Hanieh Mirzaee Jennifer K. Ryan Robert M. Kirby

Transcription

1 J Sci Comput (2010) 45: DOI /s Quantification of Errors Introduced in te Numerical Approximation and Implementation of Smootness-Increasing Accuracy Conserving (SIAC) Filtering of Discontinuous Galerkin (DG) Fields Hanie Mirzaee Jennifer K. Ryan Robert M. Kirby Received: 15 June 2009 / Revised: 8 December 2009 / Accepted: 14 December 2009 / Publised online: 8 January 2010 Te Autor(s) Tis article is publised wit open access at Springerlink.com Abstract Te discontinuous Galerkin (DG) metod continues to maintain eigtened levels of interest witin te simulation community because of te discretization flexibility it provides. Altoug one of te fundamental properties of te DG metodology and arguably its most powerful property is te ability to combine ig-order discretizations on an interelement level wile allowing discontinuities between elements, tis flexibility generates a pletora of difficulties wen one attempts to post-process DG fields for analysis and evaluation of scientific results. Smootness-increasing accuracy-conserving (SIAC) filtering enances te smootness of te field by eliminating te discontinuity between elements in a way tat is consistent wit te DG metodology; in particular, ig-order accuracy is preserved and in many cases increased. Fundamental to te post-processing approac is te convolution of a spline-based kernel against a DG field. Tis paper presents a study of te impact of numerical quadrature approximations on te resulting convolution. We discuss bot teoretical estimates as well as empirical results wic demonstrate te efficacy of te post-processing approac wen different levels and types of quadrature approximation are used. Finally, we provide some guidelines for effective use of SIAC filtering of DG fields wen used as input to common post-processing and visualization tecniques. Tis paper is in onor of our mentor and collaborator Professor David Gottlieb, wo taugt us tat implementations require approximations, approximations lead to numerical crimes, and as wit all crimes, numerical crimes come at a cost. H. Mirzaee R.M. Kirby ( ) Scool of Computing, University of Uta, Salt Lake City, UT, USA kirby@sci.uta.edu H. Mirzaee mirzaee@cs.uta.edu J.K. Ryan Delft Institute of Applied Matematics, Delft University of Tecnology, Mekelweg 4, 2628 CD Delft, Te Neterlands J.K.Ryan@tudelft.nl

2 448 J Sci Comput (2010) 45: Keywords Hig-order metods Discontinuous Galerkin SIAC filtering Accuracy enancement 1 Introduction Te discontinuous Galerkin (DG) metods provide a ig-order extension of te finite volume metod in muc te same way as ig-order or spectral/p elements [15, 21] extended standard finite elements. Te DG metodology allows for a dual pat to convergence troug bot elemental and polynomial p refinement, making it igly desirable for computational problems wic require resolution fidelity. In te overview of te development of te discontinuous Galerkin metod, Cockburn et al. [4] trace te developments of DG and provide a succinct discussion of te merits of tis extension to finite volumes. Te primary matematical advantage of DG is tat unlike classic continuous Galerkin FEM, tat seeks approximations wic are piecewise continuous, te DG metodology only requires functions wic are L 2 integrable. Muc like FEM, DG uses te variational form; owever, instead of constraining te solution to being continuous across element interfaces, te DG metod merely requires weak constraints on te fluxes between elements. Tis feature provides a discretization flexibility tat is difficult to matc wit conventional continuous Galerkin metods. Lack of inter-element continuity, owever, is often contrary to te smootness assumptions upon wic many post-processing algoritms suc as tose used in visualization are based. A class of post-processing tecniques were introduced in [7, 18] as a means of gaining increased accuracy from DG solutions troug te exploitation of te superior convergence rates of DG in te negative norm; tese filters ad as a secondary consequence tat tey increased te smootness of te output solution. Building upon tese concepts, in [20, 22] smootness-increasing accuracy-conserving (SIAC) filters were proposed as a means of ameliorating te callenges introduced by te lack of regularity at element interfaces wile at te same time maintaining accuracy constraints tat are consistent wit te verification process used in te original simulation. In essence, in te application domain, one seeks to increase smootness witout destroying (i.e. by maintaining) te order of accuracy of te original input DG solution. Te basic operation performed to gain te smootness and accuracy benefits is convolution of te DG solution against a judiciously constructed B-spline based kernel. Te goal of tis paper is to ascertain and quantify te impact of quadrature errors witin tis convolution process. All of te matematical proofs concerning accuracy and smootness assume exact integration. All te empirical numerical examples bot in te matematical literature [7, 18] and te engineering literature [20, 22] employed consistent integration wit Gaussian quadrature to guarantee tat te numerical errors witin te convolution operator could be driven below macine precision. In tis paper we seek to quantify te impact of inexact quadrature on te filtering process and to assess weter it greatly impacts its use as intermediary stage between simulation and visualization in te scientific pipeline. In tis paper we examine a collection of common scenarios tat migt arise wen one seeks to implement te aforementioned post-processing algoritms in te engineering context. We focus on one-dimensional and two-dimensional quadrilateral implementations over periodic domains as for tese cases we can derive teoretical estimates as to te impact of te numerical crimes committed troug te different quadrature strategies. We use as our gold-standard te solving of te convolution operation wit consistent integration (integration tat partitions te domain so as to respect all breaks in regularity) combined wit

