[PPT] - A Generalized Fifth Order WENO Finite Difference Scheme with Z-Type PowerPoint Presentation

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A Generalized Fifth Order WENO Finite Difference Scheme with Z-Type Nonlinear Weights

International Conference Advances in Applied Mathematics in memorial of Professor Saul Abarbanel December 18 - 20, 2018, Tel Aviv University Wai-Sun Don

School of Mathematical Sciences, Ocean University of China, Qingdao, China

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 1 / 47

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Collaborators and Research Funding

Ocean University of China, Qingdao, China. Ying-Hua Wang, Bao-Shan Wang This research is supported by National Natural Science Foundation of China (11871443), National Science and Technology Major Project(20101010), Shandong Provincial Natural Science Foundation (ZR2017MA016), Fundamental Research Funds for the Central Universities (201562012).

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 2 / 47

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Outline

1 High Order WENO Finite Difference Scheme 2 The WENO-Z Type Nonlinear Weight 3 The Generalized WENO Scheme with Z-Type Nonlinear

Weights

◮ Smooth Function: Issues with Critical Points ◮ Discontinuous Function: Essentially Non-Oscillatory

4 Numerical Examples 5 Conclusion And Future Work

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 3 / 47

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High Order WENO Finite Difference Scheme

Consider the hyperbolic conservation laws ∂Q ∂t + ∇ · F(Q) = 0. The semi-discretized of the equation, by method of lines, on a uni- formly sized cell, in a conservative manner d ¯ Qi(t) dt = 1 ∆x

hi+ 1

2 − hi− 1 2

,

h = h( ¯ Qi−r, · · · , ¯ Qi+l). where h(x) is defined implicitly as f(x) =

1 ∆x

x+ ∆x

2

x− ∆x

2

h(ξ)dξ.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 4 / 47

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Fifth order WENO Reconstruction Procedure

Nonlinear spatial adaptive combination of THREE Lagrange poly- nomials qk(x) of degree 2 in Sk, where k = 0, 1, 2 is the shift parameter, ˆ f(x) =

2

k=0

ωk qk(x) ≈ h(x) + O(∆xM) (1) at xi± 1

2 , such that, when the solution is

◮ SMOOTH, becomes a M = 5 order central upwinded scheme. ◮ NON-SMOOTH, becomes a M = 3 order Upwinded scheme

by assigning the nonlinear weight ωk ≈ 0 in Sk containing discontinuity = ⇒ essentially no Gibbs oscillations.

xi xi+1 xi+2 xi-1 xi-2 xi+1/2

S2 S0 S1 S

5

τ5 β0 β2 β1 ω1 ω0 ω2

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 5 / 47

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The Classical WENO-JS Scheme

The nonlinear weights of the classical WENO-JS scheme (Jiang and Shu) are αk = dk (βk + ε)p , ωk = αk 2

l=0 αl

, with two user defined parameters : (1) power parameter p ≥ 1 and (2) the sensitivity parameter ε > 0 (Usually a fixed small real number). The lower order local smoothness indicators βk =

2

l=1

∆x2l−1 xi+ 1

2

xi− 1

2

dl dxl qk(x) 2 dx. (2) measure the normalized modified Sobolov norm of the second de- gree polynomials qk(x) in the substencil Sk at xi in the cell Ii = [xi− 1

2 , xi+ 1 2 ]. Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 6 / 47

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The Improved WENO-Z Scheme

In the (2r−1) order WENO scheme with Z-type weights (WENO-Z), the nonlinear weights are αk = dk

1 +

τ2r−1 βk + ε p , ωZ

k =

αk r−1

j=0 αj

, k = 0, . . . , r−1. where the global smoothness indicator is τ2r−1 =

r−1
k=0

ckβk

,

where ck are given constants1. For example, τ5 of the fifth order WENO-Z scheme is τ5 = |β0 − β2| . Its leading truncation error has been shown to be O(∆x5).

