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

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

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

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

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 ¯ Q i ( t ) 1 � � h = h ( ¯ Q i − r , · · · , ¯ = h i + 1 2 − h i − 1 , Q i + l ) . dt ∆ x 2 � x + ∆ x 1 where h ( x ) is defined implicitly as f ( x ) = 2 h ( ξ ) dξ . ∆ x x − ∆ x 2 Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 4 / 47

Fifth order WENO Reconstruction Procedure Nonlinear spatial adaptive combination of THREE Lagrange poly- nomials q k ( x ) of degree 2 in S k , where k = 0 , 1 , 2 is the shift parameter, 2 ˆ � ω k q k ( x ) ≈ h ( x ) + O (∆ x M ) f ( x ) = (1) k =0 at x i ± 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 ω k ≈ 0 in S k containing discontinuity = ⇒ essentially no Gibbs oscillations . x i-2 x i-1 x i x i+1/2 x i+1 x i+2 τ 5 5 S β 0 ω 0 S 0 β 1 ω 1 S 1 β 2 ω 2 S 2 Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 5 / 47

The Classical WENO-JS Scheme The nonlinear weights of the classical WENO-JS scheme (Jiang and Shu) are d k α k α k = ( β k + ε ) p , ω k = , � 2 l =0 α l 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 � d l � x i + 1 2 � 2 � ∆ x 2 l − 1 2 dx l q k ( x ) β k = dx. (2) x i − 1 l =1 2 measure the normalized modified Sobolov norm of the second de- gree polynomials q k ( x ) in the substencil S k at x i in the cell I i = [ x i − 1 2 , x i + 1 2 ] . Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 6 / 47

The Improved WENO-Z Scheme In the (2 r − 1) order WENO scheme with Z-type weights (WENO-Z), the nonlinear weights are � τ 2 r − 1 � p � � α k ω Z α k = d k 1 + , k = , k = 0 , . . . , r − 1 . � r − 1 β k + ε j =0 α j where the global smoothness indicator is � r − 1 � � � � τ 2 r − 1 = c k β k � , � � � � � k =0 where c k are given constants 1 . For example, τ 5 of the fifth order WENO-Z scheme is τ 5 = | β 0 − β 2 | . Its leading truncation error has been shown to be O (∆ x 5 ) . 1 Castro. Costa. and Don. J. Comput. Phys. 230, 1766–1792, 2011 Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 7 / 47

Definition of Critical Points Definition If a function f ( x c ) = f ′ ( x c ) = . . . = f ( n cp ) ( x c ) = 0 but f ( n cp +1) ( x c ) � = 0 , the function f ( x ) is said to have a critical point of order n cp ≥ 0 at x c . For example, f ( x ) = x 3 , f ′ (0) = f ′′ (0) = 0 , f ′′′ (0) � = 0 , then f ( x ) has n cp = 2 at x = 0 . Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 8 / 47

Optimal Order At Critical Point � τ 2 r − 1 � p � � α k = d k 1 + . β k + ε The nonlinear weights α k 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 α k . Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 9 / 47

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 β k as in ( β k + ε ). ◮ However, a very small ε could reduce the formal order of accuracy of WENO schemes of a smooth function in the presence of high order critical points. n cp = 2 Z5p2edx3 -2 10 Z5p2edx4* 10 Z5p2edx5 Z5p2edx6 9 8 10 -7 7 L_max Error 6 Order -12 10 5 4 -17 10 3 2 10 -22 1 10 -4 10 -3 10 -2 10 -1 dx Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 10 / 47

Optimal Order At Critical Point To mitigate the critical point problem, there are many recent works on ◮ 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

Optimal Order At Critical Point To mitigate the critical point problem, there are many recent works on ◮ 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

Optimal Order At Critical Point To mitigate the critical point problem, there are many recent works on ◮ 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

The WENO-CU6 scheme The WENO nonlinear weights of WENO-CU6 scheme are � � τ 6 α k α k = d k C + , ω k = , � 3 β k + ε k =0 α k ◮ ε = 10 − 40 . (A very small number). ◮ Large constant C ≫ 1 increases the contribution of the optimal weights. (Usually, C = 20 or larger). ◮ The global smoothness indicator � � β 6 − 1 � � + O (∆ x 6 ) , � � τ 6 = 6( β 0 + 4 β 1 + β 2 ) (3) � � with a long and complex expression of β 6 . Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 12 / 47

The WENO- η scheme The WENO nonlinear weights of WENO- η are � τ � α k α k = d k 1 + , ω k = . (4) � 2 η k + ε k =0 α k ◮ The local smoothness indicators r − 1 [∆ x m P ( m ) � i − r +1+ k ( x i )] 2 , k = 0 , 1 , 2 . η k = (5) m =1 where P ( m ) ( x ) is the m th derivative of the Lagrangian i interpolation polynomial for approximating the value of the function f ( x ) based on the values ( f i − r +1+ k , . . . , f i − r +1+ k ) . ◮ the global smoothness indicator τ , such as Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 13 / 47

The WENO- η scheme ◮ the global smoothness indicator τ , such as | η 0 − η 2 | + O (∆ x 6 ) . τ 5 = (6) | 6 η 5 − (4 η 1 + η 0 + η 2 ) | / 6 + O (∆ x 6 ) . τ 6 = (7) � � � ( | P (1) 0 | − | P (1) 2 | )( P (2) + P (2) − 2 P (2) � + O (∆ x 8 ) . (8) τ 8 = 1 ) � � 0 2 WENO- η ( τ 8 ) , n cp = 2 , ε = 10 − 40 p1ncp2 p1ncp2 -1 10 p2ncp2 6 p2ncp2 -2 10 -3 10 5 -4 10 L_Max Error 10 -5 Order -6 10 4 -7 10 10 -8 3 -9 10 0.02 0.04 0.06 dx Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 14 / 47

The WENO-D Scheme The improved WENO-Z scheme, which can guarantee the optimal order of ac- curacy in the presence of critical points, the nonlinear weights are � τ 2 r − 1 � p � � α k α k = d k 1 + Φ , ω k = , k = 0 , . . . , r − 1 . � r − 1 β k + ε j =0 α j � Φ = min { 1 , φ } , φ = | β 0 − 2 β 1 + β 2 | . (9) Remark ◮ Φ , as a linear combination of the β k , 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, { f i − 1 , f i , f i +1 } (See WENO- η ). ◮ The key point is that the following condition must be satisfied, namely, � p � τ 5 ∼ O (∆ x r − 1 ) . φ (10) β k + ε to guarantee the formal order of accuracy regardless critical points. Wai-Sun Don A Generalized Fifth Order WENO-Z type weights 15 / 47