3 J Sci Comput (2010) 45: Gaussian integration tat integrates te kernel times te DG-based polynomial exactly to double-precision macine zero. We first examine te case wen consistent integration wit inexact Gaussian quadrature (under-integration) is used. Because consistent integration requires solving te callenging geometric problem of find all te places in wic regularity is decreased and generating a super-mes based on tis data, we consider wat appens wen only te original DG mes is used as te underlying support mes for integration. Under tis scenario, we examine te use of Gaussian quadrature and of midpoint quadrature. Tis coice igligts te difference between polynomial-based ig-order and adaptive loworder quadrature implementations. We empasis tat tis study is primarily for engineering circumstances wen te trade-offs between time, resources and accuracy are important. Altoug te case against committing suc numerical crimes is well-known, te repercussions ave not been well documented for te use of tis filter as a visualization tool. It is tis specific crime tat we wis to address. Te paper is organized as follows. In Sect. 2 we discuss te discontinuous Galerkin metod for systems of yperbolic equations. In Sect. 3 we present te matematical background on te smootness-increasing accuracy-conserving filters being examined in tis paper. In Sect. 4 we present te different implementation strategies one migt employ, in particular: (1) te consistent integration approac wit exact and inexact Gaussian quadrature, (2) te input mes based Gaussian quadrature approac and (3) te input mes based midpoint quadrature approac. In Sect. 5 we present analysis wic provides teoretical estimates wic bound te numerical crimes committed wen using te tree aforementioned approaces. In Sect. 6 we present an empirical study wic corroborates te error estimates we ave derived. In Sect. 7 we summarize our results and provide guidelines based upon our study concerning under wic circumstances one tecnique sould be used versus anoter. 2 Te Discontinuous Galerkin Metod In tis paper, we focus our attention on simulation results tat arise as solutions of te linear yperbolic equation u t + (a(x,t)u) = 0, (1) were x and t R. Te DG formulation for tis equation as been well studied in te series of papers [2, 3, 5, 9 12]. Here we present a brief introduction. We use te weak form of (1) to derive our discontinuous Galerkin approximation. Tat is, we multiply by a test function v to obtain d u(x,t)vdx + (a(x,t)u) ˆnv dɣ (a(x,t)u) v dx = 0, (2) dt Ɣ were ˆn denotes te unit outward normal to te boundary and Ɣ te boundary of our domain,. We can now define our discontinuous Galerkin approximation to (1) using(2). Begin by defining a suitable tessellation of te domain, T ( ) =. We note tat te current form of te post-processor requires using a rectangular domain in two-dimensions. Secondly, we define an approximation space, V, consisting of piecewise polynomials of degree less tan or equal to k. Our discontinuous Galerkin approximation will ten be of order k + 1. Using te variational formulation and taking our test function v from our approximation space we

4 450 J Sci Comput (2010) 45: obtain d u(x,t)v dx + dt K e K e (a(x,t)u) ˆn e,k v dɣ (a(x,t)u) v dx = 0, (3) K were ˆn e,k denotes te outward unit normal to edge e. We ten obtain te numerical sceme d u (x,t)v dx + (u (x,t),u (x +,t))v dɣ dt K e K e (a(x,t)u ) v dx = 0 (4) K for all test functions v V were (u(x,t),u(x +,t)) represents te numerical flux and u is te DG approximation of degree k. 3 Te Post-Processor We concentrate on quantifying and improving te computational cost of te convolution for te Smootness-Increasing Accuracy-Conserving (SIAC) filter tat were first introduced as a class of post-processors for te discontinuous Galerkin metod in [6, 7]. Te filtering tecnique was extended to a broader set of applications, including as a streamline visualization filter, in [8, 14, 17, 18, 20, 22]. As a visualization filter, it is unique in tat it takes into account te information about te discontinuous Galerkin approximation. For an approximation of degree k, we convolve it against a filter containing a linear combination of B-splines of degree k. Tis works to smoot te oscillations in te error and improve te order of te filtered solution to 2k + 1. Here we only present a brief introduction to tis post-processing tecnique. For a more detailed discussion of te post-processor see [1, 7, 14, 18]. Te post-processor itself is simply te discontinuous Galerkin solution convolved against a linear combination of B-splines. Tat is, in one-dimension, u (x) = 1 ( ) y x K 2(k+1),k+1 u (y) dy, (5) were u is te post-processed solution, is te uniform element size, u te DG solution of degree k and K 2(k+1),k+1 (x) = k γ = k c 2(k+1),k+1 γ ψ (k+1) (x γ). (6) Te B-splines used in te post-processor are well studied and can be computed using te recurrence relation ψ (1) = χ [ 1/2,1/2], (7) ψ (k+1) = 1 (( x + k + 1 ( )ψ (k) x + 1 ) ( ( k x )ψ (k) x 1 )), k k 1, (8)

