topic course on numerical methods in computational fluid ...jingqiu/math817_2019/l5.pdfeno and weno...

Comp. Math.&

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Jingmei Qiu

Outline

ENO andWENOschemes

Schemes fornonlinearsystems

Schemes forhighdimensionalproblems

Summary

Topic Course on Numerical Methods inComputational Fluid Dynamics

Lecture 5: ENO and WENO schemes

Jingmei Qiu

Department of Mathematical ScienceUniversity of Delaware

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Jingmei Qiu

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ENO andWENOschemes



Summary

Outline

1 Finite volume schemes.• Essentially non-oscillatory (ENO) and weighted ENO

(WENO) reconstructions.• 1D systems.• High dimensional problems: 2D.

2 Finite difference schemes.• Scheme formulation with ENO and WENO reconstructions.• 1D systems.• 2D problems.

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ENO andWENOschemes



Summary

Recap of TVD and TVB schemes

For nonlinear hyperbolic equations.• TV stability for a fully discretized scheme

• spatial reconstruction with minmod or modified minmodlimiters

• time discretization by TVD or SSP Runge-Kutta method• Non-oscillatory resolution of discontinuities due to the TV

stability.

• Accuracy• Full accuracy in smooth and monotone regions• Around extrema.

• TVD: Accuracy degeneracy.• TVB: problem dependent tuning parameter M.

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Outline

ENO andWENOschemes



Summary

ENO and WENO schemes

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ENO andWENOschemes



Summary

ENO schemesRecall the finite volume scheme

d

dtuj +

1

∆x

(f (u−

j+ 12

, u+j+ 1

2

)− f (u−j− 1

2

, u+j− 1

2

)

)= 0,

• MUSCL: the 2-point stencil with smaller variation ischosen to reconstruction u−

j+ 12

u−j+ 1

2

= unj +

1

2minmod(un

j − unj−1, u

nj+1 − un

j ).

• ENO: consider three three-point stencils

{Ij−2, Ij−1, Ij}, {Ij−1, Ij , Ij+1}, {Ij , Ij+1, Ij+2}

Stencils with smallest variation (measured by divideddifference) is chosen to reconstruction u−

j+ 12

.

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ENO andWENOschemes



Summary

ENO stencil

Ij−2 Ij−1 Ij Ij+1 Ij+2

Figure: ENO choose stencil {Ij , Ij+1, Ij+2} with smallest variation(thus avoid the discontinuity) to reconstruct u−

j+ 12

.

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ENO andWENOschemes



Summary

ENO stencil selection

• Solution cell averages:

u[xj ] = uj , ∀j

• First order divided difference:

u[xj , xj+1] =u[xj+1]− u[xj ]

xj+1 − xj, ∀j

• Second order divided difference:

u[xj , xj+1, xj+2] =u[xj+2, xj+1]− u[xj+1, xj ]

xj+2 − xj, ∀j .

· · ·

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ENO andWENOschemes



Summary

ENO stencil

Ij−2 Ij−1 Ij Ij+1 Ij+2

Figure: To reconstruct u−j+ 1

2

: {uj} ⇒ {uj , uj+1} ⇒ {uj , uj+1, uj+2}.

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ENO andWENOschemes



Summary

ENO reconstruction of u−j+ 1

2

1 Choose ENO stencil containing 3 cells

a starting with uj

b choose 3 cells in the ENO stencil, in an adaptive mannerguided by divided differences.

2 Construction polynomials of degree 2, whose cell averagesagree with the given cell averages.∫

Il

P2(x)dx = ul , l ∈ ENO stencil.

3 Evaluate P2(x) at xj+ 12

approximating u−j+ 1

2

.

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ENO andWENOschemes



Summary

Example for computation

Stencil {Ij−1, Ij , Ij+1} :

u−j+ 1

2

≈ −1

6uj−1 +

5

6uj +

1

3uj+1 +O(∆x3).

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ENO andWENOschemes



Summary

ENO reconstruction of u+j+ 1

2

• Choose ENO stencil containing 3 cells• starting with uj+1

• choose 3 cells in the ENO stencil, in an adaptive mannerguided by divided differences.

This maybe a different ENO stencil as that in reconstructingu−

j+ 12

.

• Construction polynomials of degree 2, whose cell averages agreewith the given cell averages.∫

Il


This maybe a different P2 polynomial as that in reconstructingu−

j+ 12

.

