Lecture 16Wednesday, May 25, 2005Consider the linear advection equationφt+~V · ∇φ = 0 (1)or in 1Dφt+ uφx= 0.• In upwinding, we choose the discretization of the spatial derivative basedon the sign of u.– if u > 0, we use φ−x(e.g., D−φ =φi−φi−14x)– if u < 0, we use φ+x(e.g., D+φ =φi+1−φi4x)– if u = 0, do nothing• Using ENO we construct higher order approximations to φ−xand φ+x.• Here we describe the WENO (weighted essentially non-oscillatory) method,which gives a better approximation of φxthan ENO.The following is Osher and Fedkiw, §3.4.1 Hamilton-Jacobi WENOWhen calculating (φ−x)i, the third order accurate HJ ENO scheme uses a subsetof {φi−3, φi−2, φi−1, φi, φi+1, φi+2} depending on how the stencil is chosen. Infact, there are exactly three possible HJ ENO approximations to (φ−x)i. Definingv1= D−φi−2, v2= D−φi−1, v3= D−φi, v4= D−φi+1and v5= D−φi+2allowsus to writeφ1x=v13−7v26+11v36(2)φ2x= −v26+5v36+v43(3)1andφ3x=v33+5v46−v56(4)as the three potential HJ ENO approximations to φ−x. The goal of HJ ENOis the choose the single approximation with the least error by choosing thesmoothest p os sible polynomial interpolation of φ.In [3], Liu et. al. pointed out that the ENO philosophy of picking exactly oneof three candidate stencils is overkill in smooth regions where the data is wellbehaved. They proposed a We ighted ENO (WENO) m ethod that takes a convexcombination of the three ENO approximations. Of course, if any of the threeapproximations interpolate across a discontinuity, it is given minimal weight inthe convex combination in order to minimize its contribution and the resultingerrors. Otherwise, in smo oth regions of the flow, all three approximations areallowed to make a significant contribution in a way that improves the localaccuracy from third order to fourth order. Later, Jiang and Shu [2] improvedthe WENO method by choosing the convex combination weights in order toobtain the optimal fifth order accuracy in smooth regions of the flow. In [1],following the work on HJ ENO in [5], Jiang and Peng extended WENO tothe Hamilton-Jacobi framework. This Hamilton-Jacobi WENO or HJ WENOscheme turns out to be very useful for solving equation 1 as it reduces the errorsby more than an order of magnitude over the third order accurate HJ ENOscheme for typical applications.The HJ WENO approximation of (φ−x)iis a convex combination of the ap-proximations in equations 2, 3 and 4 given byφx= ω1φ1x+ ω2φ2x+ ω3φ3x(5)where the 0 ≤ ωk≤ 1 are the weights with ω1+ω2+ω3= 1. The key observationfor obtaining high order accuracy in smooth regions is that weights of ω1= .1,ω2= .6 and ω3= .3 give the optimal fifth order accurate approximation toφx. While this is the optimal approximation, it is only valid in smooth regions.In nonsmooth regions, this optimal weighting can be very inaccurate and weare better off with digital (ωk= 0 or ωk= 1) weights that choose a singleapproximation to φx, i.e. the HJ ENO approximation.Reference [2] pointed out that setting ω1= .1 + O((4x)2), ω2= .6 +O((4x)2) and ω3= .3 + O((4x)2) still gives the optimal fifth order accuracyin smooth regions. In order to see this, we rewrite these as ω1= .1 + C1(4x)2,ω2= .6 + C2(4x)2and ω3= .3 + C3(4x)2and plug them into equation 5 toobtain.1φ1x+ .6φ2x+ .3φ3x(6)andC1(4x)2φ1x+ C2(4x)2φ2x+ C3(4x)2φ3x(7)2as the two terms that are added up to give the HJ WENO approximation toφx. The term given by equation 6 is the optimal approximation which givesthe exact value of φxplus an O((4x)5) error term. Thus, if the term given byequation 7 is O((4x)5), then the entire HJ WENO approximation is O((4x)5)in smooth regions. To see that this is the case, first note that each of theHJ ENO φkxapproximations gives the exact value of φx, denoted φEx, plus anO((4x)3) error term (in smooth regions). Thus, the term in equation 7 isC1(4x)2φEx+ C2(4x)2φEx+ C3(4x)2φEx(8)plus an O((4x)2)O((4x)3) = O((4x)5) term. Since, each of the Ckare O(1),as is φEx, this appears to be an O((4x)2) term at first glance. However, sinceω1+ω2+ω3= 1, we have C1+C2+C3= 0 implying that the term in equation 8is identically zero. Thus, the HJ WENO approximation is O((4x)5) in smoothregions. Note that [3] obtained only fourth order accuracy, since they choseω1= .1 + O(4x), ω2= .6 + O(4x) and ω3= .3 + O(4x).In order to define the weights, ωk, we follow [1] and estimate the smoothnessof the stencils in equations 2, 3 and 4 asS1=1312(v1− 2v2+ v3)2+14(v1− 4v2+ 3v3)2(9)S2=1312(v2− 2v3+ v4)2+14(v2− v4)2(10)andS3=1312(v3− 2v4+ v5)2+14(3v3− 4v4+ v5)2(11)respectively. Using these smoothness estimates, we defineα1=.1(S1+ )2(12)α2=.6(S2+ )2(13)andα3=.3(S3+ )2(14)with = 10−6maxv21, v22, v23, v24, v25+ 10−99(15)where the 10−99term is set to avoid division by zero in the definition of theαk. This value for epsilon was first proposed by Fedkiw et al. [4] where the3first te rm is a scaling term that aids in the balance between the optimal fifthorder acc urate stencil and the digital HJ ENO weights. In the case that φis an approximate signed distance function, the vkwhich approximate φxareapproximately equal to one so that the first term in equation 15 can be set to10−6. This first term can then absorb the second term yielding  = 10−6inplace of equation 15. Since the first term in equation 15 is only a scaling term,it is valid to make this vk≈ 1 estimate in multidimensions as well.A smooth solution has small variation leading to small Sk. If the Skaresmall enough compared to  then equations 12, 13 and 14 become α1≈ .1−2,α2≈ .6−2and α3≈ .3−2exhibiting the proper ratios for the optimal fifthorder accuracy. That is, normalizing the αkto obtain the weightsω1=α1α1+ α2+ α3(16)ω2=α2α1+ α2+ α3(17)andω3=α3α1+ α2+ α3(18)gives (approximately) the optimal weights of ω1= .1, ω2= .6 and ω3= .3 whenthe Skare small enough to be dominated by . Nearly optimal weights are alsoobtained when the Skare larger than  as long as all the Skare approximatelythe same size as is the case for sufficiently smooth data. On the other hand,if the data is not smooth as indicated by large Sk, then the corresponding αkwill be small c ompared to the other αk’s giving that particular stencil limitedinfluence. If two of the Skare relatively large, then their corresponding αk’s willboth be small and