5 J Sci Comput (2010) 45: see [19]. Te coefficients of te kernel, cγ 2(k+1),k+1, can be found by using te property tat te kernel must not destroy te accuracy of te approximation. More specifically, tey reproduce polynomials of degree 2k by convolution. Tis leads to a system of equations (k+1) c 2(k+1),k+1 = X (9) were te (i, γ )-t element of (k+1) is given by ψ (k+1) (x y γ)y i dy, i = 0,...,2k, γ = k,...,k, R and X i = x i. Wen using a symmetric B-spline kernel, it is only necessary to solve tis system once and store te resulting coefficient vector, c 2(k+1),k+1. We note tat te coefficients are symmetric in tis case, tat is c 2(k+1),k+1 j = c 2(k+1),k+1 j, j = 1,...,k. We note tat te two-dimensional case is tensor product of te one-dimensional case. After solving for te coefficients, we ten need to carry out te convolution C i,l,k = 1 k γ = k ( ) ( ) y x y cγ 2(k+1),k+1 ψ (k+1) γ φ (l) xi dy, l = 0,...,k, (10) I i were I i denotes te element and φ (l) represents te l t -piecewise polynomial from our approximation space scaled to te interval ( 1/2, 1/2). Using tis notation, we ten ave tat te post-processed solution is u = k k i=j k l=0 C i,l,k u (l) i (11) were k = 3k+1 and u (l) 2 i are te polynomial modes on I i. Te outer sum is over te elements tat fall witin te support of te post-processor. We can see from tis notation tat if we coose to compute te post-processing coefficients aead of time at specific points witin an element, te post-processor simply becomes a series of small matrix-vector multiplications. However, if we coose to compute te convolution directly, it can become computationally expensive. Te latter implementation, in wic we take care of te convolution witin te definition of C i,l,k, is te one we concentrate on for te purposes of tis paper. We will, in te next section, go into more detail as to ow one implements te post-processor and te corresponding computational costs. 4 Implementation In tis section we present te different implementation strategies used to calculate te convolution operator. 4.1 Gaussian Quadrature Approaces Te post-processed solution, u (x), wic is a piecewise polynomial of degree 2k + 1, can be evaluated exactly. As we mentioned in Sect. 3, u (x) is defined by (5), were u (y) =

6 452 J Sci Comput (2010) 45: k l=0 u i (l) φ (l) i (y), andφ (l) i are te basis functions of te projected function on cell I i = (x i 2,x i + 2 ). Terefore, for x I i,weave u (x) = 1 ( ) y x K 2(k+1),k+1 u (y) dy = 1 = 1 k j= k k j= k I i+j K 2(k+1),k+1 I i+j K 2(k+1),k+1 ( y x ) u (y) dy ( ) ( ) y x k u (l) i+j φ(l) i+j (y) dy. (12) We ave used te compact support property of K 2(k+1),k+1 to reduce te area of te integral to a total support of 2k + 1 elements were k = 3k+1. 2 As mentioned earlier in Sect. 3, te kernel in te expression above consists of a linear combination of B-splines. Terefore, in order to calculate te above integral exactly, we need to decompose te interval I i+j into subintervals tat respect te kernel knots (wic we refer to as breaks); te resulting integral is calculated as te summation of te integrals over eac subinterval. Te term consistent integration is used to denote integration tat respects tese breaks by finding te necessary subintervals suc tat te integral on eac subinterval can be done exactly to macine precision. To compute te post-processed solution for x I i,te algoritm is as follows: For j = k,...,k : Find te kernel breaks (if any) tat lie in te interval I i+j. Use te kernel breaks to identify te subintervals. Note tat in te general case te number of breaks can be zero up to several breaks witin an element. In te case of uniform meses, one can sow tat tere is at most one break per input mes element. Evaluate te integral over eac of te subintervals using Gaussian quadrature. For te case of an exact quadrature, we are required to evaluate te integrand at k + 1 Gauss points were k is te approximation degree of te DG solution. We ave used k Gauss points (one less tan te required) for te inexact quadrature experiments presented in te results section. Sum te resulting values from eac subinterval to gain te overall value of te integral on element I i+j. Tis general algoritm can be used in one of several ways wic we will mention ere. First, tis algoritm olds for any x I i, and ence can be used for isolated post-processing of te solution at some arbitrary point. Te only additional cost not previously mentioned is te searc time needed to find te element I i containing te point x of interest. Building upon tis usage, te second strategy is to post-process an entire element (i.e. find te postprocessed polynomial of degree 2k + 1 on a element) by repeating te above procedure for a collection of collocation points or at quadrature points so tat a transform to a modal representation can be done. Te tird usage, wic is often implemented for uniform meses, is to rewrite te above equation using small matrix multiplications, u (x) = k k j= k l=0 l=0 u (l) i+j C j,l,k(x) (13)