• Evaluate P2(x) at xj+ 12

approximating u+j+ 1

2

.

Evaluation location is at xj+ 12.

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ENO andWENOschemes



Summary

ENO reconstruction of u+j− 1

2

• Choose ENO stencil containing 3 cells• starting with uj

• choose 3 cells in the ENO stencil, in an adaptive mannerguided by divided differences.

This is the same ENO stencil as that in reconstructing u−j+ 1

2

.

• Construction polynomials of degree 2, whose cell averages agreewith the given cell averages.∫

Il


This is the same P2 polynomial as that in reconstructing u−j+ 1

2

.

• Evaluate P2(x) at xj− 12

approximating u+j− 1

2

.

Evaluation location is at xj− 12.

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ENO andWENOschemes



Summary

ENO schemes

Given {uj}j

1 ENO reconstruction u±j+ 1

2

, ∀j .

2 Evaluate numerical fluxes

fj+ 12

= f (u−j+ 1

2

, u+j+ 1

2

), ∀j .

3 Evolve the ODE system by method-of-lines

d

dtuj +

1

∆x

(fj+ 1

2− fj− 1

2

)= 0,

to update uj for each RK stage, and finally un+1j .

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ENO andWENOschemes



Summary

From ENO to WENO

• In an ENO scheme, 5 pieces of information in theneighborhood are called, yet only 3 pieces of information isused in the reconstruction process for u−

j+ 12

.

• WENO: to make full use of all five cell averages in theneighborhood for u−

j+ 12

by performing a weighted sum

reconstruction.

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ENO andWENOschemes



Summary

WENO schemes: Finite volumemethods

• Stencil {Ij−2, Ij−1, Ij} : u(1)

j+ 12

= 13 uj−2 − 7

6 uj−1 + 116 uj .

• Stencil {Ij−1, Ij , Ij+1} : u(2)

j+ 12

= − 16 uj−1 + 5

6 uj + 13 uj+1.

• Stencil {Ij , Ij+1, Ij+2, } : u(3)

j+ 12

= 13 uj + 5

6 uj+1 − 16 uj+2.

• Stencil {Ij−2, Ij−1, Ij , Ij+1, Ij+2} :

u(5)

j+ 12

=1

30uj−2 −

13

60uj−1 +

47

60uj +

9

20uj+1 −

1

20uj+2.

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Outline

ENO andWENOschemes



Summary

Finite volume scheme

It is found that

u(5)

j+ 12

= γ1u(1)

j+ 12

+ γ2u(2)

j+ 12

+ γ3u(3)

j+ 12

with linear weights

γ1 =1

10, γ2 =

3

5, γ3 =

3

10.

Fifth order WENO approximation of u−j+ 1

2

u−j+ 1

2

= ω1u(1)

j+ 12

+ ω2u(2)

j+ 12

+ ω3u(3)

j+ 12

,

where nonlinear weights ωj ≥ 0 satisfying

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ENO andWENOschemes



Summary

• ω1 + ω2 + ω3 = 1

• ωj = γj + O(∆x2), j = 1, 2, 3

— ωj should be close to γj in smooth region, to maintainthe fifth order accuracy.

• ωj should be around 0 for stencils containing adiscontinuity.

— taking almost zero weight from the stencil thatcontains discontinuities, thus avoid numerical oscillations.

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ENO andWENOschemes



Summary

Nonlinear weights

Define

ωj =ωj

ω1 + ω2 + ω3, with ωj =

γj

(ε+ βj )2,

j = 1, 2, 3. Here ε = 10−6 in practice and the smoothnessindicator

βj =2∑

l=1

∆x2l−1

∫ xi+ 1

2

xi− 1

2

(d l

dx lpj (x)

)2

dx .

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ENO andWENOschemes



Summary

Smoothness indicator βj

The explicit formulas of the smoothness indicators are

β1 =13

12(uj−2 − 2uj−1 + uj )

2 +1

4(uj−2 − 4uj−1 + 3uj )

2,

β2 =13

12(uj−1 − 2uj + uj+1)2 +

1

4(uj−1 − uj+1)2,

β3 =13

12(uj − 2uj+1 + uj+2)2 +

1

4(3uj − 4uj+1 + uj+2)2.

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ENO andWENOschemes



Summary

FV WENO flowchart

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ENO andWENOschemes



Summary

Summary

• WENO is a robust and practical procedure for nonlinearCFD applications.

