, it can be concluded
that further increasing the relevance of less-smooth stencils can give less dissipative simulation results around discontinuities and smooth flows as demonstrated in [1].
4.1.4 Complex Wave
A more complex initial solution is tested in [14] given by
⎧
−1 ≤ x < −1/3
⎨ −x sin 3π x 2 /2 ,
u 0 (x) = |sin (2π x)| ,
−1/3 ≤ x < 1/3
⎩
2x − 1 − sin (3π x)/6, other wise
123
(52)
J Sci Comput
1.0
u
0.5
0.0
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-P+3
WENO-JS5
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
x
Fig. 3 Numerical results with the initial condition (52), t = 6
The solution of this problem contains contact discontinuities, corner singularities and smooth
areas. We solve the equation up to t = 6 with grid number N = 200. The numerical results
are provided in Fig. 3. Comparison of the results demonstrates the advantages of the proposed
WENO-P3 and WENO-P+3 schemes to approximate the exact solution near discontinuities
as well as critical points. However, the WENO-P+3 shows worse performances compared
with the WENO-JS5 at the critical points and the sharp points due to the accuracy loss.
4.1.5 Gaussian-Square-Triangle-Ellipse Linear Test
Then, a more challenging test case [12] that contains a smooth combination of Gaussians, a
square wave, a sharp triangle wave, and a half ellipse is calculated.
⎧1
⎪
6 (G (x, β, z − δ) + G (x, β, z + δ) + 4G (x, β, z)) , − 0.8 ≤ x ≤ − 0.6
⎪
⎪
⎪
− 0.4 ≤ x ≤ − 0.2
⎨ 1,
0 ≤ x ≤ 0.2
(53)
u 0 (x) = 1 − |10 (x − 0.1)| ,
⎪
1
⎪
⎪
(F (x, α, a − δ) + F (x, α, a + δ) + 4F (x, α, a)) , − 0.8 ≤ x ≤ − 0.6
⎪
6
⎩
0, other wise
where
G (x, β, z) = e−β(x−z) , F (x, α, a) =
2
max 1 − α 2 x − a 2 , 0
(54)
The constants are taken as a = 0.5, z = − 0.7, d = 0.005, α = 10, β = 1og2/(36δ 2 ).
We solve the problem on a 200 grid points uniform mesh and the results of WENO-P3 and
WENO-P+3 schemes are compared against the WENO-JS3, WENO-N3 and WENO-JS5
schemes until the final time t = 8 as shown in Fig. 4. It can be observed that the WENO-P+3
scheme achieves better resolution of waves than the WENO-JS3, WENO-N3 schemes. The
WENO-P+3 gives poorer results than the WENO-JS5 particularly at the sharp points due to
the accuracy loss.
Figure 5 compares the results of this test with N = 200 points for the WENO-P+3 scheme
with five different values for the parameter λ: h 1/6 , h 1/4 , h 1/2 , h 2/3 , and h 3/3 . From Fig. 5,
123
J Sci Comput
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-P+3
WENO-JS5
1.0
0.8
u
0.6
0.4
0.2
0.0
-1.0
-0.5
0.0
0.5
1.0
x
Fig. 4 Numerical solution of the advection equation with the discontinuous initial condition (53)
Exact
λ = h0
λ = h1/6
λ = h1/4
λ = h1/2
λ = h2/3
λ = h3/3
1.0
0.8
u
0.6
0.4
0.2
0.0
-1.0
-0.5
0.0
0.5
1.0
x
Fig. 5 Numerical solution of the advection equation with the discontinuous initial condition (53) using the
WENO-P+3 scheme with different values of λ
we can see that the WENO-P+3 scheme with parameter λ = h 1/6 performs the best. It should
be mentioned that the WENO-P+3 scheme will become unstable with more larger λ. That is
to say that, the added term should be kept small enough for avoiding spurious oscillations
and instabilities.
From above simulation results, it is clear that WENO-P+3 scheme is less dissipative with
larger value of the parameter λ, which indicates that WENO-P+3 scheme generally places
a larger weight to less-smooth substencils with larger value of λ. Let us now verify this
inference.
