ICES REPORT Accuracy of WENO and Adaptive Order WENO Reconstructions for Solving Conservation Laws

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1 ICES REPORT October 2017 Accuracy of WENO and Adaptive Order WENO Reconstructions for Solving Conservation Laws by Todd Arbogast, Chieh-Sen Huang, Xikai Zhao The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas Reference: Todd Arbogast, Chieh-Sen Huang, Xikai Zhao, "Accuracy of WENO and Adaptive Order WENO Reconstructions for Solving Conservation Laws," ICES REPORT 17-29, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, October 2017.

2 ACCURACY OF WENO AND ADAPTIVE ORDER WENO RECONSTRUCTIONS FOR SOLVING CONSERVATION LAWS TODD ARBOGAST, CHIEH-SEN HUANG, AND XIKAI ZHAO Abstract. In this paper, we analyze standard WENO reconstructions and multilevel WENO reconstructions with adaptive order (WENO-AO using both WENO-JS and WENO-Z weighting. We also present a new WENO-AO reconstruction. We give conditions under which the reconstructions achieve optimal order accuracy for both smooth solutions and solutions with shocks. The old WENO-AO reconstruction drops to a fixed, base level of approximation when there are shocks in the solution, but the new one maintains the accuracy of the largest stencil over which the solution is smooth. Our analysis in the discontinuous case requires that the smoothness indicators do not approach zero as the grid is refined. We provide a condition to ensure this result, but we also show an example where this can fail to occur. That is, we show that WENO reconstructions can fail to maintain the order of approximation of the smallest stencil over which the solution is smooth. We also present numerical results confirming the convergence theory of the old and new WENO-AO reconstructions, and compare their performance in solving conservation laws. Key words. Weighted essentially non-oscillatory, WENO-AO, polynomial reconstruction, hyperbolic equation, approximation, shock discontinuity AMS subect classifications. 65D05, 65M08, 65M06, 65M12, 76M12, 76M20 1. Introduction. For solving of a system of hyperbolic conservation laws, (1.1 u t + f(u x = 0, t > 0, x R, u R d, d 1, weighted essentially non-oscillatory (WENO schemes [5, 11, 10] are a popular choice. They allow one to reconstruct a high order version of the solution merely from approximations of cell averages (in finite volume schemes or point values (in finite difference schemes. The key is to average approximations defined on various stencils, and to weight them so as to avoid stencils containing a shock discontinuity in the solution. The idea is that high order accuracy may be achieved by the approximation on the big stencil where the solution is smooth, and yet reduce to the order of the smaller stencils when there is a shock to avoid. It appears that standard WENO reconstructions were not proved to have this property until 2011, when Aràndiga, Baeza, Belda, and Mulet gave a proof [1]. They clarified the delicate nature of the parameters ɛ and η in the standard nonlinear weighting procedure, WENO-JS, of Jiang and Shu [7]. In particular, ɛ needs to be proportional to h 2, where h is the the grid spacing, and η > s/2, where the low order polynomials approximate to order s. Kolb [8] later analyzed the CWENO3 scheme of Levy, Puppo, and Russo [9]. This latter paper provided a new way to obtain WENO reconstructions involving combining polynomials of different degrees. It was This work was supported in part by the U.S. National Science Foundation grant DMS , the Taiwan Ministry of Science and Technology grant MOST M MY2 through the Multidisciplinary and Data Science Research Center of the National Sun Yat-sen University, and the National Center for Theoretical Sciences, Taiwan. Department of Mathematics, University of Texas, 2515 Speedway, C1200, Austin, TX and Institute for Computational Engineering and Sciences, University of Texas, 201 East 24th St., C0200, Austin, TX (arbogast@ices.utexas.edu, xzhao@math.utexas.edu Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan, R.O.C. (huangcs@math.nsysu.edu.tw 1

3 2 ARBOGAST, HUANG, AND ZHAO generalized by Zhu and Qiu [13] and Balsara, Garain, and Shu [2] to define the class of WENO reconstructions with adaptive order (WENO-AO. In this paper, we analyze the accuracy of the standard WENO and WENO-AO reconstructions. We include results for when the WENO-Z weighting procedure of Castro, Costa, and Don [4, 3] is used. Our results are summarized in Sections and 5.3 below. The standard WENO reconstruction behaves as desired, even with WENO-Z weights. It is high order accurate when the solution is smooth, and it drops to low order s when there is a discontinuity (not on the central cell, provided only that η s/2. We show that this condition is sharp. The two-level WENO-AO(r, s reconstructions behave similarly. They can achieve higher order r accuracy in the smooth case and otherwise maintain at least order s accuracy, provided that when using WENO-JS weights, r 2s 1 and η s/2. WENO-Z weights are more complex, because the accuracy of the reconstruction depends more strongly on the choice of η, as we will show. Multilevel WENO-AO s (r l,..., r 1, s reconstructions [2] are based on a base state with approximation order s. When the solution is smooth the approximation attains the highest order r l, but when it is discontinuous, it usually drops to the base level s. When using WENO-JS weights, one requires r l 2s 1 and η s/2, which is equivalent to the two-level case. That is, from the point of view of approximation order, there is no point in using the multilevel reconstruction. Again, WENO-Z weights are more complex. There is a restriction on the size of the gap between successive approximation levels, but a careful choice of η can allow any order. However, the accuracy almost always drops to order s when the solution is discontinuous. We present a new multilevel WENO-AO(r l,..., r 1, r 0 reconstruction that has no base level. When the solution is smooth the approximation attains the highest order r l, but when it is discontinuous, it drops to the order of the largest stencil that does not contain the discontinuity. When the solution has a discontinuity, our convergence results require the smoothness indicators σ 0 as the grid is refined, as is the case in [1, 8]. We show that this hypothesis can fail, and replace it by the hypothesis that the discontinuity is bounded away from the gridpoints as the grid is refined (Definition 5.1. We further show that there are sequences of grids for which even this hypothesis fails. In that case, it is possible that a WENO reconstruction fails to maintain the lowest order of approximation. This seems contrary to the prevailing understanding of WENO reconstructions appearing in the literature. The paper is structured as follows. In the next section, we give the background needed to understand WENO reconstruction techniques. For the knowledgable reader, this section, and the next, set our notation. In Section 3, we define the various WENO reconstructions and define our new one. Section 4 presents a rigorous analysis of the errors in the reconstructions when the solution is smooth. We provide conditions needed to ensure that the reconstruction achieves the order of accuracy of the big stencil. In Section 5, we give our analysis for the case when the solution has a discontinuity. Our cautionary example of a situation where a WENO reconstruction fails to maintain the lowest order of approximation appears in Section 6. Finally, numerical results comparing the old and new WENO-AO reconstructions are given in the last section. 2. Background on WENO reconstructions. We first review the background setting required for WENO reconstructions. For simplicity of exposition, we consider the finite volume framework. The finite difference reconstruction is similar.

