Superconvergence of energy-conserving discontinuous Galerkin methods for. linear hyperbolic equations. Abstract

Transcription

1 Superconvergence of energy-conserving discontinuous Galerkin metods for linear yperbolic equations Yong Liu, Ci-Wang Su and Mengping Zang 3 Abstract In tis paper, we study superconvergence properties of te energy-conserving discontinuous Galerkin (DG) metod in [8] for one-dimensional linear yperbolic equations. We prove te approximate solution superconverges to a particular projection of te exact solution. Te order of tis superconvergence is proved to be k + wen piecewise P k polynomials wit k are used. Te proof is valid for arbitrary non-uniform regular meses and for piecewise P k polynomials wit arbitrary k. Furtermore, we find tat te derivative and function value approximations of te DG solution are superconvergent at a class of special points, wit an order of k + and k +, respectively. We also prove, under suitable coice of initial discretization, a (k + )-t order superconvergence rate of te DG solution for te numerical fluxes and te cell averages. Numerical experiments are given to demonstrate tese teoretical results. Key Words: energy-conserving discontinuous Galerkin metods, superconvergence, linear yperbolic equations. Scool of Matematical Sciences, University of Science and Tecnology of Cina, Hefei, Anui 3006, P.R. Cina. yong3@mail.ustc.edu.cn Division of Applied Matematics, Brown University, Providence, RI 09, USA. su@dam.brown.edu. Researc supported by DOE grant DE-FG0-08ER5863 and NSF grant DMS Scool of Matematical Sciences, University of Science and Tecnology of Cina, Hefei, Anui 3006, P.R. Cina. mpzang@ustc.edu.cn. Researc supported by NSFC grant

2 Introduction Wave propagation is a fundamental form of energy transmission, wic arises in many fields of science, engineering and industry, suc as geoscience, petroleum engineering, telecommunication, and te defense industry (see [7, ] and te references terein). Efficient and accurate numerical metods to solve wave propagation problems are of fundamental importance to tese applications. Experience reveals tat energy conserving numerical metods, wic conserve te discrete approximation of energy, are favorable because tey are able to maintain te pase and sape of te waves accurately, especially for long time simulation. A vast amount of literature can be found on te numerical approximation of wave problems modeled by linear yperbolic systems. Recently, Fu and Su [8] developed energy-conserving discontinuous Galerkin (DG) metods for symmetric linear yperbolic systems on general meses. Tey obtain te optimal convergence accuracy of order k + for te semidiscrete scemes. We will study superconvergence properties of tis class of DG scemes in tis paper. DG metods are a class of finite element metods devised to solve yperbolic conservation laws and related equations, e.g. [4,, 3, 5, 6]. Tey use completely discontinuous polynomial space for te test and trial functions in te spatial variables and coupled wit explicit and nonlinearly stable ig order Runge-Kutta time discretization, wit a compact stencil, and wit te ability to easily accommodate arbitrary -p adaptivity. In te past few years, tere as been considerable interest in studying superconvergence properties of DG metods. We refer to [, 4, 6] for ordinary differential equations, [, 3] for multidimensional first order yperbolic systems, [9, 0, 5, 3] for one-dimensional yperbolic conservation laws and time-dependent convection-diffusion equations, and [0] for one- and two- dimensional yperbolic equations based on te Fourier approac. In 04, Cao et al. [7] introduced an approac to study te superconvergence of DG metods for linear yperbolic equations wit upwind numerical fluxes.

