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Exponential WENO schemes for Ideal MHD equations on unstructured meshes

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) February 25, 2014

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

ABSTRACT

◮ This study proposes a new 6th order weighted essentially

non-oscillatory (WENO) finite difference schemes.

◮ The interpolation method is implemented by using

exponential polynomials with shape (or tension) parameters such that they can be tuned to the characteristics of a given data, yielding better approximation than the classical ENO schemes at the same computational cost.

◮ We provide a new smoothness indicator to generate the

weight of the WENO schemes.

◮ Some numerical experiments are conducted to demonstrate

the performance, particularly near discontinuities, of the proposed schemes.

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

Hyperbolic Conservation Law and Conservative Schemes

◮ Hyperbolic conservation Law system(Euler & MHD Equation.)

qt + f(q)x = 0, t ≥ 0, x ∈ Ω(⊂ Rn, n = 1, 2, 3)

◮ Conservative (numerical) semidiscrete scheme(1D):

d ¯ q dt = − ˆ fj+1/2 − ˆ fj−1/2 ∆x f (q) = 1 ∆x x+∆x/2

x−∆x/2

h(ξ)dξ ¯ hj = 1 ∆x xj+1/2

xj−1/2

h(ξ)dξ

◮ Goal: To obtain numerical flux ˆ

f to approximate h at the cell boundaries with a suitable convergence order.

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

Sixth order WENO scheme

◮ 6th order WENO Stencil

ˆ fj+1/2

• xj− 5

2

| ¯ hj−2

• xj− 3

2

| ¯ hj−1

• xj− 1

2

| ¯ hj

• xj+ 1

2

| ¯ hj+1

• xj+ 3

2

| ¯ hj+2

• xj+ 5

2

| ¯ hj+3

• xj+ 7

2

| S5(j − r) |

S5

|

S3(j − 2) |

S0

|

S3(j − 1) |

S1

|

S3(j − 0) |

S2

|

S3(j + 1) |

S3

|

S6(j − 2) |

S6

|

◮ ˆ

fj+1/2 =

3

• k=0

ωk ˆ f k

j+1/2, where ωk are nolinear wieghts obtained

by WENO schemes.

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

◮ 3rd order approximation on substencil S0, S1, S2, S3

ˆ f 0

j+ 1

2 =

1 3fj−2 − 7 6fj−1 + 11 6 fj ˆ f 1

j+ 1

2 = −1

6fj−1 + 5 6fj + 1 3fj+1 ˆ f 2

j+ 1

2 =

1 3fj + 5 6fj+1 − 1 6fj+2 ˆ f 3

j+ 1

2 = 11

6 fj+1 − 7 6fj+2 + 1 3fj+3. (1)

◮ 6th order (polynomial) WENO and it smoothness indicators

ωk = αk αℓ , αk = dk(C + τ6 ǫ + βk ), k = 0, · · · , 3 τ6 = β6 − 1 6(β0 + 4β1 + β2), d0 = d3 = 1 20, d1 = d2 = 9 20 βk =

2

• ℓ=1

xj+1/2

xj−1/2

∆x2ℓ−1 dℓ dxℓ ˆ f k

• dx

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

Construction of Numerical Flux Using Exponential Functions

◮ Polynomials are the most natural and well-established building

blocks to construct the numerical flux ˆ f .

◮ The polynomial approximation space cannot be adjusted

according to the characteristic of a given data.

◮ As a result, when interpolating data with rapid gradients, it

has a limitation in producing sharp edges such that interpolation errors becomes large.

◮ For a better approximation to data around non-smooth

regions, we employ interpolation method based on exponential polynomials to adapt a new parameter sharpening the edges while keeping the same approximation orders.

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

◮ For the sixth-order Exponential WENO scheme, we can

consider the following spaces.

• Γ = span{xn, eγx, e−γx : n = 0, . . . , 4}
• Γ = span{xn, eiγx, e−iγx, eγx, e−γx : n = 0, . . . , 2}
• S5 := span{1, x, eγx, e−γx, eνx, e−νx}

◮ symetric, shift–invariant spaces ◮ In order for an interpolation kernel to constitute a partition of

unity, which means that the polynomial p(x) = 1 is in spaces.

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

◮ Generally, let Γ = {ϕ0, . . . , ϕm},

H(x) = x

−∞

h(ξ) dξ

◮ Interpolation QH to H at the cell-boundaries

xj−r−1/2, . . . , xj−r−1/2+m is defined by a linear combination Γ.

◮ QH can be written as Lagrange-type representation.