123
J Sci Comput
Proposition Suppose SC and S D are two substencils of the same stencil, such that the solution
is smoother at SC than at S D (meaning that βC
<
β D ). Then
λ
ω D1
λ
ωC1
>
λ
ω D2
λ
ωC2
, (λ1 > λ2 ).
Proof Firstly we have the following relation disregarding ε
ωλD1
ωλD2
Since β D
>
=
α λD1
−
α λD2
=
ωCλ2
αCλ1
αCλ2
⎫
⎧
βD
τ
1 + βτD + λ2 βτD ⎬
d D ⎨ 1 + β D + λ1 τ
− =
dC ⎩ 1 + τ + λ βC
1 + βτC + λ2 βτC ⎭
1
βC
τ
ωCλ1
−
(55)
βC , we have
a=
τ
τ
,b =
⇒0<a<b
βD
βC
Therefore
1+a+
1+b+
λ1
a
λ1
b
−
1+a+
1+b+
λ2
a
λ2
b
=
(b − a) (b + a + 1)
(λ1 − λ2 )
>0
ab
1 + b + λ1 1 + b + λ2
b
(56)
b
From Eqs. (55) to (56), we can obtain the conclusion
ωλD1
ωCλ1
>
ωλD2
ωCλ2
, (λ1 > λ2 )
(57)
4.2 One-Dimensional Euler Systems
In this subsection, we present numerical experiments with the one dimensional system of the
Euler equations for gas dynamics in strong conservation form:
∂U
∂F
+
=0
∂t
∂x
with
(58)
⎤
⎤
⎡
ρu
ρ
1
U = ⎣ ρu ⎦ , F = ⎣ ρu 2 + p ⎦ , p = (γ − 1) E − ρu 2 , γ = 1.4
2
E
u (E + p)
⎡
And ρ, u, p, and E are the density, velocity, pressure and total energy, respectively. γ is the
ratio of specific heats. The eigenvalues of Jacobian matric A (U ) = ∂ F/∂U can be written
as follows:
√
λ1 = u − c, λ2 = u, λ3 = u + c
where c = γ p/ρ is the sound speed. The characteristic decomposition with Roe’s approximation is used at the cell faces and the Lax–Friedrichs formulation is used for the numerical
fluxes to generalize the WENO schemes to the one dimensional Euler systems.
4.2.1 Sod Problem
The initial data for the Sod problem [16] are given by
123
J Sci Comput
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-JS5
WENO-P+3
1.0
Density
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
x
(a)
0.45
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-JS5
WENO-P+3
Density
0.40
0.35
0.30
0.25
0.62
0.64
0.66
0.68
0.70
0.72
x
(b)
Fig. 6 Sod problem: a density distribution and b the enlarged portion near the discontinuity
(ρ, u, p) =
(1, 0, 1),
(0.125, 0, 0.1),
0 ≤ x < 0.5
0.5 ≤ x ≤ 1
(59)
We compute the equation until t = 0.18 with N = 200 points. The simulated density
and enlarged portion near the discontinuity are shown in Fig. 6. The ’exact’ solution is the
result computed on N = 10,000 points using the WENO-JS3 scheme. The performance of
WENO-P3 yields better results than WENO-N3 and WENO-JS3 scheme. The WENO-P+3
scheme gives the least dissipative results even better than the WENO-JS5 scheme.
123
J Sci Comput
4.2.2 Lax Problem
For the Lax problem [10], the initial conditions are given by
(0.445, 0.698, 3.528), − 5 ≤ x < 0
(ρ, u, p) =
(0.5, 0, 0.571),
0≤x ≤5
(60)
and the final time is t = 1.3. The WENO-P+3 scheme is less dissipative than the other
schemes and provides better resolution of the contact discontinuities and the shock waves,
while maintaining the ENO property as shown in Fig. 7.
4.2.3 Shu–Osher Problem
For the 1D Shu–Osher problem, the initial condition is
(3.857143, 2.629369, 10.33333) , − 5 ≤ x < −4
(ρ, u, p) =
−4 ≤ x ≤ 5
(1 + 0.2 sin 5x, 0, 1) ,
(61)
We solve this problem up to time t = 1.8. This problem describes the phenomena that a
shock-wave (Mach 3) propagating to the right interacts with a perturbed density disturbance.