7 J Sci Comput (2010) 45: were C j,l,k (x) is a polynomial of degree 2k + 1andu (l) i+j are te coefficients in te discontinuous Galerkin approximation. In previous implementations, one computes te convolution of te basis functions of te DG solution and B-splines togeter for a given mes once so tat post-processing can be accomplised repetitiously for different DG solutions on te same mes troug mere matrix-vector multiplication. As te kernel is translation invariant wen a uniform mes is used, one needs only to compute a small set of matrices tat can be used for post-processing all elements. As is presented in [14], it takes O(N) operations to filter te entire domain, were N is te total number of elements. However, te purpose of tis paper is to quantify te numerical crimes committed wen using different quadrature scemes. Terefore, it is useful to understand te dominant costs in te above algoritm so tat one can appreciate wy different engineering implementation coices migt be made. Te possibly dominating cost in te above algoritm is te finding of te consistent integration mes. Tis is a geometric projection and intersection problem tat, in general, is quite complex in two and tree dimensions as it requires finding te intersection of mes elements against knots of te B-spline kernel, and possibly te furter refined tessellation of te subdomains into elements upon wic integration can be performed (for instance, partitioning polyedra into a collection of tetraedra). Wen filtering based upon te input mes, we disregard te position of te kernel breaks and evaluate te integrand over te entire element. Tat is, we skip te first step in te consistent integration approac (and ence its associated cost); in te second step tere is only one interval wic is te entire element. We ten proceed by using Q k + 1 Gauss points for computing te Gaussian quadrature. Considering te computational cost, as we increase te number of quadrature points, Q, filtering based upon te input mes will use more floating point operations for Q>2k + 2 to calculate te integral over I i+j compared to te consistent integration approac; owever, te algoritmic scaling is still O(M) were M is a number related to te extent of te local filter. Two furter notes are wort mentioning before proceeding. Te first is tat te algoritm above extends easily to te case of two-dimensional and tree-dimensional post-processing as te convolution kernel is merely a tensor-product of te one dimensional kernels. Secondly, in te case of a non-uniform mes since te kernel is no longer translation invariant te post-processing coefficients need to be recomputed for eac element as is mentioned in [14]. In order to avoid tis re-computation, Curtis et al. proposed two strategies for postprocessing over non-uniform meses: one based upon te local L 2 -projection of te solution to a uniform scratc-pad mes and one based upon te caracteristic lengt. Te algoritmic scaling for bot algoritms is O(N), wit N being te size of te domain. 4.2 Midpoint Quadrature Approac In tis section we examine an alternative strategy to using Gaussian quadrature for te approximation of te convolution integral. For a given x I i we try to approximate te integral u (x) = 1 ( ) y x K 2(k+1),k+1 u (y) dy (14) using midpoint integration. For a complete overview of te derivation and implementation of te midpoint rule we refer te reader to [16]. In tis case we evaluate te post-processed solution u (x) using te midpoint rule to compute te integral in (14) over te entire kernel support at once, i.e., we are not following te element-by-element approac mentioned in te previous section. In oter words, we proceed as follows:

8 454 J Sci Comput (2010) 45: For x I i determine were te limits of te kernel lie on te DG mes. Tat specifies te integration area, wic we denote by [x left,x rigt ]. Set te level of te midpoint integration. Assuming x is te size of eac of te n equal cells involved in te discretization we ave x = (x rigt x left )/2 level wit 2 level being te number of evaluation points used in te midpoint rule. Typically, te support of te kernel centered around zero will be te case were x left = 3k+1 and x 2 rigt = 3k+1.Tisgives 2 x = 3k+1. 2 level Perform te midpoint integration, u (x) = 1 ( ) y x K 2(k+1),k+1 u (y) dy = 1 xrigt ( ) y x K 2(k+1),k+1 u (y) dy x left ( = 1 n x K i 1/2 u,i 1/2 ). (15) i=1 Again, we follow a similar process in case of a 2D post-processor along eac direction. As it is understood from te aforementioned steps, we bot disregard te kernel breaks and te element interfaces in te input mes for tis quadrature approac. 5 Quadrature Approximations of te Convolution Operator In tis section we analyze te crimes committed wen performing a non-consistent, inexact quadrature. Altoug we discuss te simplified case of a uniform one-dimensional mes, many of te concepts extend to post-processing over non-uniform meses. We begin by discussing te ideal case of an exact, consistent quadrature, ten an inexact quadrature on a consistent integration mes. We next move to implementing an inconsistent quadrature wic takes only te DG mes into account and lastly, te midpoint quadrature on te DG mes. 5.1 Gaussian Quadrature on a Consistent Integration Mes Exact, Consistent Gaussian Quadrature Gaussian Quadrature is well-known to integrate polynomials of degree 2m 1 exactly by using m points, x 1,x 0,...,x m,andm weigts w 1,w 2,...,w m [13]. Te formula for te quadrature is given by b m f(x)dx= f(y j )w j, (16) a were y j = b a 2 x j + b+a 2,andw j is te associated weigt function. Tis means tat for our convolution, wic consists of integrals of polynomial degree at most 2k, weneedtouse k + 1 points and weigts in our quadrature in order to be exact. Te main point of tis discussion is to point out tat in order to post-process one element tat contains a discontinuous Galerkin approximation of degree k, we ave a support size j=1