QH(x) =

m

• n=0

uj−r,n(x)H(xj−r+n−1/2), H(xk+1/2) = ∆x

k

• ℓ=−∞

¯ hℓ. satisfying uj−r,n(xj−r+k−1/2) = δk,n for k, n = 0, . . . , m.

◮ The numerical flux ˆ

f is defined by ˆ f (x) = Q′

H(x) = m

• n=0

u′

j−r,n(x)H(xj−r+n−1/2)

ˆ fj+1/2 = ∆x

m−1

• ℓ=0
• m
• n=ℓ+1

u′

n(xj+1/2)

• Cm,j−r+ℓ

¯ hj−r+ℓ.

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

Construction of optimal weight for the exponential WENO

◮ The numerical flux ˆ

fj+1/2 approximating h(xj+1/2) on the stencil S6 can be defined by ˆ fj+1/2 =

5

• ℓ=0

Cm,j−2+ℓ¯ hj−2+ℓ, Cm,j−2+ℓ := ∆x

6

• n=ℓ+1

u′

n(xj+1/2). ◮ The local numerical flux associated to the sub-stencil Sk 3 is

given as a linear combination of the cell averages ¯ hj+n ˆ f k

j+1/2 = 2

• ℓ=0

C k

3,j−2+k+ℓ¯

hj−2+k+ℓ, C k

3,j−2+k+ℓ = ∆x 3

• n=ℓ+1

u′

k,n(xj+1 ◮ Ideal Weights based on Exponential Polynomial Interpolation

d0 = C6,j−2/C 0

3,j−2

d1 = (C6,j−1 − d0 · C 0

3,j−1)/C 1 3,j−1

d2 = 1 − d0 − d1 − d3 d3 = C6,j+3/C 3

3,j+3.

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

New smoothness indicator

◮ A new smoothness indicator is constructed by the (weighted)

L1-sums of the generalized undivided differences.

◮ First, for each fixed n = 1, 2, we find the coefficient vector

c[n]

k

:= (c[n]

k,ℓ : xℓ ∈ S)T,

n = 1, 2, by solving the linear system

• xℓ∈Sk

j−r+k

c[n]

k,ℓ

(xℓ − xj+1/2)m m! = ∆xnδn,m, m = 0, . . . , |S| − 1.

◮ Generalized undivided differences

Dn,kfj+1/2 :=

• xℓ∈Sk

j−r+k

c[n]

k,ℓf (xℓ),

n = 1, 2,

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

Non linear Weights for exponential WENO with 6th-order accuracy

◮ The new smoothness indicator βk (k = 0, . . . , 3):

βk := ξk |D1,kf | + |D2,kf | , ξk ∈ (0, 1]. (2) D1,kf := (1 − k)fj−2+k + (2k − 3)fj−1+k + (2 − k)fj+k D2,kf := fj−2+k − 2fj−1+k + fj+k, D1,3f := (−23fj + 2fj+1 + 3fj+2 − fj+3)/24 D2,3f := (3fj − 7fj+1 + 5fj+2 − fj+3)/2,

◮ A new version of WENO weights with the 6th-order accuracy.

ωk = αk 2

ℓ=0 αℓ

, ζ = |β0 − β3| (3) αk = dk

• C +

ζ2 (ε + βk)2

• ,

(4)

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

Application to the ideal MHD equation

◮ Ideal MHD equation

∂t     ρ ρu E B     + ∇ ·     ρu ρu ⊗ u + ˜ PI − B ⊗ B u(E + ˜ P) − B(u · B) u ⊗ B − B ⊗ u     = 0 where ˜ P = P + 1

2|B|2 and P = (γ − 1)

• E − 1

2ρ|u|2 − 1 2|B|2 ◮ B = O

= ⇒ Euler Equation

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

◮ Cloud Shock(Left:WENO-JS, Right: Proposed )

Density at time t = 0.06 ( 20 ) 50 100 150 200 250 300 350 50 100 150 200 250 300 350 5 10 15 20 25 30 35 40 Density at time t = 0.06 ( 20 ) 50 100 150 200 250 300 350 50 100 150 200 250 300 350 5 10 15 20 25 30 35 40 45

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured

Conclusion and Future work

◮ we have introduced the modified finite difference WENO

schemes based on exponential polynomials for the approximate solutions of ideal MHD equation.

◮ Numerical experiments show that the WENO-CNS(presented)

schemes resolve discontinuities sharply while keeping an essential non-oscillatory performance.

◮ The improvement is attributed to the ability of WENO-CNS

to detect the complicated solution structures by selecting the parameter λ in the exponential polynomial.

◮ The idea of WENO construction method(WENO-CNS) can be

applyed to nonuniform mesh.

Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured