Fourier Type Super Convergence Study on DDGIC and Symmetric DDG Methods

Transcription

1 DOI 0.007/s Fourier Type Super Convergence Study on DDGIC and Symmetric DDG Metods Mengping Zang Jue Yan Received: 7 December 06 / Revised: 7 April 07 / Accepted: April 07 Springer Science+Business Media New York 07 Abstract In tis paper, using Fourier analysis tecnique, we study te super convergence property of te DDGIC (Liu and Yan in Commun Comput Pys 8():54 564, 00) and te symmetric DDG (Vidden and Yan in J Comput Mat (6):68 66, 0) metods for diffusion equation. Wit kt degree piecewise polynomials applied, te convergence to te solution s spatial derivative is kt order measured under regular norms. On te oter and wen measuring te error in te weak sense or in its moment format, te error is super convergent wit (k + )t and (k + )t orders for its first two moments wit even order degree polynomial approximations. We carry out Fourier type (Von Neumann) error analysis and obtain te desired super convergent orders for te case of P quadratic polynomial approximations. Te teoretical predicted errors agree well wit te numerical results. Keywords Discontinuous Galerkin metod Diffusion equation Stability Consistency Convergence Super convergence Introduction In tis article we apply Fourier analysis tecnique to investigate solution gradient s super convergence property for diffusion equations. We focus on te simple Heat equation under one-dimensional setting, Mengping Zang: Te researc of tis autor is supported by NSFC Grant Jue Yan: Te researc of tis autor is supported by NSF Grant DMS-605. B Jue Yan jyan@iastate.edu Mengping Zang mpzang@ustc.edu.cn Department of Matematics, University of Science and Tecnology of Cina, Hefei 006, Anui, People s Republic of Cina Department of Matematics, Iowa State University, Ames, IA 500, USA

2 u t u xx = 0, for x [0, π], (.) wit zero Neumann boundary condition and initial condition u(x, 0) = cos(x). Understanding te analysis for te Heat equation is important for oter applications suc as te Keller Segal equations of cemotaxis. Recently in [] we carry out te study on cemotaxis Keller Segel equations wit DDGIC [4] and symmetric DDG [0] metods and obtain some interesting results. Let s write out te one-dimensional Keller Segel equations, { ρt + (χc x ρ) x = ρ xx, x (a, b) and t > 0, (.) c t = c xx c + ρ to illustrate te major difficulties and issues solving (.) wic in return motivate te super convergence study in tis article. We ave two solution variables wit ρ(x, t) denoting te cell density and c(x, t) denoting te cemoattractant concentration. And χ = iste cemotactic sensitivity constant. We ave zero Neumann boundary conditions associated wit problem (.). We see te equation for ρ(x, t) involves te spatial derivative of te concentration variable c x (x, t), tus direct spatial discretization of (.) leads to one order loss for te approximation of te cell density ρ(, t). It is natural to introduce one extra variable to approximate c x (, t) separately and plug it into te equation for ρ(x, t) to obtain uniform optimal convergence of (.), see [7,8,]. In [] we apply te DDGIC metod [4] and te symmetric DDG metod [0] to directly discretize te system (.) wit no extra variable introduced and numerically we obtain optimal convergence for bot ρ(, t) and c(, t). Notice tat te classical SIPG metod [] directly applied to (.) leads to te expected one order loss, see [7]. It turns out tat our diffusion solver te DDGIC metod or te symmetric DDG metod ave te idden super convergence property on it s approximation to te solution s spatial derivative c x (, t).even te super convergence is in te weak sense or in its moment format, it is sufficient to guarantee te optimal convergence of ρ(x, t) in (.). In tis paper we consider to perform Fourier type error analysis to study solution gradient s super convergence property of te DDGIC and symmetric DDG metods. Fourier analysis is a tecnique to study stability and error estimates for discontinuous Galerkin metod and oter related scemes, especially in some cases were standard finite element tecnique can not be applied. Te Fourier analysis does ave several advantages over te standard finite element tecniques. It can be used to analyze some of te bad scemes []; it can be used for stability analysis for some of te non-standard metods suc as te special volume (SV) metod [], wic belongs to te class of Petrov Galerkin metods and cannot be easily amended to te standard finite element analysis framework; it can provide quantitative error comparisons among different scemes [8,9]; and it can be used to prove superconvergence and time evolution of errors for te DG metod [5,6,9,5]. In [4] we apply Fourier analysis tecnique to study te original DDG metod [] and prove tat optimal order convergence can be obtained wit te proper coice of (β 0,β ) coefficients cosen in te numerical flux. In [4] te Fourier type analysis also sows tat non-symmetric DDG metod [] converges wit optimal order, wic improves te order loss of Baummann Oden metod [] and NIPG metod [5]. In tis article we carry out Fourier type error estimate on approximating te solution s spatial derivative, namely u x (, t) of (.) wit te SIPG metod [], te DDGIC metod [4] and te symmetric DDG metod [0]. Due to limited symbolic computing power, we focus on te case of P quadratic polynomial approximations wic is good enoug to illustrate te super convergence properties of te DDG metods. Here we use u (, t) to denote te