9 J Sci Comput (2010) 45: of 2k + 1 elements for te post-processor, were k = 3k+1 2. Tere are two integral evaluations per element. Terefore we are performing 4k + 2 Gaussian Quadratures of degree 2k + 1. Altoug Scumaker gives a simplified formula for integrating a B-spline against a polynomial [19], we sould point out tat we are convolving te B-splines against a piecewise polynomial. To begin, let us look at te kernel performed using exact integration over a uniform mes. In tis case, to post-process element i,weave u (x) = k k j= k l=0 u i+j k γ = k ( ) ( ) 1 y x y c γ ψ (k+1) γ φ (l) xi+j dy. (17) I i+j Setting η = y x and ξ i = y x i, tis becomes u (x) = k k j= k l=0 u i+j k ξi +j+1/2 c γ ψ (k+1) (η γ)φ (l) (ξ i + η j)dη. (18) γ = k ξ i +j 1/2 On eac element, I i+j, define te function in our integral to be f(η,ξ i ) = k ψ (k+1) (η γ)φ (l) (ξ i + η j). (19) l=0 For purposes of error analysis, we note tat tis function is C k 1 over eac DG element and discontinuous at element boundaries. Tis is easily seen by examining te case of piecewise monomials. In tis case, φ (l) (ξ i + η j)= (ξ i + η j) l. If we consider te boundary were η ( ξ i 1/2), ten we ave, from te left of te elemental boundary: lim η (ξ i +1/2) f(η,ξ i) = k l=0 ( ψ (k+1) ( ξ i 1/2 γ) 3 ) l, 2 since we are approacing from te element I i 1. However, approacing te limit from te rigt, we would ave lim f(η,ξ i) = η (ξ i +1/2) + k l=0 ( ψ (k+1) ( ξ i 1/2 γ) 1 ) l, 2 as we are approacing from element I i. Te limits of tese two functions are continuous for piecewise constants, but oterwise tey are discontinuous. In order to analyze te error for te case of consistent integration wit reduced quadrature and quadrature based on te input DG mes it is useful to review te existing literature. Here we follow te work of de Boor [13]forw(x) = 1. Tat is, if we were to use exact quadrature over a consistent integration mes using k + 1 Gauss points per integral, te error is given by b a f(η,ξ i )dη b a p k (η) dη = b a f [x 0,...,x 2k+1 ]g 2k+1 (x) dx (20) were p k is some polynomial tat interpolates f(η,ξ i ) at k + 1 points and g 2k+1 (x) =[(x x 0 ) (x x k )] 2.

10 456 J Sci Comput (2010) 45: If f(η,ξ i ) C 2k+2, ten we ave te error estimate b a f(η,ξ i )dη b a p k (η) dη = C(ξ i,y 0,...,y k+1 )f (2k+2) (z, ξ i ), z (a, b). (21) Note tat f(η,ξ i ) is a polynomial of degree 2k, and te integral is exact Inexact, Consistent Gaussian Quadrature Now tat we ave discussed te necessary components for exact, consistent quadrature, let us examine te case were we simply use less Gauss points. In tis case, we are respecting bot te kernel breaks and te DG elemental boundaries and use only k gauss points. From te error formula above (21), we ave te following error estimate for one integral b a f(η,ξ i )dη b a p k (η) dη = C(ξ i,y 0,...,y k+1 )f (2k) (z, ξ i ) = C. (22) Te constant in te error is not necessarily less tan one, but is certainly bounded due to te smootness of f(η,ξ i ). We also note tat te constant depends upon k. 5.2 Gaussian Quadrature on te DG Mes Next, we look at te case were we are using exact quadrature over a discontinuous Galerkin mes. Tat is, we are ignoring te knots of te B-splines and disregarding te level of smootness. For tis case, we may only use te error estimate given in (20) since te function on a given element is only C k 1.Tatis,teerrorisgivenby b a f(η,ξ i )dη b a p k (η) dη = 5.3 Midpoint Quadrature on te DG Mes b a f [x 0,...,x 2k 1 ]g 2k+1 (x) dx. (23) Lastly, we consider te problem of using midpoint integration. In tis case our formula is given by b a f(x)dx= (b a)f(c) (24) were c = (a + b)/2 andf is as given in (19). It is well known tat te midpoint error is given by Error(midpoint) = 1 24 f (z)(b a) 3, z (a, b) [13]. In our case, we are carrying out m = level midpoint quadratures. Terefore, we ave for te error m 1 j= f (z) x 3 C x 3 (25) were x = 3k+1 2 m since x left = 3k+1, x 2 rigt = 3k+1 and x = (x 2 rigt x left )/2 m. We sould note two tings. First, tat te constant tat bounds f (z)/24 depends upon te polynomial

11 J Sci Comput (2010) 45: order. Secondly, tat tis estimate of te error does not improve for iger polynomial orders. Terefore, it does not matter weter we increase te polynomial order of our B-spline or our DG solution. Te only time were implementing te midpoint quadrature may be effective is for piecewise linear polynomial approximations. 6 Results In tis section we present te results for te various coices of quadrature. We examine te L 2 -errors, and in te case of one-dimension, te smootness of te error. Lastly, we present a test example to demonstrate te usefulness for visualization purposes. 6.1 Consistent Integration wit Inexact Gaussian Quadrature Approac We consider te one dimensional projection of te function u(x) = sin(2πx), x (0, 1) (26) onto a uniform mes to mimic a DG solution. We sould note tat we always use exact integration to calculate te L 2 projections and we only focus on different integration strategies for te convolution operator used in te post-processor. We assume tat we ave a periodic interval in order to simplify te application of te post-processor. For tis example, consistent integration is used, but te number of quadrature points used in te computation of te integrals is one less tan wat is required for exact integration. Te results are presented in Fig. 1. In tis set of plots, we can see tat te aliasing error is more sensitive at low orders and tat te iger-order coefficients in te approximation are important. Additionally, notice tat increasing N for P 2 -polynomial approximations does not lower te convergence rate as expected. 6.2 Input Mes Based Gaussian Quadrature Approac In tis section we examine te results of not using a consistent integration mes (i.e., ignoring knots of te B-spline kernel), but instead attempt to overcome te numerical crimes committed by integrating over te jump by increasing te number of quadrature points One-Dimensional DG For tis section, we again consider te L 2 -projection of te function u(x) = sin(2πx), x (0, 1) (27) to a uniform mes. We assume a periodic interval. Figures 2, 3 and 4 sow te point-wise errors wen post-processing on te DG mes for P 2, P 3 and P 4 polynomials, respectively. Considering P k polynomials, we need k + 1 Gauss points to exactly evaluate te inner products involved in post-processing of a DG solution on a consistent integration mes. Using te same number of points wen post-processing on te DG input mes, we observe tat te accuracy in terms of error is improved but tat oscillations still exist (see Fig. 2 Q = 3, Fig. 3 Q = 4 and Fig. 4 Q = 5).

