Yong Liu ·Chi-Wang Shu ·Mengping Zhang

Abstract

Keywords Energy-conserving discontinuous Galerkin methods·Superconvergence·Linear hyperbolic equations

1 Introduction

Wave propagation is a fundamental form of energy transmission,which arises in many fields of science,engineering and industry,such as geoscience,petroleum engineering,telecommunication,and the defense industry (see [17,21] and the references therein).Efficient and accurate numerical methods to solve wave propagation problems are of fundamental importance to these applications.Experience reveals that energy-conserving numerical methods,which conserve the discrete approximation of energy,are favorable because they are able to maintain the phase and shape of the waves accurately,especially for long-time simulation.

A vast amount of literature can be found on the numerical approximation of wave problems modeled by linear hyperbolic systems.Recently,Fu and Shu [18] developed energyconserving discontinuous Galerkin (DG) methods for symmetric linear hyperbolic systems on general meshes.They obtain the optimal convergence accuracy of orderk+1 for the semidiscrete schemes.We will study the superconvergence properties of this class of DG schemes in this paper.DG methods are a class of finite element methods devised to solve the hyperbolic conservation laws and related equations,e.g.,[12– 16].They use completely discontinuous polynomial space for the test and trial functions in the spatial variables and coupled with explicit and nonlinearly stable high-order Runge–Kutta time discretization,with a compact stencil,and with the ability to easily accommodate arbitraryh-padaptivity.

In the past few years,there has been considerable interest in studying the superconvergence properties of the DG methods.We refer to [1,24,26] for ordinary diff erential equations,[2,3] for multidimensional first order hyperbolic systems,[9,10,23,25] for one-dimensional hyperbolic conservation laws and time-dependent convection-diff usion equations,and [20] for one- and two- dimensional hyperbolic equations based on the Fourier approach.In 2014,Cao et al.[8] introduced an approach to study the superconvergence of DG methods for linear hyperbolic equations with upwind numerical fluxes.They constructed a suitable correction function to correct the error between the exact solutionuand its Radau projection to obtain a (2k+1)-th superconvergence rate at the downwind points for piecewise polynomials of degreek.Later,Cao et al.extended this work to study upwind-biased numerical fluxes,degenerate variable coefficients and nonlinear cases for hyperbolic conservation laws [4,6,7].For one-dimensional linear convection-diff usion equations,Cao et al.[5] studied the superconvergence of the numerical solution by the direct discontinuous Galerkin methods for convection-diff usion equations.Superconvergence results can help us to design trouble cell indicators such as the KXRCF trouble cell indicator [22],which is a key point for adaptive DG schemes.Thus,we are interested in studying the superconvergence properties of the recently devised energy-conserving DG methods in [18].

In this paper,we use the techniques in Cao et al.[8] to construct a suitable correction function for correcting the error between the special projection of the exact solution and the exact solution.The main contribution of this paper is that we prove the approximate solution superconverges to a particular projection of the exact solution.The order of this superconvergence is proved to bek+2 when piecewise ℙkpolynomials withk≥1 are used.Numerical examples demonstrate this result is optimal.Furthermore,we find that the function value approximations of the DG solution are superconvergent at all Gaussian points with orderk+2 ,and the derivative approximations are superconvergent at zeros of a special polynomial with orderk+1 .We also prove,under a suitable choice of the initial discretization,a (2k+1)-th order superconvergence rate of the DG solution for the numerical fluxes and cell averages.Numerical examples show that this analysis is optimal.

The organization of this paper is as follows.In Sect.2,we first recall the energy-conserving discontinuous Galerkin method for the linear hyperbolic equations.Then,we construct a special interpolation function and study its optimal approximation properties to show the superconvergence results in Sect.3.We provide numerical examples to demonstrate our theoretical results in Sect.4.In Sect.5,we give concluding remarks and perspectives for future work.Finally,in the appendix we provide proofs for some of the more technical results of the superconvergence estimates.

2 Energy-Conserving DG Schemes

In this section,we consider the one-dimensional scalar linear equation

with periodic boundary condition.