3 DG polynomial numerical solution. We carry out Fourier type error analysis on eac metod and analytically calculate te following moment error: ME m [u x (u ) x ] = max j N I j (u x (u ) x ) v(x) dx v L, (.) between u x (, t) and (u ) x (, t). HereI j is te jt indexed computational cell and v(x) is a m degree polynomial on cell I j. Wit P approximations, we expect te errors of (.) are on te level of nd (kt) order. For m = 0 of te moment error (.), corresponding to te average error, we obtain te expected nd order convergence wit te SIPG metod []. On te oter and, we obtain (k + ) = 4t order super convergence wit te DDGIC [4] and te symmetric DDG metod [0] metods. For m = case of (.), we obtain (k + ) = 5t order super convergence wit te DDGIC and te symmetric DDG metods. All super convergence errors and orders are teoretically verified troug Fourier analysis tecnique and te predicted errors matc well wit te numerical results. We sould igligt tat our diffusion solver te DDGIC metod [4] is closely related to te SIPG metod []. Te DDGIC metod is different to te SIPG metod only wen ig order polynomials (P k wit k ) are considered. Notice tat te errors of u x (u ) x measured under strong sense, for example under te standard L, L and L norms, are all of nd order wit P polynomial approximations. Te super convergence to te solution s spatial derivative, even in te weak sense as listed in (.), is sufficient to guarantee te optimal convergence of te cell density variable ρ(, t) in te Keller Segel system (.). For te average error corresponding to m = 0 in (.), numerically we obtain (k + )t order super convergence wit all k = even degree polynomial approximations. Te super convergence result is observed wit β = /k(k +) cosen in te numerical flux. Wit k = odd degree polynomials approximations, we obtain (k + )t order super convergence for te average error and te super convergence is not sensitive to te coice of β coefficient. In a word, any admissible coefficients in te numerical flux lead to te super convergence penomena. Tis rest of te paper is organized as follows. In Sect. we describe te tree DG metods for te model Heat equation (.). In Sect. we write out te Fourier analysis tecnique and lay out te details of te ODE system for eac metod. Ten we symbolically calculate te moment errors and sow te analytical predictions matc well wit te numerical simulations. Finally some concluding remarks are given in Sect. 4. Tree Discontinuous Galerkin Metods Let s first introduce te notations. We ave I j =[x j, x j+ ], j =,...,N denoting a partition of te domain [0, π], wit x = 0andx N+ = π. We denote te center of eac cell by x j = ( ) x j + x j+ and te size of eac cell by j = x j+ x j.te computational cells are not supposed to be uniform for te numerical metods. For simplicity of analysis we only consider uniform meses in tis paper and we use to denote te uniform mes size. We ave P k (I j ) representing te polynomial space on cell I j wit degree at most k. Te DG solution space is defined as: } V k {v := L (0, π): v I j P k (I j ), j =,...,N.