4 Accuracy of WENO and Adaptive Order WENO Reconstructions 3 Partition space by grid points < x 1 < x 0 < x 1 <. Define the cell I i = [x i, x i+1 ], its length x i = x i+1 x i, and its midpoint x i+ 1 = (x i + x i+1. Let h = max i x i and assume that the grid is quasiuniform (i.e., there is some ρ > 0 such that ρh min i x i. Let ū i be the average of u(x on the cell I i, i.e., (2.1 ū i = 1 u(x dx. x i I i 2.1. Polynomial approximation on stencils. Now assume that u is smooth. For an rth order approximation of u on a given cell I i, we consider the ordered stencil S r (i, which contains r cells, including I i, and is defined by (2.2 S r (i = I i+ r 2,..., I i,..., I i++ r 1 }, 2 where r 1 2 r 2. Moreover, S r 0 denotes the central stencil. From now and for the remainder of the paper, we fix a value of i and drop it from the notation. For each S r, we obtain the rth order stencil polynomial P r (a polynomial of degree r 1 by imposing the interpolation conditions 1 (2.3 P r (x dx = ū k I k S r. x k I k 2.2. Smoothness indicators. The smoothness indicator σ defined by Jiang and Shu in [7] is normally used to measure the smoothness of stencil polynomials on the cell I i. For the stencil S r, it is given by r 1 ( d l (2.4 σ r = ( x i 2l 1 P r(x 2 I i dx l dx. l=1 If the grid is uniform in the cell size x = h, then in regions where u is smooth, the Taylor expansion of (2.4 gives (2.5 σ r = (u h 2 + O(h 4, which is O(h 2 or O(h 4 at a critical point. If there are discontinuities in u within the stencil S r, then σr = O(1. Summarizing, σ r O(h 2, if u is smooth on S r (2.6 =, O(1, if u has a ump discontinuity on S r Nonlinear weights. A WENO reconstruction is a weighted sum of stencil polynomials. For the stencil S r, let us denote its (constant weight as αr. We refer to this weight as a linear weight. For the linear weight α r, we define its nonlinear weight α r, as discussed below. For some collection of stencils, we merely require that the linear weights sum to one, i.e.,,r αr = 1. (We also desire that αr > 0. When the solution u is smooth, the linear weights should be chosen so as to give a high order WENO reconstruction. However, when there are discontinuities in the data over some stencils, we want to bias the weighted sum so as to exclude those stencils. The idea is to make α r αr when u is smooth on S r and to make αr 0 when u is not smooth on Sr.

5 4 ARBOGAST, HUANG, AND ZHAO WENO-JS weights. For complete generality, let the linear weights be αk s for various stencils indexed by k of size s. To define the nonlinear weights, Jiang and Shu [7] scale each linear weight α r as (2.7 ˆα r = and then normalize (2.8 α r = α r (σ r + ɛ h η, ˆα r, k,s ˆαs k for some exponent η 1 and ɛ h > 0, which avoids any possibility of division by zero. Usually, one takes η = 2 and ɛ h = ɛ Aràndiga et al. [1] show that, in fact, η must be chosen carefully and that it is valuable to take ɛ h = Kh 2, for some fixed K > 0. We will see this in Sections WENO-Z weights. For standard WENO, Castro et al. give the general formula for the WENO-Z reconstruction in [4], for r 3. First let m = r 2 and use the smoothness indicators to define σ m r σ r (2.9 τ = m, if r is odd, σ m+1 r σ m+2 r σm 1 r + σm, r if r is even. Then the un-normalized nonlinear weights are, for η 1, ( ( (2.10 ˆα r = α r τ η 1 + σ r + ɛ, h and the normalized weights are given by ( WENO reconstructions. In this section, we review the standard and adaptive order WENO reconstructions. We also present our new adaptive order WENO reconstruction Standard WENO reconstruction. Suppose we are interested in an r = (2s 1st order standard WENO reconstruction for s 2. First consider all the small stencils with s cells containing the given cell I i. For each S s, we construct P s. Moreover, we define the big stencil Sr 0 = Ss and construct a higher order polynomial P0 r on it. At a fixed point x, the polynomial P0 r can often be written as a convex combination of P s, so (3.1 P r 0 (x = α s P s (x, where αs = 1. We refer to αs as an exact linear weight. These weights can be precomputed for the given x (if they do indeed exist. When there are discontinuities in the data over the big stencil S0, r we want to make use of the relatively small stencils on which u is smooth in order to achieve the essentially non-oscillatory property. The standard WENO reconstruction, valid only for x = x, is (3.2 R r (x = α s P s (x.

6 Accuracy of WENO and Adaptive Order WENO Reconstructions WENO reconstructions with adaptive order (WENO-AO. In [9], Levy, Puppo and Russo describe a third order compact, Central WENO scheme (CWENO. They use a somewhat different WENO reconstruction than the standard one (3.2, because the linear weights in (3.1 fail to exist when r = 2 (2r 1 = 3 and x = x i For the given cell I i, they combine the optimal quadratic polynomial P 3 0 and two linear polynomials P 2 0 and P 2 1. Three advantages of their approach are that exact linear weights are not required, the weights can be taken to be positive, and the reconstruction holds for any point x I i. The disadvantage is that the big stencil polynomial must be computed. Zhu and Qiu describe a fifth order WENO reconstruction in [13], where they combined the fifth order stencil polynomial with two linear stencil polynomials. Later, Balsara, Garain and Shu introduced a new class of WENO reconstructions with adaptive order in [2], which we will briefly recall below. The idea is to combine the three quadratic stencil polynomials with some higher order stencil polynomials Two-level WENO-AO reconstruction. The fifth order reconstruction WENO-AO(5,3 is based on the large stencil S 5 0 = I i 2, I i 1, I i, I i+1, I i+2 } and the three small stencils S 3 1 = I i 2, I i 1, I i }, S 3 0 = I i 1, I i, I i+1 }, S 3 1 = I i, I i+1, I i+2 }, from which we obtain the stencil polynomials P 5 0, P 3 1, P 3 0, and, P 3 1, respectively. The reconstruction is given by [ (3.3 R 5,3 (x = α5 0 α0 5 P0 5 (x 1 = 1 ] αp 3 3 (x + 1 = 1 α 3 P 3 (x, where α0 5 and α 3, = 1, 0, 1, are arbitrary positive linear weights such that α5 0 + α3 = 1. Using the same idea, WENO-AO(7,3 is defined on the big stencil S0 7 and the same small stencils S 1, 3 S0, 3 S1, 3 and WENO-AO(9,3 is given on S0, 9 S 1, 3 S0, 3 S1. 3 In analogy with (3.3, the general formulation of WENO-AO(r, s reconstruction, r > s 2, can be written as [ (3.4 R r,s (x = Rs r,s (x = αr 0 α0 r P0 r (x ] αp s s (x + α s P s (x, where S0 r S s. The positive linear weights αr 0 and α s can be chosen arbitrarily up to requiring α0 r + α s = 1. If WENO-Z weights are used, following [2], the definition of τ is generalized to (3.5 τ = 1 # of σ0 r σ s Multilevel WENO-AO s reconstruction. It is possible that the solution on S0 7 is non-smooth but S0 5 gives a smooth solution, so Balsara et al. [2] combine R 7,3 3 and R 5,3 3. The algorithm is given by (3.6 R 7,5,3 3 (x = γ7,3 γ 7,3 [ R 7,3 3 (x γ5,3 R 5,3 3 (x] + γ 5,3 R 5,3 3 (x, where γ 7,3 + γ 5,3 = 1 and γ 7,3 > 0, γ 5,3 > 0 are arbitrary. Similarly, WENO-AO 3 (9,5,3 is defined by R 9,3 3 and R 5,3 3. Using this recursive process WENO-AO 3 (9,7,5,3 combines R 9,3 3 and R 7,5,3 3 [2], where each reconstruction