• Yet, little theoretical properties (e.g. TVD or TVB) can beproved.

• TVD RK methods could be used for time discretization.

We have been considering finite volume schemes evolve the cellaverages {uj}, and approximate integral form of the equation.Another framework, that offers computational simplicity andlower computational cost for high dimensional problems, is thefinite difference scheme. It evolves point values andapproximates the differential form of the equation.

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ENO andWENOschemes



Summary

Finite difference scheme

The scheme in conservative form

d

dtuj +

1

∆x

(fj+ 1

2− fj− 1

2

)= 0,

where the numerical flux fj+ 12

= f (uj−p, · · · , uj+q).

Consideration of accuracy: The scheme is rth order accurate if

1

∆x

(fj+ 1

2− fj− 1

2

)= f (u)x |x=xj + O(∆x r ),

when u is smooth in the stencil.

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ENO andWENOschemes



Summary

Sliding average function andnumerical flux

Lemma. If there is a function h(x) such that

1

∆x

∫ x+ ∆x2

x−∆x2

h(ξ) dξ = f (u(x)),

then,1

∆x

(h(xj+ 1

2)− h(xj− 1

2))

= f (u)x |x=xj

• h is called sliding average function.

• The sliding average function h is a man-made function,that is designed so that the flux difference offers highorder accuracy in the finite difference framework.

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ENO andWENOschemes



Summary

Sliding average function andnumerical flux

Lemma. If there is a function h(x) such that

1

∆x

∫ x+ ∆x2

x−∆x2

h(ξ) dξ = f (u(x)),

then,1

∆x

(h(xj+ 1

2)− h(xj− 1

2))

= f (u)x |x=xj

The lemma indicates that the numerical flux can be chosen as

fj+ 12

= h(xj+ 12),

while hj = 1∆x

∫ xj+ 1

2x

j− 12

h(ξ) dξ = f (uj ) is known.

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Outline

ENO andWENOschemes



Summary

Reconstruction of flux

The reconstruction from:

cell averages {hj} ⇒ point values of h(xj+1/2)

is the same as that in a finite volume scheme,

cell averages {uj} ⇒ point values of u(xj+1/2).

Notice that in the finite difference framework:

• hj = f (uj )

• h(xj+1/2) = fj+1/2.

point values {f (uj )} ⇒ numerical flux fj+1/2.

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ENO andWENOschemes



Summary

How to respect the upwindprinciple

If f ′(u) ≥ 0, the polynomial reconstruction of fi+1/2 uses aleft-biased stencil.

• First order scheme: {f (uj )}

fj+1/2 = f (uj ).

• Third order scheme: {f (uj−1), f (uj ), f (uj+1)}

fj+1/2 = −1

6f (uj−1) +

5

6f (uj ) +

1

3f (uj+1).

• Fifth order scheme: {f (uj−2), · · · , f (uj+2)}

fj+1/2 =1

30f (uj−2)− 13

60f (uj−1) +

47

60f (uj )

+9

20f (uj+1)− 1

20f (uj+2)

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Outline

ENO andWENOschemes



Summary

How to respect the upwindprinciple

If f ′(u) ≤ 0, the polynomial reconstruction of fi+1/2 uses aright-biased stencil.• First order scheme: {f (uj+1)}

fj+1/2 = f (uj+1).

• Third order scheme: {f (uj ), f (uj+1), f (uj+2)}

fj+1/2 =1

3f (uj ) +

5

6f (uj+1)− 1

6f (uj+2).

• Fifth order scheme: {f (uj−1), · · · , f (uj+3)}

fj+1/2 =− 1

20f (uj−1) +

9

20f (uj ) +

47

60f (uj+1)

− 13

60f (uj+2) +

1

30f (uj+3)

The reconstruction coefficients are symmetric to that from theleft-biased stencil.

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Outline

ENO andWENOschemes



Summary

When f ′(u) changes sign, a flux splitting is needed,

f (u) =1

2f (u) +

α

2u︸︷︷︸

f +(u)

+1

2f (u)− α

2u︸︷︷︸

f −(u)

with α = maxu|f ′(u)|.

d

duf +(u) ≥ 0,

d

duf −(u) ≤ 0.

1 The reconstruction of f +j+ 1

2

is based on a left-biased stencil

from {f +(uj )}.2 The reconstruction of f −

j+ 12

is based on a right-biased

stencil from {f −(uj )}.3 fj+ 1

2= f +

j+ 12

+ f −j+ 1

2

.

