ERROR ESTIMATE TO SMOOTH SOLUTIONS OF HIGH ORDER …

L2 ERROR ESTIMATE TO SMOOTH SOLUTIONS OFHIGH ORDER RUNGE–KUTTA DISCONTINUOUSGALERKIN METHOD FOR SCALAR NONLINEAR

CONSERVATION LAWS WITH AND WITHOUT SONICPOINTS

JINGQI AI∗, YUAN XU† , CHI-WANG SHU‡ , AND QIANG ZHANG§

Abstract. In this paper we shall establish an a priori L2-norm error estimate of the fourthorder Runge–Kutta discontinuous Galerkin method for solving sufficiently smooth solutions of one-dimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtainedunder the standard Courant-Friedrichs-Lewy condition, and the quasi-optimal and/or optimal orderof accuracy in space are achieved for many widely-used numerical fluxes, no matter whether thesolution contains sonic points or not. The main tools used in this paper are the matrix transferringprocess and the generalized Gauss-Radau projection of the reference functions, depending on therelative upwind effect. Finally some numerical experiments are given to support our theoreticalresults.Keywords. error estimate, Runge–Kutta discontinuous Galerkin method, nonlinear conservationlaw, numerical flux, sonic point.AMS Subject Classification. 65M12, 65M15

1. Introduction. The discontinuous Galerkin (DG) method was firstly proposed by Reed andHill [22] for steady linear transport problem, and then it was developed into the Runge–Kutta dis-continuous Galerkin (RKDG) method to solve unsteady nonlinear conservation law by Cockburn etal. [7–11]. After that, this method has been widely used in many fields, for example, for solving theEuler equation, the shallow water equation and so on. Numerical experiments show good stability,high-order accuracy, and the ability to sharply capture discontinuous jumps. For more details, thereaders are referred to the review paper [24] and the references therein.

Compared with ample successful applications in numerics, there are fewer theoretical analysesfor the DG method. Limited in the error estimate, many works in the literature are focused on thesemidiscrete DG method. For linear hyperbolic equations, the L2-norm stability and error estimateshave been set up in [13,15], where the optimal order of accuracy is obtained when the upwind numericalflux is used. For nonlinear conservation laws, it is well known that the L2-norm of the numericalsolution does not increase with time, when the monotone numerical flux is used, owing to the localcell entropy inequality [12]. It is implicitly proved in [20, 29] that the semidiscrete DG method hasthe quasi-optimal and optimal error estimate for most of monotone numerical fluxes and the upwindnumerical flux, respectively. In recent years, more attention has been paid to the DG method withgeneralized numerical fluxes, which may be not monotone fluxes. For example, the upwind-biasednumerical flux has been studied for linear hyperbolic equations [19], as an extension of the upwindflux; and the generalized local Lax-Friedrichs (GLLF) flux has been studied for nonlinear conservationlaws [17], as an extension of the local Lax-Friedrichs (LLF) flux. The optimal convergence orders areboth obtained by the help of the generalized Gauss-Radau (GGR) projection that is independent of

∗Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China. E-mail:[email protected]. Research is partially supported by NSFC grant 12071214.†Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China. E-mail:

[email protected]. Research is partially supported by NSFC grant 12071214 and Postgraduate Research andPractice Innovation Program of Jiangsu Province grant KYCX21 0028.‡Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: chi-wang [email protected].

Research is partially supported by NSF grant DMS-2010107 and AFOSR grant FA9550-20-1-0055.§Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China. E-mail:

[email protected]. Research is partially supported by NSFC grants 12071214 and 11671199.

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the time.As for the fully discrete RKDG method, there are not enough theoretical results to support the

numerical performance, for instance, how to strictly prove the accuracy order in space and in timefor the sufficiently smooth solution of nonlinear conservation laws. The second order and the thirdorder (in time) RKDG methods with the monotone flux have been proved to have quasi-optimaland/or optimal error estimates in [32,34], under suitable temporal-spatial conditions. In general, thestandard Courant-Friedrichs-Lewy (CFL) condition is expected, namely, the ratio of the time stepover the spatial mesh size is bounded by a fixed constant. The second order RKDG method achievesthis goal only for k ≤ 1, while the third order RKDG method does it for any k. Here and below k isthe degree of piecewise polynomials used in the discontinuous finite element space. The above resultshave been extended to systems of conservation laws in [18,33]. Similar works can also be found in [3].

Actually, the fourth or higher order time-marching is also widely used in the DG framework[1, 14, 27]. In this paper we turn our attention to the error estimate of the higher order RKDGmethods. However, this task is difficult because we encounter a serious trouble in setting up an energyequation facilitating the error estimate. Recently, this obstacle is overcome by the matrix transferringprocess based on the temporal difference of stage solutions [30, 31]. With the help of this technique,the authors have systematically carried out the L2-norm stability analysis for the RKDG methodswith the upwind-biased numerical flux to solve linear constant-coefficient hyperbolic equations. Thereare three different stability performances, including the monotonicity stability, strong stability (multi-steps monotonicity stability) and the weak stability. After that, this technique has been successfullyapplied to the error estimate [30], the superconvergence analysis [28] and so on, for linear hyperbolicequations.

In this paper we continue the work in [30] and carry out the error estimates of high order RKDGmethod to solve the one-dimensional scalar nonlinear conservation law

(1.1) ut + f(u)x = 0, (x, t) ∈ [0, 1]× (0, T ],

where the unknown solution u(x, t) is sufficiently smooth till the final time T . In particular, theinitial condition u(x, 0) = u0(x) is smooth and hence bounded. For simplicity, the periodic boundarycondition is enforced. For convenience of statement, we would like in this paper to demand thefollowing assumptions:F1. The flux function f(·) is sufficiently smooth in R = (−∞,+∞). Specially, f ′′(·) is continuous

and bounded everywhere. This hypothesis is acceptable since the maximum principle allowsus to modify f(·) outside the range of u0(x); see [32] for more details.

F2. The number of sonic points such that f ′(·) = 0 is finite. Due to the continuity, f ′(·) keepseither nonpositive or nonnegative in each closed subinterval of R, which is divided by thesonic points. We remark that the subinterval is just R if no sonic point appears.

In this paper we would like to show the result for a fourth-order RKDG method as an example. Theother high order RKDG methods can be similarly studied.

Generally speaking, although the error estimate is technical and complicated, the line of analysisis almost the same as that in [30] for the linear case and that in [32, 34] for the nonlinear case. Inthis paper we show that the matrix transferring process is independent of the detailed definition ofthe spatial discretization, and thus can be successfully extended to the nonlinear case. Based on thisdevelopment, we are able to achieve an useful energy equation to carry out the error estimate of thefourth order RKDG method for nonlinear conservation laws.

Furthermore, we want to get the optimal time order for this method under the standard CFLcondition. To this end, we have to fully use the stability performance of this method for the linearcase. For the four stages and fourth order RKDG method with the upwind-biased numerical flux,two-step L2-norm monotonicity stability under the standard CFL condition is proved in [31]. This is amain difference with the lower order RKDG methods. Inspired by this, we would like in this paper tocarry out the error estimate on every two steps of time marching. If we directly perform the analysison every step of time marching, we would need to require the time step to be high order of the spatial

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mesh size, which does not coincide with the numerical experimental results.For nonlinear conservation laws, many numerical experiments show that the RKDG method with

the same degree k, sometimes, has different convergence orders for different numerical fluxes. Theoptimal order is always observed on quasi-optimal mesh for many examples, even when the numericalflux is not purely upwind and/or when the sonic points appear. Nevertheless, one-half order reductioncould also happen in some cases even on uniform meshes when sonic points appear. In this paper wewould like to establish a sharp estimate so that the theoretical convergence order perfectly coincideswith the numerical experiments. The result proposed in this paper will show that this order reductionis resulted from the incompatibility between the flow speed and the strength of numerical viscosity atthe sonic point position.

