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1 Journal of Computational and Applied Matematics 94 (6) Contents lists available at ScienceDirect Journal of Computational and Applied Matematics journal omepage: Smootness-Increasing Accuracy-Conserving (SIAC) filters for derivative approimations of discontinuous Galerkin (DG) solutions over nonuniform meses and near boundaries X. Li a, J.K. Ryan b,, R.M. Kirby c, C. Vuik a a Delft Institute of Applied Matematics, Delft University of Tecnology, Delft, Neterlands b Scool of Matematics, University of East Anglia, Norwic, UK c Scool of Computing, University of Uta, Salt Lake City, Uta, USA a r t i c l e i n f o a b s t r a c t Article istory: Received 5 December 4 Received in revised form June 5 Keywords: Discontinuous Galerkin metod Post-processing SIAC filtering Superconvergence Nonuniform meses Boundaries Accurate approimations for te derivatives are usually required in many application areas suc as biomecanics, cemistry and visualization applications. Wit te elp of Smootness-Increasing Accuracy-Conserving (SIAC) filtering, one can enance te derivatives of a discontinuous Galerkin solution. owever, current investigations of derivative filtering are limited to uniform meses and periodic boundary conditions, wic do not meet practical requirements. Te purpose of tis paper is twofold: to etend derivative filtering to nonuniform meses and propose position-dependent derivative filters to andle filtering near te boundaries. Troug analyzing te error estimates for SIAC filtering, we etend derivative filtering to nonuniform meses by canging te scaling of te filter. For filtering near boundaries, we discuss te advantages and disadvantages of two eisting position-dependent filters and ten etend tem to position-dependent derivative filters, respectively. Furter, we prove tat wit te position-dependent derivative filters, te filtered solutions can obtain a better accuracy rate compared to te original discontinuous Galerkin approimation wit arbitrary derivative orders over nonuniform meses. One- and two-dimensional numerical results are provided to support te teoretical results and demonstrate tat te position-dependent derivative filters, in general, enance te accuracy of te solution for bot uniform and nonuniform meses. 5 Elsevier B.V. All rigts reserved.. Introduction In many cases, one can argue persuasively tat te canges in values of a function are often more import tan te values temselves, suc as eibited by streamline integration of fields. Terefore, an accurate derivative approimation is often required in many areas suc as biomecanics, optimization, cemistry and visualization applications. owever, computing derivatives of discontinuous Galerkin approimations is callenging because te DG solution only as weak continuity at element boundaries. Tis means tat te strong form of derivatives for a DG solution tecnically does not old at element boundaries, and computing te derivative directly does not always produce accurate results. For eample, naive and careless use of te derivatives of te discontinuous Galerkin solution directly to streamline integration can produce inconsistent Corresponding autor. addresses: X.Li-@tudelft.nl (X. Li), Jennifer.Ryan@uea.ac.uk (J.K. Ryan), kirby@cs.uta.edu (R.M. Kirby), C.Vuik@tudelft.nl (C. Vuik). ttp://d.doi.org/.6/j.cam / 5 Elsevier B.V. All rigts reserved.

2 76 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) results wit te eact solution []. Once derivatives are needed near te boundaries, te difficulty increases since te solution often as less regularity in tose regions. In order to obtain accurate approimations for te derivatives of discontinuous Galerkin (DG) solutions, tis paper focuses on using te so-called Smootness-Increasing Accuracy-Conserving class of filters. As te name implies, SIAC filtering can increase te smootness of DG solutions, and tis smootness-increasing property elps to enance te accuracy of derivatives of DG solutions. Before giving contet to wat we will propose, we first review te currently eisting SIAC and derivative SIAC filters. Te source of SIAC filtering is considered to be te superconvergence etraction tecnique introduced by [] for finite element solutions of elliptic problems. Te etension for DG metods of linear yperbolic equations given in [3]. In an ideal situation, by applying SIAC filtering, te accuracy order of te filtered DG approimation can improve from k + to k +. Te concept of te derivative filter was introduced in [4] for finite element metods and [5] for DG metods. Wit te derivative filter, te filtered solution maintains an accuracy order of k + regardless of te derivative order. owever, te previous investigations of derivative filtering ave two major limitations: te requirement of a uniform meses and periodic boundary conditions. Te purpose of tis paper is to overcome tese two limitations. We propose position-dependent derivative filters to approimate te derivatives of te discontinuous Galerkin solution over nonuniform meses and near boundaries. Our main contributions are: Nonuniform meses. Filtering over nonuniform meses as always been a significant callenge for SIAC filtering since te k + accuracy order is no longer guaranteed in general. Most of previous work for nonuniform meses (suc as [6 8]) only considered a particular family of nonuniform meses, smootly-varying meses. Among tese works, only [7] mentioned derivatives over nonuniform meses. It discussed te callenges of derivative filtering over nonuniform meses and presented preliminary results concerning smoot-varying meses. In tis paper, we propose a metod for arbitrary nonuniform meses: using te scaling = µ for filtering over nonuniform meses. We cannot guarantee tat te derivative filtering can improve te derivatives of DG solutions to accuracy order of k+, but we prove tat a iger convergence rate (compared to DG solution) is still obtained. Furter, te numerical eamples suggest tat te accuracy is improved once te mes is sufficiently refined. Boundaries. First, we point out tat previously tere was no derivative filter tat could be used near boundaries ecept for periodic meses. Witout considering derivatives, tere are tree eisting position-dependent filters tat can be used to andle boundary regions, see [8 ]. Two of tem, [9,], are constructed by only using central B-splines. Tey sowed good performance over uniform meses. Te position-dependent filter recently introduced in [8] was aimed at nonuniform meses. It uses k + central B-splines and an etra general B-spline. Te results in [8] suggested tat adding te etra general B-spline improves te performance of te position-dependent filter over nonuniform meses compared to using only central B-splines. In tis paper, we etend te position-dependent filter [] (referred to as te SRV filter) and te new position-dependent filter [8] (referred to as te RLKV filter) to position-dependent derivative filters. Ten, we discuss te advantages and disadvantages of tese position-dependent derivative filters over uniform and nonuniform meses. For nonuniform meses, we prove tat by using te position-dependent derivative filtering, te convergence rate of te derivatives of te DG solution can be improved. Numerical comparisons over uniform and nonuniform meses also demonstrate tat te derivative filtered solutions are more accurate tan te derivatives of DG approimations. Our new contributions are: Testing te position-dependent derivative filters for uniform meses, wic as never been done before; Applying te symmetric and position-dependent derivative filters over different nonuniform meses. Tis paper is organized as follows. In Section, we first review properties of te discontinuous Galerkin solution and its derivatives. Ten, we introduce te symmetric and position-dependent filters. Lastly, we sow te symmetric derivative filter over uniform meses. Te symmetric and position-dependent derivative filters over nonuniform meses are presented in Section 3 wit teoretical error estimations. Numerical results over uniform and nonuniform one-dimensional meses are given in Section 4. Nonuniform two-dimensional quadrilateral meses are considered in Section 5. We present conclusions in Section 6.. Background In tis section, we first briefly review te basic features of discontinuous Galerkin metods and te properties related to derivatives. Ten, we present te relevant background of Smootness-Increasing Accuracy-Conserving filters... DG approimation and its derivatives Consider a multi-dimensional linear yperbolic equation d u t + A i u i + A u =, u(, ) = u (), (, t) Ω (, T], Ω R d, (.) i=

3 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) were te initial condition u () is a sufficiently smoot function and te coefficients A i are constants. Te details of discontinuous Galerkin metods for yperbolic equations can be found in [,]. ere, we skip te details and write te DG approimation directly as k u (, t) = u l K (t)ϕl K (), for K, l = were k is te igest degree of approimation polynomial, and ϕ l K is te multi-dimensional piecewise polynomial basis function of degree l, polynomial inside element K and zero outside K, l = (l,..., l d ) is a multi-inde. ere we coose scaled Legendre polynomials for simplicity. Te mes elements K can be eiter uniform or nonuniform, and te mes size is represented as K ( for uniform meses). A well-known feature of te DG approimation is tat te use of a piecewise polynomial basis leads to superconvergence. owever, by using a piecewise polynomial basis, te solution u is discontinuous at te interface of te elements, wic is te main callenge to defining te global derivatives of te DG approimation. Alternatively, we can define te local derivatives of u in te interior of eac element by α u (, t) = k l = u l K (t) α ϕl K (), for K, (.) were α = (α,..., α d ) is an arbitrary multi-inde. Eq. (.) is an approimation of α u, but it is only a local derivative, and te convergence rate in Eq. (.3) is not satisfied. Unlike te general DG approimation, wic as te convergence rate of k +, wit eac successive derivative, one order of accuracy is lost, α u α u O(k+ α ). (.3) Tis means α k, since te (k + )t derivative is just zero. In order to obtain a global and accurate derivative of te DG approimation, we can increase its smootness by using SIAC filtering. Since te filtered solutions are more smoot and ave iger accuracy order, tis filtering tecnique allows us to obtain derivatives for α > k. We empasize tat te ability of calculating derivatives for α > k is a unique benefit of using SIAC filtering... SIAC filter Smootness-Increasing Accuracy-Conserving filtering is based on a postprocessing tecnique first demonstrated by Bramble and Scatz [] for finite element metods to obtain a better approimation. Later, in [3], tis tecnique was etended to DG metods. In te one-dimensional case, te filtered approimation u is given by and u (r+,k+) () = K u (), u u C k+, were te symmetric filter K (r+,k+) is a linear combination of te central B-splines, r K (r+,k+) () = c γ (r+,k+) ψ (k+) + r γ γ = and te scaled filter K is given by formula K () = K( ). In [,3], te number of B-splines is r + = k+. Te coefficients are calculated by requiring te filter to reproduce te polynomials by convolution c (r+,k+) γ K (r+,k+) p = p, p =,,..., r. ere, te central B-splines are constructed recursively by ψ () () = χ [ /,/) (), ψ (l+) () = ψ () ψ (l), l. We note tat tis symmetric filter is suitable only for te interior region, (r + k + )/ elements away from te boundaries, since it uses symmetric information around te evaluated point. Wen filtering near te domain boundaries, we need to use position-dependent filters. In te multi-dimensional case, te filter is a tensor product of te one-dimensional filters (.5) and can be written as d K (r+,k+) () = i= K (r+,k+) ( i ), = (,..., d ) R d, wit te scaled filter K (r+,k+) () = K (r+,k+) d. (.4) (.5)

4 78 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Position-dependent filter Tere are tree position-dependent filters tat ave been introduced in previous work. Two of te tree, [9,], use only central B-splines. Tese central B-spline filters ave similar structures. ere we only discuss te one wit better performance, [], referred to as te SRV filter, in tis section. Te last position-dependent filter use central B-splines, and an etra noncentral B-spline. It was recently introduced in [8], and te error of te filtered solution near te boundary is reduced over nonuniform meses compared to te SRV filter. In te following contet, we refer to position-dependent filter [8] as te RLKV filter..3.. SRV filter Te SRV filter using 4k+ central B-splines was introduced in [] for uniform meses. Tis boundary filter demonstrated better beavior in terms of error tan te original position-dependent filter wic uses k + given by Ryan and Su in [9]. It canges its support according to te location of te point being filtered. For eample, at te left boundary, a translation of te filter sould be done so tat te support of te filter remains inside te domain (up to te left boundary). Te SRV filter for filtering near te boundaries can ten be written as 4k K (4k+,k+) () = c γ (4k+,k+) ψ (k+) γ ( ), (.6) γ = were γ depends on te location of te evaluation point and is given by γ ( ) = k + γ + λ( ), wit min, 5k + + left, left, left + rigt, λ( ) = ma, 5k + + rigt left + rigt,, rigt. (.7) ere left and rigt are te left and rigt boundaries, respectively. In te interior, te symmetric filter uses k + central B-splines. In order to provide a smoot transition between te SRV filter and te symmetric filter, a conve combination was used: u () = θ() K (k+,k+) u () + ( θ()) K (4k+,k+) u (), (.8) were θ() C k suc tat θ = in te interior and θ = in te boundary regions. Te SRV filter sowed good performance over uniform meses in [] wen a multi-precision package was used. owever, recent work [8] sowed tat te SRV filter was not suitable for nonuniform meses..3.. RLKV filter Te performance of te SRV filter strongly depends on tree conditions: te problem is linear, te mes is uniform and te computations are carried out wit a multi-precision package (at least quadruple precision). Wen one of tese conditions is not satisfied, te good performance of te SRV filter can no longer be guaranteed. In order to overcome te aforementioned limitations, te RLKV filter was proposed in [8]. Tis RLKV filter uses k + central B-splines and an etra general B-spline. Near te left boundary, te RLKV filter as te formula K (k+,k+) T () = k c γ (k+,k+) ψ (k+) T(γ ) γ = () Position-dependent filter wit k+ central B-splines + c (k+,k+) k+ ψ (k+) T(k+) () General B-spline, (.9) were T is te knot matri defined in [8]. Tis knot matri is a (k + ) (k + ) matri suc tat eac row, T(i) = [T(i, ),..., T(i, k + )] (i =,..., k + ), of te matri is a knot sequence wit k + elements tat are used to create te B-spline ψ (k+) T(γ ) (). Let T(i, j : j ) represents te knot sequence [T(i, j ), T(i, j + ),..., T(i, j )], ten te B-spline can be constructed by te following recurrence relation [3]: χ [T(i,),T(i,)) for l = ; ψ (l) T(i,:l) () = T(i, ) T(i, l ) T(i, ) ψ (l ) T(i,:l ) () + T(i, ) ψ (l ) T(i,:l) () for l >. T(i, l) T(i, )

5 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) We note tat te definition above provides more fleibility tan te previous central B-spline notation, were T(i) are sequences of equidistant knots. Assume tat we want to filter te point wic is located near te left boundary, ten T is given by, 3k + j + i + left, i k, j k + ; T(i, j) = (.) left + min{j, }, i = k +, j k + ; similarly near te rigt boundary te knot sequence is given by rigt + ma{j k, }, i =, j k + ; T(i, j) = i + j + rigt, i k +, j k +. More details of tis position-dependent filter can be found in [8]..4. Symmetric derivative filter: uniform meses Smootness-Increasing Accuracy-Conserving filtering is named after its improvement of te smootness of te filtered approimation. Using te filter in Eq. (.5), te filtered solution is a C k function. One can see tat te smootness is significantly improved from te original weakly continuous solution. By taking advantage of te improved smootness, we can obtain better derivative approimations. Derivative filtering over uniform meses was introduced in [5,4]. In tese papers, te autors identified two ways to calculate te derivatives. Te first metod is a direct calculation of te derivatives of te filtered solution (.4). Te convergence rate of te filtered solution is iger tan te derivatives of te DG approimation itself, but te accuracy order decreases and oscillations in te error increase wit eac successive derivative. Te second metod is employed to maintain a fied accuracy order regardless of te derivative order. In order to calculate te αt derivative of te DG approimation witout losing any accuracy order, we ave to use iger-order central B-splines to construct te filter, K (r+,k++α) () = r γ = (.) c γ (r+,k++α) ψ (k++α) + r γ. (.) Notice tat te order of te B-splines is now k + + α instead of k + in (.5), and te filtered solution becomes a C k +α function. Ten we can write te αt derivative of te symmetric kernel as dα d α K (r+,k++α) () = α (r+,k+,α) K, were r K (r+,k+,α) = c γ (r+,k++α) ψ (k+) + r γ. γ = By te property of convolution, α u = α K (r+,k++α) d α u = d α K (r+,k++α) u = α (r+,k+,α) K u. (.3) For uniform meses, [5] sowed te filtered solution (.3) as k + superconvergence rate regardless of te derivative order α α u α u O( k+ ). Unfortunately, tese metods are limited to uniform meses. For nonuniform meses, SIAC filtering becomes complicated, and derivative SIAC filtering is more difficult. If we naively apply te same derivative filtering tecnique over nonuniform meses, we will lose accuracy from O( k+ ) to O( k+ α ) since over nonuniform meses te divided differences of te DG solution no longer ave te superconvergence property. In te following section, we will address nonuniform meses by adjusting te scaling of te SIAC filter. 3. Derivative filter: nonuniform meses and near boundaries 3.. Symmetric derivative filter: nonuniform meses A brief introductory description of symmetric derivative filtering over nonuniform meses can be found in [7]. It discusses te callenges of symmetric derivative filtering over nonuniform meses and gives preliminary results for smootly-varying meses (an affine mapping of a uniform mes [6]). In order to develop derivative filtering for arbitrary nonuniform meses, we first present some useful lemmas.

6 8 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Lemma 3. (Bramble and Scatz []). Let Ω Ω and s be an arbitrary but fied nonnegative integer. Ten for u s (Ω ), tere eists a constant C suc tat u,ω C D α u s,ω. α s Lemma 3.. Let u be te eact solution to te linear yperbolic equation u t + d A i u i + A u =, Ω (, T], (3.) i= u(, ) = u (), Ω, were te initial condition u () is a sufficiently smoot function and te coefficients A i are constants. ere, Ω R d. Let u be te DG approimation over a nonuniform mes wit periodic boundary conditions. Denote Ω Ω Ω, l k +. Te negative order norm estimation of u u satisfies, and (u u )(T) l,ω C k+, α (u u )(T) l,ω C α k+ α, were α = (α,..., α d ) is an arbitrary multi-inde and is te scaling of te divided difference operator α. Proof. Te proof of te negative order norm estimation was given in [3] and te divided difference estimation was presented as a ypoteses. Te proofis trivial and terefore we only give a proof for d = case. Set Ω suc tat Ω + α, α Ω. Consider te first divided difference, by te definition of te negative order norm, we ave (u u ) +, Φ (u u ), Φ (u u ) l,ω = sup, Φ C (Ω ) Φ l,ω (u u ) +, Φ (u u ), Φ By induction, we ave sup Φ C (Ω ) u u l,ω. α (u u )(T) l,ω C α k+ α, Φ l,ω + sup Φ C (Ω ) Φ l,ω, were C α = α C. Te proof is similar for d > case. Lemma 3. demonstrates te optimal accuracy order estimation of te divided differences of te DG approimation in te sense tat te nonuniform mes is arbitrary [3,4]. Teorem 3.3. Under te same conditions as in Lemma 3., let K (r+,k++α) be te symmetric derivative filter given in (.). Denote Ω + supp(k (r+,k++α) ) Ω Ω. Ten, for general nonuniform meses, we ave α u α K (r+,k++α) u,ω C r+k++α r+ (k+), were = µ and µ = k+ r+k++α. Proof. Set Ω / suc tat Ω + supp(k (r+,k++α) ) Ω /, and Ω / + supp(k (r+,k++α) ) Ω. By applying Lemmas 3. and 3., we ave α u α K (r+,k++α) u,ω α u K (r+,k++α) α u + α,ω C r+ + C α+β β k+ K (r+,k++α) (u u ) K (r+,k++α) (u u ) (k+),ω/,ω

7 Let te scaling = µ suc tat r+ = k+ (k++α). X. Li et al. / Journal of Computational and Applied Matematics 94 (6) K (r+,k+ β,α+β) = C r+ + C β k+ = C r+ + C β k+ C r+ + C k+ (k++α). (r+,k+ β,α+β) K α+β (u u ) α+β L (u u ) (k+),ω/ (k+),ω We ten ave tat µ = k+ r+k++α and α u α K (r+,k++α) u C r+k++α r+ (k+).,ω Remark 3. (Discussion of te Number of B-Splines). Te filter given in (.) uses (r + ) B-splines. Teorem 3.3 implies r+ tat by increasing te value of r, one can increase te value of, and ten approimate te superconvergence rate r+k++α k + as close as we want and regardless of te derivative order α. owever, increasing te value of r presents a serious inconvenience for computational implementation. For eample, wile r k, a multi-precision package is required during te computation process, [8]. Anoter disadvantage is tat te support size of te filter, (r +k++α) µ, increases wit r [3]. Te increased support size means te convolution involves more DG elements and tat te computational cost increases as well. For nonderivative filtering, we usually keep r = k, but tere is anoter consideration for derivative filtering. We notice tat te accuracy order decreases wit te derivative order α if we keep r = k. One solution is to eliminate te negative effect of te derivative order α is to use r = (k + α) instead of r = k. owever, our eperience sows tat te benefit of using r = (k + α) is limited. It sligtly improves te accuracy and smootness, but increases te computational cost. In tis paper, we will focus on using r = k for nonuniform meses. 3.. Position-dependent derivative filter In te previous section, we discussed te symmetric derivative filtering over nonuniform meses. As we mentioned before, in order to andle boundary regions, we need to use position-dependent filters. In tis section, we etend two position-dependent filters to position-dependent derivative filters Derivative SRV filter Since te SRV filter uses only central B-splines, we can easily etend it to te derivative SRV filter by increasing te order of B-splines from k + in (.6) to k + + α 4k K (4k+,k++α) () = c γ (4k+,k++α) ψ (k++α) ( γ ), (3.) γ = and adjusting te sift function λ( ) (.7) to min, 5k + + α + left, left, left + rigt, λ( ) = ma, 5k + + α + rigt left + rigt,, rigt. (3.3) For eample, Fig. 3. sows te derivative SRV filters wit k = for te first and second derivatives at te left boundary. Remark 3.. Te teoretical analysis of te derivative SRV filter remains te same as Teorem 3.3 wit r = 4k. Te difference between te derivative SRV filter and te symmetric derivative filter is te scaling = µ. Te scaling of te derivative SRV filter is = 5k++α k+, wic is muc larger tan te scaling of te symmetric derivative filter, = k+ 3k++α Derivative RLKV filter For te RLKV filter, we need to sift te k + central B-splines and ten cange te etra general B-spline according to te derivative order α. To complete tese canges, we ave to cange te knot sequence (.), wic is used only for

8 8 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Fig. 3.. Te derivative SRV filters (first and second derivatives) before convolution at te left boundary wit k = and scaling =. Te boundary is represented by =. te DG approimation u witout derivatives. For te derivative RLKV filter near te left boundary (similar for te rigt boundary), we need to redistribute te knots in te knot matri T to meet te derivative requirement by 3k α + j + γ + left, γ k, j k + + α; T(γ, j) = (3.4) left + min{j α, }, γ = k +, j k + + α, and te position-dependent derivative filter is given by K (k+,k++α) T () = k c γ (k+,k+) ψ (k++α) T(γ ) γ = () Position-dependent filter wit k+ central B-splines + c (k+,k++α) k+ ψ (k++α) T(k+), (3.5) General B-spline Remark 3.3. It is necessary to use a B-spline of order k + + α instead of k + wen α > k. In formula (3.5), if we keep te order of B-spline as k +, wen α > k te knot sequence T(k + ) becomes a uniformly spaced knot sequence, and ten te general B-spline ψ (k++α) T(k+) added at te boundary reduces to a central B-spline. Ten, te purpose of adding a special B-spline at te boundary fails, and tis special B-spline is needed to place more weigt on te filtered point. We note tat te derivative RLKV filter allows us to approimate arbitrary order of derivatives near boundaries teoretically. For eample, Fig. 3. sows te derivative RLKV filters wit k = for te first and second derivatives at te left boundary. Compared to te derivative SRV filter in Fig. 3., te derivative RLKV filter clearly as reduced support and magnitude (range from 4 to 6 versus 4 to 6). Teorem 3.4. Under te same conditions as in Lemma 3., let K (k+,k++α) T α u α were = µ, µ = k+ 3k+3+α. Proof. K (k+,k++α) T u,ω C µ(k+), α u α K (k+,k++α) T u C k+ +,Ω + α k γ = be te derivative RLKV filter (3.5). We ave c γ ψ (k++α) T(γ ) (u u ) α c k+ ψ (k++α) T(k+) (u u ),Ω For te second term on te left side of te above inequality, wic only involves central B-splines, similar to Teorem 3.3, we ave k α γ = c γ ψ (k++α) T(γ ) (u u ) C k+ (k++α).,ω.,ω