7 6 ARBOGAST, HUANG, AND ZHAO includes the base order 3 polynomials. The generalized recursion formula of multilevel WENO-AO s (r l, r l 1,..., r 1, s, l 1, for approximation levels r l > r l 1 > > r 1 > s 2 and base level s, is (3.4 for R r k,s s (x for all 1 k l and (3.7 R r l,r l 1,...,r 1,s s (x = γr l,s [ R r l,s γ r l,s s (x γ r l 1,...,r 1,s R r l 1,...,r 1,s s (x ] + γ r l 1,...,r 1,s s R r l 1,...,r 1,s s (x, l 2, where S r l 0 S r l 1 0 S r1 0 S s include all pertinent stencils, i.e., all S rm. We could further generalize (3.7 to S r l 0, but we omit the details. The linear weights γ r l,s > 0 and γ r l 1,...,r 1,s > 0 are arbitrary such that γ r l,s + γ r l 1,...,r 1,s = 1. We define the WENO-JS nonlinear weights through the un-normalized weighting γ r l,s (3.8 ˆγ rl,s = (σ r l 0 + ɛ h η, ˆγr l 1,...,r 1,s = γrl 1,...,r1,s (σ r l ɛ h, η and the normalized nonlinear weights are then (3.9 γ r l,s = ˆγ r l,s ˆγ r l,s + ˆγ r l 1,...,r 1,s, γr l 1,...,r 1,s = For WENO-Z weights, when l 2, define (3.10 τ = σ r l 0 σr l 1 0, the un-normalized weights (3.11 ( ( ˆγ rl,s = γ r l,s 1 + σ r l ( 0 + ɛ h ( ˆγ r l 1,...,r 1,s = γ r l 1,...,r 1,s 1 + τ ˆγ r l 1,...,r 1,s ˆγ r l,s + ˆγ r l 1,...,r 1,s. η, τ σ r l ɛ h η, and the normalized nonlinear weights by (3.9. Note that σ r l 0 and σ r l 1 0 from the larger stencils are used in (3.8 and (3.11. We can certainly use different values of η at each stage of the reconstruction. We will find this useful for the WENO-Z weights. In this case, we use η 0,k in the initial stage (3.4 for R r k,s s (x, and we use η l in ( A new multilevel WENO-AO reconstruction. The original multilevel WENO-AO s reconstructions R r l,r l 1,...,r 1,s s in (3.7 are based on R r k,s s, 1 k l; that is, at each level, the reconstructions may revert back to the base level s. As we will see, for each k, when u is smooth on S r k 0, r k and s need to satisfy Theorem 4.4 below to have order r k accuracy. Moreover, when u has a discontinuity on the two biggest stencils, Theorem 5.7 below shows that the order of accuracy is at best the base level s. Our goal is to define a new reconstruction that has no base level, and thereby has relaxed constraints on the levels needed for accuracy and achieves a higher order of accuracy near discontinuities. Define R r1,r0 as in (3.4, which uses the stencils S r1 0 and Sr0 for several. The new multilevel WENO-AO reconstruction has no base level, and it is denoted R r l,r l 1,...,r 0,

8 Accuracy of WENO and Adaptive Order WENO Reconstructions 7 where r l > r l 1 > > r 0 2. It is given recursively for l 2 by (3.12 R r l,r l 1,...,r 0 (x = αr l [ 0 α r P r l l 0 ( + 0 (x ( α r0 R r l 1,...,r 0 (x, ] α r0 R r l 1,...,r 0 (x where S r l 0 Sr l 1 0 S r1 0 S s. Again, we could generalize this to include all pertinent stencils, but we do not pursue this here. The linear weights α r l 0 > 0 and α r0 > 0, for all, are arbitrary such that α r l 0 + αr0 = 1. For WENO-Z weights, we define τ by (3.10. Compared to (3.7, note that here we use P r l 0 instead of R r l,r 0 and we use the smoothness indicators σ r l and σ r0 for the nonlinear weighting. For example, the new multilevel WENO-AO(7,5,3 is defined as (3.13 [ ( 1 ] ( 1 R 7,5,3 (x = α7 0 α0 7 P0 7 (x α 3 R 5,3 (x + α 3 R 5,3 (x, = 1 = 1 where α α3 = 1 and R5,3 (x is given by ( Accuracy analysis when u is smooth. In this section, we give a rigorous analysis of accuracy of WENO reconstructions in the case where u is smooth on the large stencil. We show under what conditions they give the desired accuracy. Our results can be viewed as generalizations of those in [1], where the authors analyzed standard WENO reconstructions, and in [8], where the author proved the accuracy for the compact CWENO3 scheme and its reconstruction. Both papers considered only WENO-JS weights. At times, we require a tight assessment of asymptotic behavior. Recall that for a function f(h, (4.1 f = O(h r h r f C as h 0, for some constant C > 0. The notation f = Θ(h provides upper and lower bounds: (4.2 f = Θ(h r C 1 h r f C 2 h r as h 0, for some positive constants C 1 and C Smoothness indicators. We begin by looking at the smoothness indicators. The following lemma appears in [1] when comparing smoothness indicators on the same size stencils. When the stencil sizes differ, we have the result of Kolb [8], which deals only with the compact CWENO3 reconstruction. We provide a simple proof that covers all cases. Lemma 4.1. Let cell I i and any stencils S r I i and S s I i be given (actually S r and Ss k, but the offsets and k are immaterial. For r s 2, assume P r and P s are stencil polynomials from S r and S s, respectively. If the smoothness indicators σ s and σ r are given by (2.4, then (4.3 σ r σ s = O(h s+1, provided that u is smooth on S r S s.