The optimal energy-conserving DG method designed in [18] for the advection Eq.(2.1) needs to introduce an auxiliaryzerofunction(x,t)=0 ,which is the solution of an advection equation using the opposite speed as that foru(x,t),but withzeroinitial data.Then,we obtain the following 2×2 system:

with the initial conditionu(x,0)=u0(x) and(x,0)=0 .We first introduce the usual notation of the DG method.For a given interval Ω=[a,b] ,we divide it intoNcells as follows:

We denote

and

We also assume the mesh is regular,i.e.,the ratio between the maximum and minimum mesh sizes shall stay bounded during mesh refinements.We define the finite element space as

Here,ℙk(Ij) denotes the set of all polynomials of degree at mostkonIj.For a functionvh∈Vkh,we useto refer to the value ofvhatform the left cellIjand the right cellIj+1,respectively.[vh] is used to denotev+h−v−h,i.e.,the jump ofvhat cell interfaces.denotes the average of the left and right boundary values.We also introduce some standard Sobolev spaces notations.For any integerm>0 ,letWm,p(D) be the standard Sobolev spaces on the sub-domainD ⊂Ω equipped with the norm ‖⋅‖m,p,Dand semi-norm |⋅|m,p,D.WhenD=Ω ,we omit the indexD; and ifp=2 ,we setWm,p(D)=Hm(D) ,‖⋅‖m,p,D=‖⋅‖m,D,and |⋅|m,p,D=|⋅|m,D

The semi-discrete DG method for (2.2) reads as follows: findsuch that

withbeing any real constant.In [18],the following energy-conserving result is obtained.

Theorem 2.1The energy

is conserved by the semi-discrete scheme(2.7)with the numerical f lux(2.8)for all time.

Also,in [18],it is shown,with simply takingfor allj,an optimal convergence result with a clean proof.We will prove our superconvergence results also under this setting.We first introduce a set of projections.The left and right Gauss–Radau projectionsare defined by

For these projections,the following inequality holds [11]:

We also need the following basic facts.For any functionthe following inequalities hold [11]:

We shall also use the coupled projection specifically designed for the DG scheme (2.7) in [18].For any functionu,∈H1(Ω) ,define the following coupled auxiliary projection

for allj.It was shown in Lemma 2.6 in [18] that the projection is actually an optimal local projection,

Let us introduce a few notations.We define

where

We denote

Then,it is straightforward to deduce from (2.7),

Obviously,the exact solutionuandalso satisfies

Subtracting (2.25) from (2.26),we obtain the error equations

By (2.27),the estimate for ‖uh−P∗hu‖ can usually be reduced to estimating

A straightforward analysis using the definitions ofa(⋅,⋅) andb(⋅,⋅) results in

leading to the optimal error bound

where here and in the following,A ≲Bdenotes thatAcan be bounded byBmultiplied by a constant independent of the mesh sizeh.This rate is of course still far from our superconvergence goal.Similar to [8],in which Cao et al.constructed a correction function to help obtain the desired superconvergence results,we would also like to find a series of suitable correction functionsto improve the error betweenandsuch that

3 Construction of a Special Interpolation Function

We now define the correction functionsDenotethen we define

Note that the definition of the correction functions is recursive,thus we can firstly analyze the first correction functionsand then obtain similar properties by induction.We have the following properties for the correction functions.

Lemma 3.1For anyk≥1 ,supposeare defined by(3.1)–(3.4).Then,

andare some bounded constants,andLj,m()are the standard Legendre polynomials on intervalIj.Furthermore,ifthe following holds:

ProofThe proof of this lemma is provided in the appendix; see Sect.A.1.

From Lemma 3.1,we have the straightforward corollary as follows.

Corollary 3.1For anyk≥1 ,suppose(u,)∈[H2k+2]2are the solutions of the equations(2.2),and the correction functions are defined by(3.1)–(3.4),then

wherea(⋅,⋅)and b(⋅,⋅)are defined by(2.20)–(2.21).

3.1 Superconvergence of the Interpolation

Now,if we take our correction functions aswe have the following superconvergence results.

Theorem 3.1For anyk≥1 ,supposeare the solutions of the equa-tions(2.2),the correction functions are defined by(3.1)–(3.4)and(uh,)are the DG solutions of(2.7).Then,for1 ≤l≤k,

ProofThe proof of this theorem is provided in the appendix; see Sect.A.2.