9 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Fig. 3.. Te derivative RLKV filters (first and second derivatives) before convolution at te left boundary wit k = and scaling =. Te boundary is represented by =. For te tird term wit a general B-spline, we ave α c k+ ψ (k++α) T(k+) (u u ) C d α+β c (k+) k+ ψ,ω dα+β T(k+) (u u ) β k+ (k+),ω/ C d α+β c (k+) k+ ψ dα+β T(k+) u u (k+),ω β k+ L d C 3 (α+β) α+β (k+) ψ dα+β T(k+) u u (k+),ω β k+ L were C 4 k+ (k++α), Ω + supp(k (k+,k++α) T ) Ω /, and Ω / + supp(k (k+,k++α) T ) Ω. Ten, we ave α u α K (k+,k++α) T u,ω C k+ + C 5 k+ (k++α). Similar to te symmetric filter case in Teorem 3.3, we require te scaling satisfies k+ = k+ (k++α) and finally, we ave α u α K (k+,k++α) T u C µ(k+),,ω were = µ and µ = k+ 3k+3+α. Remark 3.4 (Discussion of Support Size of te Filters). Teorems 3.3 and 3.4 give convergence rates of te symmetric and position-dependent derivative filters, respectively. One can easily verify tat te convergence rates are better tan calculating te derivatives of te DG approimation directly, k + α. For te scaling = µ, for convenience we let te degree k, ten te symmetric derivative filter and derivative RLKV filters ave te scaling = /3 and te derivative SRV filter as te scaling = /5. In order to sow te difference of support size of te different filters, we present Fig. 3.3 to sow a direct comparison. From Fig. 3.3, we can see tat te SRV filter requires a muc larger support size tan te RLKV filter. Te large support size usually will lead to computational problems (increased flop counts, round-off error, etc.). owever, we notice tat te scaling = µ is still quite large compared to. A large support usually as negative effects on te accuracy over coarse meses. Let te domain be Ω = [, ] and = /N, were N is te number of elements. In order to guarantee te conclusions in Teorems 3.3 and 3.4, we must coose N large enoug so tat te support size of filters is less tan te domain size, wic requires (r + k + α + ) µ = N (r + k + α + ) /µ, ere r = k for te symmetric and derivative RLKV filters and r = 4k for te derivative SRV filter. Table 3. gives te minimum number of elements for different filters. We note tat for te SRV filter, te required number of elements is always too large. Tis is one important reason tat te SRV filter performs poorly over nonuniform meses in te numerical eamples. Once N is smaller tan te minimum requirement given in Table 3., we ave to rescale te filter by using scaling

10 84 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) α = α = α =3 Kernel Support Size 5 5 Kernel Support Size 5 5 Kernel Support Size 5 5 k =, SRV Filter k =, New Filter k =, SRV Filter k =, New Filter k = 3, SRV Filter k = 3, New Filter Fig Comparison of support sizes of te derivative SRV filter and te derivative RLKV filters. Te symmetric derivative filter as te same support size as te derivative RLKV filter. Table 3. Te minimum requirement of element number N according to te derivative order α. ere, N is used for te symmetric and derivative RLKV filters and N is used for te derivative SRV filter. = /(r + k + α + ). owever, tis rescaling tecnique normally as negative effects on te accuracy order. Tere are two ways by wic to overcome tese drawbacks. One is to keep te order of B-splines as k + as te nonderivative filter and ten calculate te derivatives directly as mentioned in [9]. Tis metod can decrease te support size of te filters from (r + k + α + ) to (r + k + ). In te net section, we will present a numerical comparison of using te order of B-splines of k + and k + + α. Te oter way will be presented in our future work; it will give an alternative version of Lemma 3. according to te given nonuniform mes. It allows us to coose a smaller scaling (or larger µ) tan tat in Teorems 3.3 and Numerical results In te previous section, we proposed two position-dependent derivative filters and investigated ow to coose te proper scaling of te filters over nonuniform meses. We also proved tat te filtered solutions ave better accuracy order and smootness compared to te original DG approimations regardless of te derivative order α. We now turn to te numerical eamples of te position-dependent derivative filtering. Te aims of tis section are:. Testing te position-dependent derivative filters (te SRV and RLKV filter) for uniform meses, wic as never been done before;. Applying te symmetric and position-dependent derivative filters over different nonuniform meses; 3. Comparing te derivative filters wit different order B-splines. For convenience, we introduce te following notation: te derivative of te filtered solution, α u. Tis filtered solution uses te standard filter and ten takes te derivative. te filtered derivative, α K u, wic uses te iger order derivative filter K (r+,k++α) for filtering te DG solution. We note tat te DG approimation makes sense only wen α k. In addition, te derivative of te filtered solution α u loses te wanted accuracy order wen α > k (u = K r+,k+ u is a C k function only). Terefore, we mainly present comparison eamples wit α k in tis section. Wenα > k, we only present te results of te filtered derivative α K u, and we point out tat te filtered solution α K u as a teoretical meaning for an arbitrary α, but te accuracy deteriorates wit eac successive derivative. owever, we also note tat once α > k, te nonuniform meses ave to be sufficiently refined in order to see te accuracy improvement. Because of tese reasons, we only present te α = k + case for nonuniform meses. Also, since te symmetric derivative filter is applied in te interior region of eac eample, we do not present it separately. Remark 4.. For te following numerical eamples: wen te number of elements is less tan te minimum requirement in Table 3., a rescaling tecnique is used; quadruple precision is used for te SRV filter, and double precision is used for te RLKV filter and all two-dimensional eamples;