To achieve the above aim, we have to resolve several issues, to be detailed below. It is worthy topoint out that we can apply these techniques to the semidiscrete DG method and obtain the samespatial orders as in this paper.

From our experiments, the L2-norm stability mechanism in the RKDG method is often explicitlyexpressed by the numerical viscosities associated with some temporal differences at different timestages. Generally speaking, the numerical viscosity is proportional to the square of the jump at theelement endpoint, and its strength strongly depends on the numerical flux being used. To measurethe strength of the numerical viscosity, we would like to follow the context in [32, 34] and introducean important quantity; see (2.8) in this paper. For nonlinear conservation laws, this quantity usuallydepends on the numerical solution under consideration. For the purpose of error estimation, we movethe dependence from the numerical solution to the exact solution. Distinguished with the linearproblem, the relationship between the flow speed and the strength of numerical viscosity plays animportant role to determine the accuracy order in space. In our previous works [32, 34], we havecarried out error estimates under the assumption on their disparity. This description works well forthe RKDG methods with the monotone flux and gives the quasi-optimal order in space. However,this description makes the proof complicated and nonintuitive when obtaining the optimal order forthe upwind flux with the help of the Gauss-Radau (GR) projection. In this paper we would like tocarry out error estimate of the RKDG method with general numerical fluxes, and turn to the ratio ofthe flow speed over the strength of numerical viscosity. We remark that, for some numerical fluxes,this ratio may be zero (or infinity) at the sonic point. This incompatibility, we think, represents toomuch (or too little) numerical viscosity and leads to the reduction of the optimal order of accuracy inspace.

It is well-known that a suitably-defined projection is important to get good error estimates.The local L2-projection is widely used to get the quasi-optimal (k + 1/2)-th order in space when thecorrection of the flow speed (2.10) is bounded. However, in order to obtain the optimal (k+1)-th orderin a similar line of proof, we propose in Section 3.4 a novel projection based on the GGR projection.This projection is motivated by two series of studies on the DG methods. One is the upwind-biased fluxfor linear hyperbolic equations [5,16,19] and the GLLF flux for nonlinear conservation laws [17]. Theother is the upwind flux for nonlinear conservation laws [32,34], where the GR projection is proposedsimilarly. The projection proposed in this paper strongly depends on the ratio of the flow speed overthe strength of numerical viscosity, and hence is time dependent in many cases. If the strength ofnumerical viscosity is compatible with the flow speed, we can easily prove that the projection errorhas optimal order. Otherwise, if the incompatibility happens at the sonic point, the projection erroronly has the sub-optimal k-th order in general. However, by the help of the superconvergence on theuniform mesh, we make rigorous matrix analysis and successfully recover the optimal projection errororder, for the global Lax-Friedrich (GLF) flux and the global FORCE (GFORCE) flux when k is odd,and for the Richtmyer flux when k is even. We refer to Section 2 and the appendix for more details.

The remaining context of this paper is organized as follows. In Section 2 we first present theRKDG methods and then show the main conclusion for the four stages and fourth order RKDGmethod. The quasi-optimal and/or optimal error estimates are given under some assumptions on thenumerical flux. In Section 3 we set up some preliminaries to carry out the energy analysis for the

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RKDG methods. Firstly we recall the matrix transferring process and set up the important energyequation for the nonlinear problem. Then we present some inverse inequalities and make some abstractdiscussions on the difference of the DG discretization. Lastly we propose the GGR projection andset up sharp approximation properties. The main conclusion is proved in Section 4, and numericalexperiments are given in Section 5. Concluding remarks and some technical proofs are given in Section6 and in the appendix, respectively.

2. The RKDG method and the main conclusion. In this section we first present the RKDGmethod following the notations in [30,34], and then show the error estimates under some assumptionson the numerical flux.

2.1. The RKDG method. Let Ih = ∪1≤j≤JIj be a quasi-uniform partition of I = (0, 1), whereJ is an integer and Ij = (xj−1/2, xj+1/2) has the length hj = xj+1/2−xj−1/2. Here h is the maximumof hj for 1 ≤ j ≤ J . The discontinuous finite element space is defined as

(2.1) Vh = v ∈ L2(I) : v|Ij ∈ Pk(Ij), 1 ≤ j ≤ J,

where Pk(Ij) denotes the space of polynomials in Ij of degree at most k ≥ 1. The jump and theaverage at the element boundaries are respectively denoted by

(2.2) [[v]] = v+ − v−, v =1

2(v− + v+),

where v− and v+ are the traces along the left and the right-directions, respectively. The subscript isomitted here for simplicity.

To define the RKDG method of (1.1), we would like to begin with the semi-discrete DG method:find the map uh(t) : [0, T ]→ Vh such that

(2.3)(

(uh)t, vh

)Ih

= H(uh, vh), ∀ vh ∈ Vh, t ∈ (0, T ],

where (·, ·)Ih is the inner product in L2(Ih) as usual, and

(2.4) H(uh, vh) =∑

1≤j≤J

∫Ij

f(uh)(vh)xdx︸︷︷︸(f(uh),(vh)x

)Ih

+∑

1≤j≤J

(f(u−h , u

+h )[[vh]]

)j+ 1

2︸︷︷︸⟨f(u−h ,u

+h ),[[vh]]

⟩Γh

is the DG spatial discretization associated with the periodic boundary condition. Here Γh denotesthe element boundaries, and f(u−h , u

+h ) is the numerical flux. As usual, we demand that f(·, ·) is

consistent to f(·) and Lipschitz continuous with respect to its two arguments. More discussions onthe numerical fluxes will be given later.

Let N be a positive integer, and take the uniform time step τ = T/N for simplicity. The RKDGmethod employs the explicit Runge–Kutta (RK) algorithm to (2.3) and gives the numerical solutionat every time level tn = nτ , denoted by unh ∈ Vh, for 0 ≤ n ≤ N . The time-marching from tn to tn+1

is traditionally given in the Shu-Osher representation:• Let un,0h = unh;

• For ` = 0, 1, . . . , s− 1, successively find the stage solution un,`+1h ∈ Vh by the variational form

(2.5)(un,`+1h , vh

)Ih

=∑

0≤κ≤`

[c`κ

(un,κh , vh

)Ih

+ τd`κH(un,κh , vh)], ∀ vh ∈ Vh,

where c`κ and d`κ are the given parameters by the explicit RK algorithm with s stages andrth order;

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• Let un+1h = un,sh .

In this paper we take the four stages and fourth order RKDG method as an example. Specially, weset s = r = 4 and the parameters [25]

(2.6) c`κ =

1

1/2 1/21/9 2/9 2/30 1/3 1/3 1/3

, d`κ =

1/2−1/4 1/2−1/9 −1/3 1

0 1/6 0 1/6

,where the arguments ` and κ are taken from 0, 1, 2, 3 in natural order.

The initial solution can be taken as any approximation of u0(x). In this paper we use the localL2-projection and define u0h = Phu0 ∈ Vh. That is to say, there holds

(2.7)(u0h − u0, vh

)Ih

= 0, ∀ vh ∈ Vh.

We have now completed the definition of the RKDG scheme.

2.2. Main conclusion. Now we present the error estimate to the smooth solution of the abovefourth order RKDG method. To show the relationship between the convergence order in space andthe numerical flux, we introduce a quantity as in [18,32,34]

(2.8) α(f ;w−, w+) ≡12 [f(w−) + f(w+)]− f(w−, w+)

w+ − w−,

to measure the strength of the numerical viscosity due to f(w−, w+). Here w− and w+ respectivelyare the left- and the right- status. If the denominator is equal to zero, this formulation is understoodas the limit.