3 Tey constructed a suitable correction function to correct te error between te exact solution u and its Radau projection to obtain a (k + )-t superconvergence rate at te downwind points for piecewise polynomials of degree k. Later, Cao et al. extended tis work to study upwind-biased numerical fluxes, degenerate variable coefficients and nonlinear cases for yperbolic conservation laws [8, 6, 5]. For one-dimensional linear convection-diffusion equations, Cao et al. [4] studied superconvergence of te numerical solution by te direct discontinuous Galerkin metods for convection-diffusion equations. Superconvergence results can elp us to design trouble cell indicators suc as te KXRCF trouble-cell indicator [], wic is a key point for adaptive DG scemes. Tus we are interested in studying te superconvergence properties of te recently devised energy-conserving DG metods in [8]. In tis paper, we use te tecniques in Cao et al. [7] to construct a suitable correction function for correcting te error between te special projection of te exact solution and te exact solution. Te main contribution of tis paper is tat we prove te approximate solution superconverges to a particular projection of te exact solution. Te order of tis superconvergence is proved to be k + wen piecewise P k polynomials wit k are used. Numerical examples demonstrate tis result is optimal. Furtermore, we find tat te function value approximations of te DG solution are superconvergent at all Gaussian points wit order k +, and te derivative approximations are superconvergent at zeros of a special polynomial wit order k +. We also prove, under a suitable coice of te initial discretization, a (k + )-t order superconvergence rate of te DG solution for te numerical fluxes and cell averages. Numerical examples sow tat tis analysis is optimal. Te organization of tis paper is as follows. In section, we first recall te energyconserving discontinuous Galerkin metod for te linear yperbolic equations. Ten, we construct a special interpolation function, and study its optimal approximation properties to sow te superconvergence results in section 3. We provide numerical examples 3

4 to demonstrate our teoretical results in section 4. In section 5, we give concluding remarks and perspectives for future work. Finally, in te appendix we provide proofs for some of te more tecnical results of te superconvergence estimates. Energy-conserving DG scemes In tis section, we consider te one-dimensional scalar linear equation u t + u x = 0, x [a, b], t 0 u(x, 0) = u 0 (x), x [a, b] (.) wit periodic boundary condition. Te optimal energy-conserving DG metod designed in [8] for te advection equation (.) needs to introduce an auxiliary zero function φ(x, t) = 0, wic is te solution of an advection equation using te opposite speed as tat for u(x, t), but wit zero initial data. Ten, we obtain te following system: u t + u x = 0, x [a, b], t 0 φ t φ x = 0, x [a, b], t 0 (.) wit te initial condition u(x, 0) = u 0 (x) and φ(x, 0) = 0. We first introduce te usual notation of te DG metod. For a given interval Ω = [a, b], we divide it into N cells as follows: We denote a = x < x3 <... < x N+ = b. (.3) and I j = (x j, x j+ x ), j = (x j + x j+ ) (.4) j = x j+ x j, j = j / = max j. (.5) j 4

5 We also assume te mes is regular, i.e., te ratio between te maximum and minimum mes sizes sall stay bounded during mes refinements. We define te finite element space as V k = {v : (v ) Ij P k (I j ), j =,...,N}. (.6) Here P k (I j ) denotes te set of all polynomials of degree at most k on I j. For a function v V k, we use (v ) j+ and (v ) + j+ to refer to te value of v at x j+ form te left cell I j and te rigt cell I j+, respectively. [v ] is used to denote v + v, i.e. te jump of v at cell interfaces. {v } = (v + v+ ) denotes te average of te left and rigt boundary values. We also introduce some standard Sobolev spaces notations. For any integer m > 0, let W m,p (D) be te standard Sobolev spaces on sub-domain D Ω equipped wit te norm m,p,d and semi-norm m,p,d. Wen D = Ω, we omit te index D; and if p =, we set W m,p (D) = H m (D), m,p,d = m,d, and m,p,d = m,d. Te semi-discrete DG metod for (.) reads as follows: find (u, φ ) V k V k, suc tat (u ) t v dx u (v ) x dx + (û v ) j+ (û v + ) j = 0, (.7a) I j I j (φ ) t ψ dx + φ (ψ ) x dx ( ˆφ ψ ) j+ + ( ˆφ ψ + ) j = 0 (.7b) I j I j olds for any (v, ψ ) V k V k and j =,...,N. Here û, ˆφ are te energy conserving numerical fluxes aving te following form û j ˆφ j = {u } + α j [φ ], (.8a) = {φ } + α j [u ] (.8b) wit α j obtained. being any real constant. In [8], te following energy-conserving result is Teorem.. Te energy E (t) = (u + φ ) dx (.9) Ω 5

