EXTENSION OF A POSTPROCESSING TECHNIQUE FOR THE DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS WITH APPLICATION TO AN AEROACOUSTIC PROBLEM

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1 SIAM J. SCI. COMPUT. Vol. 26, No. 3, pp c 2005 Society for Industrial and Applied Matematics ETENSION OF A POSTPROCESSING TECHNIQUE FOR THE DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS WITH APPLICATION TO AN AEROACOUSTIC PROBLEM JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Abstract. In tis paper we furter explore a local postprocessing tecnique, originally developed by Bramble and Scatz [Mat. Comp., 31 (1977), pp ] using continuous finite element metods for elliptic problems and later by Cockburn et al. [Mat. Comp., 72 (2003), pp ] using discontinuous Galerkin metods for yperbolic equations. We investigate te tecnique in te context of superconvergence of te derivatives of te numerical solution, two space dimensions for bot tensor product local basis and te usual kt degree polynomials basis, multidomain problems wit different mes sizes, variable coefficient linear problems including tose wit discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We demonstrate troug extensive numerical examples tat te tecnique is very effective in all tese situations in enancing te accuracy of te discontinuous Galerkin solutions. Key words. accuracy enancement, postprocessing, discontinuous Galerkin metod, yperbolic equations AMS subject classification. 65M60 DOI /S Introduction. In tis paper we furter explore a local postprocessing tecnique, originally developed by Bramble and Scatz [3] in te context of continuous finite element metods for elliptic problems, and later by Cockburn et al. [10] in te context of te discontinuous Galerkin metods for yperbolic equations. Two key ingredients of tis postprocessing tecnique are a negative norm estimate for te numerical solution, wic sould be of iger order tan te L 2 error estimate, and a local translation invariance of te mes. Te main advantages of tis tecnique, compared to oter postprocessing tecniques, include its local feature, ence its efficiency and its easiness in te parallel implementation framework, and its effectiveness in almost doubling te order of accuracy rater tan increasing te order of accuracy by one or two. We investigate te tecnique in te context of superconvergence of te derivatives of te numerical solution, two space dimensions for bot tensor product local basis and te usual kt degree polynomials basis, multidomain problems wit different mes sizes, variable coefficient linear problems including tose wit discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. In [10] (see also [9]), Cockburn et al. establised a framework to prove te negative norm estimates for discontinuous Galerkin metods applied to linear yperbolic Received by te editors Marc 8, 2003; accepted for publication (in revised form) April 29, 2004; publised electronically January 12, ttp:// Division of Applied Matematics, Brown University, Providence, RI (ryanjk@ornl.gov, su@dam.brown.edu). Te researc of te first autor was supported by NASA Langley grant NGT Current address for first autor: Oak Ridge National Lab, P.O. Box 2008, MS6367, Oak Ridge, TN Te researc of te second autor was supported by ARO grant DAAD , NSF grant DMS , and NASA Langley grant NCC Computational Modeling and Simulations Branc, NASA Langley Researc Center, Hampton, VA (.l.atkins@larc.nasa.gov). 821

2 822 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS equations wit locally smoot solutions. Using suc negative norm error estimates, and a locally translation invariant property of te mes and te numerical metod, a igly efficient and local postprocessor, originally due to Bramble and Scatz [3] in te context of continuous finite element metods for elliptic problems, is suggested to enance te accuracy of te discontinuous Galerkin metod applied to yperbolic equations. Instead of getting te usual (k + 1)t order accuracy using P k (piecewise polynomials of degree up to k) elements, one can get (2k + 1)t order accuracy in te L 2 norm after postprocessing. Numerical experiments were carried out in [9, 10] confirming tis teoretical prediction. Tis approac as great potential in making te discontinuous Galerkin metod more effective in many applications, suc as for aeroacoustic simulations, were usually linearized Euler equations are solved for very long times, and for solving Maxwell equations in electromagnetism. Te requirement for te mes is only locally uniform, i.e., uniform in te support of te postprocessor, wic as a widt of k = (3k+1)/2 cells on eiter side for P k elements, were te symbol r denotes te smallest integer greater tan or equal to r. Tis is realistic for many applications including aerodynamic and aeroacoustic problems. In tis paper we include a preliminary result in aeroacoustic simulations. We would like to point out tat, altoug te postprocessed solutions are (2k + 1)t order accurate, te magnitudes of te error on te same mes are usually iger tan tose using a discontinuous Galerkin metod wit k elements. In tis paper we do not attempt to address te issue of optimal order k in te discontinuous Galerkin metod since tis is problem dependent. We would assume tat k is fixed and address te issue of ow muc te accuracy can be improved after postprocessing. We would also like to point out tat, even toug te postprocessed solutions are of iger order in accuracy tan te original solutions, tis does not mean tat on a fixed mes, te postprocessed solution is always more accurate tan te original solution. On a very coarse mes, it may appen tat te postprocessed solution as actually a larger error tan te original solution. Tis is similar to te situation tat a iger order metod may ave larger error on a fixed mes tan a lower order metod, altoug asymptotically te errors decay muc faster for iger order metods. We would like to mention briefly oter postprocessing tecniques available in te literature. Te tecnique of Adjerid et al. [1] and Krivodonova and Flaerty [16] as no uniform mes assumption and works for te discontinuous Galerkin metod applied to yperbolic conservation laws wit linear fluxes. Based on a superconvergence of order 2k +1 at te Radau points of te outflow edge, a global accuracy of order 2k +1 can be obtained after interpolation. Te metod of Zienkiewicz and Zu [23, 24] as been sown to be robust and includes a variety of mes types for continuous finite element metods solving elliptic equations. Te recovered solution depends only on information from its cell neigbors, and an improvement of one order in accuracy can be observed and proved [21, 22] in many cases. For tis postprocessing tecnique, only computational results ave been obtained using te discontinuous Galerkin formulation for elliptic equations [5]. A comparison of all tese different postprocessing tecniques constitutes future work and will not be included in tis paper Te discontinuous Galerkin metod. We will only briefly review te discontinuous Galerkin metod for solving yperbolic conservation laws. For more details, we refer te reader to te series of papers of Cockburn et al. [12, 11, 8, 7, 13],