WENO procedure can be used for the reconstructions of f ±j+ 1

2

.

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Outline

ENO andWENOschemes



Summary

FD WENO flowchart

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Outline

ENO andWENOschemes



Summary

Comparison of FV and FD schemesA finite volume WENO scheme

• is based on {uj} and uses the integral form of the PDE;

• {uj}WENO−−−−−−−−→

reconstruction{u±

j+ 12

};• any monotone flux f (u−, u+);

• ∆x can be nonuniform.

A finite difference WENO scheme

• is based on {uj} and uses the original PDE directly;

• {f ±uj} WENO−−−−−−−−→

reconstruction{f ±

j+ 12

}• can only use monotone fluxes which correspond to smooth

flux splitting f (u) = f +(u) + f −(u) satisfying

d

duf +(u) ≥ 0,

d

duf −(u) ≤ 0;

• ∆x has to be uniform.

FV and FD WENO are equally good for 1D problems.29 / 55

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ENO andWENOschemes



Summary

Two dimensional problems

ut + aux + buy = 0.

ut + f (u)x + g(u)y = 0.

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ENO andWENOschemes



Summary

FV for 2D problem

We have the integral form

d ˜ui,j (t)

dt=−

1

∆x∆y

∫ yj+ 1

2

yj− 1

2

f (u(xi+ 12, y , t))dy −

∫ yj+ 1

2

yj− 1

2

f (u(xi− 12, y , t))dy

+

∫ xi+ 1

2

xi− 1

2

g(u(x , yj+ 12, t))dx −

∫ xi+ 1

2

xi− 1

2

g(u(x , yj− 12, t))dx

• Iij = [xi− 12, xi+ 1

2]× [yj− 1

2, yj+ 1

2].

• ˜ui ,j = 1∆x∆y

∫Iij

u(x , y)dxdy ;

• ·: average over [xi− 12, xi+ 1

2];

• ·: average over [yj− 12, yj+ 1

2].

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ENO andWENOschemes



Summary

FV for 2D problem (cont.)

Approximate the integral form by FV scheme

d ˜ui ,j (t)

dt= − 1

∆x(fi+ 1

2,j − fi− 1

2,j )−

1

∆y(gi ,j+ 1

2− gi ,j− 1

2),

where the numerical fluxes are given by

fi+ 12,j ≈

1

∆y

∫ yj+ 1

2

yj− 1

2

f (u(xi+ 12, y , t))dy ≡ fi+ 1

2,j ,

gi ,j+ 12≈ 1

∆x

∫ xi+ 1

2

xi− 1

2

g(u(x , yj+ 12, t))dx ≡ gi ,j+ 1

2.

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ENO andWENOschemes



Summary

FV for 2D linear constantcoefficient case

Considerut + aux + buy = 0

for which

fi+ 12,j = aui+ 1

2,j , gi ,j+ 1

2= bui ,j+ 1

2.

In this case, only two one-dimensional WENO reconstructionsare needed

{˜ui ,j} → {ui+ 12,j} for fixed j

and{˜ui ,j} → {ui ,j+ 1

2} for fixed i .

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ENO andWENOschemes



Summary

FV for 2D nonlinear case

If f (u) and g(u) are nonlinear functions, then

f (u) 6= f (u).

Thus, we would need to do

{˜ui ,j} → {u±i+ 12,j}︸︷︷︸

WENO reconstruction

WENO−→ {u±i+ 1

2,j+jα}αg

α=1 → {fxj+ 1

2,j}︸︷︷︸

numerical flux and intergration

,

where

• {j + jα}αg

α=1 are the Gaussian quadrature points for theinterval [yj− 1

2, yj+ 1

2]. We can reconstruct them with

sufficiently high order of accuracy and with WENO.

• likewise for gi ,j+ 12.

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ENO andWENOschemes



Summary

Computational cost analysis

Assume there are three Gaussian quadrature points per side,per construction of u±

i+ 12,j

• 1 WENO: {˜ui ,j} → {u±i+ 12,j}.

• 3 WENO: {u±i+ 1

2,j} → {u±

i+ 12,j+jα}αg

α=1.

• Evaluation of numerical flux and numerical integration(less costly).