6 is conserved by te semi-discrete sceme (.7) wit te numerical flux (.8) for all time. Also, in [8], it is sown, wit simply taking α j = for all j, an optimal convergence result wit a clean proof. We will prove our superconvergence results also under tis setting. We first introduce a set of projections. Te left and rigt Gauss-Radau projections P ± are defined by P ± u(x)v dx = I j u(x)v dx v P k (I j ), I j (.0) (P ± u)± = u ± at x j. (.) For tese projections, te following inequality olds []: w e + w e Γ C k+ (.) were w e = P ± w w, Γ denotes te set of boundary points of all elements I j, and te constant C depends on k and te standard Sobolev k+ semi-norm k+ of te smoot function w. Here and below, an unmarked norm denotes te L norm. We also need te following basic facts. For any function v V k, te following inequalities old []: (i) (v ) x C v, (ii) v Γ C v, (.3) (iii) v C v. We sall also use te coupled projection specifically designed for te DG sceme (.7) in [8]. For any function u, φ H (Ω), define te following coupled auxiliary projection (P,, k u, P φ) [V ] : P, u(x)v dx = u(x)v dx v P k (I j ), (.4) I j I j P, φ(x)v dx = φ(x)v dx v P k (I j ), (.5) I j I j ({P, u} +, [P φ]) j = u(x j ), (.6) 6

7 ({P, φ} +, [P u]) j = φ(x j ), (.7) for all j. It was sown in Lemma.6 in [8] tat te projection is actually an optimal local projection, Let us introduce a few notations. We define P, u(x) = (P + (u + φ) + P (u φ)), (.8) P, φ(x) = (P + (u + φ) P (u φ)) (.9) a(u, v) = b(φ, v) = N a j (u, v), (.0) j= N b j (φ, v) (.) j= were a j (u, v) =(u t, v) (u, v x ) + (ûv ) j+ b j (φ, v) =(φ t, v) + (φ, v x ) (ˆφv ) j+ (ûv + ) j, (.) + (ˆφv + ) j, (.3) We denote ε u := u P, u, η u := u P, u Ten it is straigtforward to deduce from (.7) ε φ := φ P, φ, η φ := φ P, φ (.4) a j (u, v ) = 0, b j (φ, ψ ) = 0, (v, ψ ) [V k ] (.5) Obviously, te exact solution u and φ also satisfies a j (u, v ) = 0, b j (φ, ψ ) = 0, (v, ψ ) [V k ] (.6) Subtracting (.5) from (.6), we obtain te error equations a j (u u, v ) = 0, b j (φ φ, ψ ) = 0, (v, ψ ) [V k ] (.7) 7

8 By (.7), te estimate for u P u can usually be reduced to estimating a(η u, η u ) + b(η φ, η φ ) = a(ε u, η u ) + b(ε φ, η φ ) A straigtforward analysis using te definitions of a(, ) and b(, ) results in a(ε u, v ) + b(ε φ, ψ ) = ((ε u ) t, v ) + ((ε φ ) t, ψ ) k+ ( v + ψ ) leading to te optimal error bound (ε u ) t + (ε φ ) t k+ were ere and in te following, A B denotes tat A can be bounded by B multiplied by a constant independent of te mes size. Tis rate is of course still far from our superconvergence goal. Similar to [7], in wic Cao et al. constructed a correction function to elp obtaining te desired superconvergence results, we would also like to find a series suitable correction functions (ω (i) u, ω (i) φ ) [V k ], i k to improve te error between (u, φ) and (P, a(u P, u + l i= ω (i), u, P φ) suc tat l u, v ) + b(φ P, φ + ω (i) φ, ψ ) k++l ( v + ψ ), i= (v, ψ ) [V k ], l k 3 Construction of a special interpolation function We now define te correction functions (ω u (i), ω (i) φ ). Denote (ω(0) u, ω (0) φ ) = (ε u, ε φ ), ten we define (ω (i) u, (v ) x ) = ((ω (i ) u ) t, v ), v P k (I j ), (3.) (ω (i) φ, (ψ ) x ) = ((ω (i ) φ ) t, ψ ), ψ P k (I j ), (3.) (ω (i) u (ω (i) u + ω(i) φ )+ j ω(i) φ ) j+ = 0, (3.3) = 0, i =,..., k, j, (3.4) 8