3 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD 823 te lecture notes [6], and te review paper [14]. Consider te conservation law (1.1) u t + f(u) x =0. In order to approximate te solution to tis equation, we coose a basis for te approximation space, V, consisting of piecewise polynomials of degree less tan or equal to k, were k + 1 is te order of accuracy of te approximation. For te purposes of our studies, we ave cosen te basis to be te monomials, ξ i = x xi x i, on te interval I i =(x i xi 2,x i + xi 2 ). Te approximation is based on a weak formulation of (1.1): u t vdx = f(u)v x dx f(u i+1/2 )v i+1/2 + f(u i 1/2 )v i 1/2 I i I i for all smoot functions v. Te sceme is ten given by te following: Find u (x) V suc tat (1.2) (u ) t vdx = I i f(u )v x dx ˆf i+1/2 v i+1/2 + ˆf i 1/2 v + i 1/2 I i for all test functions v V. Te numerical flux ˆf i+1/2 = ˆf(u i+1/2,u+ i+1/2 )iscosen so tat it is an upwind monotone flux; namely, it is a nondecreasing function of te first argument u and a nonincreasing function of te second argument u +, and v is taken from inside te cell. Te approximation is a linear combination of te basis functions. Te time discretization in (1.2) is given by te ig order TVD Runge Kutta metods in [19], and te tird order version is used in tis paper unless oterwise indicated. During accuracy tests, te time step is suitably reduced to guarantee tat time accuracy is comparable wit spatial accuracy Te postprocessor. In order to improve te accuracy of tis approximation, we are implementing a postprocessor. Te Galerkin metod allows us to obtain negative norm error estimates, wic are of iger order tan te error estimates for te usual L 2 norm; see [10] for te discontinuous Galerkin metod applied to linear yperbolic equations. Te negative norm error estimates give us information on te oscillatory nature of te error. Te postprocessor extracts tis information and works to filter out oscillations in te error and to enance te accuracy in te usual L 2 norm, up to te order of te error estimates in te negative norm. If we work under a local uniform mes assumption and ence a local translation invariant property of te mes and numerical metod, te postprocessor can be made local. Te postprocessor tat we use was introduced by Cockburn et al. [9, 10] for solving yperbolic PDEs using te discontinuous Galerkin metod. Te framework was initially establised by Bramble and Scatz [3] and Mock and Lax [18]. In [9, 10] te autors considered te postprocessing of te discontinuous Galerkin approximation to time dependent linear yperbolic systems. In tis case, te autors sow tat te postprocessor improves te accuracy from order k + 1 to order 2k + 1 for linear yperbolic systems solved over a locally uniform mes, namely, = x i for all i in te support of te postprocessor. Te postprocessor consists of a convolution kernel applied to te approximation only once, at te final time, and is independent of te PDE under consideration as long as te necessary negative norm error estimate can be proven. Te convolution kernel essentially works as a smooting operator by exploiting te oscillatory nature of te error to enance te quality of te approximation.