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ENO andWENOschemes



Summary

FV for 2D problem: flowchart

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ENO andWENOschemes



Summary

FD for 2D problem

The scheme is

duij (t)

dt= − 1

∆x(fi+ 1

2,j − fi− 1

2,j )−

1

∆y(gi ,j+ 1

2− gi ,j− 1

2).

• ui ,j = u(xi , yj , t);

• the numerical flux fi+ 12,j can be

computed from {ui ,j} with fixedj in exactly the same way as inthe one-dimensional case, andlikewise for gi ,j+ 1

2;

• FD scheme approximates thePDE form directly and can beproceed dimension by dimension.

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Outline

ENO andWENOschemes



Summary

Computational cost analysis

FD WENO scheme: per construction of f ±i+ 1

2,j

• 1 WENO: {f ±(ui ,j )} → {f ±i+ 12,j}.

Compared with that in FV scheme: per construction of u±i+ 1

2,j

• 1 WENO: {˜ui ,j} → {u±i+ 12,j}.

• 3 WENO: {u±i+ 1

2,j} → {u±

i+ 12,j+jα}αg

α=1.

• Evaluation of numerical flux and numerical integration(less costly).

Not only the computational cost is saved, but also theimplementation efforts.

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Outline

ENO andWENOschemes



Summary

FD for 2D problem: flowchart

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Outline

ENO andWENOschemes



Summary

Comparison of FV and FD for 2Dproblems

A FV WENO scheme

• is based on {˜ui ,j} and uses the integral form of the PDE;

• {˜ui,j}WENO−→ {ui+ 1

2 ,j} WENO−→ {ui+ 1

2 ,j+jα}αg

α=1

Integration−→ {fxj+ 1

2,j};

• any monotone flux f and g ;

A FD WENO scheme

• is based on {uij} and uses the original PDE directly;

• {f ±(ui ,j )}WENO−→ {f ±

i ,j+ 12

};• can only use monotone fluxes which correspond to smooth

flux splitting.

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Outline

ENO andWENOschemes



Summary

Comparison of performance andCPU cost

• Second order accuracy:

• Comparable performance, in accuracy and in

computational complexity.

• FV is more physically meaningful, as the scheme is based

on an integral form and mass conservation is naturally

enforced.

• Third order accuracy or higher.

• FD offers great savings in implementation complexity and

computational cost.

• Expect even more significant savings for higher

dimensional problems.

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Outline

ENO andWENOschemes



Summary

Linear and nonlinear systems

Finite Difference schemes

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Outline

ENO andWENOschemes



Summary

Linear systems

Ut + AUx = 0 (1)

U(x , 0) = U0(x) (2)

where A has the eigen-decomposition

A = RΛR−1 (3)

where Λ is a diagonal matrix of real eigenvalues

Λ = diag(λ1, λ2, · · · , λs)

and R is the matrix of right eigenvectors

R = [r1|r2| · · · |rs ].

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ENO andWENOschemes



Summary

Linear systems

Multiplying (1) with R−1 and using (3) give

R−1Ut + ΛR−1Ux = 0

Let V = R−1U, then

Vt + ΛVx = 0

which can be decoupled into s independent scalar equations

(Vp)t + λp(Vp)x = 0, p = 1, 2, · · · , s.

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ENO andWENOschemes



Summary

FD WENO scheme for V

For each component Vp, p = 1, 2, · · · , s• Depending on the sign of λp, construct the numerical flux

Vp,j+ 12

base on upwind principle (stencils) and WENO procedures.

• Evolve, by a third order TVD Runge-Kutta method, thefollowing time-dependent ODEs

d

dtVp,j (t) +

λp

∆x(Vp,j+ 1

2− Vp,j− 1

2) = 0.

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Outline

ENO andWENOschemes



Summary

FD WENO for linear system1 Perform the eigendecomposition of

A = A+ + A− = RΛ+R−1 + RΛ−R−1.

2 At each point, perform the flux splitting

AUj = A+Uj + A−Uj.

= F +(Uj ) + F−(Uj ).

3 WENO reconstruction of fluxesa Compute characteristics variables CF±

j = R−1F±(Uj )

b Perform WENO reconstruction of CF±j+ 1

2along

characteristics directions CF±j based on a left/right-biased

stencil.c Compute the original flux F±

j+ 12

= RCF±j+ 1

2.

d Fj+ 12

= F +j+ 1

2

+ F−j+ 1

2

.

4 Evolve, by a third order TVD Runge-Kutta method, thefollowing time-dependent ODEs

d

dtUj (t) +

1

∆x(Fj+ 1

2− Fj− 1

2) = 0.