9 Note tat te definition of te correction functions is recursive, tus we can firstly analyze te first correction functions (ω u (), ω () ) and ten obtain similar properties by induction. We ave te following properties for te correction functions. Lemma 3.. For any k, suppose (ω u (i), ω (i) ) are defined by (3.)-(3.4). Ten ω (i) u I j = k m=k i c i j,m L j,m(ξ), ω (i) φ I j = k m=k i φ φ d i j,m L j,m(ξ), ξ = (x x j) j [, ] (3.5) were c i j,m and di j,m are some bounded constants, and L j,m(ξ) are te standard Legendre polynomials on interval I j. Furtermore, if (u, φ) [H k++r+i ], r = 0,, i =,...k, tere olds r t ω(i) u + r t ω(i) φ k++i ( u k++r+i + φ k++r+i ) (3.6) Proof. Te proof of tis lemma is provided in te appendix; see section A. From Lemma 3., we ave te straigtforward corollary as following. Corollary 3.. For any k, suppose (u, φ) [H k+ ] are te solutions of te equations (.), and te correction functions are defined by (3.)-(3.4), ten a(u P, u + l i= ω (i) u, v ) + b(φ P, φ + l i= were a(, ) and b(, ) are defined by (.0)-(.). 3. Superconvergence of te interpolation ω (i) φ, ψ ) k++l ( v + ψ ), (3.7) (v, ψ ) [V k ], Now if we take our correction functions as (u l I, φl I ) = ( l i= ω(i) u, l te following superconvergence results. i= ω(i) φ l k (3.8) ), we ave Teorem 3.. For any k, suppose (u, φ) [H k+ ] are te solutions of te equations (.), te correction functions are defined by (3.)-(3.4) and (u, φ ) are te DG solutions of (.7). Ten, for l k (P, u ul I u )(, t) + (P, φ φl I φ )(, t) 9

10 (P, u ul I u )(, 0) + (P, φ φl I φ )(, 0) + t k++l sup u(, τ) k++l (3.9) τ [0,t] Proof. Te proof of tis teorem is provided in te appendix; see section A. As a direct consequence of (3.9) and (3.6), we ave te following superconvergence result of u towards te specially designed projection P, u. Corollary 3.. For any k, suppose (u, φ) [H k+ ] are te solutions of te equations (.), te correction functions are defined by (3.)-(3.4) and (u, φ ) are te DG solutions of (.7). Te initial discretizations are taken as u (, 0) = P, u 0( ) and φ (, 0) = P, φ 0( ). Ten u P, u + φ P, φ (t + )k+ sup u(, τ) k+ (3.0) τ [0,t] 3. Superconvergence for te numerical fluxes and te cell averages We denote by e u,f and e u,c te errors of te fluxes and te cell-averages, respectively. Tat is, ( e u,f = N e u,c = N N j= j= ) ((u û )(x j+, t)) + ((φ ˆφ )(x j+, t)), (3.) ( N ) ( ) (u u )(x) dx + (φ φ )(x) dx j I j j I j We ave te following superconvergence results.. (3.) Teorem 3.. For any k, suppose (u, φ) [H k+ ] are te solutions of te equations (.), te correction functions are defined by (3.)-(3.4) and (u, φ ) are te DG solutions of (.7). Te initial discretizations are taken as u (, 0) = P, u 0( ) u k I (, 0) and φ (, 0) = P, φ 0( ) φ k I (, 0). Ten tere olds (e u,f + e u,c ) t k+ sup u(, τ) k+ (3.3) τ [0,t] Proof. Te proof of tis teorem is provided in te appendix; see section A.3 0