11 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Fig. 4.. Comparison of te point-wise errors in log scale of te first and second derivatives of te DG approimation α u togeter wit te two filtered solutions (te SRV and RLKV filters) for te linear convection equation (4.), over uniform meses. Te B-spline order is k + + α, te filter scaling is taken as =, and k =. te blending function θ() in (.8) is no longer needed for te RLKV filter (see [8]), terefore te function θ() is not used in te following eamples. 4.. Uniform mes Before approacing nonuniform meses, we first apply te position-dependent derivative filters over uniform meses. ere we present results of using bot te SRV filter and te RLKV filter since eac of tem as its advantages over uniform meses tat we address in te following eamples. Consider a linear convection equation u t + u =, [, ], u(, ) = sin(π), at time T = wit periodic boundary conditions. For uniform meses, we can also use scaling = µ and obtain results as Teorems 3.3 and 3.4 described. owever, according to te analysis in [3,8,], in order to maimize te benefits of using central B-splines over uniform meses, we coose te uniform mes size,, as te filter scaling. ere, we compare te derivatives of te DG approimation, te filtered solutions (te SRV and RLKV filter) wit using B-splines of order k + + α (Table 4. and Fig. 4.) and using B-splines of order k + (Table 4. and Fig. 4.). From te tables, we can see tat te filtered solutions α K u and α u ave better accuracy compared to te original DG solutions. Wit te scaling =, te SRV filter clearly as an advantage for uniform meses. Because te SRV filter is constructed using only central B-splines, it was proven to ave k + accuracy order regardless of te derivative order α for linear equations over uniform meses in [9]. In Tables 4. and 4., te SRV filter sows muc better accuracy compared to te RLKV filter near te boundaries, especially wen α is large. For te RLKV position-dependent derivative filter, we notice tat te filtered solutions only ave accuracy of order k+ α in Tables 4. and 4.. Tis is because we use scaling = instead of scaling = µ in Teorem 3.4. We note tat if using a multi-precision package is acceptable, ten te SRV filter is more advantageous for te accuracy order. owever, if only double precision is available during computation (for eample, GPU computing), ten in order to avoid round-off error, te RLKV filter is a better coice, see [8]. owever, wen α > k, te optimal coice is still te SRV filter wit B-splines of order k + + α because only tis filter does not lose te accuracy wit eac successive derivative. We note tat te derivative of filtered solution α u also performs well near boundaries for uniform meses. owever, for te derivative order α > k, we still need to use iger-order B-splines to construct te derivative filters. Figs. 4. and 4. present te point-wise error plots in log scale using te DG approimation of degree k =. After filtering, te filtered approimations are muc smooter tan te DG solution, but in order to reduce oscillations in te interior regions, we still ave to use B-splines of order k + + α. (4.)

12 86 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Table 4. L - and L -errors for te αt derivative of te DG approimation α u togeter wit te two filtered solutions (te SRV and RLKV filters) for te linear convection Eq. (4.), over uniform meses. Te B-spline order is k + + α and te filter scaling is taken as =. Remark 4.. For uniform meses, we coose to use te scaling = instead of te scaling = µ in Teorem 3.3. Tis is because for uniform meses, te scaling = can provide a better accuracy order of k + compared to te conclusion in Teorem 3.3, especially in te interior region. Also, te SRV filter benefits of te scaling = in te boundary region once quadruple precision is used. If te scaling = µ is used for uniform meses, te accuracy order will decrease and te error magnitude will increase in te interior region. owever, te RLKV filter will ave better accuracy order in te boundary region, and te error magnitude will improve once te mes is sufficiently refined Smootly-varying mes As mentioned in [6,8], tere is a particular family of nonuniform meses, smootly-varying meses for wic te filter beaves similar to te uniform case. In [8], te autors proved tat te filtered solutions (bot te SRV and RLKV filters) ave a similar performance over smootly-varying meses compared to uniform meses. owever, it would be of practical interests to sow te performance of te position-dependent derivative filters over smootly-varying meses, especially for smootly increasing/decreasing meses. Consider a linear convection equation wit Diriclet boundary conditions u t + u =, [, ], (4.)

13 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Table 4. L - and L -errors for te αt derivative of te DG approimation α u togeter wit te two filtered solutions (te SRV and RLKV filters) for te linear convection equation (4.), over uniform meses. Te B-spline order is k + and te filter scaling is taken as =. u(, ) = sin(π), u(, t) = sin( πt), at time T = over a smootly decreasing mes defined in [8]: = ξ.(ξ )ξ, were is te smootly decreasing mes variable and ξ is te variable of te uniform mes over domain [, ]. Similar to te uniform mes case, we coose te local mes size as te filter scaling, = j according to [8]. ere, in order to save space, we present results for te filtered derivative α K u only. Te L and L errors are presented in Table 4.3 wit te first tree derivatives over te above smootly decreasing mes. Te respective point-wise error plots (k = case) are given in Fig We point out one penomenon tat is very different to te uniform mes case. Te RLKV filter provides better accuracy compared to te SRV filter near te boundaries, see Table 4.3. owever, bot te SRV and RLKV filters can improve te accuracy of te original DG solutions once te mes is sufficiently refined.