This quantity can be directly discussed for every numerical flux. To be more general, however,we would like in this paper to propose the following assumptions:A1. The function α(f ;w−, w+) is Lipschitz continuous everywhere in the sense

(2.9) |α(f ;w−, w+)− α(f ; w, w)| ≤ L|[[w]]|, ∀w−, w+ ∈ R,

where L is a fixed constant.A2. Denote α(w) ≡ α(f ;w,w) for convenience. This function is non-negative, bounded and Lipschitz

continuous everywhere. Furthermore, the correction of the flow speed

(2.10) π(w) ≡ |f′(w)|α(w)

|f ′(w)|

is bounded everywhere and the relative upwind effect

(2.11) β(w) ≡12 |f′(w)| − α(w)

12 |f ′(w)|+ α(w)

∈ [−1, 1]

is Lipschitz continuous on the closure of every subinterval divided by the sonic points, wherethe flow speed f ′(w) keeps either nonpositive or nonnegative.

If the denominators in (2.10) and (2.11) are equal to zero, the corresponding formulations are alsounderstood as the limits.

Remark 2.1. Since π(·) is bounded, we allow α(w) = 0 only at the sonic point. This statementcoincides with the upwind mechanism in the DG methods.

Remark 2.2. The boundedness of α(·) implies that β(w) may achieve the critical status ±1 onlyat the sonic point. Moreover, β(w) is allowed to be discontinuous at the sonic point; see the GLLFflux below.

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Assumptions A1 and A2 hold for many widely-used numerical fluxes [17, 21, 26], since the fluxfunction f(·) satisfies the aforementioned assumptions F1 and F2 in the introduction. Some examplesare listed below:

• The GLF flux is defined in the form

(2.12) f(w−, w+) =1

2

[f(w−) + f(w+)− CGLF(w+ − w−)

],

where CGLF is usually taken as the maximum of |f ′(·)| over the range of the initial solution, toensure (2.12) to be the monotone flux or the E-flux. Actually, CGLF can be any fixed positiveconstant independent of the numerical solution and the mesh size. It is easy to see that

α(f ;w−, w+) = α(w) =CGLF

2,

which reflects the upwind mechanism. All assumptions are easily verified, except that theLipschitz continuity of β(·) is implied by the boundedness of f ′′(·) and

|β(w)− β(v)| ≤∣∣∣∣ |f ′(w)|α(w)

− |f′(v)|α(v)

∣∣∣∣ .It is important to point out that only one critical status β(w) = −1 is achieved for the sonicpoint.

• The GLLF flux was firstly proposed in [17], as an extension of the LLF flux. It is defined inthe form

f(w−, w+) =

(1

2+ %1

)f(w−) +

(1

2− %1

)f(w+)− %2%0[[w]],

where %1 and %2 are two given weight parameters satisfying %2 > |%1|, and %0 is the maximumof |f ′(·)| between w− and w+. When %2 = 1/2 and %1 = 0, the GLLF flux is just the LLF

flux. It is easy to get α(f ;w−, w+) = %2%0 + %1f(w+)−f(w−)

w+−w− , and hence

α(w) = [%2 + %1sgnf ′(w)]|f ′(w)|.

All assumptions can be verified similarly and trivially, so the verification is omitted here. Notethat β(w) is piecewise constant in (−1, 1) and discontinuous at the sonic point (or subintervalinterface), due to the sign function in α(w).

• The Godunov flux is an upwind numerical flux, defined in the form

f(w−, w+) =

minw−≤v≤w+ f(v), w− ≤ w+,

maxw+≤v≤w− f(v), w− > w+.

It is easy to get α(w) = 12 |f′(w)| and to verify the assumptions, since f ′′(·) is continuous and

bounded. Note that β(w) ≡ 0, which means the best upwind effect under our viewpoint.• The Richtmyer flux [21,23,26] is given in the form

(2.13) f(w−, w+) = f

(1

2

[w− + w+ − CR(f(w+)− f(w−))

]),

where CR is originally defined as the ratio of time step over the mesh size. In this paper weonly demand CR to be any fixed positive constant. Since f ′′(·) is bounded, it follows fromL’Hospital’s rule that

(2.14) α(w) =CR

2|f ′(w)|2,

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and all assumptions can be trivially verified by simple calculations. For example,

|β(w)− β(v)| ≤ 4

∣∣∣∣ α(w)

|f ′(w)|− α(v)

|f ′(v)|

∣∣∣∣ ,which implies the Lipschitz continuity of β(·). It is worthy to emphasize that only one criticalstatus β(w) = 1 is achieved for the sonic point.

• The GFORCE (or LFORCE) flux [21, 26] is the mathematical average of the GLF (or LLF)flux and the Richtmyer flux. Consequently, all assumptions are easily checked with

(2.15) α(w) =1

4

[α(w) + CR|f ′(w)|2

].

For the GFORCE flux α(w) = CGLF and only one critical status β(w) = −1 is achieved forthe sonic point. For the LFORCE flux α(w) = |f ′(w)| and no critical status happens.

Throughout this paper the usual notations of Sobolev spaces are used. For example, for integersm1 and m2 and the spatial domain Ω, let Wm2,∞(Ω) be the Sobolev space that the function itselfup to the m2-th order spatial derivatives all belong to L∞(Ω) = W 0,∞(Ω), and Wm1,∞(Wm2,∞(Ω))denotes the space-time Sobolev space that the function itself up to the m1-th order time derivativesat t ∈ [0, T ] are all bounded in Wm2,∞(Ω).

Now we are ready to present the error estimate in the following theorem, whose detailed proofwill be given in the next two sections.

Theorem 2.1. Let uh be the numerical solution of the fourth order RKDG method (2.5), wherethe parameters are given in (2.6) and assumptions A1 and A2 hold. Assume the exact solution of theconservation law (1.1) is sufficiently smooth, for example

(2.16) u ∈W 3,∞(W k+2,∞(I)), uttttt ∈ L∞(L2(I)),

then for sufficiently small h there holds the L2-norm error estimate

(2.17) max0≤n≤N

‖u(x, tn)− unh(x)‖ ≤ Cbnd(hk+ζ + τ4),

under the standard temporal-spatial condition λ ≡ τ/h ≤ λmax, where λmax and Cbnd are positiveconstants independent of h and τ . Moreover,

1. ζ = 1/2 for k ≥ 2 on the quasi-uniform partition, regardless of the range of β(·);2. ζ = 1 for k ≥ 1 for either of the following status:

(a) |β(w)| < 1 holds for any w ∈ R and the partition is quasi-uniform;(b) β(w) = (−1)k is achieved at every sonic point and the partition is uniform.

We would like here to give some remarks on the conclusions. For the problem without sonic points,the optimal error estimate holds for the aforementioned numerical fluxes, no matter whether they areupwind flux or not. However, for the problem with sonic points, the convergence order stronglydepends on the numerical flux. The optimal order on the quasi-uniform mesh is still preserved formost numerical fluxes, except the GLF flux, the GFORCE flux, and the Richtmyer flux. Althoughthe incompatibility may cause the reduction of accuracy order, the optimal order in space can berecovered on the uniform mesh for the above three numerical fluxes. Case 2(b) shows that the resultdepends on the parity of k. For the GLF flux and the GFORCE flux, we demand k to be odd. Forthe Richtmyer flux we demand k to be even.

Remark 2.3. The smoothness assumption of the exact solution can be weakened. It depends onthe projection technique on the reference functions. We refer to Section 3.4 and the appendix.

Remark 2.4. There is a restriction on k ≥ 2 for the first item in Theorem 2.1. This is resultedfrom the proof technique to deal with the nonlinearity of f(·). For the linear hyperbolic equation, thisrestriction is not necessary. See Remark 5.1 in [34] for more details.

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3. Preliminaries. In this section we present some basic conclusions that will be used in theupcoming energy analysis.

3.1. Matrix transferring process. The matrix transferring process provides a good strategyto obtain the useful energy equation. It was first proposed in [31], in order to carry out a systematicanalysis on the L2-norm stability for arbitrary order RKDG methods. Limited by the length of thispaper, we would like to briefly present the related context for the considered fourth order RKDGmethod.

To be more general, we make some extensions and write (2.5) into an abstract formulation

(3.1)(un,`+1h , vh

)Ih

=∑

0≤κ≤`

[c`κ

(un,κh , vh

)Ih

+ τd`κHn,κ(vh)], ∀ vh ∈ Vh,

where H(un,κh , vh) is replaced by a linear functional Hn,κ(vh). Here un,κh should be looked upon as anarbitrary stage solution belonging to Vh, not limited as the solution of (2.5).

Below we give a short description on the matrix transferring process. Following [30,31], throughoutthis paper we introduce the generalized notation

un,4κ+`h = un+κ,`h

for any integer κ > 0 and ` = 0, 1, 2, 3. Since the fourth order RKDG methods has been proved in [31]to have two-step monotonicity stability for linear constant-coefficient hyperbolic equations, we wouldlike to define the temporal differences of stage solutions for every two steps of time marching

(3.2) D`unh =∑

0≤κ≤`

σ`κun,κh , ` = 0, 1, . . . , 8,

where σ`κ solely depends on the used RK algorithm, showing in the ninth order matrix

(3.3) σ`κ =

1−2 2

0 −4 44 0 −8 48 0 −16 −16 24

−32 48 64 16 −144 48−64 −96 32 32 288 −288 96352 −192 −320 −32 96 576 −576 96−64 768 −256 −256 −768 0 768 −768 576

.

The first row is fixed forever, namely, D0unh = unh. The other row sums are equal to zero.

In fact, the data in (3.3) are easily obtained by an inductive process. Assume that the `-th rowhave been obtained, a linear combination of stage marching (3.1) for κ ≤ ` yields

(3.4) τD`Hn(vh) ≡ τ∑

0≤κ≤`

σ`κHn,κ(vh) =∑

0≤κ≤`+1

σ`+1,κ

(un,κh , vh

)Ih.

The right hand side gives the coefficients σ`+1,κ to define the new temporal difference D`+1unh. In a

short notation, (3.4) implies the relationship

(3.5)(D`+1u

nh, vh

)Ih

= τD`Hn(vh), ∀ vh ∈ Vh.

Since σ`κ is reversible, definition (3.2) yields the evolution identity

(3.6) γ0un+2h =

∑0≤`≤8

γ`D`unh,

8

with (γ0, γ1, · · · , γ8) = (576, 1152, 1152, 768, 384, 144, 40, 8, 1). By virtue of the matrix transferringprocess, we are able to set up a series of energy equations

(3.7) γ20

(‖un+2

h ‖2 − ‖unh‖2)

=∑

0≤i,j≤8

a(`)ij

(Diunh,Djunh

)Ih

+ τ∑

0≤i,j≤8

b(`)ij DiH

n(Djunh),

where ` ≥ 0 is the sequence number. The two terms on the right hand side represent the temporalinformation and the spatial information, which can be uniquely shown by the ninth-order symmetric

matrices A(`) = a(`)ij and B(`) = b(`)ij , respectively. For ` = 0, identity (3.6) implies

a(0)ij =

0, if (i, j) = (0, 0),

γiγj , otherwise,and b

(0)ij = 0.

Owing to the matrix symmetry, in what follows we only show the manipulation on the upper triangularpart. Let ` ≥ 1. The `-th matrix transform eliminates the (` − 1)-th column of A(`−1), by using

relationship (3.5) again and again. For example, a(`−1)`−1,j+1 is erased by the help of a

(`−1)`,j , since(

D`unh,Djunh)Ih

+(Dj+1u

nh,D`−1unh

)Ih

= τ[D`−1Hn(Djunh) + DjHn(D`−1unh)

].

If j ≥ `+ 1, the relevant actions are given in the form

a(`)`−1,j+1 = 0, a

(`)`,j = a

(`−1)`,j − a(`−1)`−1,j+1, b

(`)`−1,j = b

(`−1)`−1,j + 2a

(`−1)`−1,j+1.

The other entries can be similarly dealt with. We refer to [30,31] for more details.The above discussion clearly shows that the temporal difference coefficients (3.3), consequently,

the definition (3.2), the relationship (3.5), and the above matrix transform process are independentof the detailed definition of Hn,`(vh).

To shorten the length of this paper, we skip the tedious manipulations. The matrix transferringprocess stops at the third step, with the final matrices

(3.8) A(3) =

O3×3 O3×1 O3×5O1×3 a33 ?O5×3 ? ?

, B(3) =

B3×3 ? O3×1? O5×5 O5×1

O1×3 O1×5 O1×1

,where the entries’ superscripts are omitted, and the symbol ? represents the nonzero numbers. Herea33 = −9216 < 0 and

(3.9) B3×3 =

1327104 1327104 8847361327104 1769472 1327104884736 1327104 1050624

is a symmetric positive definition (SPD) matrix with the minimum eigenvalue ε ≈ 8581.66 > 0. By theterminologies used in [30, 31], the termination index and the contribution index are both equal to 3,and the central objective is negative. Hence, for the linear constant-coefficient hyperbolic equation, thefourth order RKDG method coupled with the upwind-biased flux has two-step monotonicity stability.

3.2. Inverse inequalities. Some inverse properties [2,6] are used in this paper. For any vh ∈ Vh,there exists an inverse constant µ > 0 independent of h and vh, such that

(3.10) ‖(vh)x‖ ≤ µh−1‖vh‖, ‖vh‖Γh ≤ µh−12 ‖vh‖, ‖vh‖∞ ≤ µh−

12 ‖vh‖.

All norms should be understood element by element. Here ‖ · ‖∞ is the maximum norm, and

‖vh‖Γh =

∑1≤j≤J

1

2

[(vh)+

j− 12

]2+

1

2

[(vh)−

j+ 12

]212

is the L2 norm in the space L2(Γh).

9

3.3. Basic conclusions in the DG discretization. Let z = z(x) be a continuous and periodicfunction with bounded derivative. For any w, v ∈ H1(Ih), define the bilinear functional

(3.11) Hlinz (w, v) =

(f ′(z)w, vx

)Ih

+⟨f ′(z)w − α(z)[[w]], [[v]]

⟩Γh.

Here H1(Ih) denotes the broken Sobolev space whose function belongs to H1(Ij) for 1 ≤ j ≤ J . SinceVh ⊂ H1(Ih), those notations used in Vh can be trivially extended to H1(Ih).

For convenience of statements, throughout the paper we would like to use the symbols C and C?to denote the generic positive constant independent of h, τ and n. They may have different values atdifferent occurrences. Here C? is used to emphasize the nonlinearity of the considered problem. Thatis to say, C? = 0 if f(·) is a linear function.

A simple manipulation yields

(3.12) Hlinz (w, v) +Hlin

z (v, w) = −(

[f ′(z)]xw, v)Ih− 2⟨α(z)[[w]], [[v]]

⟩Γh,

as the extension of the approximate skew-symmetric property. Along the same line as in [30, 31, 34],we have the following properties.

