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.) t ≥ 0 , x ∈ Ω( ⊂ R n , q t + f ( q ) x = 0 , n = 1 , 2 , 3) ◮ Conservative (numerical) semidiscrete scheme(1D): f j +1 / 2 − ˆ ˆ f j − 1 / 2 d ¯ q dt = − ∆ x � x +∆ x / 2 � x j +1 / 2 1 1 ¯ f ( q ) = h ( ξ ) d ξ h j = h ( ξ ) d ξ ∆ x ∆ x x − ∆ x / 2 x j − 1 / 2 ◮ 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 f j +1 / 2 ¯ ¯ ¯ ¯ ¯ ¯ h j − 2 h j − 1 h j h j +1 h j +2 h j +3 | | | | | | | � � � � � � x j − 5 x j − 3 x j − 1 x j + 1 x j + 3 x j + 5 x j + 7 2 2 2 2 2 2 2 � | S 5 ( j − r ) | � S 5 � | S 3 ( j − 2) | � S 0 � | S 3 ( j − 1) | � S 1 � | S 3 ( j − 0) | � S 2 � | S 3 ( j + 1) | � S 3 � | S 6 ( j − 2) | � S 6 3 � ◮ ˆ ω k ˆ f k f j +1 / 2 = j +1 / 2 , where ω k are nolinear wieghts obtained k =0 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 S 0 , S 1 , S 2 , S 3 1 3 f j − 2 − 7 6 f j − 1 + 11 f 0 ˆ 2 = 6 f j j + 1 2 = − 1 6 f j − 1 + 5 + 1 f 1 ˆ 6 f j 3 f j +1 j + 1 (1) 1 + 5 6 f j +1 − 1 f 2 ˆ 2 = 3 f j 6 f j +2 j + 1 2 = 11 6 f j +1 − 7 6 f j +2 + 1 ˆ f 3 3 f j +3 . j + 1 ◮ 6th order (polynomial) WENO and it smoothness indicators α k τ 6 � α ℓ ω k = , α k = d k ( C + ) , k = 0 , · · · , 3 ǫ + β k τ 6 = β 6 − 1 d 0 = d 3 = 1 20 , d 1 = d 2 = 9 6( β 0 + 4 β 1 + β 2 ) , 20 � d ℓ 2 � x j +1 / 2 � � dx ℓ ˆ ∆ x 2 ℓ − 1 f k β k = dx x j − 1 / 2 ℓ =1 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 { x n , e γ x , e − γ x : n = 0 , . . . , 4 } • Γ = span { x n , e i γ x , e − i γ x , e γ x , e − γ x : n = 0 , . . . , 2 } • S 5 := span { 1 , x , e γ x , e − γ x , e ν x , e − ν x } ◮ 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
� x ◮ Generally, let Γ = { ϕ 0 , . . . , ϕ m } , H ( x ) = h ( ξ ) d ξ −∞ ◮ Interpolation Q H to H at the cell-boundaries x j − r − 1 / 2 , . . . , x j − r − 1 / 2+ m is defined by a linear combination Γ. ◮ Q H can be written as Lagrange-type representation. m k � � ¯ Q H ( x ) = u j − r , n ( x ) H ( x j − r + n − 1 / 2 ) , H ( x k +1 / 2 ) = ∆ x h ℓ . n =0 ℓ = −∞ satisfying u j − r , n ( x j − r + k − 1 / 2 ) = δ k , n for k , n = 0 , . . . , m . ◮ The numerical flux ˆ f is defined by m � ˆ f ( x ) = Q ′ u ′ H ( x ) = j − r , n ( x ) H ( x j − r + n − 1 / 2 ) n =0 m − 1 m � � � � ˆ ¯ u ′ f j +1 / 2 = ∆ x n ( x j +1 / 2 ) h j − r + ℓ . ℓ =0 n = ℓ +1 � �� � C m , j − 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 f j +1 / 2 approximating h ( x j +1 / 2 ) on the stencil S 6 can be defined by 5 6 � � ˆ C m , j − 2+ ℓ ¯ u ′ f j +1 / 2 = h j − 2+ ℓ , C m , j − 2+ ℓ := ∆ x n ( x j +1 / 2 ) . ℓ =0 n = ℓ +1 ◮ The local numerical flux associated to the sub-stencil S k 3 is given as a linear combination of the cell averages ¯ h j + n 2 3 � � ˆ 3 , j − 2+ k + ℓ ¯ f k C k C k u ′ j +1 / 2 = h j − 2+ k + ℓ , 3 , j − 2+ k + ℓ = ∆ x k , n ( x j +1 ℓ =0 n = ℓ +1 ◮ Ideal Weights based on Exponential Polynomial Interpolation d 0 = C 6 , j − 2 / C 0 d 1 = ( C 6 , j − 1 − d 0 · C 0 3 , j − 1 ) / C 1 3 , j − 2 3 , j − 1 d 3 = C 6 , j +3 / C 3 d 2 = 1 − d 0 − d 1 − d 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) L 1 -sums of the generalized undivided differences. ◮ First, for each fixed n = 1 , 2, we find the coefficient vector c [ n ] := ( c [ n ] k ,ℓ : x ℓ ∈ S ) T , n = 1 , 2 , k by solving the linear system ( x ℓ − x j +1 / 2 ) m � c [ n ] = ∆ x n δ n , m , m = 0 , . . . , |S| − 1 . k ,ℓ m ! x ℓ ∈S k j − r + k ◮ Generalized undivided differences � c [ n ] D n , k f j +1 / 2 := k ,ℓ f ( x ℓ ) , n = 1 , 2 , x ℓ ∈S k j − r + k 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 β k ( k = 0 , . . . , 3): β k := ξ k | D 1 , k f | + | D 2 , k f | , ξ k ∈ (0 , 1] . (2) D 1 , k f := (1 − k ) f j − 2+ k + (2 k − 3) f j − 1+ k + (2 − k ) f j + k D 2 , k f := f j − 2+ k − 2 f j − 1+ k + f j + k , D 1 , 3 f := ( − 23 f j + 2 f j +1 + 3 f j +2 − f j +3 ) / 24 D 2 , 3 f := (3 f j − 7 f j +1 + 5 f j +2 − f j +3 ) / 2 , ◮ A new version of WENO weights with the 6th-order accuracy. α k ω k = , ζ = | β 0 − β 3 | (3) � 2 ℓ =0 α ℓ � � ζ 2 α k = d k C + , (4) ( ε + β k ) 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
Application to the ideal MHD equation ◮ Ideal MHD equation ρ ρ u ρ u ⊗ u + ˜ ρ u P I − B ⊗ B ∂ t + ∇ · = 0 u ( E + ˜ E P ) − B ( u · B ) B u ⊗ B − B ⊗ u � 2 | B | 2 � 2 | B | 2 and P = ( γ − 1) 2 ρ | u | 2 − 1 where ˜ P = P + 1 E − 1 ◮ 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 ) Density at time t = 0.06 ( 20 ) 45 40 350 350 40 35 300 300 35 30 250 250 30 25 25 200 200 20 20 150 150 15 15 100 100 10 10 50 50 5 5 50 100 150 200 250 300 350 50 100 150 200 250 300 350 Changho Kim ( Konkuk University) Youngsoo Ha, Jungho Yoon (Ewha Womans University) Kwang-Il You (NFRI) Exponential WENO schemes for Ideal MHD equations on unstructured
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