Figure 8 provides a comparison for all schemes with N = 600 points. The reference solution
has been computed on a fine grid of 10,000 points using the WENO-JS3 scheme. It can
be seen that the WENO-P+3 scheme captures the fine scale structures, especially at the
high-frequency waves behind the shock, better than the WENO-JS3 and WENO-N3 scheme.
However, WENO-JS5 scheme shows more accuracy results than the WENO-P+3 scheme
particularly at the high-frequency waves. This is mainly due to the accuracy loss of the
WENO-P+3 scheme at the critical points.
Figure 9 compares the results of the Shu–Osher test with N = 600 points for the WENOP+3 scheme with five different values for the parameter λ: h 1/6 , h 1/4 , h 1/2 , h 2/3 , and h 3/3 .
From Fig. 9, we can see that the choice λ = h 1/6 gives less dissipative results while the
choice λ = h 3/3 gives more dissipative results.
4.2.4 Titarev–Toro Problem
This is a variation of the Shu–Osher test, with a different initial condition
(1.515695, 0.523346, 1.805000) , −5 ≤ x < −4.5
(ρ, u, p) =
−4.5 ≤ x ≤ 5
(1 + 0.1 sin (20π x) , 0, 1) ,
(62)
which consists of a right-facing shock wave of Mach number 1.1 running into a highfrequency density perturbation. The flow contains physical oscillations which have to be
resolved by the numerical method. We compute the solution at the output time t = 5.
Figure 10 provides graphical results for all schemes on a mesh of 3000 cells. The reference
solution is obtained by applying the WENO-P+3 scheme on a fine mesh of 8000 cells and
is shown by the solid line on all figures. It can be seen from Fig. 10 that the WENO-P+3
scheme resolves most of the waves with a good approximation to their amplitudes better than
the WENO-JS3 and WENO-N3 schemes. Again, WENO-JS5 scheme shows more accuracy
results than the WENO-P+3 scheme particularly at the high-frequency waves.
Figure 11 compares the results of the Titarev–Toro test with N = 3000 cells for the
WENO-P+3 scheme with the five different values for the parameter λ: h 1/6 ,h 1/4 h 1/2 , h 2/3 ,
and h 3/3 . We see that the choice λ = h 1/6 gives the best results, resolving most of the waves
with a good approximation to their amplitudes.
123
J Sci Comput
1.4
1.2
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-JS5
WENO-P+3
Density
1.0
0.8
0.6
0.4
0.2
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
(a)
1.4
1.2
Density
1.0
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-JS5
WENO-P+3
0.8
0.6
0.4
1.2
1.6
2.0
2.4
2.8
3.2
3.6
x
(b)
Fig. 7 Lax problem: a density distribution and b the enlarged portion near the discontinuity
4.2.5 Interacting Blast Wave
The one dimensional blast waves interaction problem of Woodward and Collela [18] has the
following initial condition, with reflective boundary conditions on both ends
⎧
⎨ (1, 0, 1000) ,
(ρ, u, p) = (1, 0, 0.01) ,
⎩
(1, 0, 100) ,
123
0 ≤ x < 0.1
0.1 ≤ x ≤ 0.9
0.9 ≤ x ≤ 1
(63)
J Sci Comput
5.0
4.5
4.0
3.5
Density
3.0
2.5
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-P+3
WENO-JS5
2.0
1.5
1.0
0.5
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
(a)
4.8
4.4
4.0
Density
3.6
3.2
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-P+3
WENO-JS5
2.8
2.4
2.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
x
(b)
Fig. 8 Shu–Osher problem: a density distribution and b the enlarged portion of the high frequency waves
region
The initial pressure gradients generate two density shock waves that collide and interact later
in time, forming a profile as shown in Fig. 12 on uniform grid of 600 at t = 0.038. The
reference solutions are the numerical solutions of the WENO-JS3 scheme with grid points of
N = 10,000. Careful examination of Fig. 12, it can be concluded that WENO-P+3 solution
are much better resolved as compared with other schemes counterpart.