Proposition 3.1. For any w ∈ H1(Ih), there holds

(3.13) |Hlinz (w, vh)| ≤ Ch−1‖w‖?‖vh‖, ∀ vh ∈ Vh,

where ‖w‖? =(‖w‖2 + h‖w‖2Γh

) 12

is a mesh-dependent norm. Note that ‖w‖? ≤ C‖w‖ if w ∈ Vh.

Proof. The proof is trivial by using the Cauchy-Schwartz inequality and the first two inverseinequalities in (3.10), since f ′(z) and α(z) are bounded.

Proposition 3.2. Let G be an index set, and giji,j∈G forms an SPD matrix with the minimaleigenvalue εmin > 0. Then there holds

(3.14)∑i,j∈G

gijHlinz (w

(i)h , w

(j)h ) ≤ −εmin

∑i∈G

⟨α(z)[[w

(i)h ]], [[w

(i)h ]]⟩Γh

+ C?∑i∈G‖w(i)

h ‖2,

for any sequence of functions w(i)h ∈ Vh with i ∈ G.

Proof. As an extension of the approximating non-positivity property, the line of proof is the sameas that in [31]. The conclusion follows from (3.12), since [f ′(z)]x is bounded.

In the error estimate we will repeatedly encounter the term likeH(z, vh)−H(zh, vh), where zh ∈ Vhapproximates z, and vh ∈ Vh is an arbitrary test function. Split it in the form

(3.15) H(z, vh)−H(zh, vh) = Hlinz (z − zh, vh) +Hnls

z,zh(vh),

where the main body is the bilinear functional defined in (3.11), and the remainder is a linear functional

(3.16) Hnlsz,zh

(vh) =(Rz,zhf , (vh)x

)Ih

+⟨Rz,zhf −Rz,zhα [[z − zh]], [[vh]]

⟩Γh.

Here Rz,zhf = f(z)− f(zh)− f ′(z)(z − zh) and Rz,zhα = α(f ; z−h , z+h )− α(z).

The detailed analysis on the main body is very important to achieve the quasi-optimal and/oroptimal error estimates. The results depend on two projections given in the next subsection. However,a trivial boundedness to the remainder is good enough to prove Theorem 2.1.

Lemma 3.1. For any vh ∈ Vh, there holds |Hnlsz,zh

(vh)| ≤ C?h−1‖z − zh‖∞‖z − zh‖?‖vh‖.Proof. The line of proof is the same as those in the previous works [32,34]. Since f ′′(·) is bounded

and α(f ; ·, ·) is Lipschitz continuous, it is easy to see that

|Rz,zhf |+ |Rz,zhα [[z − zh]]| ≤ C?|z − zh|2.

Notice that Rz,zhf means either of its traces at the element boundaries. Simple applications of Cauchy-Schwartz inequality and the first two inverse inequalities in (3.10) can yield this lemma.

10

3.4. Two projections. In this subsection we would like to consider two projections. The firstone is the local L2-projection Ph(t) = Ph, independent of the time t. It has been defined in Section2.1. The second one is a new projection Gh(t), depending on the exact solution u(x, t) and the relativeupwind effect. For convenience, we still name it as the GGR projection.

Denote all sonic point positions at the time t by

S(t) = x : f ′(u(x, t)) = 0.

Since the exact solution is sufficiently smooth, the characteristic (straight) lines do not intersect witheach other. Hence the sonic point positions do not change with time, i.e., S(t) = S(0) ≡ S. Forsimplicity, we would like to assume that S, if not empty, contains a finite number of isolate points.Otherwise, if S has an interval with positive length, then the exact solution is constant there for anytime. All elements in Ih are then divided into two groups of sub-partitions, namely

Kg = Ij : Ij ∩ S 6= ∅ for Kming ≤ j ≤ Kmax

g , g = 1, 2, . . . , NK ,(3.17a)

Gg = Ij : Ij ∩ S = ∅ for Gming ≤ j ≤ Gmax

g , g = 1, 2, . . . , NG,(3.17b)

where NK and NG are two fixed integers. Recalling the subintervals mentioned in F2, we can concludethat f ′(u(x, t)) keeps either positive or negative in each sub-partition Gg for t ∈ [0, T ].

We denote (·, ·)Ij as the inner product in L2(Ij). Associated with the exact solution u(x, t) andthe relative upwind effect (2.11), we define

q(x, t) = β(u(x, t)).

Here and below we denote qj±1/2(t) = q(xj±1/2, t).Let w = w(x) be a given function. For any time t ∈ [0, T ], denote by η(t) = w − Gh(t)w the

projection error, where Gh(t)w ∈ Vh is the GGR projection of w. For convenience, the argument x isomitted. The detailed definition depends on whether sonic points appear or not.

• If S 6= ∅, the projection is defined in two steps. First, compute the L2-projection in every Kg,namely, for Kmin

g ≤ j ≤ Kmaxg there holds

(3.18) (η(t), vh)Ij = 0, ∀ vh ∈ Pk(Ij).

Then compute the GGR projection in every Gg, namely, for Gming ≤ j ≤ Gmax

g there holds

(3.19) (η(t), vh)Ij = 0, ∀ vh ∈ Pk−1(Ij),

together with some suitable restrictions on the element boundaries. If f ′(u(x, t)) keeps positivein Gg, they read

(3.20a) η−j+ 1

2

(t) + qj+ 12(t)η+

j+ 12

(t) = 0;

otherwise, if f ′(u(x, t)) keeps negative in Gg, they read

(3.20b) η+j− 1

2

(t) + qj− 12(t)η−

j− 12

(t) = 0.

• If S = ∅, we skip the first step and compute the GGR projection by (3.19) and (3.20) forj = 1, . . . , J .

We have now completed the definition of the GGR projection. Following the line of proof for Lemma3.2, we know that this projection is well defined.

For convenience, we also use the same notation η(t) to represent the projection error for the localL2-projection, namely, η(t) = w − Ph(t)w with Ph(t) ≡ Ph.

11

Lemma 3.2. Assume w is sufficiently smooth such that w ∈ W k+2,∞(I). Under the same condi-tions as in Theorem 2.1, for any t and t+ τ lying in [0, T ] we have

(3.21) ‖η(t)‖∞ ≤ Chk+1, ‖∂tη(t)‖∞ ≤ Chk+1,

where ∂tη(t) = [η(t+ τ)− η(t)]/τ is the first order difference quotient.

The proof for the L2-projection can be found in many literatures. However, the proof for the GGRprojection is technical and lengthy, especially when sonic points appear. We postpone the detailedproof to the appendix.

Notice that (3.21) implies

(3.22) ‖η(t)‖? + ‖∂tη(t)‖? ≤ Chk+1, t ∈ [0, T ].

For many cases, this inequality actually holds under a slightly weaker regularity. In this paper we donot pay further attention on this issue.

Lemma 3.3. Under the same conditions as in Theorem 2.1, we have

(3.23) |Hlinu(t)(η(t), vh)| ≤ Chk+σ|||vh|||α(u(t)), ∀ vh ∈ Vh,

with σ = 1/2 for the L2 projection and σ = 1 for the GGR projection. Here u(t) = u(x, t) and

|||vh|||α(u(t)) =(‖vh‖2 +

⟨α(u(t))[[vh]], [[vh]]

⟩Γh

)1/2is the norm depending on the strength of numerical viscosity.

Proof. Since both projection errors in each element are orthogonal to the polynomials of degreeup to k − 1, we have the division Hlin

u (η, vh) = Int(vh) + Bry(vh), where

Int(vh) =∑

1≤j≤J

([f ′(u)− f ′(uj)]η, (vh)x

)Ij,(3.24a)

Bry(vh) =⟨f ′(u)η, [[vh]]

⟩Γh−⟨α(u)[[η]], [[vh]]

⟩Γh,(3.24b)

respectively represent the terms in the interior of the element and on the element endpoints. Here theargument t is dropped and uj is the value of u(x, t) at the middle point of Ij .