1Castro. Costa. and Don. J. Comput. Phys. 230, 1766–1792, 2011

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 7 / 47

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Definition of Critical Points

Definition If a function f(xc) = f′(xc) = . . . = f(ncp)(xc) = 0 but f(ncp+1)(xc) = 0, the function f(x) is said to have a critical point

f order ncp ≥ 0 at xc.

For example, f(x) = x3, f′(0) = f′′(0) = 0, f′′′(0) = 0, then f(x) has ncp = 2 at x = 0.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 8 / 47

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Optimal Order At Critical Point

αk = dk

1 +

τ2r−1 βk + ε p . The nonlinear weights αk have two important free parameters:

◮ Power p: increases the separation of scales, and controls the

amount of numerical dissipation.

◮ Sensitivity ε: avoids a division by zero in the denominator of

αk.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 9 / 47

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The issue of critical points

◮ In general, a very small ε, say O(10−40), is highly desirable for

capturing shock in an essentially non-oscillatory manner because

◮ ε does not over-dominate over the size of the local smoothness

indicators βk as in (βk + ε).

◮ However, a very small ε could reduce the formal order of accuracy

f WENO schemes of a smooth function in the presence of high
rder critical points.

ncp = 2

dx L_max Error Order

10

4

10

3

10

2

10

1

10

22

10

17

10

12

10

7

10

2

1 2 3 4 5 6 7 8 9 10 Z5p2edx3 Z5p2edx4* Z5p2edx5 Z5p2edx6

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 10 / 47

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Optimal Order At Critical Point

To mitigate the critical point problem, there are many recent works

n

◮ applying a mapping on the nonlinear weights such as the

WENO-M by Henrick et al. (J. Comput. Phys. 207, 2005).

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 11 / 47

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Optimal Order At Critical Point

To mitigate the critical point problem, there are many recent works

n

◮ applying a mapping on the nonlinear weights such as the

WENO-M by Henrick et al. (J. Comput. Phys. 207, 2005).

◮ reformulating the WENO-Z type weights such as the

WENO-CU6 by Hu et al. (J. Comput. Phys., 229, 2010). and WENO-η by Fan et al. (J. Comput. Phys. 269, 2014).

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 11 / 47

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Optimal Order At Critical Point

To mitigate the critical point problem, there are many recent works

n

◮ applying a mapping on the nonlinear weights such as the

WENO-M by Henrick et al. (J. Comput. Phys. 207, 2005).

◮ reformulating the WENO-Z type weights such as the

WENO-CU6 by Hu et al. (J. Comput. Phys., 229, 2010). and WENO-η by Fan et al. (J. Comput. Phys. 269, 2014).

◮ setting the lower bound on the sensitivity parameters ε by

Don et al. (J. Comput. Phys. 250, 2013).

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 11 / 47

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The WENO-CU6 scheme

The WENO nonlinear weights of WENO-CU6 scheme are αk = dk

C +

τ6 βk + ε

,

ωk = αk 3

k=0 αk

,

◮ ε = 10−40. (A very small number). ◮ Large constant C ≫ 1 increases the contribution of the

ptimal weights. (Usually, C = 20 or larger).

◮ The global smoothness indicator

τ6 =

β6 − 1

6(β0 + 4β1 + β2)

+ O(∆x6),

(3) with a long and complex expression of β6.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 12 / 47

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The WENO-η scheme

The WENO nonlinear weights of WENO-η are αk = dk

1 +

τ ηk + ε

,

ωk = αk 2

k=0 αk

. (4)

◮ The local smoothness indicators

ηk =

r−1

m=1

[∆xmP (m)

i−r+1+k(xi)]2, k = 0, 1, 2.

(5) where P (m)

i

(x) is the m th derivative of the Lagrangian interpolation polynomial for approximating the value of the function f(x) based on the values (fi−r+1+k, . . . , fi−r+1+k).

◮ the global smoothness indicator τ, such as

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 13 / 47

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The WENO-η scheme

◮ the global smoothness indicator τ, such as

τ5 = |η0 − η2| + O(∆x6). (6) τ6 = |6η5 − (4η1 + η0 + η2)| /6 + O(∆x6). (7) τ8 =

(|P (1)

0 | − |P (1) 2 |)(P (2)

+ P (2)

2

− 2P (2)

1 )

+ O(∆x8). (8)

WENO-η(τ8), ncp = 2, ε = 10−40

dx

L_Max Error Order 0.02 0.04 0.06 10

9

10

8

10

7

10

6

10

5

10

4

10

3

10

2

10

1

3 4 5 6 p1ncp2 p1ncp2 p2ncp2 p2ncp2

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 14 / 47

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The WENO-D Scheme

The improved WENO-Z scheme, which can guarantee the optimal order of accuracy in the presence of critical points, the nonlinear weights are αk = dk

1 + Φ

τ2r−1 βk + ε p , ωk = αk r−1

j=0 αj

, k = 0, . . . , r − 1. Φ = min{1, φ}, φ =

|β0 − 2β1 + β2|.

(9) Remark

◮ Φ, as a linear combination of the βk, can be treat as a shock sensor.

◮ when the solution is smooth, Φ = φ. ◮ around the shock, Φ = 1, the new weights become the

WENO-Z scheme.

◮ It can also be derived in other form, for example, a weighted linear

combination of function values, says, {fi−1, fi, fi+1} (See WENO-η).

◮ The key point is that the following condition must be satisfied, namely,

φ

τ5

βk + ε p ∼ O(∆xr−1). (10) to guarantee the formal order of accuracy regardless critical points.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 15 / 47

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The WENO-D Scheme

In order to guarantee that the formal order of accuracy can be achieved regardless of critical points, the following condition must be satisfied, namely, φ

τ5

βk + ε p ∼ O(∆xr−1). (11) By the Taylor expansions, one has φ2 and τ5 at xi as

φ2 =

a13f (1)

i

f (3)

i

∆x4 +
a15f (1)

i

f (5)

i

+ a24f (2)

i

f (4)

i

+ a33(f (3)

i

)2 ∆x6 +

a17f (1)

i

f (7)

i

+ a26f (2)

i

f (6)

i

+ a35f (3)

i

f (5)

i

+ a44(f (4)

i

)2 ∆x8 + O(∆x9)

.

(12) τ5 =

a14f (1)

i

f (4)

i

+ a23f (2)

i

f (3)

i

∆x5

+

a16f (1)

i

f (6)

i

+ a25f (2)

i

f (5)

i

+ a34f (3)

i

f (4)

i

∆x7 + O(∆x9)
.

(13)

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 16 / 47

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Definition of Order θ

Definition The notation θ (g(∆x)) denotes the power of ∆x in the leading term of the Taylor series expansion of g(∆x), that is, θ (g) = n ⇐ ⇒ g(∆x) = Θ(∆xn). For instance, if g(∆x) = 5∆x2 + ∆x3, then θ (g) = 2.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 17 / 47

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Analysis of Order of Accuracy

According to Don et al. 2, the nonlinear component of the Z-type weights must satisfy φ τ2r−1 βk + ε p ≥ ∆xr−1, (14) to guarantee the formal order of accuracy regardless of critical points,

r

θ

φ

τ2r−1 βk + ε p ≥ r − 1. (15) θ(φ) + pθ(τ2r−1) − pθ(βk + ε) ≥ r − 1, (16) θ(βk + ε) ≤ θ(τ2r−1) + θ(φ) − (r − 1) p , (17) min{θ(βk), θ(ε)} ≤ θ(τ2r−1) + θ(φ) − (r − 1) p . (18)

2Don and Borges, J. Comput. Phys. 250, 2013 Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 18 / 47

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Analysis of Order of Accuracy

◮ If θ(ε) > θ(βk) implies ε < βk, the non-linear component is

dominated by βk, the accuracy of the scheme won’t be affected by ε.