9 8 ARBOGAST, HUANG, AND ZHAO (4.4 Since Proof. First we have that, for any l = 0, 1,..., s, we have that d l dx l ( P r P s = dl dx l ( P r u dl dx l ( P s u = O(h s l. ( d l dx l P r 2 ( d l = dx l P s + dl 2 dx l (P r P s ( d l = dx l P s 2 ( d l + dx l (P r P s σ r σ s = = = r l=1 r l=1 r l= d l [( d x 2l 1 l i I i dx l P r 2 ( d l dx l P s 2] dx [( d l x 2l 1 i x 2l 1 i = O(h s+1. I i I i 2 dx l (P r P s d l + 2 [ ] O(h s l O(1 O(h s l dx dx l P s dl dx l (P r P s, dx l P s dl dx l (P r P s ] dx 4.2. Nonlinear weights. The following theorem quantifies the perturbation of the nonlinear weights from the linear ones. Theorem 4.2. Let η 1, ɛ > 0, and K > 0. Let cell I i and l 1 be given. For a collection of l + 1 stencils S r k I i, k = 0, 1,..., l, where 2 r 0 r 1... r l, let α r k be positive linear weights such that k αr k = 1, and let σ r k be the corresponding smoothness indicators. If u is smooth on l k=0 Sr k, then the following hold. 1. WENO-JS weights satisfy, for all k = 0, 1,..., l, (4.5 α r k = α r k O(h r0+1, if ɛ h = ɛ, + O(h r0 1, if ɛ h = Kh WENO-Z weights satisfy, for all k = 0, 1,..., l, (4.6 α r k = α r k O(h r0+1+(rm+1η, if ɛ h = ɛ, + O(h r0 1+(rm 1η, if ɛ h = Kh 2, where τ = σ r l σ rm = O(h rm+1 for some 0 m l. Proof. We first prove the results for the WENO-JS weights. For any k, we have (4.7 α r k = Now by (2.6, α r k (σ r k + ɛh η l =0 α r (σ r + ɛ h η = (4.8 σ r + ɛ h = α r k = l (σ r k + ɛ h η α r (σ r + ɛ h η =0 α r k. l ( α r 1 + σr k σ r η σ r + ɛ h =0 Θ(1, if ɛ h = ɛ, Θ(h 2, if ɛ h = Kh 2,

10 Accuracy of WENO and Adaptive Order WENO Reconstructions 9 and by Lemma 4.1, we have (4.9 σ r k σ r = O(h min(r k,r +1. Hence (4.10 l =0 ( α r 1 + σr k σ r η 1 + O(h r0+1, if ɛ h = ɛ, = σ r + ɛ h 1 + O(h r0 1, if ɛ h = Kh 2. Combining this with (4.7 and recalling that r 0 2 shows that the result (4.5 holds for the WENO-JS weights. Now for the WENO-Z weights, let ρ r = τ/(σ r + ɛ h and write (4.11 α r k = αr k (1 + ρ η r k α r (1 + ρ η r = αrk (1 + ρ η r k 1 + α r ρ η. r For any, since τ = σ r l σ rm = O(h rm+1, (4.12 ρ r = τ σ r + ɛ h = O(h rm+1, if ɛ h = ɛ, O(h rm 1, if ɛ h = Kh 2, by (4.8 (4.9. Since r 0 2, ρ r 0 as h 0, and so (4.13 ( α r k α r k (1 + ρ η r k 1 ( α r k 1 + ρ η r k ( = α r k 1 + α r ρ η r α r ρ η r α r (ρ η r k ρ η r. The mean value theorem shows that (4.14 ρ η r k ρ η r = η ξ η 1 (ρ rk ρ r, for some ξ between ρ rk and ρ r. Furthermore, by (4.8 (4.9 and (4.12, for all k and, (4.15 ( 1 1 ρ rk ρ r = τ σ r k + ɛh σ r + ɛ h σ r k σ r = τ (σ r k + ɛh (σ r + ɛ h O(h rm+r0+2, if ɛ h = ɛ, = O(h rm+r0 2, if ɛ h = Kh 2. Combining (4.12 (4.15 gives the conclusion ( Accuracy. We now present our results on the accuracy of the various WENO reconstructions when u is smooth. After this presentation, we provide a discussion of the results.

11 10 ARBOGAST, HUANG, AND ZHAO Standard WENO. For standard WENO, we generalize the results in [1] as follows. Theorem 4.3. Let η 1, ɛ > 0, and K > 0. When u is smooth on S r 0, r = 2s 1, s 2, the standard WENO reconstruction R r (x is order r accurate at the point x defined in (3.2, using ɛ h = ɛ or Kh 2 and either WENO-JS or WENO-Z weights. Proof. We consider only the case of WENO-Z weights, since the case of WENO- JS weights is smilar and can be found in [1]. By Lemma 4.1, τ = O(h s+1, so (4.6 is valid. We compute that R r (x u(x = α s (P s (x u(x (4.16 = α(p s s (x u(x + ( α s α(p s s (x u(x = (P0 r (x u(x + ( α s α(p s s (x u(x = O(h r + O(h (s 1(η+1 O(h s, using (4.6 with r = r m = r 0 = s WENO-AO(r, s. The next theorem gives the accuracy of convergence of two-level WENO-AO(r, s given in (3.4. Theorem 4.4. Let η 1, ɛ > 0, and K > 0. For r > s 2, WENO-AO(r, s has order of accuracy min(r, r max on I i if u is smooth on S0, r where for WENO-JS weights, 2s + 1, if ɛ h = ɛ, (4.17 r max = 2s 1, if ɛ h = Kh 2, and for WENO-Z weights, (4.18 r max = r max (η = (4.19 2s (s + 1η, if ɛ h = ɛ, 2s 1 + (s 1η, if ɛ h = Kh 2. Proof. Because α r 0 + αs = αr 0 + αs = 1, we have on I i that [ R r,s u = αr 0 α0 r (P0 r u = αr 0 α0 r (P0 r u ] α(p s s u + α (P s s u [ α r 0 α0 r ] α s ( α s α s (P s u α r 0 = O(h r + [ O( α r 0 α r 0 + O( α s α s ] O(h s. Applying Theorem 4.2, we determine the value of r max WENO-AO s (r l, r l 1,..., r 1, s. We can extend the above theorem to the multilevel WENO-AO s (r l, r l 1,..., r 1, s given in (3.7. Theorem 4.5. Let η 1, ɛ > 0, K > 0, and l 1. Let r l > r l 1 > > r 1 > s 2, and assume that u is smooth on S r l 0. Then WENO-AO s(r l,..., r 1, s has order

12 Accuracy of WENO and Adaptive Order WENO Reconstructions 11 of accuracy min(r l, r max on I i, where r max is given by (4.17 for WENO-JS weights and (4.18 for WENO-Z weights when using a constant value for η. Moreover, if WENO-Z weights are used with variable η, then the reconstruction WENO-AO s (r l,..., r 1, s has order of accuracy r l on I i, provided that (4.20 r k r max (η 0,k, for all 1 k l, and, for all 2 k l, s r k 1 + (r k 1 + 1η k, if ɛ h = ɛ, (4.21 r k s 1 + r k 1 + (r k 1 1η k, if ɛ h = Kh 2, 3s (r k 1 + 1η k + (s + 1η l 1, if ɛ h = ɛ, (4.22 r k 3s 2 + (r k 1 1η k + (s 1η l 1, if ɛ h = Kh 2, where r 0 = s and η 1 = η 0,1. Proof. For fixed s 2, the proof proceeds by induction on l 1. The result holds for l = 1 by Theorem 4.4. Assume the result holds for l 1 1. We write the argument for variable η, since we can simply fix the value for the first part of the theorem. We compute on I i that (4.23 R r l,r l 1,...,r 1,s s = γr l,s γ r l,s u [ (R r l,s u γ r l 1,...,r 1,s (R r l 1,...,r 1,s u ] + γ r l 1,...,r 1,s (R r l 1,...,r 1,s u = γr l,s ( R r l,s γ r u [ γ r l,s γ r l,s l,s γ r γ r l 1,...,r 1,s l,s ] ( γ r l 1,...,r 1,s γ r l 1,...,r 1,s (R r l 1,...,r 1,s u = O(h min(r l,r max(η 0,l + [ O( γ r l,s γ r l,s + O( γ r l 1,...,r 1,s γ r l 1,...,r 1,s ] O(h min(r l 1,r max(η l 1, using Theorem 4.4 and induction. If the perturbation of the linear weights is written as O(h w l, then we have (4.24 R r l,r l 1,...,r 1,s s u = O ( h min(r l,r max(η 0,l,w l +r l 1,w l +r max(η l 1, where w l is given in Theorem 4.2 with r m = r l 1, r 0 = s, and η = η l as s ± 1 for WENO-JS weights, w l = s ± 1 + (r l 1 ± 1η l for WENO-Z weights, respectively for ɛ h = ɛ (+ sign and ɛ h = Kh 2 ( sign. For the first part of the theorem, WENO-Z weights use a constant η, and so both types of weights lead to w l + r l 1 w l + s r max (η. Thus, min(r l, r max (η, w l + r l 1, w l + r max (η = min(r l, r max (η. For the second part of the theorem, i.e., when WENO-Z weights are used with variable η, note that w l + r l 1 = s ± 1 + (r l 1 ± 1η l + r l 1 r l,