Figure 13 compares the results of the Interacting blast wave problem with N = 600 points
for the WENO-P+3 scheme with the five different values for the parameter λ: h 1/6 ,h 1/4 h 1/2 ,
h 2/3 , and h 3/3 . It is clear that the choice λ = h 1/6 gives the best results, resolving physical
123
J Sci Comput
5.0
4.5
4.0
Density
3.5
3.0
Exact
2.5
λ =h1/6
λ =h1/4
λ =h1/2
λ =h2/3
λ =h3/3
2.0
1.5
1.0
0.5
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
(a)
4.8
4.4
4.0
Density
3.6
3.2
Exact
λ =h1/6
λ =h1/4
λ =h1/2
λ =h2/3
λ =h3/3
2.8
2.4
2.0
1.6
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
x
(b)
Fig. 9 a Comparison of the five choices of λ. b Comparison of the solutions in the high-frequency waves.
Numerical solution of the Shu–Osher problem by the WENO-P+3 scheme with different λ
extrema at x = 0.647, 0.748, 0.777 and the contact discontinuities at x = 0.594 and 0.765
better.
The results in Sects. 4.1.5, 4.2.3, 4.2.4, and 4.2.5 indicate that λ = h 1/6 is a good
choice considering stability and numerical resolution, at least in the above four typical tested
problems. This is the reason for determining the parameter λ to be h 1/6 , and all the numerical
tests in the present paper also shows that the WENO-P+3 scheme with the parameter λ = h 1/6
outperforms better.
123
J Sci Comput
1.8
1.6
Density
1.4
Exact
WENO-JS5
WENO-P+3
WENO-P3
WENO-N3
WENO-JS3
1.2
1.0
0.8
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
(a)
1.7
Density
1.6
1.5
1.4
1.3
-2.5
Exact
WENO-JS5
WENO-P+3
WENO-P3
WENO-N3
WENO-JS3
-2.0
-1.5
-1.0
x
(b)
Fig. 10 Titarev–Toro problem: a density distribution and b the enlarged portion of the high frequency waves
region
4.3 Two-dimensional Euler Systems
In this subsection, we apply the proposed scheme to 2D problem for two-dimensional gas
dynamics, Rayleigh–Taylor instability and Double Mach reflection of a strong shock in
Cartesian coordinates. The governing two-dimensional compressible Euler equations are
given by
∂U
∂F
∂G
+
+
=0
∂t
∂x
∂y
(64)
123
J Sci Comput
1.8
1.7
1.6
1.5
Density
1.4
1.3
Exact
λ = h1/6
λ = h1/4
λ = h1/2
λ = h2/3
λ = h3/3
1.2
1.1
1.0
0.9
0.8
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
(a)
1.7
Density
1.6
1.5
Exact
λ = h1/6
λ = h1/4
λ = h1/2
λ = h2/3
λ = h3/3
1.4
1.3
-2.5
-2.0
-1.5
-1.0
x
(b)
Fig. 11 a Comparison of the five choices of λ. b Comparison of the solutions in the high-frequency waves.
Numerical solution of the Titarev–Toro problem by the WENO-P+3 scheme with different λ
where
⎞
⎛
⎞
⎛
⎞
ρu
ρv
ρ
⎜ ρu 2 + p ⎟
⎜ ρvu ⎟
⎜ ρu ⎟
⎟
⎜
⎟
⎜
⎟
U =⎜
⎝ ρv ⎠ F = ⎝ ρuv ⎠ G = ⎝ ρv 2 + p ⎠
E
u (E + p)
v (E + p)
⎛
The total energy E and pressure p is defined by
1
1
p = (γ − 1) E − ρu 2 − ρv 2
2
2
123
(65)
(66)
J Sci Comput
7
6
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-JS5
WENO-P+3
Density
5
4
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
x
(a)
6.6
Exact
WENO-JS3
WENO-N3
WENO-P3
WENO-JS5
WENO-P+3
6.0
5.4
Density
4.8
4.2
3.6
3.0
2.4
1.8
0.60
0.64
0.68
0.72
0.76
0.80
0.84
x
(b)
Fig. 12 Interacting blast wave problem: a density distribution and b the enlarged portion of the uppermost
part
where ρ, u, v are the density, x-velocity, y-velocity, respectively. γ is the ratio of specific
heats.