11 3.3 Superconvergence at some special points To study te superconvergence of te DG solution at special points. We firstly denote e u,r and e u,d te maximum error of u u at tese special points G j,l I j and derivative points G d j,m I j, respectively. Here G j,l are zeros of polynomials L j,k+, i.e. te standard Gaussian points, and G d j,m are zeros of te polynomials x L j,k+ on te interval I j. To be more precise, e u,r = max (u u )(G j,l ), j,l e u,d = max x(u u )(G d j,m ). (3.4) j,m Teorem 3.3. For any k, suppose (u, φ) [W k+, ] are te solutions of te equations (.), and (u, φ ) are te DG solutions of (.7). Te initial discretizations are taken as u (, 0) = P, u 0( ) and φ (, 0) = P, φ 0( ). Ten e u,r k+ sup u(, τ) k+,, τ [0,t] e u,d k+ sup u(, τ) k+, (3.5) τ [0,t] Proof. Te proof of tis teorem is provided in te appendix; see section A.4 Remark 3.. In our superconvergence analysis, te initial discretizations are of great significance. To obtain te (k + )-t order superconvergence rate at special points and towards te special projection of te exact solution, te initial value (u (x, 0), φ (x, 0)) = (P, u 0, P, φ 0) is a valid coice. However, to acieve te goal of (k + )-t order superconvergence for te numerical fluxes and cell averages, it is indicated in Teorem 3. tat te initial error sould also reac te same superconvergence rate, wic imposes a stronger condition on te initial discretization. A natural way of te initial discretization to obtain tis is to coose (u (x, 0), φ (x, 0)) = (P, u 0 u k, I (x, 0), P φ 0 φ k I (x, 0)).

12 4 Numerical examples In tis section, we give a numerical example. We solve te linear equation given by u t + u x = 0, φ t φ x = 0, u(x, 0) = sin(x), φ(x, 0) = 0 u(0, t) = u(π, t) (4.) Te exact solution is u(x, t) = sin(x t), φ(x, t) = 0 (4.) We use te fifteent order SSP time discretization [9] wit te time step t = CFL wit CFL = 0. to reduce te influence of te temporal error. Our computation is based on te numerical flux coice (.8) for te augmented system (.) on a uniform mes of N cells. Te numerical initial condition is taken by u (x, 0) = P, u uk I, φ (x, 0) = P, u φk I. Te various errors and numerical order of accuracy for Pk elements wit k 4 are listed in Tables We observe tat u P, u as (k +)-t order accuracy wic confirms our teoretical results, and te maximum function value error at Gaussian points e u,r and derivative error at te special points e u,d ave (k +)-t and (k + )-t order convergence rates respectively, and average errors of te fluxes e u,f and te cell averages e u,c are bot (k + )-t order, also confirming our teoretical results. Recall tat tese errors are defined by e u,r = max (u u )(G j,l ), j,l ( e u,f = N e u,c = N N j= j= ((u û )(x j+ e u,d = max j,m x(u u )(G d j,m), (4.3) ), t)) + ((φ ˆφ )(x j+, t)), (4.4) ( N ) ( ) (u u )(x) dx + (φ φ )(x) dx j I j j I j. (4.5)