◮ If θ(ε) ≤ θ(βk), then equation (18) becomes

θ(ε) ≤ θ(τ2r−1) + θ(φ) − (r − 1) p . (19) The integer parts of the optimal sensitivity order θ(ε), m, for the fifth order WENO schemencp = 1, 2, 3 and p = 1, 2, 3 are given in the table. WENO-Z(m) WENO-D(m) ncp θ(τ5) θ(φ) p = 1 p = 2 p = 3 p = 1 p = 2 p = 3 1 5 3 3 4 4 6 5 5 2 7 3 5 6 6 8 7 7 3 9 4 7 8 8 11 10 9

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 19 / 47

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The New WENO-D Scheme

Based on the WENO-D scheme, a small modification on the non- linear weights is applied to the WENO-D scheme, namely, αk = dk

max
1, Φ

τ2r−1 βk + ε p . (20) This modification do not affect the optimal order at the presence of critical points as analysed above for the WENO-D scheme.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 20 / 47

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The WENO-A Scheme

We name this improved WENO-D scheme as WENO-A scheme with A stands for Abarbanel.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 21 / 47

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Analysis of Order of Accuracy

We will examine the performance of the WENO-Z, WENO-D and WENO-A scheme in achieving the formal order of accuracy for a smooth function in the presence of critical points. Consider the following test function f(x) = xke0.75x, x ∈ [−1, 1], (21) in which its first k − 1 derivatives f(j)(0) = 0, j = 0, 1, ..., k − 1. That is, this function has a critical point of order ncp = k − 1 at x = 0.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 22 / 47

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Analysis of Order of Accuracy Optimal Variable ε

ncp = 1 ncp = 2 ncp = 3

dx

L_Max Error Order 10

5

10

4

10

3

10

2

10

1

10

27

10

22

10

17

10

12

10

7

10

2

3 4 5 6 7 8 Z5p2dx4 Z5p2dx4 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

dx

L_Max Error Order 10

5

10

4

10

3

10

2

10

1

10

27

10

22

10

17

10

12

10

7

10

2

3 4 5 6 7 8 Z5p2dx4 Z5p2dx4 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

dx

L_Max Error Order 10

5

10

4

10

3

10

2

10

1

10

27

10

22

10

17

10

12

10

7

10

2

3 4 5 6 7 8 Z5p2dx4 Z5p2dx4 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

◮ Notice that for various ε = ∆xm ( m = 4 for WENO-Z scheme, m = 5

for WENO-D and WENO-A scheme )

◮ All three methods achieve the optimal order asymptotically. ◮ WENO-A and WENO-D have a smaller L∞ error than

WENO-Z.

◮ For coarse mesh, WENO-A has smaller L∞ error than

WENO-D, and

◮ WENO-A has a convergence quicker than WENO-D and

WENO-Z schemes.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 23 / 47

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Analysis of Order of Accuracy Optimal Variable ε

ncp = 1 ncp = 3

dx

L_Max Error Order 0.02 0.04 0.06 0.08 0.1 10

18

10

16

10

14

10

12

10

10

10

8

10

6

10

4

10

2

3 4 5 6 7 8 Z5p2dx4 Z5p2dx4 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

dx

L_Max Error Order 0.02 0.04 0.060.08 0.1 10

16

10

14

10

12

10

10

10

8

10

6

10

4

10

2

4 5 6 7 8 Z5p2dx4 Z5p2dx4 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

◮ The Zoomed in figure.

◮ For coarse mesh, WENO-A has a smaller L∞ error than

WENO-D.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 24 / 47

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Analysis of Order of Accuracy Fixed Variable ε

ncp = 1 ncp = 2 ncp = 3

dx

L_Max Error Order 10

5

10

4

10

3

10

2

10

1

10

27

10

22

10

17

10

12

10

7

10

2

3 4 5 6 7 8 Z5p2dx5 Z5p2dx5 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

dx

L_Max Error Order 10

5

10

4

10

3

10

2

10

1

10

27

10

22

10

17

10

12

10

7

10

2

2 3 4 5 6 7 8 Z5p2dx5 Z5p2dx5 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

dx

L_Max Error Order 10

5

10

4

10

3

10

2

10

1

10

27

10

22

10

17

10

12

10

7

10

2

3 4 5 6 7 8 Z5p2dx5 Z5p2dx5 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