4 824 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Te convolution kernel as tree main properties. First, it as compact support. Tis makes postprocessing computationally advantageous. Second, te kernel can reproduce polynomials of degree 2k+1 by convolution. Tis ensures tat te accuracy is not destroyed by te postprocessor. Tird, te kernel is a linear combination of B-splines, wic allows us to express derivatives of te convolution kernel in terms of simple difference quotients. Wit te convolution kernel, te postprocessed solution is given by ) u (y) dy, u (x) = 1 ( y x (1.3) K 2(k+1),k+1 were K 2(k+1),k+1 is a linear combination of B-splines. It can be sown tat te postprocessed solution u (x) is a piecewise polynomial of degree 2k + 1 for te discontinuous Galerkin solution u (x) using P k elements. Te B-splines are obtained by convolving te caracteristic function ( 1, x 1 χ(x) = 2, 1 ), 2 0, oterwise wit itself k times. Te form of K for te discontinuous Galerkin approximation using P k elements in one dimension is ten k (1.4) K 2(k+1),k+1 (x) = c 2(k+1),k+1 γ ψ (k+1) (x γ), γ= k were ψ (1) = χ and ψ (n) = ψ (n 1) χ for n 2. Te postprocessor uses information from its cell neigbors, te kernel is scaled by in (1.3), and c 2(k+1),k+1 γ are real constants. As an example, suppose u (x) is a (second order accurate) discontinuous Galerkin approximation to te solution of some PDE at te final time T. Te postprocessed solution would be of te form (1.3) wit K 4,2 ( y x )=c4,2 1 ψ(2) ( y x +1)+ c 4,2 0 ψ(2) ( y x )+c4,2 1 ψ(2) ( y x 1). Using te definition above, ψ(2) is found to be { ψ (2) 1 x, x 1, (x) = 0, oterwise. Te coefficients of te kernel are found by implementing te second property of te convolution kernel. Tat is tat te kernel must reproduce polynomials of degree 3 by convolution. Te problem is ten to find c 4,2 γ for γ = 1, 0, 1 suc tat (1.3) leaves invariant u (x) =u (x) wen u (x) =1,x,x 2 globally in all cells (it will ten also old automatically for u (x) =x 3 ). Tis gives te system x +1 x x 1 c 1 1 x 2 +2x + 7 x x 2 2x + 7 c 0 = x. c 1 x Solving for te coefficients, we find te form of te convolution kernel to be ( ) y x K 4,2 = 1 ( ) y x 12 ψ(2) ( ) y x 6 ψ(2) 1 ( ) y x 12 ψ(2) 1 for te piecewise linear polynomial discontinuous Galerkin approximation.

5 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD 825 It is important to note ere tat we are considering a uniform mes wic gives te kernel a particularly simple form. Also, note tat te kernel uses information only from its nearest neigbors. Te postprocessed solution, u (x), wic is a piecewise polynomial of degree 2k + 1, can be evaluated exactly. We know tat u (x) is defined by (1.3), were u (x) = k ( x xi )l for x I i and K 2(k+1),k+1 (x) is given by (1.4). Terefore, l=0 u(l) i for x I i,weave (1.5) u (x) = 1 = 1 = 1 = k j= k k j= k k ) u (y) dy ( y x K 2(k+1),k+1 ( y x k j= k l=0 I i+j K 2(k+1),k+1 I i+j K 2(k+1),k+1 u (l) i+j C(j, l, k, x), ) u (y) dy ( ) k y x l=0 u (l) i+j ( ) l y xi+j dy were k = (3k+1)/2, and in te second equality we ave used te compact support property of K to write u (x) as a finite local sum. Eac C(j, l, k, x) is a polynomial of degree 2k + 1 for x I i and is given by C(j, l, k, x) = 1 k γ= k ( )( y x y c 2(k+1),k+1 γ ψ (k+1) xi+j γ I i+j ) l dy. Clearly, tese polynomials C(j, l, k, x) can be computed one time and stored. In particular, wen we measure errors in tis paper we use a six point Gaussian quadrature in eac cell I i. For x equaling any of tese six Gaussian points, C(j, l, k, x) are just constants wic can be computed once and stored. Te postprocessing is ten implemented in a simple manner by doing small matrix vector multiplications of te in neigboring cells I i+j. Tis is done only once, at te end of a time dependent calculation; tus it involves negligible additional computational cost. We discuss superconvergence of te derivatives of te numerical solution in section 2 and generalize te postprocessing to two dimensions in section 3. Section 4 discusses te postprocessor for multidomains wit different mes sizes. In section 5 we discuss te postprocessor for linear equations wit variable coefficients, including discontinuous coefficients. Section 6 covers linearized Euler equations wit an example in aeroacoustic applications. Concluding remarks are given in section 7. prestored C(j, l, k, x) and te coefficients of te numerical solution u (l) i+j 2. Superconvergence of te derivatives of te postprocessed solution. Since te postprocessed solution u (x) defined by (1.3) is a piecewise polynomial of degree 2k + 1, it makes sense to ask ow well its derivatives approximate tat of te exact solution. For tis purpose we compute te discontinuous Galerkin solution to te advection equation u t + u x = 0 wit an initial condition u(x, 0) = sin(x). Weusea2π-periodic boundary condition on a uniform mes and calculate up to a final time T =12.5, wic is approximately two periods in time, as in [10]. In Table 2.1 we list te L 2 and L errors of te first and second derivatives before