4 For v V k, we adopt following notations to denote te jump and average of v across te cell interface x j+/ : v ± = v(x ± 0, t), v = v + v, {{v}} = v+ + v. Now we are ready to formulate te tree DG scemes, namely te SIPG metod [], te DDGIC metod [4] and te symmetric DDG metod [0]tosolve(.). In [] we introduce a numerical flux concept (u ) x to approximate u x (, t) at te cell interface x j+/ and obtain a new diffusion solver called direct discontinuous Galerkin (DDG) metod. It is ard to identify suitable coefficients in te numerical flux formula wit iger order approximations (k 4), tus we modify te DDG metod wit extra interface terms included and obtain te DDGIC metod in [4]. To obtain L (L ) error estimate, we furter introduce same format numerical flux for te test function ṽ x and obtain te symmetric DDG metod in [0]. Te symmetric DDG metod is te one tat ave symmetric structure for te bilinear form. It turns out tat te DDGIC metod is closely related to te classical SIPG metod []. Te major difference of te two is tat we ave te numerical flux (u ) x introduced and formulated as: u (u ) x = β 0 +{{(u ) x }} + β x (u ) xx. (.) In [4] we sow any admissible coefficient pair (β 0,β ) leads to te optimal convergence of te DDGIC metod. Wit te second derivative jump term (u ) xx dropped out from (.), te DDGIC metod degenerates to te SIPG metod. On te oter and, wit te second derivative jump term included in te numerical flux, we do obtain quite a few advantages of DDGIC metod over te SIPG metod. Numerically we observe tat only a small fixed penalty coefficient (β 0 = ) is needed to stabilize te DDGIC sceme for P k (k 9) polynomial approximations. It is well known tat te penalty coefficient [β 0 in (.)] of te SIPG metod needs to be large enoug, rougly β 0 k /, to stabilize te sceme. Under te topic of maximum principle, tird order DDGIC numerical solution can be proved to satisfy strict maximum principle even on unstructured triangular meses [4], wile for SIPG metod only second order piecewise linear polynomial solution can be proved to satisfy maximum principle. Including te second derivative jump term (u ) xx in (.) does give DDGIC or symmetric DDG metods extra flexibility. In tis paper we sow tat wit te proper coice of β coefficient in te numerical flux (.), extra super convergence property can be obtained wit DDGIC and symmetric DDG metods. Now we adopt te notation of te numerical flux (u ) x of (.) to uniformly write out te SIPG, DDGIC and symmetric DDG metods solving (.).. SIPG Metod [] We multiply te Heat equation (.) wit an arbitrary test function v V x, integrate over te cell I j, ave te integration by parts and add test function interface terms to symmetrize te diffusion term, formally we ave te SIPG metod written out as: I j (u ) t vdx (u ) x v j+ + (u ) x v + j + I j (u ) x v x dx + u (v x ) + j+ u (v x ) + = 0, j (.) u (u ) x = β 0 +{{(u ) x }}, at x j±/.

5 Summing (.) over all I j, we ave te primary formulation of te SIPG metod as, π wit te bilinear form laid out as: B(u,v)= N j= 0 I j (u ) x v x dx + (u ) t vdx + B(u,v)= 0, N j= ( ) u v {{(u ) x }} v +{{v x }} u + β 0. j+/ Suppose periodic boundary condition is considered ere. Obviously te bilinear form is symmetric wit B(u,v)= B(v, u ).. DDGIC Metod [4] To solve te model Heat equation (.) wit te DDGIC metod [4], we multiply te equation wit test function v V x and integrate over te computational cell I j. We perform te integration by parts, add interface terms involving test function derivative {{v x }} and we obtain te DDGIC sceme formulation as below, I j (u ) t vdx (u ) x v j+ + (u ) x v + j + I j (u ) x v x dx + u (v x ) + j+ u (v x ) + = 0, j (.) u (u ) x = β 0 +{{(u ) x }} + β (u ) xx, at x j±/. We see te only difference between te DDGIC metod (.) and te SIPG metod (.) is tat we include te solution s second derivative jump term (u ) xx in te numerical flux (u ) x. In a word, te DDGIC metod is only different to te SIPG metod wen ig order P k (k ) polynomials are applied. Te DDGIC metod degenerates to te SIPG metod wen piecewise constant and linear polynomials are considered.. Symmetric DDG Metod [0] Now we introduce test function numerical flux ṽ x and refine te interface correction terms to obtain a symmetric DDG metod as, (u ) t vdx (u ) x v j+ + (u ) x v + I j + (u ) x v x dx j I j + ṽ x u j+ + ṽ x u j = 0, (.4) wit numerical flux, { u (u ) x = β 0u +{{(u ) x }} + β (u ) xx v ṽ x = β 0v +{{v, at x j±/. (.5) x}} + β v xx Notice tat te test function v is taken to be non-zero only inside te cell I j. Wen evaluating ṽ x at cell interfaces x j±/, only te quantities from te side of I j contribute to te calculation of ṽ x. For example at x j+ we ave, ṽ x j+ = β 0v v + v x + β ( v xx ).