13 12 ARBOGAST, HUANG, AND ZHAO by (4.21. Moreover, (4.22 shows that w l + r max (η l 1 = 3s ± 2 + (r l 1 ± 1η l + (s ± 1η l 1 r l, so, with (4.20 min(r l, r max (η 0,l, w l + r l 1, w l + r max (η l 1 = r l, and the proof is complete WENO-AO(r l, r l 1,..., r 0. The following theorem discusses the new reconstruction (3.12. Theorem 4.6. Let ɛ > 0, K > 0, and l 1. Let r l > r l 1 > > r 0 2, and assume that u is smooth on S r l 0. Let WENO-JS weights or WENO-Z weights be used with parameter η k 1 on level r k, and define r max,0 = r 0 and, for 1 k l, r 0 ± 1 for WENO-JS weights, (4.25 r max,k = min(r k 1, r max,k 1 + (r 0 ± 1(η k + 1 for WENO-Z weights, respectively for ɛ h = ɛ (+ sign and ɛ h = Kh 2 ( sign. Then WENO-AO(r l,..., r 0 has order of accuracy min(r l, r max,l on I i. Moreover, WENO-AO(r l,..., r 0 has order of accuracy r l on I i if, for all 1 k l, r k 1 + r 0 ± 1 for WENO-JS weights, (4.26 r k r k 1 + (r 0 ± 1(η k + 1 for WENO-Z weights. Proof. The proof is similar to the inductive proof of Theorem 4.5. The result holds for l = 1 by Theorem 4.4, so assume the result holds for l 1. We compute on I i that (4.27 R r l,...,r 0 u = αr l 0 α r l 0 = αr l 0 α r l 0 [ (P r l 0 u ( [ (P r l α r l 0 u ] ( α r0 (R r l 1,...,r 0 u + ( 0 αr l 0 α r l 0 α r0 By Theorem 4.4 and induction, we have [ R r l,...,r 0 u = O(h r l + O( α r l 0 αr l 0 + O( α r0 ( α r0 α r0 (R r l 1,...,r 0 u α r0 ](R r l 1,...,r 0 u. α r0 ]O(h min(r l 1,r max,l 1. The perturbation of nonlinear weights is given by Theorem 4.2 with r m = r 0, and the main result follows. The result (4.26 is given by requiring r k r max,k for all k Discussion. Standard WENO reconstructions have a simple convergence theory. They give the optimal high-order convergence rate whenever u is smooth. The two-level WENO-AO(r, s achieves the optimal convergence O(h r when the base level s is sufficiently high. In terms of the gap r s between levels, one needs s ± 1 for WENO-JS weights, (4.28 r s (s ± 1(η + 1 for WENO-Z weights,

14 Accuracy of WENO and Adaptive Order WENO Reconstructions 13 respectively for ɛ h = ɛ (+ sign and ɛ h = Kh 2 ( sign. The WENO-Z weights are interesting in that one can adust the value of η to reduce the constraint. For the multilevel WENO reconstructions with adaptive order, the weights used has a marked effect on the results. The WENO-JS weights have a simple convergence theory. The optimal convergence O(h r l is attained by WENO-AO s (r l,..., r 1, s with base level s when (4.29 r l s s ± 1, but the new WENO-AO(r l,..., r 1, r 0 requires only that (4.30 r k r k 1 r 0 ± 1, 1 k l. The condition for WENO-AO s is that the largest gap r l s must be bounded by s ± 1, independent of the intermediate levels. In contrast, the new WENO-AO merely requires that each intermediate gap be bounded by this number, i.e., r 0 ± 1. Obtaining optimal accuracy with WENO-Z weights is a much more complex proposition. WENO-AO s (r l,..., r 1, s has the three conditions (4.20 (4.22. The first condition is (4.31 r k s (s ± 1(η 0,k k l. That is, each two-level approximation must be accurate, and then the gaps in the levels must satisfy the two relatively relaxed bounds (4.21 (4.22. The new WENO- AO(r l,..., r 1, r 0 has only the condition (4.26, i.e., (4.32 r k r k 1 (r 0 ± 1(η k ± 1 1 k l, This condition is worse than (4.21 (4.22, but the very stringent condition (4.31 is removed. A careful choice of η s can recover the full accuracy when using WENO-Z weights for any chosen approximation levels. As (4.32 shows, the new WENO-AO can use a bounded set of η s, whatever value for r l is taken. The condition (4.31 for WENO- AO s requires very large values of η when r l is taken very large. While large values of η improve the convergence rates, they do so by strongly biasing the values of the nonlinear weights to that of the linear ones. This has a tendency to diminish the essentially nonoscillatory property of WENO schemes for solving problems with shocks. This was noticed in [2]: the authors remarked that WENO-JS weights were more stable, while WENO-Z weights gave better convergence results in the smooth case. 5. Accuracy analysis in the discontinuous case. We now consider the case when u is not smooth over the big stencil, but u is smooth on some of the smaller stencils. We consider only the case that u is smooth on a stencil or has a ump discontinuity somewhere in its interior. That is, we do not consider the intermediate case where u is continuous but pertinent derivatives are discontinuous, nor the case of multiple discontinuities, because we are interested in reconstructions involving a single shock discontinuity Smoothness indicators in the discontinuous case. In general, when there is an actual discontinuity, it is obvious that the smoothness indicator is O(1, as noted in (2.6. However, it is far from obvious that σ = Θ(1, and this is in general not true, as we will see in Example 5.3 below. The result σ = Θ(1 holds for some particular sequences of grids.