6 826 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Table 2.1 Errors in te first and second derivatives for te discontinuous Galerkin metod applied to te one-dimensional linear advection equation, before and after postprocessing. Errors in first derivative E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Errors in second derivative 10 N/A N/A 3.39E E N/A N/A 5.68E E N/A N/A 1.16E E N/A N/A 2.72E E N/A N/A 6.68E E E E E E E E E E E E E E E E E E E E E E and after postprocessing. It can be seen tat, before postprocessing, te error for te first derivative is of order k and tat for te second derivative is of te order k 1, as expected. Normally, we would expect one order lower in accuracy for te postprocessed derivative compared to tat of te postprocessed solution. However, after postprocessing, te order of accuracy for te first derivative appears to be 2k +1, wic is te same as te order of accuracy for te postprocessed solution, and for te second derivative it seems to be 2k. Tis is actually one order iger tan wat we would expect. It sould be noted tat te postprocessed solution u (x) is a piecewise polynomial of degree 2k + 1; ence it as te potential to yield an order of accuracy of 2k + 2 for te postprocessed solution if te negative norm error estimates allow. We do see tis 2k + 2 order of accuracy for te discontinuous Galerkin metods applied to some PDEs, suc as te time dependent biarmonic equations and a PDE involving fift spatial derivatives in [20]. Tus te (2k + 2 d)t order accuracy for te postprocessed dt derivative (we ave sown te case for d = 1 and 2) is not completely surprising. For a representative case of te approximations, we plot

7 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD : Before Post-Processing First Derivative N=10 N= : After Post-Processing N= N= : Before Post-Processing Second Derivative N= : After Post-Processing N=160 N= N=160 Fig Pointwise errors for te first derivative (top) and te second derivative (bottom) in logaritmic scale wen te discontinuous Galerkin metod is used to solve te linear advection equation. Left: Errors before postprocessing. Rigt: Errors after postprocessing. te errors for te first and second derivatives in logaritmic scale bot before and after postprocessing in Figure 2.1. We can see tat te postprocessed errors are muc less oscillatory. 3. Postprocessing in two dimensions. For two dimensions te convolution kernel is a tensor product of te one-dimensional kernels (3.1) K 2(k+1),k+1 (x, y)= k k γ x= k γ y= k c 2(k+1),k+1 γ x c 2(k+1),k+1 γ y ψ (k+1) (x γ x )ψ (k+1) (y γ y ). Again, te kernel will be suitably scaled by, were we assume, for simplicity of notation, tat te uniform mes sizes are te same in x and in y, and is applied bot to te case of piecewise tensor product two-dimensional Q k polynomials and to te usual piecewise P k polynomials. Te postprocessing is expected to work in bot cases provided te negative error estimates can be obtained and te mes is translation invariant. In tis section we present numerical examples to confirm tis.

8 828 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Table 3.1 Errors for te two-dimensional Q k polynomial approximations of te discontinuous Galerkin metod applied to te linear advection equation (3.2) before and after postprocessing E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E (3.2) 3.1. Q k polynomials. We use te simple linear advection PDE u t + u x + u y =0, u(x, y, 0) = sin(x + y) wit 2π-periodic boundary conditions in bot directions and compute te solution up to T =12.5, wic is about two periods in time. We first perform numerical experiments on te two-dimensional linear advection equation (3.2) for te piecewise tensor product Q k polynomials. Notice tat, altoug Q k polynomials are more natural wit te tensor product postprocessing kernel (3.1), tey ave more degrees of freedom for te same order of accuracy and ence may not be te best coice for discontinuous Galerkin metods. Te L 2 and L errors before and after postprocessing are listed in Table 3.1 for k =1, 2, and 3. We can see tat te postprocessing acieved te designed order of accuracy 2k P k polynomials. Next we perform numerical experiments on te twodimensional linear advection equation (3.2) for te usual piecewise P k polynomials. Tis is te usual coice for te basis of te discontinuous Galerkin metods as tey acieve te same k + 1 order of accuracy wit far fewer degrees of freedom tan piecewise Q k polynomials. It migt seem tat tis is not a good fit wit te tensor product postprocessing kernel (3.1); owever, te L 2 and L errors before and after te postprocessing, sown in Table 3.2 for k =1, 2, and 3, again sow tat te postprocessing acieved te designed order of accuracy 2k + 1. It sould be noted toug, as expected, for te same mes, te errors for te piecewise P k approximations are larger tan tose for te piecewise Q k approximations, bot before and after te postprocessing Systems. To test te postprocessor furter, we consider a two-dimensional linear system (3.3) ( u v ) t + ( )( u v ) x + ( )( u v ) y = ( 0 0 )

9 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD 829 Table 3.2 Errors for te two-dimensional P k polynomial approximations of te discontinuous Galerkin metod applied to te linear advection equation (3.2) before and after postprocessing E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E Table 3.3 Errors for u(x, y, t) using te P k polynomial approximation of te discontinuous Galerkin metod applied to te two-dimensional linear system (3.3) before and after postprocessing E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E wit initial conditions u(x, y, 0) = 1 2 (sin(x + y) cos(x + y)), 2 v(x, y, 0) = (( 2 1) sin(x + y)+(1+ 2) cos(x + y)) and 2π-periodic boundary conditions in bot directions. Tis equation is te traditional second order wave equation written as a first order linear system. Te solutions are smoot wit te given initial and boundary conditions. In Table 3.3 we list te