◮ Notice that for ε = ∆x5

◮ Not all three methods can get optimal order. ◮ WENO-A and WENO-D have smaller L∞ error than WENO-Z. ◮ WENO-A has a better convergence rate than the other two

schemes.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 25 / 47

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Analysis of Order of Accuracy Fixed Variable ε

ncp = 1 ncp = 2

dx

L_Max Error Order 0.02 0.04 0.06 0.08 0.1 10

17

10

15

10

13

10

11

10

9

10

7

10

5

10

3

4 5 6 7 8 Z5p2dx5 Z5p2dx5 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

dx

L_Max Error Order 10

4

10

3

10

2

10

17

10

15

10

13

10

11

10

9

10

7

10

5

10

3

2 3 4 5 6 7 8 9 Z5p2dx5 Z5p2dx5 D5p2dx5 D5p2dx5 A5p2dx5 A5p2dx5

◮ The Zoomed in figure.

◮ For coarse mesh, WENO-A has smaller L∞ error than

WENO-D.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 26 / 47

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WENO-A Scheme

Generally Speaking, WENO-A Scheme has

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 27 / 47

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WENO-A Scheme

Generally Speaking, WENO-A Scheme has

◮ a quicker convergence rate in the presence of high order

critical points,

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 27 / 47

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WENO-A Scheme

Generally Speaking, WENO-A Scheme has

◮ a quicker convergence rate in the presence of high order

critical points,

◮ for fine meshes, the errors are the same as WENO-D scheme,

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 27 / 47

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WENO-A Scheme

Generally Speaking, WENO-A Scheme has

◮ a quicker convergence rate in the presence of high order

critical points,

◮ for fine meshes, the errors are the same as WENO-D scheme, ◮ for coarse meshes, a smaller errors than WENO-D scheme.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 27 / 47

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WENO-A Scheme

Generally Speaking, WENO-A Scheme has

◮ a quicker convergence rate in the presence of high order

critical points,

◮ for fine meshes, the errors are the same as WENO-D scheme, ◮ for coarse meshes, a smaller errors than WENO-D scheme.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 27 / 47

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Numerical Result

We compare the numerical performance of WENO-A, WENO-D and classical WENO-Z scheme. For those numerical examples, the flow is describes by the Euler equations         ρ ρu ρv ρw E        

t

+         ρu ρu2 + p ρuv ρuw (E + p)u        

x

+         ρv ρvu ρv2 + p ρvw (E + p)v        

y

+         ρw ρwu ρwv ρw2 + p (E + p)w        

z

= 0. This set of equations describes the conservation laws expressed by mass density ρ, momentum density ρv ≡ (ρu, ρv, ρw) and total energy density E = ρe + 1

2ρv2, where e is the internal energy per

unit mass. To close this set of equations, the ideal-gas equation of state p = (γ − 1)ρe with γ = 1.4 is used.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 28 / 47

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Numerical Result

For those numerical experiments, the Euler equations are solved by following the general WENO methodology.

◮ Characteristic projection by Roe averaged eigensystem.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 29 / 47

SLIDE 36

Numerical Result

For those numerical experiments, the Euler equations are solved by following the general WENO methodology.

◮ Characteristic projection by Roe averaged eigensystem. ◮ Flux splitting by local Lax-Friedrichs.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 29 / 47

SLIDE 37

Numerical Result

For those numerical experiments, the Euler equations are solved by following the general WENO methodology.