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Jingmei Qiu

Outline

ENO andWENOschemes



Summary

Nonlinear system

Consider the equation

Ut + F (U)x = 0.

For example, the Euler system with equation of stateE = 1

2ρu2 + pγ−1 ρ

ρuE

t

+

ρuρu2 + pEu + pu

x

= 0.

• There is essentially no theory.

• The numerical procedure is essentially identical to that forthe linear system.

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Comp. Math.&

Applications

Jingmei Qiu

Outline

ENO andWENOschemes



Summary

Nonlinear system

Un+1j = Un

j −∆t

∆x(Fj+ 1

2− Fj− 1

2)

Flux splitting for a nonlinear system:

Fj+ 12

=1

2(F (U−

j+ 12

) + F (U+j+ 1

2

)) +α

2(U−

j+ 12

− U+j+ 1

2

)

where α = maxλi ,u|λi (FU)|.

48 / 55

Comp. Math.&

Applications

Jingmei Qiu

Outline

ENO andWENOschemes



Summary

FD WENO for nonlinear system1 Perform the flux splitting

F (Uj ) = F +(Uj ) + F−(Uj ).

2 WENO reconstruction of fluxes at xj , Fj+ 12

a Compute a local Jacobian matrix FU |Uj and perform

localized eigen-decomposition FU |Uj = R−1j Λj Rj .

b Compute characteristics variables CF±j = R−1

j F±(Uj ).

c Perform WENO reconstruction of CF±j+ 1

2along

characteristics directions CF±j based on a left/right-biased

stencil.d Compute the original flux F±

j+ 12

= Rj CF±j+ 1

2.

e Fj+ 12

= F +j+ 1

2

+ F−j+ 1

2

.

3 Evolve, by a third order TVD Runge-Kutta method, thefollowing time-dependent ODEs

d

dtUj (t) +

1

∆x(Fj+ 1

2− Fj− 1

2) = 0.

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Comp. Math.&

Applications

Jingmei Qiu

Outline

ENO andWENOschemes



Summary

High dimensional systems

Finite difference WENO scheme

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Comp. Math.&

Applications

Jingmei Qiu

Outline

ENO andWENOschemes



Summary

3D Euler system for air flowdynamics

ρt +∇ · (ρu) = 0,

(ρu)t +∇ · (ρu⊗ u) +∇p = 0,

Et +∇ · ((E + P)u) = 0.

• U = (ρ, ρu,E )T

• F (U) = (ρu, ρu⊗ u + pI,Eu + pu).

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Comp. Math.&

Applications

Jingmei Qiu

Outline

ENO andWENOschemes



Summary

FD WENO for high-D nonlinearsystem: key components

An organic integration of numerical techniques previouslyintroduced.

• Base scheme: finite difference WENO scheme.

• High dimensional problem: dimension-by-dimensiontreatment.• Nonlinear system

• Flux splitting• Local characteristics decomposition• WENO reconstructions of characteristics variables.

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Comp. Math.&

Applications

Jingmei Qiu

Outline

ENO andWENOschemes



Summary

Summary of WENO

• Adaptive nonlinear localized treatment for convectionterms.

• Mainly for compressible convection terms:handling shocks

• Some work for incompressible convection terms.

• Application to kinetic equation:— main challenge: high D and large scale computing.

• Other issues: boundary conditions etc.

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Comp. Math.&

Applications

Jingmei Qiu

Outline

ENO andWENOschemes



Summary

WENO for CFD

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Comp. Math.&

Applications

Jingmei Qiu

Outline

ENO andWENOschemes



Summary

Reference:

• B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor(Editor: A. Quarteroni), Advanced NumericalApproximation of Nonlinear Hyperbolic Equations, LectureNotes in Mathematics, volume 1697, Springer, 1998.

• C.-W. Shu, High Order Weighted EssentiallyNonoscillatory Schemes for Convection DominatedProblems, SIAM Rev., 51(1), 82-126.

Acknowledgement:

• Lecture notes of applied math course 257 by ProfessorChi-Wang Shu, when I was a graduate student at BrownUniversity.

• Special thanks to Ms. Mingchang Ding (Ph.D. student atUniversity of Delaware) for her help in preparing theseslides.

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topic course on numerical methods in computational fluid ...jingqiu/math817_2019/l5.pdfeno and weno...

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