10 830 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS errors at T =12.5 for u(x, y, t) using te P k polynomial basis. Te errors for v(x, y, t) are similar and ence are not listed. Again we can clearly see te (k +1)t order accuracy before te postprocessing and te (2k + 1)t order accuracy after postprocessing. It is evident tat te postprocessor works well for two-dimensional linear yperbolic systems using te P k polynomial basis. 4. Domains wit different mes sizes One dimension. Te motivation for looking at domains wit different mes sizes is to see te effect of te postprocessor on more complicated mes structures. To begin exploring tis case, we consider solving a yperbolic equation over a domain wit two different mes sizes; see Figure 4.1. Te domain is divided up into two parts, were one side of te domain as a finer mes structure tan te oter. Te problem is still solved over te entire domain as one and not over eac individual mes region. As an example, consider te equation u t + u x = 0, wit an initial condition u(x, 0) = sin(3x) anda2π-periodic boundary condition. For tis case, approximately two tirds of te total number of cells are contained in te interval (0,π). Te rest of te cells are in te interval (π, 2π), giving te rigt alf of te interval a coarser mes structure. Te approximation is calculated at te final time T = 12.5, wic is approximately two periods in time. Te mes is fixed over te time evolution. We listed te errors 0.6 away from te interface in bot regions before and after postprocessing in Tables For a representative case of approximations, we also plot te errors in logaritmic scale bot before and after postprocessing in Figure 4.2. Te number 0.6 ere is arbitrary; we expect to see te same asymptotic beavior wit any fixed number, but te actual mes tresold for te improvement of accuracy by te postprocessor depends on tis number. te error is igly oscillatory, and te coarser mes region on te rigt as a larger error tan te more refined mes region on te left (see Figure 4.2). At te interface of te two meses te error is not larger, wic is wy Domain 1: N 1 =[2N/3] Domain 2: N 2 = N-N Time x Fig Mes structure for solving te linear advection equation. Te left alf of te domain contains approximately two tirds of te elements; te rigt contains approximately one tird of te elements.

11 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD 831 Table 4.1 One-dimensional discontinuous Galerkin approximations to te advection equation over a domain wit two different mes sizes as sown in Figure 4.1. Errors calculated in te fine mes (left) region outside of a radius r =0.6 from te interface of two different meses E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E E E E E P E E E E E E E E E E E E E E E E te discontinuous Galerkin metod is a good metod for arbitrary triangulations., tere is smooting of te oscillations, and not only does te error even out overall, te error in eac region improves away from te interface. Tat is, te error on te coarser mes catces up to tat of te fine mes and te error improves overall. Te errors calculated in eac region support tese findings. Tat is, before postprocessing, we obtain (k + 1)t order accuracy in eac region. we obtain te (2k + 1)t order accuracy. Also note tat before postprocessing te magnitude of te error on te coarse mes and te fine mes are different. te magnitude of te errors are te same. Tis is exibited in Tables Notice tat te postprocessed solution as a larger error near te interface of te different meses. Tis is expected as we ave used te same postprocessor for uniform meses near te interface, wic is not designed to andle nonuniform meses. For te same reason, because te support of te postprocessor is wider for larger k, we can see tat te expected improvement in accuracy is observed only for more refined meses, wen te support of te postprocessor for all cells outside of a radius r =0.6 from te interface is completely contained in te region of uniform mes. A different, one-sided postprocessor (e.g., [4, 15]) migt be able to andle tis situation. We leave tis case for future work Two dimensions. We now look at te case of multidomains in two dimensions using a P k polynomial approximation. Again we use te same divisions for bot

12 832 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Table 4.2 One-dimensional discontinuous Galerkin approximations to te advection equation over a domain wit two different mes sizes as sown in Figure 4.1. Errors calculated in te coarse mes (rigt) region outside of a radius r =0.6 from te interface of two different meses E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E E E E E P E E E E E E E E E E E E E E E E : Before Post-Processing : After Post-Processing N= N= Fig Pointwise errors in logaritmic scale wen te discontinuous Galerkin metod is used to solve te linear advection equation over a domain wit two different mes sizes as sown in Figure 4.1. Left: Errors before postprocessing. Rigt: Errors after postprocessing. In eac grap te rigt side of te domain as a coarser mes structure. Te graps are scaled te same to empasize te improvement in error by te postprocessor.