Since |f ′(u)− f ′(uj)| ≤ Ch in each element Ij , an application of Cauchy-Schwartz inequality, thefirst inverse inequality in (3.10), and the approximation property (3.22) yields

(3.25) |Int(vh)| ≤ Ch‖η‖‖(vh)x‖ ≤ Chk+1‖vh‖ ≤ Chk+1|||vh|||α(u),

for both the L2 projection and the GGR projection.The estimate to Bry(vh) depends on the projection. The Cauchy-Schwartz inequality yields

|Bry(vh)| ≤[⟨π(u)η, η

⟩Γh

+⟨α(u)[[η]], [[η]]

⟩Γh

] 12[2⟨α(u)[[vh]], [[vh]]

⟩Γh

] 12

.

Due to the boundedness of π(·) and α(·), it follows from (3.22) that

(3.26) |Bry(vh)| ≤ C‖η‖Γh |||vh|||α(φ) ≤ Chk+12 |||vh|||α(φ)

for the L2-projection.Below we turn to the GGR projection. Let Γ ?h be the element endpoints of all Kg. For Γh\Γ ?h ,

the restrictions on the element endpoints yields

f ′(u)η − α(u)[[η]] =

[ 12f′(u) + α(u)][η− + β(u)η+] = 0, if f ′(·) > 0,

[ 12f′(u)− α(u)][β(u)η− + η+] = 0, if f ′(·) < 0,

12

due to the boundedness of f ′(·) and α(·). Here the subscript is omitted for simplicity. Hence Bry(vh)is equal to the sum of those terms emerged at Γ ?h . Hence

|Bry(vh)| ≤ C‖f ′(u)‖∞,Γ?h ‖η‖Γ?h ‖vh‖Γ?h +⟨α(u)[[η]], [[η]]

⟩ 12

Γ?h

⟨α(u)[[vh]], [[vh]]

⟩ 12

Γ?h

,

where the subscript Γ ?h means that the corresponding manipulation and/or notations are restrictedto this domain. The total number in Γ ?h is finite and independent of h, and the flow speed satisfies|f ′(u)| ≤ Ch at the endpoints of Γ ?h . Using Lemma 3.2 and the second inverse inequality in (3.10),we can yield

(3.27) |Bry(vh)| ≤ C‖η‖∞,Γ?h ‖vh‖+ C‖η‖∞,Γ?h⟨α(u)[[vh]], [[vh]]

⟩ 12

Γh≤ Chk+1|||vh|||α(u),

since α(·) is bounded. Summing up (3.25) with (3.26) and (3.27), respectively, we complete the proofof this lemma.

4. Proof of Theorem 2.1. In this section we are ready to prove Theorem 2.1, along the sameline as that in [30,34].

4.1. Energy equation. Let u(0)(x, t) = u(x, t), and then define a series of functions parallelingto the stage marching of the RK algorithm

(4.1) u(`+1)(x, t) =∑

0≤κ≤`

c`κu

(κ)(x, t)− τd`κ[f(u(κ)(x, t))

]x

, ` = 0, 1, 2.

The reference functions at time stage are then defined as

(4.2) un,`(x) = u(`)(x, tn), ∀n and ` = 0, 1, 2, 3.

It follows from the Sobolev embedding theorem that the above reference functions are continuous,due to the smoothness regularity (2.16). Noticing the consistency of the numerical flux, after somemanipulations we can derive for any n and ` = 0, 1, 2, 3,

(4.3)(un,`+1, vh

)Ih

=∑

0≤κ≤`

c`κ

(un,κ, vh

)Ih

+ d`κτ[H(un,κ, vh) +

(%n,κ, vh

)Ih

], ∀vh ∈ Vh.

Here %n,0 = %n,1 = %n,2 = 0, and %n,3 is the local truncation error in time, satisfying

(4.4) ‖%n,3‖ ≤ Cτ4, ∀n.

The stage error is defined by en,` = un,`−un,`h . As the usual treatment in finite element analysis,the stage error can be divided in the form

(4.5) en,` =[Qh(tn)un,` − un,`h

]−[Qh(tn)un,` − un,`

]≡ ξn,` − ηn,`,

where Qh(tn) is either the L2-projection or the GGR projection. Since the projection error ηn,` hasbeen estimated by (3.21), say,

(4.6) ‖ηn,`‖? + ‖ηn,`‖∞ ≤ Chk+1,

we only need to estimate ξn,` ∈ Vh sharply in what follows.To this end, we first set up the error equations about ξn,`. Subtracting (2.5) from (4.3) and using

the error decomposition (4.5) yield the error equation

(4.7)(ξn,`+1, vh

)Ih

=∑

0≤κ≤`

[c`κ

(ξn,κ, vh

)Ih

+ τd`κFn,κ(vh)], ∀ vh ∈ Vh,

13

with

(4.8) Fn,`(vh) = H(un,`, vh)−H(un,`h , vh) +(En,` + %n,`, vh

)Ih,

Here En,` is the difference quotient of the previous `+ 1 projection errors, given in the form

En,` =1

τ

∑0≤κ≤`+1

σ`+1,κηn,κ, with

∑0≤κ≤`+1

σ`+1,κ = 0.

Notice that σ`+1,κ can be inductively expressed by those parameters in the RK algorithm. To shortenthe length of this paper, we do not show their detailed values.

As mentioned in Section 3.1, we can carry out the matrix transferring process on the above errorequations. To be specific, we first define the temporal difference

(4.9) D`ξn =∑

0≤κ≤`

σ`κξn,κ, ` = 0, 1, . . . , 8,

with the same combination coefficients as those in (3.4). Then we get from the error equation (4.7)the relationship for ` = 0, 1, . . . , 7,

(4.10)(D`+1ξ

n, vh

)Ih

= τD`Fn(vh) ≡ τ∑

0≤κ≤`

σ`κFn,κ(vh), ∀ vh ∈ Vh.

By the matrix transferring process we can finally obtain the energy equation

(4.11) γ20(‖ξn+2‖2 − ‖ξn‖2

)=

∑0≤i,j≤8

aij

(Diξn,Djξn

)Ih

+ τ∑

0≤i,j≤8

bijDiFn(Djξn),

where aij and bij are the same as those in (3.8).

4.2. Some basic results. Before estimating the right hand side of (4.11), we set up some basicresults. Since u(x, t) and f(·) are sufficiently smooth, it follows from Lemma 3.2 that

(4.12) ‖En,`‖? + ‖En,`‖∞ ≤ Chk+1, ∀n, and 0 ≤ ` ≤ 7.

In fact, this conclusion is easily proved as follows. If the same projection is used for the referencesolutions, we know that En,` is just the projection error of the difference quotient

1

τ

∑0≤κ≤`+1

σ`+1,κun,κ,

which is also sufficiently smooth due to (2.16). In this case, (4.12) is true by using the first inequality inLemma 3.2. Otherwise, En,` has an additional term involving (ηn+1− ηn)/τ and/or (ηn+2− ηn+1)/τ ,which can be bounded by the second inequality in Lemma 3.2. Summing up, we complete the proofof (4.12).

Along the similar line as above, we have

(4.13) ‖D`ηn‖? ≤ Cτhk+1, ∀n, and 1 ≤ ` ≤ 8.