6 Summing (.4) over all computational cells I j, we ave te primal formulation of symmetric DDG metod solving (.)as, π were te bilinear form is defined as, B(u,v)= N j= 0 (u ) t vdx + B(u,v)= 0, I j (u ) x v x dx + N ( (u ) x v + ṽ x u ) We see te bilinear form B(u,v)= B(v, u ) is symmetric for te symmetric DDG metod. For time dependent parabolic problems, we find out te DDGIC metod [4] istemost efficient diffusion solver. For time independent elliptic type problems, te symmetric DDG metod [0] is more efficient. Among te four DDG metods [,4,0,] te symmetric DDG metod is te only one tat gives te mass matrix being symmetric. Te non-symmetric mass matrix of te DDGIC metod may cause extra issues wen fast solvers are applied. In [0] we observe symmetric DDG metod also gives te smallest condition number on te mass matrix. Wen comparing to te SIPG metod, te symmetric DDG metod saves 7 0% CPU running time, especially for ig order polynomial approximations. In [0]we also observe tat te symmetric DDG metod resolves te oscillatory wave muc better tan te DDGIC metod. Up to now, we ave taken te metod of lines approac and ave left time variable t continuous. For time discretization, TVD Runge Kutta metod [6,7] isusedtosolvete ODE to matc te accuracy in space, j= j+ (u ) t = L(u ). (.6) Specifically te tird-order TVD Runge Kutta metod tat we use in tis paper is given by, u () = u n + tl(un ), u () = 4 un + ( 4 u n+ = un + ) u () + tl(u () ), ( ) u () + tl(u () ).. Fourier Analysis for te Moment Errors In tis section we write out te SIPG metod, te DDGIC metod and te symmetric DDG metod in details and lay out te tree DG metods as finite difference scemes. We need te assumption of uniform mes and still zero Neumann boundary condition is considered. Treated as finite difference metods, we ten perform te standard Von Neumann Fourier analysis and symbolically calculate te moment errors for eac metod and finally compare te analytical errors wit te numerical ones. Weuse te SIPGmetod (.) to demonstrate te procedure of te Fourier analysis. After picking a local basis for te solution space V and inverting a local (k + ) (k + ) mass matrix (wic could be done by and), te DG metod of (.) can be rewritten out as: d dt u j = ( ) A u j + B u j + C u j+, (.)

7 were u j is a small vector of size k + containing te coefficients of te DG solution u (, t) in te local basis inside cell I j and A, B and C are (k + ) (k + ) constant matrices. If we coose te point values of te solution u (, t) inside cell I j astedegreeoffreedom, denoted by u j+ i k, i = 0,...,k, (k+) at te k + equally spaced points, ten te DG metod, rewritten in terms of tese degrees of freedom, can be considered as a finite difference sceme on a globally uniform mes (wit mes size /(k + )); owever tey are not standard finite difference scemes because eac point in te group of k + points belonging to te cell I j obeys a different form of finite difference sceme. Since we focus on P quadratic polynomial approximations, let us discuss te procedure in detail wit k =. Te degree of freedom are now te point values at te N uniformly spaced points, u j, u j, u j+, j =,...,N. Te DG polynomial solution inside cell I j is ten represented by, u (x, t) = u j (t)φ j (x) + u j (t)φ j (x) + u j+ (t)φ j+ (x), (.) were φ j (x), φ j (x), φ j+ (x) are te Lagrange interpolation polynomials at points x j, x j and x j+. Now we obtain te finite difference representation of te SIPG metod (.) asin(.) wit te numerical solution vector defined as, u j (t) u j = u j (t), on cell I j. (.) u j+ (t) As a finite difference metod, we perform te following Von Neumann Fourier analysis. Te analysis depends on te assumption of uniform mes and periodic boundary condition. We make an ansatz of te solution wit te form, u j (t) =û j (t)e ix j, (.4) and substitute te ansatz of (.4) into te SIPG sceme (.) to find te coefficient vector evolving in time as, û d j (t) û j (t) û j (t) = G() û j (t). (.5) dt û j+ (t) û j+ (t) Te amplification matrix G() is given by, G() = (Ae i + B + Ce i). (.6) Te matrices A, B, C are from (.) wic is te rewritten of te SIPG sceme (.) wit Lagrange interpolation polynomials as basis functions. Te general solution of te ODE system (.5) isgivenby, û j (t) û j (t) = a e λt V + a e λt V + a e λt V, (.7) û j+ (t)