13 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD Y Fig Two-dimensional multidomain: Mes for te two-dimensional advection equation. Table 4.3 Two-dimensional discontinuous Galerkin approximations to te advection equation over a domain wit four different mes sizes as sown in Figure 4.3. Errors calculated in te finest mes (te lower left region) outside of a radius r =0.6 from te interfaces of different meses E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E E E E E x and y as te one-dimensional case. Tis leads us to a domain wit four different mes structures, as sown in Figure 4.3. We calculate te errors for te linear advection equation u t + u x + u y = 0 wit initial condition u(x, y, 0) = sin(3(x + y)) and a 2π-periodic boundary condition in bot directions. Te errors are computed for final time T =12.5, about two periods later. In tis case, we use a sevent order Runge Kutta metod to make te time evolution faster. We list te errors away from te interfaces in two of te four regions of different mes sizes, in Tables Te errors in te oter two regions follow te same pattern and ence are not listed. As in te one-dimensional case, before postprocessing te magnitude of te error on regions wit different mes sizes is different, and tey are all on te order of k +1.

14 834 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Table 4.4 Two-dimensional discontinuous Galerkin approximations to te advection equation over a domain wit four different mes sizes as sown in Figure 4.3. Errors calculated in te coarsest mes (te top rigt region) outside of a radius r =0.6 from te interfaces of different meses E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E E E E E After te postprocessing, te magnitude of te errors in regions wit different mes sizes becomes smaller, te errors are closer to eac oter, and te order improves to 2k+1. As expected, te postprocessed solution as a larger error near te interfaces of different meses, wic are not included in te calculation of errors in Tables Hyperbolic equations wit variable coefficients. In tis section we consider linear yperbolic equations wit variable coefficients. We study problems in one and two dimensions wit smoot or nonsmoot coefficients and demonstrate tat te postprocessor works well in all tese cases. For two dimensions we consider only te P k polynomial approximations One dimension, continuous coefficients. We consider te following variable coefficient linear conservation law wit a forcing function (5.1) u(x, t) t +(a(x)u(x, t)) x = f(x, t). We first take a smoot coefficient a(x) = 2 + sin(x), wit initial condition u(x, 0) = sin(x), 2π-periodic boundary condition, and a suitable forcing function f(x, t) so tat te exact solution is u(x, t) = sin(x t). We calculate te errors over te entire domain at T = 12.5, wic is approximately two periods in time, and list tem in Table 5.1. Te errors for te case are also plotted in Figure 5.1 in logaritmic scale bot before and after postprocessing. We can clearly see tat te postprocessing effectively removes te oscillations in te errors and improves te order of accuracy from k + 1 to 2k + 1, similar to te constant coefficient cases. Next, we consider a continuous but not differentiable coefficient, given by { 2 + sin(x), x [0,π], (5.2) a(x) = 2 sin(x), x (π, 2π),

15 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD 835 Table 5.1 Errors for te discontinuous Galerkin approximation to te variable coefficient equation (5.1) wit a(x) = 2 + sin(x) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E : Before Post-Processing : After Post-Processing N= N=10 Fig Pointwise errors in logaritmic scale wen te discontinuous Galerkin metod is used to solve te variable coefficient equation (5.1) wit a(x) = 2 + sin(x). Left: Errors before postprocessing. Rigt: Errors after postprocessing. extended periodically, wit te same initial condition u(x, 0) = sin(x), 2π-periodic boundary condition, and a suitable forcing function f(x, t) so tat te exact solution is still u(x, t) = sin(x t). We again calculate te errors over te entire domain at T =12.5, approximately two periods in time, and list tem in Table 5.2. Te errors for te and P 3 cases are also plotted in Figure 5.2 in logaritmic scale bot before and after postprocessing. Notice tat, altoug te coefficient a(x) is not smoot, te singularity of its derivative is at a cell boundary and does not move wit time, and te forcing function is cosen so tat te exact solution is still smoot. For suc cases clearly te discontinuous Galerkin metod can keep te full designed order of

16 836 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Table 5.2 Errors for te discontinuous Galerkin approximation to te variable coefficient equation (5.1) wit a(x) =2+ sin(x) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E accuracy, bot before and after postprocessing, at least for te L 2 norm. It seems tat te order of accuracy of te postprocessed solution in te L norm for te case is four instead of five, one order lower tan expected. From Figure 5.2, top rigt, it can be seen tat te postprocessing removed most but not all te oscillations in te errors in te case, peraps due to te singularity in te coefficient and te forcing function of te PDE. We remark tat te proof of (2k + 1)t order of accuracy is only for te L 2 norm. However, it seems tat te postprocessing did remove all te oscillations in te errors in te P 3 case (Figure 5.2, bottom rigt) and te case (not sown) Two dimensions, smoot coefficients. Next we consider two dimensional linear conservation laws wit smoot coefficients and a forcing function: (5.3) u(x, y, t) t +(a(x, y)u(x, y, t)) x +(a(x, y)u(x, y, t)) y = f(x, y, t). We take smoot coefficients a(x, y) = 2 + sin(x + y), wit initial condition u(x, y, 0) = sin(x + y), 2π-periodic boundary conditions in bot directions, and a suitable forcing function f(x, y, t) so tat te exact solution is u(x, y, t) = sin(x+y 2t). We calculate te errors over te entire domain at T = 12.5, approximately two periods in time, and list tem in Table 5.3. We can clearly see tat te postprocessing improves te order of accuracy from k + 1 to 2k One dimension, discontinuous coefficients. We now look at te more difficult case of a linear yperbolic equation wit a discontinuous coefficient (5.4) u(x, t) t +(a(x)u(x, t)) x =0, were a(x) is discontinuous. We take a(x) to be piecewise constant for simplicity: 1, x [ a, a]\( b, b), a(x) = 1, x ( b, b), 2