For convenience we assume λ ≤ 1 in what follows. Moreover, we would like to introduce thetemporal difference of stage functionals. For example, for 0 ≤ ` ≤ 8 we denote

D`Ln(·) =∑

0≤κ≤`

σ`κLn,κ(·), with Ln,κ(·) = Hlinun,κ(ξn,κ, ·),

D`Kn(·) =∑

0≤κ≤`

σ`κKn,κ(·), with Kn,κ(·) = Hlinun,κ(ηn,κ, ·).

Similarly, we can define D`Nn(·) withNn,κ(·) = Hnlsun,κ,un,κh

(·). They form the main context in D`Fn(·).14

Lemma 4.1. Let 1 ≤ ` ≤ 7. For any vh ∈ Vh, there holds

|D`Ln(vh)| ≤ Ch−1‖D`ξn‖‖vh‖+ C?∑

1≤κ≤`

‖ξn,κ‖‖vh‖,(4.14a)

|D`Kn(vh)| ≤ Chk+1‖vh‖.(4.14b)

Proof. Take (4.14a) as an example. Owing to the linear construction, a direct calculation yields

D`Ln(vh) = Hlinun(D`ξn, vh) +

∑1≤κ≤`

σ`κ

[Hlinun,κ(ξn,κ, vh)−Hlin

un(ξn,κ, vh)].

Both terms on the right-hand side can be bounded by Proposition 3.1. For the second term, we havealso used the Lipschitz continuity of f ′(·) and α(·), as well as ‖un,κ − un‖∞ ≤ Cτ . Finally, we have

|D`Ln(vh)| ≤ Ch−1‖D`ξn‖?‖vh‖+ C?λ∑

1≤κ≤`

‖ξn,κ‖?‖vh‖.

This implies (4.14a), since ‖wh‖? ≤ C‖wh‖ for any wh ∈ Vh.Along the same line, we can prove (4.14b) by using (4.6) and (4.13).

Lemma 4.2. Let 3 ≤ ` ≤ 7. There holds

(4.15) ‖D`+1ξn‖2 ≤ Cλ2‖D`ξn‖2 + τ2

∑0≤κ≤`

[C(en,κ)]2[‖ξn,κ‖2 + h2k+2

]+ Cτ10.

Here and below we use the notation C(en,κ) = C + C?h−1‖en,κ‖∞.

Proof. Taking vh = D`+1ξn in (4.10), and we get ‖D`+1ξ

n‖2 = Π1 + Π2, where

Π1 = τ[D`Ln(D`+1ξ

n)− D`Kn(D`+1ξn) + D`Nn(D`+1ξ

n)],

Π2 = τ(D`En + D`%n,D`+1ξ

n)Ih.

The first two terms in Π1 can be estimated by Lemma 4.1 and so can the last term by Lemma 3.1and (4.6). We can easily bound Π2 by (4.12) and (4.4), as well as Cauchy-Schwartz inequality. Anapplication of Young’s inequality yields this lemma.

Remark 4.1. The factor λ in Lemma 4.2 is important to determine the standard CFL condition.

Lemma 4.3. Let 0 ≤ ` ≤ 7. We have the rough estimate ‖ξn,`+1‖2 ≤ C‖ξn‖2 + Ch2k+2 + Cτ10.

Proof. Taking vh = ξn,`+1 in (4.7), and we get ‖ξn,`+1‖2 = Π3 + Π4, where

Π3 =∑

0≤κ≤`

[c`κ

(ξn,κ, ξn,`+1

)Ih

+ τd`κ

(H(un,κ, ξn,`+1)−H(un,κh , ξn,`+1

)],

Π4 = τ∑

0≤κ≤`

d`κ

(En,` + %n,`, ξn,`+1

)Ih.

Owing to the Lipschitz continuity of f(·) and f(·, ·), an application of the inverse inequalities (3.10)and the triangle inequality implies

H(un,κ, vh)−H(un,κh , vh) ≤ Ch−1[‖ξn,κ‖? + ‖ηn,κ‖?

]‖vh‖.

Since ‖ξn,κ‖? ≤ C‖ξn,κ‖ and (4.6), some simple applications of Cauchy-Schwartz inequality and thetriangle inequality yield

Π3 ≤ C∑

0≤κ≤`

[‖ξn,κ‖+ hk+1

]‖ξn,`+1‖.

15

The estimate to Π4 is similar to Π2. Summing up the above conclusions yields

‖ξn,`+1‖2 ≤ C∑

0≤κ≤`

‖ξn,κ‖2 + Ch2k+2 + Cτ10,

which implies this lemma by an induction procedure.

4.3. Energy inequality. Below we are going to estimate the two terms on the right hand sideof the energy equation (4.11), which are respectively denoted by Υtm and Υsp.

Noticing the distribution of zeros in A(3) and using Young’s inequality, we have

(4.16) Υtm =∑

3≤i,j≤8

aij

(Diξn,Djξn

)Ih≤ 1

2a33‖D3ξ

n‖2 + C∑

4≤i≤8

‖Diξn‖2.

For the second term, the analysis starts from the separation Υsp = Θ1 + Θ2 + Θ3 + Θ4, where

Θ1 =∑

0≤i,j≤7

τbijDiLn(Djξn), Θ2 = −∑

0≤i,j≤7

τbijDiKn(Djξn),

Θ3 =∑

0≤i,j≤7

τbijDiNn(Djξn), Θ4 =∑

0≤i,j≤7

τbij

(DiEn + Di%n,Djξn

)Ih,

because bij = 0 when i = 8 or j = 8; see (3.8). Each term can be separately estimated at each step.Step 1. The first term Θ1 can be expressed by the sum of the following three terms

Θ11 =∑

0≤i≤7

∑0≤j≤7

τbij∑

1≤κ≤i

σiκ

[Hlinun,κ(ξn,κ,Djξn)−Hlin

un(ξn,κ,Djξn)],

Θ12 =∑

0≤i≤2

∑0≤j≤2

τbijHlinun(Diξn,Djξn),

Θ13 =∑

0≤i≤2

∑3≤j≤7

τbij

[Hlinun(Diξn,Djξn) +Hlin

un(Djξn,Diξn)],

where the matrix symmetry is used in the last term. Along the same line of proof as that in Lemma4.1, we have from the fact ‖un,κ − un‖∞ ≤ Cτ that

Θ11 ≤ Cτλ∑

0≤i≤7

∑0≤j≤7

‖ξn,i‖‖Djξn‖ ≤ Cτ∑

0≤i≤7

‖ξn,i‖2.

It follows from Proposition 3.2 that

Θ12 ≤ −Vn + C?τ∑

0≤i≤2

‖Diξn‖2 ≤ −Vn + C?τ∑

0≤i≤2

‖ξn,i‖2,

where ε is the minimum eigenvalue of B3×3 as mentioned in Section 3.1, and

(4.17) Vn = ετ∑

0≤i≤2

⟨α(un)[[Diξn]], [[Diξn]]

⟩Γh

explicitly shows the kernel numerical viscosity hidden in the two-step marching. An application of(3.12) yields

Θ13 = −τ∑

0≤i≤2

∑3≤j≤7

bij

([f ′(un)]xDiξn,Djξn

)Ih− 2τ

∑0≤i≤2

∑3≤j≤7

bij

⟨α(un)[[Diξn]], [[Djξn]]

⟩Γh.

16

The two terms on the right hand side are respectively denoted by Θ131 and Θ132. By Cauchy-Schwartzinequality and the triangle inequality, we have

Θ131 ≤ C?τ∑

0≤i≤2

∑3≤j≤7

‖Diξn‖‖Djξn‖ ≤ C?τ∑

0≤j≤7

‖ξn,j‖2.