13 Table 4.. Errors u P, u, e u,f and e u,c for k =,, 3, 4. T = k N u P, u order e u,f order e u,c order E-03.E-03.34E E E E k = E E E E E E E E E E E E E E E k = E E E E E E E E E E-07.66E-08.07E E E E k = E E E E E E E E E E-09.09E-.3E- 0.04E E E k = E E E E E E E E E Remark 4.. We note tat we can obtain te superconvergence results for te numerical fluxes and te cell-averages of u and φ respectively from Teorem 3., tey sould bot be (k + )-t order accurate. However, in Table 4.3, we can observe tat te orders of ( ) convergence for te numerical fluxes of u alone, namely ẽ u,f = N N j= ((u û )(x j+, t)), as fluctuations, and te errors for te cell averages of u alone, namely ( ) ( ) ẽ u,c = N N j= j I j (u u )(x) dx, ave a iger order tan k +, close to k+. Bot of tese errors are significantly smaller tan te corresponding errors for φ, ( ( namely ẽ φ,f = N N j= ((φ ˆφ ) )(x j+, t)) ( and ẽ φ,c = N N j= j I j (φ φ )(x) dx wic are bot (k + )-t order accurate as can be seen in Table 4.3. In fact, energy conserving DG scemes ave no numerical dissipation, tus any spurious numerical errors (e.g. tose from initial condition) is difficult to be damped. From Teorem 3., te superconvergence results we obtain are coupled results of u and φ. So we need to consider te sum of te errors of u and φ for teir numerical fluxes and cell averages, ) ), 3

14 Table 4.. Errors e u,r and e u,d for k =,, 3, 4. T = k N e u,r order e u,d order 0.65E E E E-03.0 k = 40.0E E E E E E E E E E k = E E E E E E E-07.69E E E k = E E E E E E E E E E k = E E E E E E wic are listed in Table 4. sowing a muc cleaner superconvergence rate as predicted by our analysis. However, te fact tat te errors for u are smaller tan tose for φ, as sown in Table 4.3, is good news, since te errors for u are wat we are really interested in. 5 Concluding remarks We ave studied te superconvergence beaviour of te energy-conserving DG solution for time dependent linear advection equation. We prove tat, wit suitable initial discretization, te error between te DG solution and te exact solution converges wit te rate of (k + )-t order for te cell averages and te numerical fluxes wen k. Moreover, we prove tat te error between te approximate solution and a special projection of te exact solution as a (k +)-t order superconvergence rate. We also proved tat te values and derivative values of numerical solution superconverge to tat of te 4

15 Table 4.3. Errors ẽ u,f, ẽ u,c, ẽ φ,f and ẽ φ,c for k =,, 3, 4. T = k N ẽ u,f order ẽ u,c order ẽ φ,f order ẽ φ,c order 0 4.6E-04.8E-04.08E-03.35E E E E E k = 40.5E E E E E E E E E E E E E E E E E E E E k = 40.99E E E E E E E E E E E E E E-0.64E-08.07E E E E E k = E E E E E E E E E E E E E- 6.57E-3.08E-.3E E E E E k = E E E E E E E E E E E E exact solution wit te rate of ((k + ))-t and (k + )-t order respectively at some special points. Numerical experiments demonstrate tese superconvergence results are optimal. In future work, we will consider te optimal analysis for nonlinear cases and use te superconvergence result of te energy-conserving DG scemes to construct new trouble cell indicators to design an adaptive solver to yperbolic equations. A Proof of a few tecnical lemmas and teorems In tis appendix, we collect te proof of some of te tecnical lemmas and teorems in te superconvergence error estimates. 5

16 A. Proof of Lemma 3. Proof. We prove tis lemma by induction. Since ω (0) u Ij P k (I j ) and ω (0) φ I j P k (I j ), ten from (3.)-(3.), we obtain ω () u I j P k (I j ) ω () φ I j P k (I j ) (A.6) Tus ω u () and ω () φ ave te following expressions on interval I j ω () u I j = k m=k c j,m L j,m(ξ), ω () φ I j = k m=k d j,m L j,m(ξ), ξ = (x x j) j [, ] Taking v = D x L j,k = x x j L j,k (ξ) dx P k (I j ) in (3.), we ave j k c j,k = (ε u ) t Dx L j,k dx I j (ε u ) t Ij D x L j,k Ij k++3 ( u k+,ij + φ k+,ij ) (A.7) Tus c j,k k+3 (A.8) By te same argument for ωφ, we obtain d j,k k+3 (A.9) To estimate c j,k and d j,k, we obtain from (3.3)-(3.4) c j,k + d j,k = ( ) k (c j,k + d j,k ) c j,k d j,k = c j,k d j,k Tus c j,k k+3, d j,k k+3 6