15 14 ARBOGAST, HUANG, AND ZHAO Definition 5.1. Let h > 0 and x h n be the gridpoints with maximal spacing h. For x fixed, let m be defined so x h m x < x h m+1. We say that x is bounded away from the gridpoints as h 0 if there exists a constant c (0, 1 such that 0 < c x m x x h m and 0 < c x m x h m+1 x for all h. We also say that the grids are bounded away from x as h 0. Lemma 5.2. Let cell I i and the stencil S r I i, where r 2, be given. Assume that u is smooth except for a ump discontinuity at x I m S r. If x is bounded away from the gridpoints as h 0, then the smoothness indicator (5.1 σ r = Θ(1 as h 0. Proof. Define the reference coordinate ξ = (x x i+ 1 /h. The stencil polynomial 2 P r can be written as r 1 ( x xi+ (5.2 P r 1 r 1 2 (x = a = a ξ = h ˆP r (ξ, =0 where a dependents only on the cell averages. By (2.4, we compute =0 r 1 ( d σ r = ( x i 2l 1 l P r 2 (x r 1 I i dx l dx = l=1 l= ( d l ˆP r 2 (ξ dξ l dξ. Hence σ r depends only on the cell averages. As h 0, if σ r 0, then all the derivatives of ˆP r converge to 0 on [ 1 2, 1 2 ], i.e., ˆP r converges to some constant, and the same is true of P r. However, as h 0, u(x, if < m, (5.3 ū = O(h + c u(x + (1 c u(x +, if = m, u(x +, if > m. Since r 2, the cell averages converge to at least two distinct values. So P r cannot converge to a constant, which is a contradiction. Thus there exists some positive constant C 1 such that σ r C 1. Clearly ˆP r is smooth, and all the cell averages are bounded, so σ r C 2 holds for some positive C 2. We remark that for a specific stencil (actually a specific sequence of stencils, the proof shows that the smoothness indicator σ r 0 only if x is near the endpoints of the stencil. That is, x must be in the left-most or right-most cell of the stencil. This is the only case in which as h 0, if we allowed c 0, then we would have only a single value for the cell averages arising in (5.3. However, WENO reconstruction involves a combination of substencils. So if the discontinuity is near the gridpoints anywhere in the big stencil, some small substencil will have this endpoint property. We thus make Definition 5.1 apply to all the gridpoints. The above lemma does not hold for all sequences of grids, as shown in the next example, where c = h. Example 5.3. Given cell I i = [h h 2, 2h h 2 ] and 1, x 0, (5.4 u(x = H( x = 0, x > 0,

16 Accuracy of WENO and Adaptive Order WENO Reconstructions 15 where H is the Heaviside function, consider the stencil S 2 0 = [ h 2, h h 2 ], I i }, for which the average of u on each cell is h and 0, respectively. The stencil polynomial is and the smoothness indicator (2.4 is P 2 0 (x = 3h 2 h2 x, σ 2 0 = h 2 = Θ(h 2 Θ(1. The literature is fraught with the belief that σ 0 as h 0 when there is a discontinuity (e.g., in [1], this is assumed as a hypothesis, and in [8], this belief is stated as being obvious WENO approximation on grids bounded away from the discontinuity. In [1], the authors showed that in the discontinuous case when the smoothness indicator σ 0, WENO approximations are expected to converge only if ɛ h = o(h, so we only consider the case ɛ h = Kh 2, K > 0 in this section. The next theorem gives the magnitude of WENO weights as h 0. The results for WENO-JS weights appear in [1] for standard WENO and in [8] for compact CWENO3. Theorem 5.4. Let η 1, K > 0, and ɛ h = Kh 2. Given cell I i, let S r I i be a stencil of size r 2, for = 0, 1,..., l. Assume that r 0 r 1... r l. Let α r be the positive linear weights such that = 1, and let σ αr r be the corresponding smoothness indicators. If u is smooth except for a single discontinuity, and if u is smooth on at least one stencil, then for grids bounded away from the discontinuity, Θ(1, if u is smooth on S r, (5.5 α r = Θ(h 2η, if u has a ump discontinuity on S r, for all = 0, 1,..., l, for both WENO-JS and WENO-Z weights provided τ = Θ(1. We remark that the WENO-Z weights defined in (2.9 for standard WENO require r 3. When r is odd, τ = Θ(1, since τ = σ k σ k, k = r 2, compares the leftmost and rightmost stencils, only one of which contains the discontinuity. It is not clear whether τ = Θ(1 when r is even in (2.9. Proof. First consider the WENO-JS weights. By (2.6, Lemmas 4.1 and 5.2, for any k, σ r σ r k σ r k + ɛh = Hence we obtain m (5.6 k=0 α r k O(h min(r,rk 1, if u is smooth on S r and S r k, Θ(1, if u is smooth on S r, but umps on S r k, Θ(h 2, if u umps on S r, but is smooth on S r k, O(1, if u umps on S r and S r k. (1 + σr σ r k η Θ(1, if u is smooth on S r, = σ r k + ɛh Θ(h 2η if u umps on S r, and so (4.7 and (5.6 imply the result (5.5. For the WENO-Z weights, since τ = Θ(1, (4.8 shows (5.7 ρ r = τ σ r + ɛ h = Θ(h 2, if u is smooth on S r, Θ(1, if u umps on S r.

17 16 ARBOGAST, HUANG, AND ZHAO Thus the denominator in (4.11 is dominated by Θ(h 2, and the result follows. We present in the next theorems the accuracy of the WENO reconstructions. The first theorem is a generalization of a result in [1] Standard WENO and WENO-AO(r, s. Theorem 5.5. Let K > 0 and ɛ h = Kh 2. Given cell I i, let u be smooth except for a ump discontinuity x I m, m i. Assume that the grids are bounded away from the discontinuity, and that WENO-JS weights or WENO-Z weights are used, where in the latter case τ = Θ(1. For the standard WENO reconstruction R r, r = 2s 1, s 2, for the point x in (3.1, where I m S 2s 1 0, (5.8 R r (x u(x = O(h s if η s/2. For the WENO-AO(r, s reconstruction R r,s, r > s 2, where I m S0 r and I i S s Sr 0, for all, on I i, (5.9 R r,s (x u(x = O(h s x I i if η s/2. Proof. For either weights, we have R r (x u(x = α (P s s (x u(x u discontinuous on S s α s P s (x u(x + = Θ(h 2η O(1 + Θ(1 O(h s. u smooth on S s α s P s (x u(x by Theorem 5.4, and (5.8 follows. Again using Theorem 5.4, we also compute, R r,s (x u(x α r [ 0 α 0 r (P0 r (x u(x O( α 0 r [O(1 + O(h s ] + ] α(p s s (x u(x + u discontinuous on S s = Θ(h 2η [O(1 + O(h s ] + u discontinuous on S s O( α s O(1 + Θ(h 2η O(1 + α s P s (x u(x u smooth on S s u smooth on S s } O( α s O(h s } Θ(1 O(h s. Therefore we conclude the result (5.9. The example below shows that when there is a ump discontinuity bounded away from the gridpoint on the big stencil, the standard WENO reconstruction R r and the WENO-AO(r, s reconstruction may not drop to order s when η < s/2. That is, the requirement that η s/2 is sharp. ] and u defined by (5.4, consider the stencil Example 5.6. Given cell I i = [ h 2, 3h 2 S0 5 = [ 3h 2, h h 2 ], [ 2, h 2 ], I i, [ 3h 2, 5h 2 ], [ 5h 2, 7h 2 ]}. The average of u on each cell is 1, 1/2, 0, 0, and 0, respectively. Consider the standard WENO reconstruction R 3 and the WENO-AO(5, 3 reconstruction with ɛ h = h 2. The stencil polynomials are P 3 1(x = 1 2 x 2h, P 3 0 = x 4h + x2 4h 2, P 3 1 = 0, P 5 0 (x = x 24h + x2 16h 2 + x3 6h 3 x4 24h 4.