14 88 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Fig. 4.. Comparison of te point-wise errors in log scale of te first derivative of te DG approimation α u togeter wit te two filtered solutions (te SRV and RLKV filters) for te linear convection equation (4.), over uniform meses. Te B-spline order is k +, te filter scaling is taken as =, and k =. Fig Comparison of te point-wise errors in log scale of te first and second derivatives of te DG approimation α u togeter wit te two filtered solutions (te SRV and RLKV filters) for te linear convection equation wit Diriclet boundary condition (4.), over smootly decreasing meses. Te B-spline order is k + + α, te filter scaling is taken as = j, and k =. 4.. Nonuniform mes Now we sow te main results of tis paper: te position-dependent derivative filtering over arbitrary nonuniform meses. Before proceeding furter, we first give te numerical setting of nonuniform meses. In order to generate a more general format for nonuniform meses, we use a random number generator to design te following two meses.

15 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Table 4.3 L - and L -errors for te αt derivative of te DG approimation α u togeter wit te two filtered solutions (te SRV and RLKV filters) for te linear convection equation wit Diriclet boundary conditions (4.), over smootly decreasing meses. Te B-spline order is k + + α and te filter scaling is taken as = j. Mes 4.3. Te first nonuniform mes tat we consider is defined by =, N+ =, j+ = j + b r j+, j =,..., N were r j+ N j= j are random numbers between (, ), and b (,.5] is a constant number. Te size of te element j = j+ b =.4 only, oter values of b suc as.,. and.45 were also calculated. is between (( b), ( + b)). In order to save space, in tis paper we present an eample wit Mes 4.4. Te second nonuniform mes is more irregular tan te first one. We distribute te element interface by j+, j =,..., N randomly for te entire domain and require only j = j+ j b, j =,..., N. In tis paper, we only present b =.5 case for tis mes, oter values of b suc as.6,.8, etc. were also calculated.

16 9 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Table 4.4 L - and L -errors for te first derivative of te DG approimation α u togeter wit te two filtered solution α u and α K u (wit te SRV filter) for te linear convection equation (4.), over Mes 4.3. Te filter scaling is taken as = /5 near boundaries and = /3 for interior regions (were te symmetric filter is applied). Mes α u α u (SRV) α K u (SRV) α = P L error Order L error Order L error Order L error Order L error Order L error Order 5.48E.76E+.85E.49E+ 5.9E.54E+ 4.8E.96.5E E..57E E.7.37E E E.8.E.3.5E E.4.38E E..57E.96.9E.7.8E E.43.3E E.99.3E.98 3.E E E E.4 P 3.56E.E.5E 8.74E 3.3E.35E E E.79.3E 3.3.9E. 4.E E E 3..6E.3.8E E E E E E E E. 3.6E E. 3.3E E 4..36E 3.3.5E..63E E. P 3.53E 3.E.33E 7.E 4.3E.58E 4.E E E E E E E E E E E E E E E E E E E E E 6.8.9E E E 5. We remark tat we ave tested various differential equations over bot kinds of nonuniform meses: Mess 4.3 and 4.4. owever, te filtered approimations ave similar performances since te performance mainly depends on te mes. In order to save space, we coose to present one equation for eac nonuniform mes Comparison of te SRV filter and RLKV filter over nonuniform Mes In [8], te autors sowed tat te SRV filter as worse performance compared to te RLKV filter over nonuniform meses for filtering te solution itself. We also mentioned tat te enormous support size of te SRV filter causes problems: we ave to rescale te SRV filter to fit te domain size. ence, we cannot guarantee neiter te accuracy order nor error reduction. Table 3. sows te minimum requirement of te number of elements for te SRV filter, and we can see tat it is difficult to satisfy. Based on tese deficiencies, we conclude tat te SRV filter is not suitable for approimating derivatives over nonuniform meses. owever, in order to provide a complete view of te position-dependent derivative filters, we still give one eample of using te SRV filter for te first derivative over Mes 4.3. Table 4.4 sows tat wit te SRV filter, te filtered solutions (no matter wat order of B-splines is used) are even worse tan te derivative of te DG approimation. In te rest of tis section, we focus on testing te RLKV filter over nonuniform meses Linear equation over Mes 4.3 For Mes 4.3, we present results for te linear convection equation (4.) wit te first, second and tird derivatives. Te L and L norm errors are given in Table 4.5 and Fig. 4.4 sows te point-wise error in log scale. Wen α k, bot te derivative of filtered solution α u and te filtered derivative α K u ave better accuracy and convergence rates tan te original DG approimation. Te filtered derivative α K u sows better smootness and teoretically as a better accuracy order tan te derivative of te filtered solution α u wen α k, but α u as better accuracy near te boundaries. For smootness, te results are similar to te uniform mes case; α K u as fewer oscillations compared to te DG solution and α u. Furtermore, we point out tat by using iger-order B-splines we can disregard te requirement tat α k. For te point-wise error plots given in Fig. 4.4, te middle column is te filtered approimation α u, wic sows more oscillations tan te α K u, especially in te interior regions. We note, owever tat te support size of te filter tat uses a iger-order B-spline increases wit te derivative order α and it sligtly increases te computational cost. Near te boundaries, te filtered solutions ave a larger error magnitude tan tose in te interior region. Because near te boundaries we cannot obtain symmetric information around te filtered points, and te general B-spline as less regularity compared to te central B-spline. We note tat te coarse meses (suc as N = or even N = 4) are not sufficient to

17 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Table 4.5 L - and L -errors for te αt derivative of te DG approimation α u togeter wit te two filtered solutions α u and α K u (wit te RLKV filter) for te linear convection equation (4.), over Mes 4.3. Te filter scaling is taken as = /3. use te position-dependent derivative filter, te filtered solution may ave worse accuracy compared to te original DG approimation.