◮ Characteristic projection by Roe averaged eigensystem. ◮ Flux splitting by local Lax-Friedrichs. ◮ Time integration by third order TVD Runge-Kutta method

with CFL number CFL= 0.45.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 29 / 47

SLIDE 38

One Dimension Shock Entropy Problem

The final time is t = 5 and resolution is N = 1500.

x Entropy

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 0.43 0.435 0.44 0.445 0.45 0.455 0.46 0.465 0.47 Reference Z5p2dx4 D5p2dx5 A5p2dx5

x Entropy

5 5.1 5.2 5.3 5.4 5.5 5.6 0.43 0.435 0.44 0.445 0.45 0.455 0.46 Reference Z5p2dx4 D5p2dx5 A5p2dx5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 30 / 47

SLIDE 39

One Dimension Shock Entropy Problem

The final time is t = 5 and resolution is N = 1600.

x Entropy

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 0.43 0.435 0.44 0.445 0.45 0.455 0.46 0.465 0.47 Reference Z5p2dx4 D5p2dx5 A5p2dx5

x Entropy

5 5.1 5.2 5.3 5.4 5.5 5.6 0.43 0.435 0.44 0.445 0.45 0.455 0.46 Reference Z5p2dx4 D5p2dx5 A5p2dx5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 31 / 47

SLIDE 40

One Dimension Shock Entropy Problem

The final time is t = 5 and resolution is N = 1700.

x Entropy

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 0.43 0.435 0.44 0.445 0.45 0.455 0.46 0.465 0.47 Reference Z5p2dx4 D5p2dx5 A5p2dx5

x Entropy

5 5.1 5.2 5.3 5.4 5.5 5.6 0.43 0.435 0.44 0.445 0.45 0.455 0.46 Reference Z5p2dx4 D5p2dx5 A5p2dx5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 32 / 47

SLIDE 41

One Dimension Shock Entropy Problem

The final time is t = 5 and resolution is N = 2000.

x Entropy

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 0.43 0.435 0.44 0.445 0.45 0.455 0.46 0.465 0.47 Reference Z5p2dx4 D5p2dx5 A5p2dx5

x Entropy

5 5.1 5.2 5.3 5.4 5.5 5.6 0.43 0.435 0.44 0.445 0.45 0.455 0.46 Reference Z5p2dx4 D5p2dx5 A5p2dx5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 33 / 47

SLIDE 42

One Dimension Shock Density Problem

The final time is t = 5 and resolution is N = 800.

x Rho

5

5 10 15 1 1.5 2 2.5 3 3.5 4 4.5 5 Reference Z5p2dx4 D5p2dx5 A5p2cp5

x Rho

9 10 11 12 13 3 3.5 4 4.5 Reference Z5p2dx4 D5p2dx5 A5p2dx5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 34 / 47

SLIDE 43

One Dimension Shock Density Problem

The final time is t = 5 and resolution is N = 700.

x Rho

5

5 10 15 1 1.5 2 2.5 3 3.5 4 4.5 5 Reference Z5p2dx4 D5p2dx5 A5p2cp5

x Rho

9 10 11 12 13 3 3.5 4 4.5 Reference Z5p2dx4 D5p2dx5 A5p2dx5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 35 / 47

SLIDE 44

One Dimension Shock Density Problem

The final time is t = 5 and resolution is N = 600.

x Rho

5

5 10 15 1 1.5 2 2.5 3 3.5 4 4.5 5 Reference Z5p2dx4 D5p2dx5 A5p2cp5

x Rho

9 10 11 12 13 3 3.5 4 4.5 Reference Z5p2dx4 D5p2dx5 A5p2dx5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 36 / 47

SLIDE 45

Interacting Blast Wave Problem

The final time is t = 0.038 and resolution is N = 400.

x Rho

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 Reference Z5p2dx4 D5p2dx5 A5p2dx5

x Rho

0.6 0.65 0.7 0.75 0.8 0.85 1 2 3 4 5 6 Reference Z5p2dx4 D5p2dx5 A5p2dx5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 37 / 47

SLIDE 46

Two Dimension Riemann Problem

The final time is t = 0.8 and resolution is N = 400 × 400. WENO-Z5 WENO-D5 WENO-A5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 38 / 47

SLIDE 47

Two Dimension DMR Problem

The final time is t = 0.2 and resolution is N = 800 × 200. WENO-Z5 WENO-D5 WENO-A5

Table: The CPU time (in seconds) of the DMR problem.