18 Accuracy of WENO and Adaptive Order WENO Reconstructions 17 Therefore, the errors at x = h/2 (the leftmost point of I i are P 3 1 ( h ( h u 2 2 ( h ( h u 2 2 P 3 1 and the smoothness indicators are = 1 4 = Θ(1, P 3 0 = 0, P 5 0 ( h ( h u = = Θ(1, ( h ( h u = = Θ(1, σ 3 1 = 1 4 = Θ(1, σ3 0 = 1 3 = Θ(1, σ3 1 = 0, σ 5 0 = = Θ(1. By Theorem 5.4, since τ = Θ(1, for both weights, we have α 3 1 = Θ(h 2η, α 3 0 = Θ(h 2η, α 5 0 = Θ(h 2η. So the error of the reconstruction R 3 at x = h/2 is R 3 ( h 2 ( h u = 2 1 = 1 On the other hand, for R 5,3 at x = h/2, R 5,3( h ( h u 2 2 α 3 [ ( h ( h ] P 3 u = O(h 2η Θ(h 2η. 2 2 [ ( = α5 0 h ( h ] α0 5 P0 5 u = 1 α 3 1 = 1 [ ( h ( h ] P 3 u 2 2 α 3 [ ( h ( h ]} P 3 u 2 2 = O(h 2η Θ(h 2η. Numerical results in Table 5.1 show that we can achieve Θ(h 2η convergence for R 3 and WENO-AO(5, 3, for η = 1, 1.5, 2, 3. When η = 1, both of the reconstructions are only second order accurate instead of third for either WENO-JS and WENO- Z weights. We used the sequence of grid spacings h n = 2 n } n=0 and α0 5 = 0.85, α 3 = 0.05 for these results Multilevel WENO-AO s (r l,..., r 1, s. We have the following result for the multilevel WENO-AO s (r l,..., r 0, s reconstructions. Theorem 5.7. Let K > 0 and ɛ h = Kh 2. Let cell I i be given, l 2, r l > r l 1 > > r 0 = s 2, and I i S r0 S r1 0 Sr l 0, for all. Let u be smooth except for a ump discontinuity at x I m S r l 0, m i. Assume that x S rn+1 0 but x / S rn, 0 n < l ( = 0 if r n 1 and that WENO-JS weights or WENO-Z weights are used with variable η k. Then for the WENO-AO s (r l,..., r 1, s reconstruction, the following hold on grids bounded away from the discontinuity. 1. If n = l 1, let q = min(r l 1, r max (η l 1, where r max is given in Theorem 4.4. If η l (q s/2, η 0,l s/2, and (4.20 (4.22 hold with l replaced by l 1, then on I i, (5.10 R r l,...,r 1,s s (x u(x = O(h q x I i. 2. If n < l 1 and η 0,k s/2, 1 k l, then on I i, (5.11 R r l,...,r 1,s s (x u(x = O(h s x I i.

19 18 ARBOGAST, HUANG, AND ZHAO η = 1 η = 1.5 η = 2 η = 3 n error order error order error order error order Standard WENO R 3, WENO-JS E E E E E E E E E E E E E E E E Standard WENO R 3, WENO-Z E E E E E E E E E E E E E E E E WENO-AO(5, 3, WENO-JS E E E E E E E E E E E E E E E E WENO-AO(5, 3, WENO-Z E E E E E E E E E E E E E E E E Table 5.1: Example 5.6, Standard WENO R 3 and WENO-AO(5, 3 error and convergence rate at x = h/2. The convergence rates are indeed Θ(h 2η. Note that WENO-AO s (r l,..., r 1, s drops to the base order s when u has a ump discontinuity on the two biggest stencils. Proof. If n = l 1, then τ = σ r l σ r l 1 = Θ(1 by (2.6 and Lemma 5.2. By Theorem 5.4, γ r l,s = Θ(h 2η l and γ r l 1,...,r 1,s = Θ(1. Following (4.23 and using Theorems 4.5 and 5.5, (5.12 R r l,...,r 1,s s γ r l,s γ r l,s (x u(x [ (R r l,s s (x u(x γ r l 1,...,r 1,s (R r l 1,...,r 1,s s + γ r l 1,...,r 1,s (R r l 1,...,r 1,s s (x u(x Θ( γ rl,s [O(h s + O(h q ] + Θ( γ r l 1,...,r 1,s O(h q = Θ(h 2η l O(h s + Θ(1 O(h q. (x u(x ] Since q s, we conclude (5.10. When 0 n < l 1, u is smooth neither on S r l 0 nor on Sr l 1 0, so Lemma 5.2 shows σ r l = Θ(1 and σ r l 1 = Θ(1. Hence, following (3.8, (3.11 and (3.9, γ rl,s = Θ(1 and γ r l 1,...,r 1,s = Θ(1. Since R r l,s s (x and R r l 1,...,r 1,s s (x is at least order s accurate when η 0,k s/2, 1 k l, then an argument similar to (5.12 shows ( The new multilevel WENO-AO(r l,..., r 0. When u is smooth only on some substencils of S r l 0, we have the following result for the new reconstruction. Theorem 5.8. Let K > 0 and ɛ h = Kh 2. Let cell I i be given, l 2, r l > r l 1 > > r 0 2, and I i S r0 S r1 0 S r l 0, for all. Let u be smooth