17 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD 837 : Before Post-Processing : After Post-Processing 10-3 N= N= P 3 : Before Post-Processing P 3 : After Post-Processing N=10 N= Fig Pointwise errors in logaritmic scale wen te (top) and P 3 (bottom) discontinuous Galerkin metod is used to solve te variable coefficient equation (5.1) wit a(x) =2+ sin(x). Left: Errors before postprocessing. Rigt: Errors after postprocessing. were 0 <b<a, extended periodically beyond [ a, a]. We first consider an example wit only two stationary socks located at x = ± 1 2 for te exact solution. Tis can be acieved by satisfying te Rankine Hugoniot jump condition wit te sock speed set to zero. Te initial condition is given by ( 2 cos(4πx), x 1 u(x, 0) = 2, 1 ), 2 cos(2πx), oterwise in [ 1, 1], and extended periodically outside [ 1, 1]. Te domain is suc tat a = 1 and b = 1 2, and te boundary condition is 2-periodic. Te errors 0.4 away from te socks at T =12.5 are listed in Table 5.4. As we can see from te table, te postprocessing gives us (2k+1)t order accuracy in smoot regions away from te socks. Te errors for te case are also plotted in Figure 5.3 in logaritmic scale bot before and after postprocessing. From Figure 5.3 we can see tat te discontinuous Galerkin metod as no problems near te socks as tey are located at cell interfaces. We can see tat te postprocessor works to smoot te oscillations in te error away from te socks and works to improve te order of accuracy.

18 838 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Table 5.3 Errors for te discontinuous Galerkin approximation to te two-dimensional variable coefficient equation (5.3) wit a(x, y) = 2 + sin(x + y) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E P E E E E E E E E Table 5.4 Errors in smoot regions 0.4 away from te socks for te discontinuous Galerkin metod applied to te linear yperbolic equation wit a discontinuous coefficient (5.4) wit two stationary socks E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E Next, we consider te more complex situation wen tere are two moving socks in addition to te two stationary socks. We still solve (5.4) wit a =2,b = 1. Te two moving socks are obtained by satisfying te Rankine Hugoniot condition wit

19 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD 839 : Before Post-Processing : After Post-Processing N=160 N= Fig Pointwise errors in logaritmic scale wen te discontinuous Galerkin metod is used to solve (5.4) wit a discontinuous coefficient, wit two stationary socks in te solution located at x = ± 1. Left: Errors before postprocessing. Rigt: Errors after postprocessing. 2 Table 5.5 Errors in smoot regions 1.1 away from te socks for te discontinuous Galerkin metod applied to te linear yperbolic equation wit a discontinuous coefficient (5.4) wit two stationary and two moving socks E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E sock speed equal to two. Te initial condition is given by ( π ) cos 2 x, x [ 2, 2]\( 1, 1), u(x, 0) = 2 sin(πx), x ( 1, 1), 3 and extended periodically outside [ 2, 2]. We use a 4-periodic boundary condition and compute up to te final time T = 1, before te socks cross eac oter. We now ave two stationary socks, located at cell interfaces, and two moving socks, located inside te cells. Te discontinuous Galerkin metod as no problem andling te stationary socks, as tey are at cell interfaces. However, te moving socks are located inside te cells and will degenerate te accuracy nearby. Te errors are calculated 1.1 away from te socks to demonstrate te (2k + 1)t order accuracy wit te postprocessor; see Table 5.5. Te errors for te case are also plotted in Figure 5.4 in logaritmic scale bot before and after postprocessing. In Figure 5.4 we