17 Consequently, ω () u I j ( ω () φ I j ( k m=k k m=k (c j,m ) ) k+ (d j,m) ) k+ (A.0) (A.) Taking time derivative on bot sides of (3.)-(3.4), te four identities still old. In oter words, we can replace (ω u (), ω () φ ) by ( tω u (), t ω () φ ) in (3.)-(3.4). Ten following te same arguments as wat we did for (ω u (), ω () ), we get φ t ω () u I j k+ t ω () φ I j k+ (A.) (A.3) By te recursion formula, (3.6) olds for all i k and r = 0,. Tis finises our proof. Remark A.. We note tat tere olds ω (i) u P 0 (I j ) and ω (i) φ P0 (I j ) for all i k, tus taking v = in (3.)-(3.) te equations become trivial. From te proof of tis teorem, we can see tat te correction functions are well-defined. A. Proof of Teorem 3. From Corollary 3., we ave a(u P, u + ul I, v ) + b(φ P, φ + φl I, ψ ) = a(u P, u + ul I, v ) + b(φ P, φ + φl I, ψ ) k++l ( v + ψ ) By taking v = P, u ul I and φ = P, φ φl I, combining wit te argument of energy-conserving, te designed estimates (3.9) are obtained. 7

18 A.3 Proof of Teorem 3. Since ω (i) u P 0 (I j ), ω (i) φ P 0 (I j ), i =,...,k and by (.8), we use Caucy- Scwartz inequality, (u u ) dx = (P, u uk I u ) dx + ω u (k) dx P, u uk I u + ω (k) u I j I j I j (φ φ ) dx = (P, φ φk I φ ) dx + I j I j Ten troug a directly calculation and by Teorem 3. (A.4) I j ω (k) φ dx P, φ φk I φ + ω (k) φ (A.5) e u,c t k+ sup u(, τ) k+ τ [0,t] (A.6) Since (P, u uk I u, P, φ φk I φ ) [V k], te inverse inequality olds, and ˆ ω u (i) j+ = 0, tus ( e u,f = N ( N N (( j= N j= ˆ u ûk I û )(x j+, t)) + (( P, ) ˆ φ ˆφ k I ˆφ )(x j+, t)) P, ( P, u uk I u,i j + P, φ φk I φ,i j ) P, u uk I u + P, φ φk I φ By Teorem 3., te final results are obtained. A.4 Proof of Teorem 3.3 ) (A.7) (A.8) (A.9) Since φ = 0, from (.8), we assume u ave expansions of te Legendre polynomials, and obtain tus k P, u = u j,m L j,m + u j,k+ P, L j,k+ + m=0 u P, u = u j,k+(l j,k+ P, L j,k+) + 8 m=k+ m=k+ u j,m P, L j,m (A.0) u j,m (L j,m P, L j,m) (A.)

19 = u j,k+ L j,k+ + m=k+ In [8], for u j,m, te following estimate is given u j,m (L j,m P, L j,m) (A.) u j,m i p u i+,p,ij, 0 i m (A.3) Tus we ave By Teorem 3. and te cosen initial value, we ave max (u P, u)(g j,l) k+ (A.4) j,l max j,m x(u P, u)(gd j,m) k+ (A.5) Since u P, u + u I ( + t)k+3 u k+4, (A.6) u I,Ij ( + t) k+ u k+3, (A.7) we obtain by te inverse inequality P, u u,ij ( + t) k+ u k+4, (A.8) P, u u,,ij ( + t) k+ u k+4, (A.9) Consequently, combining te above inequality wit (A.4)-(A.5) yields (3.5) directly. References [] S. Adjerid and T. C. Massey, Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar yperbolic problem. Computer Metods in Applied Mecanics and Engineering, 95: ,

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