8 were λ,λ,λ,andv, V, V are te eigenvalues and te corresponding eigenvectors of te amplification matrix G() respectively. To fit te initial condition u(x, 0) = cos(x), weset, u j (0) = e ix j, u j (0) = e ix j, u j+ (0) = e ix j+, wose real part is te given initial condition of te model Heat equation (.). Tus we require, at t = 0, û j (0) û j (0) = û j+ (0) e i e i e ix j. Fitting te initial condition determines te coefficients a, a,anda in te general solution (.7). Tus we can explicitly write out te solution expression (.) for te SIPG metod. Wit te exact solution spatial derivative given as u x (, t) = e t sin(x), we can symbolically calculate te moment errors of (.). Troug te simple Taylor expansions, now we are able to find out te leading error terms in te moment errors expression. In te following sections we compare te predicted errors by Fourier analysis to te errors numerically calculated from te SIPG, DDGIC and symmetric DDG metods. We sould mention tis is not a standard error estimate tecnique, yet te Fourier type error analysis is a very powerful tool and can carry out error estimate for problems tat can be not analyzed troug standard finite element tecnique. In tis paper we study te super convergence property of te DDGIC and te symmetric metods under te moment errors (.). Specifically te test function on cell I j is taken as v = ( x x j / )m wit m = 0,,...,k as te polynomial power. Te moment error of (.) is laid out in detail as te following: I j ((u ) x (, t) u x (, t))v(x) dx ME m [u x (u ) x ] = max j N v L, ( ) x x m j v =. (.8) / Here te exact solution is given wit u(x, t) = e t cos(x). We first consider to apply Fourier analysis to symbolically calculate te solution derivative moment errors (.8) wit different moment powers for te SIPG, DDGIC and symmetric DDG metods. Ten we numerically compute te moment errors (.8) wit te tree DG metods and carry out comparisons to te estimated errors troug Fourier analysis. Notice tat numerical solutions are obtained wit tird order TVD Runge Kutta metod (.6) for time discretization. We coose smaller time step size to make sure spatial errors are dominant. For te average error corresponding to m = 0in(.8), an expected rd order convergence is obtained for te SIPG metod wit P quadratic polynomial approximations. On te oter and we obtain 4t order super convergence for te average error wit te DDGIC metod and te symmetric DDG metod. For te moment error of (.8) wit m =, we obtain 5t order super convergence wit te DDGIC and te symmetric DDG metods. All super convergence orders are verified troug analytical error estimate and numerical simulations.