12 458 J Sci Comput (2010) 45: Fig. 1 Point-wise errors on a logaritmic scale before post-processing (left), after post-processing on te consistent integration mes wit inexact quadrature (middle) and after post-processing wit exact quadrature (rigt) One-Dimensional DG Non-Uniform Mes In tis section, we examine te L 2 -projection of te function u(x) = sin(2πx), x (0, 1) (28) to smootly varying mes. Te smootly varying mes is defined by x = ξ + 1 sin ξ,sotat 2 te element sizes vary by at most 50% from eac oter. We use te caracteristic lengtbased post-processor implementation introduced in [14]. In te case of a non-uniform mes, Table 1 demonstrates tat te lowest quadrature order for piecewise quadratic polynomials is not sufficient for removing te error, unlike te uniform mes case. In te case of a nonuniform mes, a quadrature te uses at least nine Gauss points seem to be required. For piecewise cubic, te errors using te lowest value of te quadrature are even worse (Table 2). And, unless exact quadrature is implemented, te errors are always worse wen using ten elements.

13 J Sci Comput (2010) 45: Fig. 2 Point-wise errors on a logaritmic scale wen post-processing on te DG input mes using P 2 polynomials. Top left: errors before post-processing. Bottom rigt: errors after post-processing on te consistent integration mes using exact quadrature. Q is te number of quadrature points Two-Dimensional DG In tis case, we consider te L 2 -projection of te function u(x,y) = sin(2π(x + y)), x (0, 1), y (0, 1) (29) to a uniform mes. Again we assume a periodic domain. Additionally, we see in Tables 3 and 4 tat altoug te convergence rates wit te post-processor are not always improved, te errors are always lower tan for te input DG solution Two-Dimensional DG Constant Coefficient Linear Advection Equation For tis example, we consider solutions of te equation, u t + u x + u y = 0, (x,y) (0, 2π) (0, 2π), T = 12.5 (30)

14 460 J Sci Comput (2010) 45: Fig. 3 Point-wise errors on a logaritmic scale using P 3 polynomials Fig. 4 Point-wise errors on a logaritmic scale using P 4 polynomials wit initial condition u(0,x,y) = sin(x + y). Observe in Table 5 tat wen we use two Gauss points for linear polynomials, we immediately obtain te desired convergence rate. Additionally te errors are sligtly better tan wat is observed for te original DG solution. For te quadratic polynomial approximation as presented in Table 6 we also immediately improve bot te errors and te convergence rate using only tree Gauss points.

15 J Sci Comput (2010) 45: Table 1 Errors for one-dimensional DG using P 2 polynomials for te non-uniform mes. Before postprocessing, after post-processing on te DG mes were Q is te number of quadrature points and finally after post-processing on te consistent integration mes. CI stands for consistent integration mes P 2 Mes L 2 error Order L error Order L 2 error Order L error Order Before post-processing After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 9 After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 21 After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 45 After post-processing, CI E E E E E E E E E E E E E E E E Two-Dimensional DG Variable Coefficient Equation In tis example, we consider solutions to a two-dimensional variable coefficient equation, u t + (au) x + (au) y = f(x,y,t), (x,y) (0, 2π) (0, 2π), T = 12.5 (31) wit te variable coefficient function a(x,y) = 2 + sin(x + y). We set te forcing function so tat te solution is u(x,y,t) = sin(x + y 2t). We observe in Tables 7 and 8 tat te convergence rate is not always optimal, but tat indeed, wit a minimum number of Gauss points we improve te errors. 6.3 Input Mes Based Midpoint Quadrature Approac In tis section te effect of using midpoint integration wile approximating te convolution operator is examined for te 1D and 2D cases One-Dimensional DG using midpoint quadrature In tis example we again consider te case of projecting sin(2πx), x (0, 1)

16 462 J Sci Comput (2010) 45: Table 2 Errors for one-dimensional DG using P 3 polynomials for te non-uniform mes. Before postprocessing, after post-processing on te DG mes were Q is te number of quadrature points and finally after post-processing on te consistent integration mes. CI stands for consistent integration mes P 3 Mes L 2 error Order L error Order L 2 error Order L error Order Before post-processing After post-processing, Q = E E E E E E E E E E E E E E E E Before post-processing, Q = 7 After post-processing, Q = E E E E E E E E E E E E E E E Before post-processing, Q = 31 After post-processing, CI E E E E E E E E E E E E E E E E Table 3 Errors for two-dimensional DG using P 2 polynomials. Before post-processing, after post-processing on te DG mes were Q is te number of quadrature points and finally after post-processing on te consistent integration mes. CI stands for consistent integration mes P 2 Mes L 2 error Order L error Order L 2 error Order L error Order Before post-processing After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 6 After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 12 After post-processing, CI E E E E E E E E E E E E E E E E