4.3.1 2D Riemann Problem
The test case 3 of the two-dimensional Riemann problems introduced by Lax and Liu [11] is
chosen here. This problem was also simulated in [7,20] to assess the resolution of improved
123
J Sci Comput
7
6
Exact
λ = h1/6
λ = h1/4
λ = h1/2
λ = h2/3
λ = h3/3
5
Density
4
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
x
(a)
6.6
Exact
λ = h1/6
λ = h1/4
λ = h1/2
λ = h2/3
λ = h3/3
6.0
5.4
Density
4.8
4.2
3.6
3.0
2.4
1.8
0.60
0.64
0.68
0.72
0.76
0.80
0.84
x
(b)
Fig. 13 Numerical solution of the Interacting blast wave problem by the WENO-P+3 scheme with different
λ. a Comparison of the five choices of λ. b Comparison of the solutions in the upper part
schemes. The problem is solved on the square domain [0, 1] × [0, 1], which is divided into
four quadrants with initial constant states by lines x = 0.8 and y = 0.8:
(ρ, u, v, p) = (1.5, 0, 0, 1.5) ,
(ρ, u, v, p) = (0.5323, 1.206, 0, 0.3) ,
(ρ, u, v, p) = (0.138, 1.206, 1.206, 0.029) ,
(ρ, u, v, p) = (0.5323, 0, 1.206, 0.3) ,
123
0.8 ≤ x ≤ 1, 0.8 ≤ y ≤ 1
0 ≤ x < 0.8, 0.8 ≤ y ≤ 1
0 ≤ x < 0.8, 0 ≤ y < 0.8
0.8 < x ≤ 1, 0 ≤ y < 0.8
(67)
J Sci Comput
Fig. 14 Density contours of two-dimensional Riemann problem computed by the a WENO-JS3, b WENOZ3, c WENO-N3, d WENO-JS5, e WENO-Z5, f WENO-Z+5, g WENO-P3, h WENO-P+3 (λ = h 1/2 ), i
WENO-P+3 (λ = h 1/4 ), and j WENO-P+3 (λ = h 1/6 ) schemes at t = 0.8. 40 contours were fit between the
range of 0.14 to 1.7
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Fig. 14 continued
The ratio of specific heats is set to be γ = 1.4. The final simulation time is t = 0.8 with
960 × 960 uniform grid points. The results of WENO-JS3, WENO-Z3, WENO-N3, WENOJS5, WENO-Z5, WENO-Z+5, WENO-P3, and WENO-P+3 with three different values of the
parameter λ: λ = h 1/2 , λ = h 1/4 , λ = h 1/6 are displayed in Fig. 14. An examination of these
results reveals that WENO-P+3 with parameter λ = h 1/6 presents a higher resolution of the
structure of discrete vortex along the slip line than the WENO-JS3, WENO-Z3, WENO-N3
and WENO-JS5 schemes.
4.3.2 Two-Dimensional Rayleigh–Taylor Instability
This problem has been simulated extensively in the literature [1,6,15,17,20] to assess the
numerical dissipations of the numerical schemes. It happens on an interface instability
between two fluid of different densities when an acceleration is directed from heavier fluid
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Fig. 15 Density contours of the Rayleigh–Taylor instability computed by the a WENO-JS3, b WENO-Z3, c
WENO-N3, d WENO-JS5, e WENO-Z5, f WENO-Z+5, g WENO-P3, h WENO-P+3 (λ = h 1/2 ), i WENOP+3 (λ = h 1/4 ), and j WENO-P+3 (λ = h 1/6 ) schemes, and 15 contours were fit between the range of
0.952269 to 2.14589
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Fig. 15 continued
to lighter one. The computational domain is [0, 0.25] × [0, 1], and the initial conditions
are
√
2, 0, − 0.025√γ p/ρ cos (8π x) , 2y + 1 , 0 ≤ y < 0.5
(68)
(ρ, u, v, p) = 1, 0, − 0.025 γ p/ρ cos (8π x) , y + 1.5 , 0.5 ≤ y < 1
with the ratio of specific heats γ = 5/3. The gravity effect is introduced by adding ρ and
ρv to the right hand side of the y-momentum equation and the energy equation, respectively.