9 . SIDG Metod of (.) In tis section we perform Fourier analysis on te SIPG metod (.) wit te numerical flux takeninteformof, u (u ) x = β 0 +{{(u ) x }}. Te matrices A, B, C in (.) for te SIPG metod are: A = 48 B = 4 ( 9 + β 0 ) (549 5β 0 ) ( 9 + 5β 0 ) 7( 9 + β 0 ) 8( 4 + 5β 0 ) 7( + 5β 0 ) ( β 0 ) ( + 5β 0 ) ( 7 5β 0 ) 9( 7 + 9β 0 ) ( β 0 ) 9( 7 + β 0 ) 7( + β 0 ) 8(7 + 5β 0 ) 7( + β 0 ) 9( 7 + β 0 ) ( β 0 ) 9( 7 + 9β 0 ), C = ( 7 5β 0 ) ( + 5β 0 ) ( β 0 ) 7( + 5β 0 ) 8( 4 + 5β 0 ) 7( 9 + β 0 ). 48 ( 9 + 5β 0 ) (549 5β 0 ) ( 9 + β 0 ), (.9) We take β 0 = 4 for te SIPG metod and te same coefficient value β 0 = 4 is applied in te following Sect.. for te te DDGIC metod and Sect.. for te symmetric DDG metod. In Table we list te analytically calculated moment errors (.8) wit te SIPG metod troug Fourier tecnique. For te average error [(.8) wit m = 0] we see te leading order is of nd order wic is an expected order for te SIPG metod wit quadratic polynomial approximations. No super convergence penomena is observed in tis case. In Table Analytically calculated moment errors (.8) wit te SIPG metod (.)and(.9) m = 0 m = m = m = β 0 = 4 ( 6 e t sin(x j ) ) + O( 4 ) ( 7 e t sin(x j ) ) + O( 5 ) ( 540 e t sin(x j ) ) + O( 4 ) ( e t sin(x j ) ) + O( 5 ) Table SIPG metod moment errors (.8) comparison: numerical solution moment errors from sceme (.)and estimated errors from Table Final time t = 0.5 Numerical solutions Predicted by analysis Error Order Error Order m = 0 N = E E 0 N = E E 0.00 N = E E 0.00 N = E E 0.00 m = N = E E 0 N = E E 0.00 N = E E N = E E 05.00

10 Table we list te moment errors numerically computed wit te SIPG metod (.)andte ones calculated from Fourier analysis, namely te leading error terms in Table. Te two groups of errors matc very well.. DDGIC Metod of (.) In tis section we perform Fourier type error analysis on te DDGIC metod (.) wit quadratic P polynomial approximations. Te matrix vector format of te DDGIC metod is given as (.) wit te matrices A, B, C laid out as te following: A = 48 B = 4 ( 9 + β β ) (549 5β 0 656β ) ( 9 + 5β β ) 7( 9 + β 0 + 4β ) 8( 4 + 5β 0 + 7β ) 7( + 5β 0 + 4β ) ( β 0 4β ) ( + 5β 0 + 7β ) ( 7 5β 0 4β ) 9( 7 + 9β β ) ( β β ) 9( 7 + β β ) 7( + β 0 + 4β ) 8(7 + 5β 0 + 7β ) 7( + β 0 + 4β ) 9( 7 + β β ) ( β β ) 9( 7 + 9β β ) C = ( 7 5β 0 4β ) ( + 5β 0 + 7β ) ( β 0 4β ) 7( + 5β 0 + 4β ) 8( 4 + 5β 0 + 7β ) 7( 9 + β 0 + 4β ). 48 ( 9 + 5β β ) (549 5β 0 656β ) ( 9 + β β ) (.0) We study te super convergence property of te DDGIC metod wit different coices of (β 0,β ) coefficient in te numerical flux (u ) x (.). Te proper coice of β coefficient leads to te super convergence on its approximation to te solution s derivative measured under te moment errors format of (.8). Specifically we study two cases wit (β 0,β ) = (4, 8 ) and (β 0,β ) = (4, ). We symbolically calculate te moment errors (.8) troug Fourier analysis tool and we list te leading errors in Table. Wit β = k(k+) = cosen in te numerical flux (u ) x of te DDGIC metod (.), we see te average error (m = 0 case) is super convergent wit (k + ) = 4t order (te leading error term in Table ). Te moment error of (.8) wit m = is super convergent wit (k + ) = 5t order. Troug finite element tecnique, autors in [] prove tat on Gaussian points (k + )t order super convergence can be obtained wit β = k(k+).sameβ coefficient formula works in our study on te moment errors. Different to [], we do not need special initialization to guarantee super convergence results. Numerical experiments sow Taylor expansion initialization, L initialization and Lagrange interpolation initialization all lead to te observed super convergence orders. We sould also mention tat for iger order moment errors, for example m = of(.8), no super convergence is observed. We apply DDGIC sceme (.) to numerically solve te Heat equation and calculate te moment errors (.8). In Table 4 we compare te numerical solution moment errors and te errors predicted by Fourier analysis. Te numerical errors matc very well wit te analytical ones.. Symmetric DDG Metod In tis section we perform Fourier analysis on te symmetric DDG metod (.4) wit numerical flux (.5). Here we ave te notation of β 0 = β 0u +β 0v. Considering te rewritten of (.4) in terms of matrix vector format (.), we ave te matrices of A, B, C for te symmetric DDG metod listed below as:,,