17 J Sci Comput (2010) 45: Table 4 Errors for two-dimensional DG using P 3 polynomials. Before post-processing, after post-processing on te DG mes were Q is te number of quadrature points and finally after post-processing on te consistent integration mes. CI stands for consistent integration mes P 3 Mes L 2 error Order L error Order L 2 error Order L error Order Before post-processing After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 7 After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 13 After post-processing, CI E E E E E E E E E E E E E E E E Table 5 Errors for one-dimensional DG using P 1 polynomials for te linear advection equation. Before postprocessing, after post-processing on te DG mes were Q is te number of quadrature points and finally after post-processing on te consistent integration mes. CI stands for consistent integration mes P 1 Mes L 2 error Order L error Order L 2 error Order L error Order Before post-processing After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 5 After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 11 After post-processing, CI E E E E E E E E E E E E E E E E

18 464 J Sci Comput (2010) 45: Table 6 Errors for two-dimensional DG using P 2 polynomials for te linear advection equation. Before postprocessing, after post-processing on te DG mes were Q is te number of quadrature points and finally after post-processing on te consistent integration mes. CI stands for consistent integration mes P 2 Mes L 2 error Order L error Order L 2 error Order L error Order Before post-processing After post-processing, Q = E E E E E E E E E E E E E E E E Before post-processing, Q = 6 After post-processing, CI E E E E E E E E E E E E E E E E Table 7 Errors for two-dimensional DG using P 1 polynomials for te variable coefficient advection equation. Before post-processing, after post-processing on te DG mes were Q is te number of quadrature points and finally after post-processing on te consistent integration mes. CI stands for consistent integration mes P 1 Mes L 2 error Order L error Order L 2 error Order L error Order Before post-processing After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 5 After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 11 After post-processing, CI E E E E E E E E E E E E E E E E onto a piecewise polynomial space. Te results are displayed in Fig. 5. Te plots sow tat it takes many applications of te midpoint rule to get better errors for te post-processed solution tan te initial projection, and tat te midpoint rule is effective wen we use

19 J Sci Comput (2010) 45: Table 8 Errors for two-dimensional DG using P 2 polynomials for te variable coefficient advection equation. Before post-processing, after post-processing on te DG mes were Q is te number of quadrature points and finally after post-processing on te consistent integration mes. CI stands for consistent integration mes P 2 Mes L 2 error Order L error Order L 2 error Order L error Order Before post-processing After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 6 After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 12 After post-processing, Q = E E E E E E E E E E E E E E E E After post-processing, Q = 18 After post-processing, CI E E E E E E E E E E E E E E E E points or greater. However, te point-wise errors are smooter tan te errors for te piecewise polynomial projection. Tis is because te breakpoints for te midpoint rule align wit te element boundaries on our mes. Additionally, in Fig. 6 we can see tat te lines level off to te same as tat for consistent integration error, wic is te best possible scenario Two-Dimensional DG using midpoint quadrature In tis section we again consider te two dimensional projection problem given by (29). It is observed tat in te case of 2D midpoint rule te convergence rate in te L 2 -norm is linear, as is sown in te sample plot in Fig. 7. We observe tis linear trend as we double te number of evaluation points. For tis particular example and based upon te slope of our convergence diagram, we would need approximately 2 13 evaluation points to get an error level similar to te DG solution. Te errors for te iger degree polynomials are not sown as tey do not provide any new information given te computational time required to compute tem.

20 466 J Sci Comput (2010) 45: Fig. 5 Point-wise errors on a logaritmic scale wen using midpoint integration for post-processing wit different number of evaluation points. Top left: before post-processing. Bottom rigt: after post-processing on te consistent integration mes Fig. 6 Convergence of te L 2 errors wen using midpoint integration for post-processing 6.4 Two-Dimensional Vector Field As it is mentioned in [20, 22], smootness-increasing, accuracy-conserving filtering can be applied to discontinuous Galerkin vector fields to enance streamline integration. In tis sec-

21 J Sci Comput (2010) 45: Fig. 7 Convergence of te L 2 errors wen using midpoint integration for post-processing tion we examine te impact of input mes based filtering of a 2D vector field on streamline calculations for visualization purposes. A two dimensional vector field was created from [ ] u(r, θ) = v(r,θ) ] cos(20θ)cos(θ) r sin(θ). (32) 1 sin(20θ)sin(θ) r cos(θ) 2 [ 1 2 Tis as streamlines wic are oscillating closed circuits. In Fig. 8 we present a sample streamline of tis vector field by projecting te function above over a uniform mes on te interval [ 1, 1] [ 1, 1] wit a starting location of (0.0, 0.3). Te field approximations are linear in bot te x- and y-directions. Streamlines were calculated using tree different time integration scemes, Euler Forward, 2nd-order Runge-Kutta (RK-2) and 4torder Runge-Kutta (RK-4), wit tree different time steps dt = 0.1, 0.01, Te true solution streamlines (denoted as solid black line in all te images) are calculated by performing RK-4 on te analytical function. Tese results corroborate tat input mes based integration wit sufficient quadrature provides sufficient post-processing benefit in terms of smootness and accuracy to be of use in data processing and visualization. 7 Summary and Conclusions Tis paper presents a study of te impact of numerical quadrature approximations used for evaluating te convolution operator in smootness-increasing accuracy-conserving (SIAC) filters. We provide bot teoretical estimates as well as empirical results wic demonstrate te efficacy of te post-processing approac wen different levels and types of quadrature approximation are used. We first examined te case wen consistent integration wit inexact Gaussian quadrature (under-integration) is used. Because consistent integration requires solving te callenging geometric problem of find all te places in wic regularity is decreased and generating a super-mes based on tis data, we considered wat appens wen only te original DG mes is used as te underlying support mes for integration. Under