As a direct consequence of (3.9) and (3.6),we have the following superconvergence result ofuhtowards the specially designed projection

Corollary 3.2For anyk≥1 ,supposeare the solutions of the equations(2.2),the correction functions are defined by(3.1)–(3.4)and(uh,)are the DG solutions of(2.7).The initial discretizations are taken asThen,

3.2 Superconvergence for the Numerical Fluxes and the Cell Averages

We denote byeu,fandeu,cthe errors of the fluxes and the cell averages,respectively.That is,

We have the following superconvergence results.

Theorem 3.2For anyk≥1 ,suppose(u,)∈[H2k+2]2are the solutions of the equations(2.2),the correction functions are defined by(3.1)–(3.4)and(uh,)are the DG solutions of(2.7).The initial discretizations are taken asuh(⋅,0)=P1,⋆h u0(⋅)−ukI(⋅,0)andThen,the following holds:

ProofThe proof of this theorem is provided in the appendix; see Sect.A.3.

3.3 Superconvergence at Some Special Points

To study the superconvergence of the DG solution at special points,we firstly denoteeu,randeu,dthe maximum error ofu−uhat these special pointsGj,l∈Ijand derivative pointsrespectively.Here,Gj,lare the zeros of polynomialsLj,k+1,i.e.,the standard Gaussian points,andare the zeros of the polynomialson the intervalIj.To be more precise,

Theorem 3.3For anyk≥2 ,supposeare the solutions of the equa-tions(2.2),and(uh,)are the DG solutions of(2.7).The initial discretizations are taken asThen,

ProofThe proof of this theorem is provided in the appendix; see Sect.A.4.

Remark 3.1In our superconvergence analysis,the initial discretizations are of great signifi-cance.To obtain the (k+2)-th order superconvergence rate at special points and towards the special projection of the exact solution,the initial valueis a valid choice.However,to achieve the goal of (2k+1)-th order superconvergence for the numerical fluxes and cell averages,it is indicated in Theorem 3.1 that the initial error should also reach the same superconvergence rate,which imposes a stronger condition on the initial discretization.A natural way of the initial discretization to obtain this is to choose

4 Numerical Examples

In this section,we give a numerical example.We solve the linear equation given by

The exact solution is

We use the 15th order SSP time discretization [19] with the time step Δt=CflhwithCFL=0.1 to reduce the influence of the temporal error.Our computation is based on the numerical flux choice (2.8) for the augmented system (2.2) on a uniform mesh ofNcells.The numerical initial condition is taken byThe various errors and numerical order of accuracy for ℙkelements with 1 ≤k≤4 are listed in Tables 1,2,3.We observe thathas (k+2)-th order accuracy which confirms our theoretical results,and the maximum function value error at Gaussian pointseu,rand derivative error at the special pointseu,dhave (k+2)-th and (k+1)-th order convergence rates,respectively,and average errors of the fluxeseu,fand the cell averageseu,care both (2k+1)-th order,also confirming our theoretical results.Recall that these errors are defined by

Table1 Errorsand eu,c for k=1,2,3,4 .T =1

Table1 Errorsand eu,c for k=1,2,3,4 .T =1

k N ‖uh −P1,⋆h u‖ Order eu,f Order eu,c Order 1 10 3.45E-03 – 2.12E-03 – 1.34E-03 – 20 4.35E-04 2.99 2.55E-04 3.05 1.64E-04 3.05 40 5.45E-05 3.00 3.16E-05 3.01 2.06E-05 3.00 80 6.82E-06 3.00 3.95E-06 3.00 2.57E-06 3.00 160 8.56E-07 3.00 4.97E-07 3.00 3.21E-07 3.00 2 10 1.03E-05 – 7.64E-06 – 5.27E-06 – 20 6.45E-07 4.00 2.68E-07 4.83 1.63E-07 5.01 40 4.02E-08 4.00 7.77E-09 5.11 5.08E-09 5.00 80 2.51E-09 4.00 2.50E-10 4.96 1.59E-10 5.00 160 1.57E-10 4.00 7.73E-12 5.02 4.96E-12 5.00 3 10 2.78E-07 – 1.66E-08 – 1.07E-08 – 20 8.71E-09 5.00 1.31E-10 6.98 8.18E-11 7.03 40 2.72E-10 5.00 1.01E-12 7.02 6.38E-13 7.00 80 8.51E-12 5.00 7.87E-15 7.01 4.99E-15 7.00 160 2.66E-13 5.00 6.12E-17 7.01 3.90E-17 7.00 4 10 6.66E-09 – 2.09E-11 – 1.31E-11 – 20 1.04E-10 6.00 3.81E-14 9.10 2.49E-14 9.04 40 1.63E-12 6.00 7.48E-17 8.99 4.86E-17 9.00 80 2.55E-14 6.00 1.48E-19 8.97 9.50E-20 9.00 160 3.98E-16 6.00 2.88E-22 9.01 1.86E-22 9.00