20 840 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS : Before Post-Processing : After Post-Processing N=160 N=160 N=320 N= N= N= Fig Pointwise errors in logaritmic scale wen te discontinuous Galerkin metod is used to solve (5.4) wit a discontinuous coefficient, wit two stationary and two moving socks in te solution. Left: Errors before postprocessing. Rigt: Errors after postprocessing. can see were te four socks are located. Te postprocessor works to smoot te oscillations in te error away from tese locations to improve te order of accuracy. 6. Linearized Euler equations and an example in aeroacoustics. In tis section we use te postprocessing tecnique for te discontinuous Galerkin solutions of te linearized Euler equations in an example of aeroacoustics. Te two-dimensional linearized Euler equations are a yperbolic system u t + Au x + Bu y =0, were u is a vector of four components (density, x-momentum, y-momentum, and total energy), and A and B are constant matrices wic are te Jacobians wit freestream flow values. Tis system is similar to (3.3); ence we expect te discontinuous Galerkin metod wit te postprocessor to work well. We apply te discontinuous Galerkin metod and te postprocessing tecnique to an aeroacoustic problem of scatter of a plane wave off of a cylinder, sown in Figure 6.1. Te domain is 10r < x < 10r, 0 < y < 10r, were r is te cylinder radius. Tere is no freestream flow oter tan an incident acoustic plane wave wit wavelengt λ = 2.5r. Te unsteady scattered wave is imposed at te cylinder and te field is initialized to te known exact solution. Results of te simulations are evaluated on a series of meses after five periods. For more details of tis problem, we refer to [2] and [17]. In order to ave a uniform Cartesian mes suitable for te postprocessing, we use meses wic are uniform Cartesian everywere except for in te immediate neigborood of te cylinder. Figures 6.1(b), (c) sow te near cylinder region for two mes resolutions. Te postprocessing is performed only in a subregion of te uniform Cartesian portion of te domain. Figure 6.2 sows te convergence of te k = 2 and k = 4 (tird and fift order) simulations as well as te convergence of te postprocessed k = 2 solution. Te error is measured over a subset of te domain were te postprocessor is applicable (avoiding boundaries), and errors are measured in te L 2 norm using quadrature rules in eac element. It can be seen clearly tat te k = 2 and k = 4 solutions converge at teir designed order of accuracy k +1, and te postprocessed k = 2 solution converges

21 POSTPROCESSING FOR THE DISCONTINUOUS GALERKIN METHOD 841 Fig Domain and typical meses for te aeroacoustic scattering problem: (a) wole domain, coarse mes; (b) near cylinder region, coarse mes; (c) near cylinder region, one refinement p=2 p=2, post-processed p=4 ε Fig Convergence of L 2 errors for te k =2and k =4solutions, as well as te postprocessed k =2solution. at its designed order of accuracy 2k + 1. It is noted tat, owever, te magnitude of te error for te postprocessed k = 2 solution is considerably iger tan tat of te k = 4 solution, even toug tey are bot fift order accurate. In Figure 6.3 we plot te errors (in logaritmic scale) for te solution on a fine mes, bot before and after te postprocessing. We can clearly see tat te postprocessing as dramatically reduced te level of te error on tis mes.

22 842 JENNIFER RYAN, CHI-WANG SHU, AND HAROLD ATKINS Fig Pointwise errors in logaritmic scale wen te discontinuous Galerkin metod is used to solve te aeroacoustic scattering problem. Left: Errors before postprocessing. Rigt: Errors after postprocessing. 7. Concluding remarks. We ave discussed te implementation and application of a local postprocessing tecnique for enancing accuracy of te discontinuous Galerkin solution to yperbolic conservation laws on locally uniform meses. Numerical examples on te superconvergence of te derivatives of te numerical solution, on two space dimensions using bot P k and Q k elements on domains wit different mes sizes, on linear yperbolic equations wit variable coefficients including discontinuous coefficients, and on linearized Euler equations wit an application to aeroacoustic problems are given to demonstrate te effectiveness of tis postprocessing tecnique. Te generalization and application of tis postprocessing tecnique to smoot varying meses, to one-sided postprocessing kernels, and to nonlinear problems, as well as a comparison of tis postprocessing tecnique wit various oter postprocessing tecniques in te literature in different contexts, constitute future work. REFERENCES [1] S. Adjerid, K.D. Devine, J.E. Flaerty, and L. Krivodonova, A posteriori error estimation for discontinuous Galerkin solutions of yperbolic problems, Comput. Metods Appl. Mec. Engrg., 191 (2002), pp [2] H.L. Atkins, Applications of discontinuous Galerkin metod to acoustic scatter problems, in 2nd Computational Aeroacoustics Worksop on Bencmark Problems, Tallaassee, FL, 1997, NASA Conference Publication 3352, Wasington, D.C. 1997, pp [3] J.H. Bramble and A.H. Scatz, Higer order local accuracy by averaging in te finite element metod, Mat. Comp., 31 (1977), pp [4] W. Cai, D. Gottlieb, and C.-W. Su, On one-sided filters for spectral Fourier approximations of discontinuous functions, SIAM J. Numer. Anal., 29 (1992), pp [5] A.V. Celani Duarte and E.G. Dutra do Carmo, Te validity of te superconvergent patc recovery in discontinuous finite element formulations, Comm. Numer. Metods Engrg., 16 (2000), pp [6] B. Cockburn, Discontinuous Galerkin metods for convection-dominated problems, in Hig- Order Metods for Computational Pysics, T.J. Bart and H. Deconinck, eds., Lect. Notes Comput. Sci. Eng. 9, Springer-Verlag, Berlin, 1999, pp [7] B. Cockburn, S. Hou, and C.-W. Su, Te Runge-Kutta local projection discontinuous Galerkin finite element metod for conservation laws IV: Te multidimensional case, Mat. Comp., 54 (1990), pp