18 9 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Fig Comparison of te point-wise errors in log scale of te first and second derivatives of DG approimation α u togeter wit te two filtered solutions α u and α K u (wit te RLKV filter) for te linear convection equation (4.) over Mes 4.3. Te filter scaling is taken as = /3, and k = Variable coefficient equation over Mes 4.4 After testing te linear convection equation (4.), we move to a variable coefficient equation wit periodic boundary conditions, u t + (au) = f, (, t) [, ] (, T] (4.3) u(, ) = sin(π), were te variable coefficient a(, t) = + sin(π( + t)) and te rigt side term f (, t) are cosen to make te eact solution u(, t) = sin(π( t)). As wit te linear convection eample, we present te L and L errors in Table 4.6 wit te first tree derivatives over Mes 4.4. Te respective point-wise error plots (k = case) are sown in Fig Te results are similar to te results for te constant coefficient case. In order to save space we no longer repeat te descriptions, wic are similar. owever, we still want to point out one penomenon. In tis variable coefficient case, te filtered solution α u sows somewat better accuracy tan te filtered solution α K u near te boundaries wen α k. Tis performance suggests tat wen α k we can consider not increasing te order of te B-splines, altoug it causes more oscillations in te error. Remark 4.5. ere we summarize te consequences of using B-splines of order k + compared to using normal order k + + α; tey are te following: it can give better accuracy near te boundaries; it can give better accuracy in te interior regions (wen α k), but it damages te smootness of filtered solution (more oscillations); it as a smaller support size; it allows te use of te symmetric filter over a larger area; and it requires α k. Remark 4.6. We notice tat in Tables 4.5 and 4.6, te accuracy order is smaller tan te conclusion in Teorem 3.4. Te reason is twofold: Te first reason is te effect of te boundary region. Te error magnitude of te filtered solution in te interior region is muc better tan te error magnitude in te boundary region. Terefore, te accuracy order in te L norm appears to be unstable and te numbers are smaller tan te teoretical epectation. If we look at Figs. 4.4 and 4.5 (rigt column), te convergence rates are stable respective to boundary and interior regions separately.

19 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Table 4.6 L - and L -errors for te αt derivative of te DG approimation α u togeter wit te two filtered solutions α u and α K u (wit te RLKV filter) for variable coefficient equation (4.3), over Mes 4.4. Te filter scaling is taken as = /3. Te second reason is tat te Mess 4.3 and 4.4 are randomly generated. Tere is no strict refinement relation among te meses. Terefore, te accuracy order is affected by te randomness of te meses, and a very stable accuracy order is

20 94 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Fig Comparison of te point-wise errors in log scale of te first and second derivative of te DG approimation α u togeter wit te two filtered solutions α u and α K u (wit te RLKV filter) for variable coefficient equation (4.3) over Mes 4.4. Te filter scaling is taken as = /3, and k =. log (error) 8.5 y log (error) 8.5 y log (error) 8.5 y Fig. 5.. Comparison of te point-wise errors in log scale of te cross-derivative DG approimation y u togeter wit te filtered solutions for te two-dimensional linear advection equation (5.) over Mes 4.3 (D, N = 6 6). not observed. One can see tat once te effect caused by randomness becomes smaller in te two-dimensional eample, te accuracy order becomes stable, see Tables 5. and Two-dimensional eample For te two-dimensional eample, we consider a D version of te linear convection equation u t + u + u y =, (, y) [, ] [, ], (5.) u(, y, ) = sin(π + πy), at time T = wit periodic boundary conditions. Te nonuniform meses we used are te D quadrilateral etension of Mess 4.3 and 4.4. ere, we sow te cross-derivative y, te first derivatives and y are omitted as tey are similar to te D results. We give te L and L error in Tables and te point-wise error plots in Figs We note tat te filtered accuracy error seems sligtly worse tan te DG approimation over coarse meses, because near te boundary regions we need sufficiently refined meses to sow te advantage of te position-dependent filter. Once te mes is sufficiently refined, we see better results. We also note tat altoug we require a relatively refined mes for boundary regions, te results in te interior regions beave better (see te point-wise error plots).

21 X. Li et al. / Journal of Computational and Applied Matematics 94 (6) Table 5. L - and L -errors for te cross-derivative DG approimation y u togeter wit te filtered solutions for te two-dimensional linear convection equation (5.) over Mes 4.3 (D). Mes y u y u (RLKV) ( y K ) u (RLKV) P L error Order L error Order L error Order L error Order L error Order L error Order 5.47E+.3E+.3E+.5E E+..33E+.8.97E.75.8E E E+.6.8E.8.56E E. 3.38E E E.9 P 3.48E.49E+ 4.68E 3.44E+ 5.66E 3.59E E.9 7.3E.8.65E E E E E.8.8E.98.38E E 4..83E E E E. 6.86E E E E P 3.54E.47E 4.E.79E 4.6E.63E E E.68.4E.85.7E.3.45E.73.4E E E E E E E E E E E E E Table 5. L - and L -errors for te cross-derivative DG approimation y u togeter wit te filtered solutions for te two-dimensional linear advection equation (5.) over Mes 4.4 (D). Mes y u y u (RLKV) ( y K ) u (RLKV) P L error Order L error Order L error Order L error Order L error Order L error Order 6.3E+.94E+.53E+.7E E+.9.95E+.59.7E.5.7E E+.99.9E E.3 4.E E. 5.8E E E.55 P 5.6E 7.E+ 4.73E 3.48E+ 5.68E 3.59E E.73.65E+.4.67E E E E E.3 5.3E.3.36E E E E E 3.4.3E E E E E P 3 4.E 3.66E 4.E.79E 4.5E.64E E 3.4.3E.83.4E.86.9E..45E.7.6E E E.6.4E E E E E E E E E E log (error) 8.5 y log (error) 8.5 y log (error) 8.5 y Fig. 5.. Comparison of te point-wise errors in log scale of te cross-derivative DG approimation y u togeter wit te filtered solutions for te two-dimensional linear advection equation (5.) over Mes 4.4 (D, N = 6 6). 6. Conclusion In tis paper, we ave proposed two position-dependent derivative filters (te SRV and RLKV filter), to approimate te derivatives of te discontinuous Galerkin solutions over uniform and nonuniform meses. Tese position-dependent