Remark 4.1We note that we can obtain the superconvergence results for the numerical fluxes and the cell averages ofuhandrespectively,from Theorem 3.2; they should both be (2k+1)-th order accurate.However,in Table 3,we can observe that the orders of convergence for the numerical fluxes ofualone,namely,has fluctuations,and the errors for the cell averages ofualone,namely,have a higher-order than 2k+1 ,close to 2k+2 .Both of these errors are significantly smaller than the corresponding errors fornamely,which are both (2k+1)-th order accurate as can be seen in Table 3.In fact,the energy-conserving DG schemes have no numerical dissipation,thus any spurious numerical errors (e.g.,those from the initial condition) are difficult to be damped.From Theorem 3.1,the superconvergence results we obtain are coupled results ofuhandSo we need to consider the sum of the errors ofuhandfor their numerical fluxes and cell averages,which are listed in Table 1,showing a much cleaner superconvergence rate as predicted by our analysis.However,the fact that the errors foruare smaller than those foras shown in Table 3,is good news,since the errors foruare what we are really interested in.

Table2 Errors eu,r and eu,d for k=1,2,3,4 .T =1

5 Concluding Remarks

We have studied the superconvergence behaviour of the energy-conserving DG solution for the time-dependent linear advection equation.We prove that,with suitable initial discretization,the error between the DG solution and the exact solution converges with the rate of (2k+1)-th order for the cell averages and the numerical fluxes whenk≥1 .Moreover,we prove that the error between the approximate solution and a special projection of the exact solution has a (k+2)-th order superconvergence rate.We also proved that the values and derivative values of the numerical solution superconverge to that of the exact solution with the rate of (k+2)-th and (k+1)-th order,respectively,at some special points.Numerical experiments demonstrate these superconvergence results are optimal.In future work,we will consider the optimal analysis for nonlinear cases and use the superconvergence result of the energy-conserving DG schemes to construct new trouble cell indicators to design an adaptive solver to hyperbolic equations.

Table3 Errors and for k=1,2,3,4 .T =1

Table3 Errors and for k=1,2,3,4 .T =1

?

Appendix: Proof of a Few Technical Lemmas and Theorems

In this appendix,we collect the proofs of some of the technical lemmas and theorems in the superconvergence error estimates.

A.1 Proof of Lemma 3.1

We prove this lemma by induction.Sinceandfrom (3.1)–(3.2),we obtain

Thus,

By the same argument forwe obtain

Thus,

Consequently,

Taking time derivative on both sides of (3.1)–(3.4),the four identities still hold.In other words,we can replacein (3.1)–(3.4).Then,following the same arguments as what we did forwe get

By the recursion formula,(3.6) holds for all 1 ≤i≤kandr=0,1 .This finishes our proof.

Remark A.1We note that there holdsfor all 1 ≤i≤k−1 ,thus takingvh=1 in (3.1)–(3.2) the equations become trivial.From the proof of this theorem,we can see that the correction functions are well defined.

A.2 Proof of Theorem 3.1

From Corollary 3.1,we have

A.3 Proof of Theorem 3.2

Then,through a direct calculation and by Theorem 3.1,

By Theorem 3.1,the final results are obtained.