N × M WENO-Z WENO-D WENO-A 800 × 200 4.1E+03 4.6E+03 4.5E+03

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SLIDE 48

Rayleigh-Taylor Instability Problem

WENO-Z5 WENO-D5 WENO-A5 N = × higher resolution will be updated in the next version, the code is still running

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SLIDE 49

Compressible Multicomponent Flows

Use WENO-A scheme for solving overestimated quasi-conservative form of the compressible multicomponent flows simulation. ∂Q ∂t + U∂F ∂x = 0, (22) Q =       ρ ρu ρe ρY1 γp       , U =       1 1 1 1 u       , F =       ρu ρu2 + P u (ρe + P) ρuY1 γp       . (23)

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 41 / 47

SLIDE 50

Richtmyer-Meshkov Instability Problem

The initial conditions are

(ρ u v P γ M) =    ( 1.4112 −0.3613 1.6272/1.4 1.4 28.8 ), x > −0.8 ( 5.04 1/1.4 1.093 145.15 ), x < x0 ( 1 1/1.4 1.4 28.8 ),

therwise.

(24)

◮ where x0 = −1.1 − 0.1 cos(2πy). ◮ The computational domain is −8 ≤ x ≤ 0 and 0 ≤ y ≤ 1.

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 42 / 47

SLIDE 51

Richtmyer-Meshkov Instability Problem

The final time is t = 8.25 and resolution is N = 1024 × 128. WENO-Z5 WENO-D5 WENO-A5

Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 43 / 47

SLIDE 52

Shock-Bubble Interaction Problem

The initial conditions are

(ρ u v P γ M) =    ( 1.3764 −0.3336 1.5698/1.4 1.4 28.80 ), x ≥ 1.0 ( 3.153 1/1.4 1.249 90.82 ),

x2 + y2 < 0.5

( 1 1/1.4 1.4 28.80 ),

therwise.

(25)

◮ The computational domain is −3.5 ≤ x ≤ 3 and −1 ≤ y ≤ 1.

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SLIDE 53

Shock-Bubble Interaction Problem

The final time is t = 7.337 and resolution is N = 650 × 178. WENO-Z5 WENO-D5 WENO-A5

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SLIDE 54

Conclusion And Future Work

◮ Summary and conclusion remark

◮ We present the WENO-D and WENO-A schemes for the

solution of nonlinear hyperbolic conservation laws.

◮ We analyzed the new schemes in resolving function with a high

rder critical point.

◮ We demonstrate that the new schemes can achieve the

ptimal order of accuracy with a greatly relaxed constraint on

the sensitivity parameter ε.

◮ The WENO-A scheme has a substantially smaller ε than the

standard WENO-Z scheme, and performs competitively for shocked flows.

◮ Future Work

◮ Extend the WENO-A scheme to higher order. ◮ Extend the WENO-A scheme to alternative WENO scheme

(AWENO) scheme for multi-components shocked flows in a general curvilinear coordination.

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SLIDE 55

References

[1] R. Borges, M. Carmona, B. Costa, and W. S. Don. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J.

Comput. Phys., 227(2008), 3101–3211.

[2] W. S. Don, and R. Borges. Accuracy of the weighted essentially non-

scillatory conservative finite difference schemes. J. Comput. Phys., 250(2013),

347–372. [3] P. Fan, Y. Shen, B. Tian and C. Yang. A new smoothness indicator for inproving the weighted essentially non-oscillatory scheme. J. Comput. Phys., 269(2014), 329–354. [4] Henrick, A.K., Aslam, T.D., Powers and J.M. Mapped weighted-essentially- non-oscillatory schemes: achieving optimal order near critical points. J.

Comput. Phys., 207(2005), 542–567.

[5] X. Y. Hu, Q. Wang and N. A. Adams. An adaptive central-upwind weighted essentially non-oscillatory scheme.

J. Comput.

Phys., 229(2010), 8952– 8965.

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