11 Table Analytically calculated moment errors (.8) wit te DDGIC metod (.)and(.0) β = 8 β = m = 0 ( 7 e t sin(x j ) ) + O( 4 ) ( 4t e t sin(x j ) ) 4 + O( 6 ) m = ( 44 e t sin(x j ) ) + O( 5 ) ( t e t sin(x j ) ) 5 + O( 7 ) ( m = e t sin(x j ) ) + O( 4 ) ( 90 e t sin(x j ) ) + O( 4 ) m = ( 8400 e t sin(x j ) ) + O( 5 ( ) 00 e t sin(x j ) ) + O( 5 ) Table 4 DDGIC metod moment errors (.8) wit β = /8orβ = / in (u ) x β = 8 β = Numerical solutions Predicted by analysis Numerical solutions Predicted by analysis Error Order Error Order Error Order Error Order m = 0 N = 0.e 0.e 0.65e 05.7e 05 N = e e e e N = 40.06e e e e N = e e e e m = N = 0.97e 0.08e 0.9e 05.55e 05 N = 0.57e e e e N = 40.5e e e e N = e e e e Numerical errors from sceme (.) and analytical errors from Table. Final time t = 0.5 A = ( 9 + β 0 + 6β ) (549 5β 0 956β ) ( 9 + 5β β ) 7( 9 + β β ) 8( 4 + 5β 0 + 7β ) 7( + 5β 0 + 4β ), 48 ( β 0 + 6β ) ( + 5β 0 8β ) ( 7 5β β ) B = 9( 7 + 9β β ) ( β β ) 9( 7 + β β ) 7( + β β ) 8(7 + 5β 0 + 7β ) 7( + β β ) 4 9( 7 + β β ) ( β β ) 9( 7 + 9β β ) C = ( 7 5β β ) ( + 5β 0 8β ) ( β 0 + 6β ) 7( + 5β 0 + 4β ) 8( 4 + 5β 0 + 7β ) 7( 9 + β β ). 48 ( 9 + 5β β ) (549 5β 0 956β ) ( 9 + β 0 + 6β ) (.) Similar to te super convergence study on te DDGIC metod, we consider two cases wit (β 0,β ) = (4, 8 ) and (β 0,β ) = (4, ) cosen in te numerical fluxes (.5). Notice tat we ave β 0 = β 0u + β 0v for te symmetric DDG metod. Again we symbolically calculate te moment errors (.8) and list te analytical results in Table 5. Wit β = case, we ave 4t order super convergence on te average error (m = 0 case) and 5t order super convergence on te moment error of (.8) wit m =. In Table 6 we list te numerical solution moment errors and te moment errors calculated from Fourier analysis. Te two groups of errors matc very well.

12 Table 5 Analytically calculated moment errors (.8) wit te symmetric DDG metod (.)and(.) β = 8 β = m = 0 7 sin(x j ) + O( 4 ) m = 88 e t sin(x j ) + O( 5 ) m = e t sin(x j ) + O( 4 ) m = 7 4t 780 e t sin(x j ) 4 + O( 6 ) 7+840t e t sin(x j ) 5 + O( 7 ) 90 e t sin(x j ) + O( 4 ) e t sin(x j ) + O( 5 ) 00 e t sin(x j ) + O( 5 ) Table 6 Symmetric DDG metod moment errors (.8) wit β = /8andβ = / in (u ) x and ṽ x β = 8 β = Numerical solutions Predicted by analysis Numerical solutions Predicted by analysis Error Order Error Order Error Order Error Order m = 0 N = 0.4e 0.e 0.78e 05.7e 05 N = e e e e N = 40.06e e e e N = e e e e m = N = e 04.04e 0 7.7e e 06 N = 0.7e e e e N = 40.6e e e e N = 80.0e e e e Numerical solution errors from sceme (.4) and estimated errors from Table 5. Final time t = Piecewise Linear Approximations As we mentioned previously in Sect. tat our diffusion solver te DDGIC and symmetric DDG metods degenerate to te SIPG metod wit low order (P k wit k ) polynomial approximations. Wit piecewise linear polynomial approximations it turns out tat te SIPG metod is also super convergent wit (k +) = a second order for te average error [m = 0 in moment errors (.8)]. We carry out Fourier analysis moment error estimate wit piecewise linear approximations wit SIPG metod and list te first two moments errors below: { MEm=0 [u x (u ) x ] = 96 e t sin(x j )(7 + 8t) + O( 4 ), ME m= [u x (u ) x ] = 6 e t cos(x j ) + O( (.) ). We also compute te moment errors (.8) numerically wit SIPG metod (.) andp approximations. We list te comparison between numerical errors and te estimated ones in Table 7 for te average error (m = 0 case). We see te SIPG metod on its approximation to te solution s spatial derivative is also super convergent wit (k + ) = nd order. Based on te super convergence studies, te SIPG metod seems not complete and our diffusion solver te DDGIC and symmetric DDG metods complements te SIPG metod in some way.