22 468 J Sci Comput (2010) 45: Fig. 8 (Color online) Streamline integration example based upon vector field mentioned in (32). Solid black streamlines denote true solution; blue streamlines were created based upon integration on an L 2 projected field; red streamlines were created based upon integration on a filtered field using consistent integration approac and dased black streamlines were created based upon integration on a filtered field using te input mes based approac tis scenario, we examined te use of Gaussian quadrature and of midpoint quadrature. Tis coice igligted te differences between polynomial-based ig-order and adaptive loworder quadrature implementations. Tere are several points tat can be draw from our results and discussions: Te major uncontrollable cost in te post-processing algoritm is finding te consistent integration mes wen given an arbitrary tessellation. In te case of uniform meses, many tings simplify to drastically cut down te cost; owever, uniform meses are not often used in general engineering practice. Because te post-processor consists of integrating a B-spline kernel against a DG solution, tere are certain tings we can state about te integrand being integrated. As te DG element discontinuities and te B-spline knot lines cannot overlap, we know tat in te worst case te integrand contains a reduction in regularity due to te product of te DG

23 J Sci Comput (2010) 45: discontinuity wit te polynomial on a B-spline knot-segment. Altoug Gauss quadrature over suc a region is not exact, it can noneteless be very effective. If te cost of finding te consistent mes is proibitive, post-processing wit input mes based post-processing provides in many cases benefits comparable wit consistent integration. Te error introduced can be controlled by increasing te number of quadrature points. Alternatives to Gaussian quadrature can be used for evaluating te convolution integrals; owever, a large number of samples are needed to obtain comparable results. Wen examined in te ligt of te our focus application area visualization of DG solutions input mes based post-processing appears to provide a convenient means of obtaining smoot solutions wit controllable accuracy. We empasis again tat our study is primarily for engineering circumstances wen te trade-offs between time, resources and accuracy are important. Altoug te case against committing suc numerical crimes is well-known, te repercussions ave not been well documented for te use of tis filter as a visualization tool. It is concerning tis specific crime to wic we ave attempted to provide bot teoretical and empirical insigt. Acknowledgements Te first and tird autors are sponsored in part by te Air Force Office of Scientific Researc (AFOSR), Computational Matematics Program (Program Manager: Dr. Fariba Faroo), under grant number FA Te second autor is sponsored by te Air Force Office of Scientific Researc, Air Force Material Command, USAF, under grant number FA Te U.S. Government is autorized to reproduce and distribute reprints for Governmental purposes notwitstanding any copyrigt notation tereon. Open Access Tis article is distributed under te terms of te Creative Commons Attribution Noncommercial License wic permits any noncommercial use, distribution, and reproduction in any medium, provided te original autor(s) and source are credited. References 1. Bramble, J., Scatz, A.: Higer order local accuracy by averaging in te finite element metod. Mat. Comput. 31, (1977) 2. Cockburn, B.: Discontinuous Galerkin metods for convection-dominated problems. In: Hig-Order Metods for Computational Pysics. Lecture Notes in Computational Science and Engineering, vol. 9. Springer, Berlin (1999) 3. Cockburn, B., Hou, S., Su, C.-W.: Te Runge-Kutta local projection discontinuous Galerkin finite element metod for conservation laws IV: te multidimensional case. Mat. Comput. 54, (1990) 4. Cockburn, B., Karniadakis, G., Su, C.-W.: Discontinuous Galerkin Metods: Teory, Computation and Applications. Springer, Berlin (2000) 5. Cockburn, B., Lin, S.-Y., Su, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element metod for conservation laws III: one dimensional systems. J. Comput. Pys. 84, (1989) 6. Cockburn, B., Luskin, M., Su, C.-W., Süli, E.: Post-processing of Galerkin metods for yperbolic problems. In: Proceedings of te International Symposium on Discontinuous Galerkin Metods. Springer, Berlin (1999) 7. Cockburn, B., Luskin, M., Su, C.-W., Suli, E.: Enanced accuracy by post-processing for finite element metods for yperbolic equations. Mat. Comput. 72, (2003) 8. Cockburn, B., Ryan, J.: Local derivative post-processing for discontinuous Galerkin metods. J. Comput. Pys. 28, (2009) 9. Cockburn, B., Su, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element metod for conservation laws II: general framework. Mat. Comput. 52, (1989) 10. Cockburn, B., Su, C.-W.: Te Runge-Kutta local projection P 1 -discontinuous-galerkin finite element metod for scalar conservation laws. Mat. Model. Numer. Anal. (M²AN) 25, (1991) 11. Cockburn, B., Su, C.-W.: Te Runge-Kutta discontinuous Galerkin metod for conservation laws V: multidimensional systems. J. Comput. Pys. 141, (1998)