20 Accuracy of WENO and Adaptive Order WENO Reconstructions 19 except for a ump discontinuity at x I m S r l 0, m i. Assume that x S rn+1 0 but x / S rn, 0 n < l ( = 0 if r n 1 and that WENO-JS weights or WENO-Z weights are used with variable η k. Then on grids bounded away from the discontinuity, the WENO-AO(r l,..., r 0 reconstruction satisfies on I i, (5.13 R r l,...,r 0 (x u(x = O(h p if η k p/2, n + 1 k l, where p = min(r n, r max,n with r max,n being given in (4.25. Moreover, if r k r max,k for each 1 k n and q > 0 is fixed, then on I i, (5.14 R r l,...,r 0 (x u(x = O(h min (q,rn if η k q/2, n + 1 k l. Thus, if q = r l 1, the reconstruction drops to the best order possible. A smaller value of q may be chosen to keep the collection of η k from becoming too large. Proof. Following (4.27, (5.15 R r l,...,r 0 (x u(x αr l [ 0 α r P r l l 0 ( + 0 (x u(x + ( α r0 R r l 1,...,r 0 (x u(x. ] α r0 R r l 1,...,r 0 (x u(x We will prove the result by induction on l n + 1. When l = n + 1, by Theorems 4.6 and 5.4, we have (5.16 R r l,...,r 0 (x u(x Θ( α r l 0 [ O(1 + O(h min(r l 1,r max,l 1 ] + = Θ(h 2η l O(1 + Θ(1 O(h p. ( Θ( α r0 O(h min(r l 1,r max,l 1 So the result holds when l = n + 1. Assume by induction that the result holds for some l 1 n + 1. Then R r l 1,...,r 0 (x u(x = O(h p, so by an argument similar to (5.16, we conclude the result (5.13 holds for l. Result (5.14 is shown in a similar way Discussion. WENO philosophy desires that our reconstructions be high order accurate when the solution is smooth and yet maintain low order accuracy when there is a discontinuity not in the central cell I i. For the latter, it is required that ɛ h = Kh 2 [1]. We summarize and discuss the results when the solution is either smooth or there is a discontinuity not in the central cell I i, but the grids are then bounded away from the discontinuity. Theorems 5.5 and 4.3 together show that, given any s, the standard WENO reconstruction R r, r = 2s 1, behaves as desired. It is high order accurate, i.e., order r = 2s 1, when the solution is smooth, and it drops to low order, i.e., order s, when there is a discontinuity not on I i, provided only that we satisfy a condition on η. This condition, given originally in [1], is that η s/2. We showed that this condition is sharp. Theorems 5.5 and 4.4 together show that the two-level WENO-AO(r, s reconstructions can achieve higher order r accuracy in the smooth case and otherwise

21 20 ARBOGAST, HUANG, AND ZHAO maintain at least order s accuracy. For WENO-JS weights, we simply require that r 2s 1 and η s/2. WENO-Z weights are more complex, and we require that τ be chosen so as to have τ = Θ(1 in the discontinuous case. Now, we require that r 2s 1 + (s 1η and η s/2, so any r and s can be used, at the expense of requiring η to be very large. For the multilevel WENO-AO s (r l,..., r 1, s, we have Theorems 4.5 and 5.7. The latter theorem tells us that in the discontinuous case, when the discontinuity lies within the two biggest stencils, the multilevel reconstruction reduces to the base order s, independently of how the multiple levels are treated. For WENO-JS weights, we require r l 2s 1 and all the base reconstructions WENO-AO s (r k, s to be accurate, so we are required to take η 0,k s/2. The multilevel reconstruction is no better than the two-level one when WENO-JS weights are used. When WENO-Z weights are used, the accuracy in the smooth case can be as high as we like, provided enough intermediate levels or large enough η are taken to satisfy (4.21 (4.22, and provided η 0,k are taken so very large that (4.20 holds. The new multilevel WENO-AO(r l,..., r 1, s behaves better. In the smooth case, we have order r l accuracy provided that the levels satisfy (4.26. Moreover, in the nonsmooth case, the reconstruction drops to a lower order depending on exactly where the discontinuity lies. So if the discontinuity is in stencil S rn+1 0 but not in S rn 0 (or some S r0 when n = 0, then we drop to order r n, provided that η k r l /2 for each k. Since this latter condition forces large values of η, one can make the reconstruction drop to order min(s, r n for any s provided only that η k s/2 for each k. 6. A cautionary example of a discontinuity not bounded away from the grid points. Now we consider grids for which the discontinuity x is not bounded away from the grid points. The following example shows that, in exact arithmetic, the WENO reconstruction may not drop to the accuracy of the smallest stencil, as the philosophy of WENO expects. Let u(x = H(x x be a simple step function with a discontinuity at x, where (6.1 x = k=0 2 2k = k=2 1 }} k=3 k=4 1 } } 1 } } 7 15 k= , in binary. We use the sequence of grid spacings h n } n=1, where h = h n = 2 n, and grids x h k = kh} k=. Let x [x h m, x h m+1 and define c(h so x = x h m + c(hh, i.e., c(h = x x h m h = 2 n x m = 2 n 2k m. The gridpoints have only n digits after the binary point, so lim inf c(h = 0 and the k=0 h 0 grid points are not bounded away from the discontinuity. Our sequence of grids gives rise to three subsequences as follows. (1 For the subsequence h n : n = 2 l, l = 0,..., }, we have c(h n = Θ(h n as l. That is, x gets closer to the gridpoints when the grid is refined. (2 For the subsequence h n : 2 l < n < 2 l+1, l = 2,..., }, we have 2 2l+1 c(h n h n 2 2l+1. For each n between its limits 2 l and 2 l+1, we abuse notation by writing c(h n = Θ(h 1 n. (3 For the subsequence h n : n = 2 l 1, l = 2,..., }, we have c(h n = Θ(1 as n. This subsequence of grids has x bounded away from the gridpoints.

22 Accuracy of WENO and Adaptive Order WENO Reconstructions 21 We consider the WENO-AO(3,2 reconstruction with WENO-JS weights and ɛ h = h 2. Let S 3 0 = [x h m, x h m+1], [x h m+1, x h m+2], [x h m+2, x h m+3]}, that is, the ump discontinuity lies in the left most cell of S0. 3 The average on each cell is c(h, 0, and 0, respectively. Then the stencil polynomials are P0 2 (x = c(h ( x x h m 3h, P1 2 (x = 0, h 2 P0 3 (x = c(h 24 c(h 2h ( x x h m 3h 2 Therefore, the errors at, say, x = x h m + h = x h m+1 are + c(h ( 2h 2 x x h m 3h 2. 2 (6.2 P 2 0 (x h m+1 u(x h m+1 = Θ(c(h, P 2 1 (x h m+1 u(x h m+1 = 0, P 3 0 (x h m+1 u(x h m+1 = Θ(c(h, and the smoothness indicators are (6.3 σ0 2 = c(h 2 = Θ(c(h 2, σ1 2 = 0, σ0 3 = 4 3 c(h2 = Θ(c(h 2. Using (2.7 and (2.8, we compute that Θ(1 if c(h = Θ(h, α 0 2 and α 0 3 = Θ(h 4η if c(h = Θ(h 1, Θ(h 2η if c(h = Θ(1, So the error of the reconstruction R 3,2 at x h m+1 is (6.4 R 3,2 (x h m+1 u(x h m+1 [ = α3 0 (P 3 α0 3 0 (x h m+1 u(x h m =0 1 =0 α 2 ( P 2 (x h m+1 u(x h m+1 O(h if c(h = Θ(h, = O(h 4η 1 if c(h = Θ(h 1, O(h 2η if c(h = Θ(1. α 2 ( P 2 (x h m+1 u(x h m+1 ] We illustrate the results (6.2 (6.4 numerically, using high precision arithmetic. In Figures , the black dots are the logarithm of the smoothness indicators and errors to base 2, respectively. The red, green, and blue dashed lines connect the grids in subsequences (1, (2, and (3, where c(h = Θ(h, Θ(h 1, and Θ(1, respectively. The negative value of the slope is the convergence rate, computed over the subsequence. Figure 6.1 shows that the smoothness indicators of σ0 2 and σ0 3 indeed have order 2 in subsequence (1, order 2 in subsequence (2, and constant order in subsequence (3. Figure 6.2 shows the stencil polynomials P0 2 and P0 3 approximate to order