A.4 Proof of Theorem 3.3

thus

In [4],foruj,m,the following estimate is given:

Thus,we have

By Theorem 3.1 and the chosen initial value,we have

Since

we obtain

by the inverse inequality

Consequently,combining the above inequality with (A.19)–(A.20) yields (3.15) directly.

References

1.Adjerid,S.,Massey,T.C.: Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem.Comput.Methods Appl.Mech.Eng.195,3331–3346 (2006)

2.Adjerid,S.,Weinhart,T.: Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems.Comput.Methods Appl.Mech.Eng.198,3113–3129 (2009)

3.Adjerid,S.,Weinhart,T.: Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems.Mathe.Comput.80,1335–1367 (2011)

4.Cao,W.,Li,D.,Yang,Y.,Zhang,Z.: Superconvergence of discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations.ESAIM.51,467–486 (2017)

5.Cao,W.,Liu,H.,Zhang,Z.: Superconvergence of the direct discontinuous Galerkin method for convection-diff usion equations.Numer.Methods Partial Diff er.Equ.33,290–317 (2017)

6.Cao,W.,Shu,C.-W.,Yang,Y.,Zhang,Z.: Superconvergence of discontinuous Galerkin method for nonlinear hyperbolic equations.SIAM J.Numer.Anal.56,732–765 (2018)

7.Cao,W.,Shu,C.-W.,Zhang,Z.: Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients.ESAIM.51,2213–2235 (2017)

8.Cao,W.,Zhang,Z.,Zou,Q.: Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations.SIAM J.Numer.Anal.52,2555–2573 (2014)

9.Cheng,Y.,Shu,C.-W.: Superconvergence and time evolution of discontinuous Galerkin finite element solutions.J.Comput.Phys.227,9612–9627 (2008)

10.Cheng,Y.,Shu,C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diff usion equations in one space dimension.SIAM J.Numer.Anal.47,4044–4072 (2010)

11.Ciarlet,P.G.: The Finite Element Method for Elliptic Problems.North-Holland Publishing Co.,Amsterdam,New York (1978)

12.Cockburn,B.,Hou,S.,Shu,C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case.Math.Comput.54,545–581 (1990)

13.Cockburn,B.,Lin,S.-Y.,Shu,C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems.J.Comput.Phys.84,90–113 (1989)

14.Cockburn,B.,Shu,C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws.II.General framework.Math.Comput.52,411–435 (1989)

15.Cockburn,B.,Shu,C.-W.: The Runge–Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws.Math.Model.Numer.Anal.25,337–361 (1991)

16.Cockburn,B.,Shu,C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems.J.Comput.Phys.141,199–224 (1998)

17.Durran,D.R.: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics.Springer-Verlag,New York (1999)

18.Fu,G.,Shu,C.-W.: Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems.Submitted to Journal of Computational Physics.arXiv:1805.04471

19.Gottlieb,S.,Shu,C.-W.,Tadmor,E.: Strong stability-preserving high-order time discretization methods.SIAM Rev.43,89–112 (2001)

20.Guo,W.,Zhong,X.,Qiu,J.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: eigen-structure analysis based on Fourier approach.J.Comput.Phys.235,458–485 (2013)

21.Kampanis,N.A.,Ekaterinaris,J.,Dougalis,V.: Eff ective Computational Methods for Wave Propagation.Chapman & Hall/CRC,Boca Raton (2008)

22.Krivodonova,L.,Xin,J.,Remacle,J.F.,Chevaugeon,N.,Flaherty,J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws.Appl.Numer.Math.48,323–338 (2004)

23.Liu,Y.,Shu,C.-W.,Zhang,M.: Optimal error estimates of the semidiscrete central discontinuous Galerkin methods for linear hyperbolic equations.SIAM J.Numer.Anal.56,520–541 (2018)

24.Xie,Z.,Zhang,Z.: Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D.Math.Comput.79,35–45 (2010)

25.Yang,Y.,Shu,C.-W.: Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations.SIAM J.Numer.Anal.50,3110–3133 (2012)

26.Zhang,Z.,Xie,Z.,Zhang,Z.: Superconvergence of discontinuous Galerkin methods for convectiondiff usion problems.J.Sci.Comput.41,70–93 (2009)