Reflective boundary conditions are imposed for the left and right boundaries, and the top
boundary are set as ( ρ, u, v, p) = (1, 0, 0, 2.5) and ( ρ, u, v, p) = (2, 0, 0, 1) are set to be
the bottom boundary condition. The final simulation time is t = 1.95.
The density contours of the solutions computed on the 240 × 960 uniform grid in Fig. 15
show than the WENO-P3 scheme proposed performs better than the WENO-N3, WENO-Z3,
and WENO-JS3 scheme. It is obvious that the contact discontinuity resolution (manifested
by the small scale structures) is significantly improved by the WENO-P+3 scheme. The
WENO-P+3 scheme with the parameter λ = h 1/6 resolves the most complex structures even
better than the WENO-JS5 scheme.
4.3.3 Double Mach Reflection of a Strong Shock
The final test considered in this paper is the two dimensional double Mach reflection of a
shock from an oblique surface. It describes the reflection of a planar Mach shock in air hitting
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Fig. 16 Double Mach reflection computed by the a WENO-JS3, b WENO-Z3, c WENO-N3, d WENO-JS5,
e WENO-Z5, f WENO-Z+5, g WENO-P3, h WENO-P+3 (λ = h 1/2 ), i WENO-P+3 (λ = h 1/4 ), and j
WENO-P+3 (λ = h 1/6 ) schemes from the top to bottom at t = 0.2
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Fig. 16 continued
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Fig. 16 continued
a wedge which was initially proposed and studied in detail by Woodward and Colella [18].
This test is widely used to verify the performance of numerical methods [6,7,17,20]. It was
calculated on [0, 4] × [0, 1] and the initial conditions are
(8.0, 7.145, −4.125, 116.5) , x < 16 + √y
3
(69)
(ρ, u, v, p) =
x ≥ 16 + √y
(1.4, 0, 0, 1) ,
3
with a Mach 10 shock reflected from the wall with an incidence angle of 60◦ .
We solve it up to time t = 0.2 using 1920 × 480 grid points with γ = 1.4. The results
in [0, 3] × [0, 1] are displayed. The density contours with 40 equally spaced contour lines
from ρ = 1.8 to ρ = 20 are drawn. The numerical results of the WENO-P+3 scheme with
three different values of the parameter λ: λ = h 1/2 , λ = h 1/4 , λ = h 1/6 are compared with
WENO-JS3, WENO-Z3, WENO-N3, WENO-JS5, WENO-Z5, WENO-Z+5 and WENO-P3
in Fig. 16. Figure 17 shows the details at the Mach stem of the density variable at the final
time. It can be concluded from the small vortices captured along the slip line that the WENOP+3 scheme with the parameter λ = h 1/6 has the highest resolution among the WENOJS3, WENO-Z3, WENO-N3, and WENO-P3 schemes, even better than the WENO-JS5
scheme.
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Fig. 17 Density contours of double Mach stems on [2, 3] × [0, 1] computed by the a WENO-JS3, b WENOZ3, c WENO-N3, d WENO-JS5, e WENO-Z5, f WENO-Z+5, g WENO-P3, h WENO-P+3 (λ = h 1/2 ), i
WENO-P+3 (λ = h 1/4 ), and j WENO-P+3 (λ = h 1/6 ) schemes at t = 0.2
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Fig. 17 continued
5 Conclusions
In this paper, we introduce a modified third-order WENO scheme to approximate solutions
of nonlinear hyperbolic conservation laws. The motivation for this study is that at relatively
coarse grids, increasing the weights of less-smooth stencils is the most relevant cause than
the order of accuracy at critical points for better resolution of waves. WENO-P3 scheme is
constructed by slightly modified the reference smoothness indicator of WENO-N3 scheme.
WENO-P+3 scheme is obtained through adding a new term to the weights of WENO-P3
scheme to further slightly increase the weight of less-smooth stencil. It is observed that the
improved WENO-P+3 scheme was able to achieve significantly better results compared with
WENO-JS3, WENO-Z3, WENO-N3 and WENO-P3 schemes, which is verified by several
benchmark problems of one- and two-dimensional Euler system of equations.
Acknowledgements The authors acknowledge the support of National Defense Fundamental Research
Project (B1420133057), National Natural Science Foundation of China (51409202 and 11502180) and the
Fundamental Research Funds for the Central Universities (2016-YB-016).
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