13 Table 7 Piecewise linear approximations moment errors (.8) comparison: numerical errors wit SIPG metod (.) and analytical errors from estimate (.) Final time t = 0.5 Numerical solutions Predicted by analysis Error Order Error Order m = 0 N = 0.4e 0.74e 0 N = e e 0.00 N = 40.69e e 0.00 N = e e Conclusion We consider to solve te model Heat equation (.) wit te classical SIPG metod [], te DDGIC metod [4] and te symmetric DDG metod [0]. Specifically we study te approximation to te solution s spatial derivative, namely u x (, t) of (.), measured under te moment errors (.8). Wit kt degree polynomial applied, te error measured under L and L strong norms is of kt order on its approximation to te solution s derivative. Measured under te weak sense or in te moment format, te error is super convergent wit (k+)t and (k+)t order for its first two moments wit te DDGIC and te symmetric DDG metods wit even order approximations. No super convergence is observed wit te classical SIPG metod, even measured under weak sense. Te observed super convergence orders are teoretically verified wit Fourier type error estimate wen P quadratic polynomials approximations is considered. Te analytically predicted errors matc well wit te numerical results. References. Arnold, D.N.: An interior penalty finite element metod wit discontinuous elements. SIAM J. Numer. Anal. 9(4), (98). Baumann, C.E., Oden, J.T.: A discontinuous p finite element metod for convection diffusion problems. Comput. Metods Appl. Mec. Eng. 75( 4), 4 (999). Cao, W.-X., Liu, H., Zang, Z.-M.: Superconvergence of te direct discontinuous Galerkin metod for convection diffusion equations. Numer. Metods Partial Differ. Equ. (), 90 7 (07) 4. Cen, Z., Huang, H., Yan, J.: Tird order maximum-principle-satisfying direct discontinuous Galerkin metods for time dependent convection diffusion equations on unstructured triangular meses. J. Comput. Pys. 08, 98 7 (06) 5. Ceng, Y., Su, C.-W.: Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Pys. 7, (008) 6. Ceng, Y., Su, C.-W.: Superconvergence of local discontinuous Galerkin metods for one-dimensional convection diffusion equations. Comput. Struct. 87, (009) 7. Epsteyn, Y.: Discontinuous Galerkin metods for te cemotaxis and aptotaxis models. J. Comput. Appl. Mat. 4(), 68 8 (009) 8. Epsteyn, Y., Kurganov, A.: New interior penalty discontinuous Galerkin metods for te Keller Segel cemotaxis model. SIAM J. Numer. Anal. 47(), (008/009) 9. Guo, W., Zong, X., Qiu, J.-M.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin metods: eigen-structure analysis based on Fourier approac. J. Comput. Pys. 5, (0) 0. Huang, H., Cen, Z., Li, J., Yan, J.: Direct discontinuous Galerkin metod and its variations for second order elliptic equations. J. Sci. Comput. 70(), (07). Huang, H., Zong, X., Yan, J.: Direct discontinuous galerkin wit interface correction and symmetric direct discontinuous galerkin metods for Keller Segel cemotaxis equation. SIAM J. Numer. Anal. to be submitted

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