By Cauchy-Schwartz inequality and Young’s inequality, we have

Θ132 ≤1

2Vn + Cτ

∑3≤j≤7

⟨α(un)[[Djξn]], [[Djξn]]

⟩Γh≤ 1

2Vn + Cλ

∑3≤j≤7

‖Djξn‖2,

where the boundedness of α(·) and the second inverse inequality in (3.10) are used at the last step.Summing up the results about Θ11,Θ12,Θ131 and Θ132, we have the expected boundedness

(4.18) Θ1 ≤ −1

2Vn + Cλ

∑3≤j≤7

‖Djξn‖2 + Cτ∑

0≤j≤7

‖ξn,j‖2.

Step 2. Now we turn to the second term Θ2, which is the sum of the following two terms,

Θ21 = −∑

1≤i≤7

∑0≤j≤7

τbijDiKn(Djξn), Θ22 = −∑

0≤j≤7

τb0jKn(Djξn).

The first term can be easily bounded by the second conclusion (4.14b) in Lemma 4.1, as well as thetriangle inequality and Young’s inequality. Namely, it reads

Θ21 ≤ Cτhk+1∑

0≤j≤7

‖Djξn‖ ≤ Cτh2k+2 + Cτ∑

0≤j≤7

‖ξn,j‖2.

The treatment to the second term is essential to get the optimal error estimate. Under the conditionof Theorem 2.1, it follows from Lemma 3.3 and Young’s inequality that

Θ22 ≤ Cτhk+σ∑

0≤j≤7

|||Djξn|||α(un)

≤ Cτh2k+2σ +ετ

2

∑0≤j≤7

‖Djξn‖2 +ετ

2

∑0≤j≤7

⟨α(un)[[Djξn]], [[Djξn]]

⟩Γh

≤ Cτh2k+2σ + Cτ∑

0≤j≤7

‖ξn,j‖2 +1

2Vn + Cλ

∑3≤j≤8

‖Djξn‖2,

where the boundedness of α(·) and the second inverse inequality in (3.10) are also used at the laststep. Summing up the estimates to Θ21 and Θ22, we have

(4.19) Θ2 ≤ Cτh2k+2ζ + Cτ∑

0≤i≤7

‖ξn,i‖2 +1

2Vn + Cλ

∑3≤j≤7

‖Djξn‖2,

where ζ = min(1, σ) = σ has been stated in Theorem 2.1.Step 3. By using Lemma 3.1, it is straightforward to obtain

(4.20) Θ3 ≤ τ∑

0≤i,j≤7

C(en,i)‖en,i‖?‖Djξn‖ ≤ τ∑

0≤i≤7

[C(en,i)]2(‖ξn,i‖2 + h2k+2

),

where ‖ξn,i‖? ≤ C‖ξn,i‖ and (4.6) are used. Note that C(en,i) is given in Lemma 4.2, and the involvedconstants C and C? are all independent of h, τ and n.

17

Step 4. According to (4.12) and (4.4), we can obtain

(4.21) Θ4 ≤ Cτ(h2k+2 + τ8) + Cτ∑

0≤i≤7

‖ξn,i‖2.

Substitute (4.16) and the estimates at four steps into the energy equation (4.11), and repeatedlyuse Lemma 4.2. Since λ ≤ 1 we have

‖ξn+2‖2 − ‖ξn‖2 ≤(1

2a33 + Cλ

)‖D3ξ

n‖2 + τ∑

0≤i≤7

[C(en,i)

]2(‖ξn,i‖2 + h2k+2ζ + τ8

).

Since a33 < 0, there is a constant λmax > 0 such that 12a33 + Cλ < 0 and hence for any n under

consideration there holds

(4.22) ‖ξn+2‖2 − ‖ξn‖2 ≤ τ∑

0≤i≤7

[Cn,i(e)]2(‖ξn‖2 + h2k+2ζ + τ8

),

provided that λ ≤ λmax. Note that Lemma 4.3 is used to bound the L2-norm errors at the intermediatestage time.

4.4. Final proof. We are now in the position to complete the proof of Theorem 2.1. In orderto conveniently deal with the nonlinearity of f(·), we would like to adopt the a priori assumption forevery even integer 0 ≤ m ≤ N : for sufficiently small h independent of m, there holds

(4.23) ‖ξn‖ ≤ h3/2, ∀ even n : 0 ≤ n ≤ m.

Since the definition of the initial solution implies ‖ξ0‖ ≤ Chk+1, due to Lemma 3.2, this assumptionobviously holds for m = 0. The reasonability of (4.23) is proved in an inductive process.

Suppose that a priori assumption (4.23) have been proved to be true for an even number m.Since k ≥ 1, it follows from Lemma 4.3 that

‖ξn,`‖ ≤ C(‖ξn‖+ hk+1 + τ4) ≤ Ch3/2, ∀ even n : n ≤ m and 0 ≤ ` ≤ 7.

Here and below the constant C is also independent of m. Owing to the approximation property (4.12)and the third inverse inequality in (3.10), we have

C(en,`) ≤ C, ∀ even n : n ≤ m and 0 ≤ ` ≤ 7.

Then it follows from (4.22) that

(4.24) ‖ξn+2‖2 − ‖ξn‖2 ≤ Cτ(‖ξn‖2 + h2k+2ζ + τ8), ∀ even n : 0 ≤ n ≤ m.

An application of discrete Gronwall’s inequality yields

(4.25) ‖ξn‖ ≤Mbnd(hk+ζ + τ4), ∀ even n : 0 ≤ n ≤ m+ 2,

where the bounding constant Mbnd > 0 is independent of h, τ and m, but may depend on the finaltime T . If k ≥ 2 for ζ = 1/2 and k ≥ 1 for ζ = 1, it is easy to see from (4.25) that, for sufficientlysmall h independent of m,

‖ξn‖ ≤ h3/2, ∀n : 0 ≤ n ≤ m+ 2.

This implies that (4.23) also holds for m + 2. Hence the a priori assumption is reasonable and theabove induction process can be carried out till the last time level, if N is even. That is to say,

(4.26) max0≤n≤N

‖ξn‖ ≤Mbnd(hk+ζ + τ4).

Due to Lemma 4.3, we can conclude that (4.26) also holds for odd N , with a little modification onthe bounding constant.

Together with the approximation property (4.12), we finally complete the proof of Theorem 2.1.

18

5. Numerical experiments. In this section we present some numerical experiments to supportTheorem 2.1. Consider the Burgers equation, i.e., f(u) = u2/2, with the periodic boundary conditionand the initial solution

u(x, 0) = a+ sin(2πx), x ∈ (0, 1).

In this paper we take a = 0, 1, 2, to represent three typical status with respect to the sonic point. Thefinal time is set as T = 0.1 before the shock emerges.

We apply the four stage fourth order RKDG method with degree k = 1, 2, 3, 4. The detailedparameters are given in (2.6). For k ≤ 3, the time step is determined by the CFL numbers not biggerthan 0.4, 0.2 and 0.1, respectively. For k = 4, we take τ = 0.1h1.25min to ensure that the spatial errordominates, where hmin is the minimum of the element length. Both uniform mesh and nonuniformmesh are considered, where the nonuniform mesh is given by perturbing the mesh nodes in the uniformmesh randomly by at most 10%.

In Tables 1 to 5, we display the numerical errors and the corresponding convergence orders (shownin the bracket) in L2-norm and L∞-norm for five typical numerical fluxes.

1. In the case a = 2, the problem does not have any sonic point. For k = 1, 2, 3, 4, we clearlyobserve the optimal order k+1 for all numerical fluxes no matter whether the mesh is uniformor not. These results are entirely consistent with case 2(a) in the theorem.

2. In the other cases, the problem has a sonic point. For a = 1 the flow speed keeps nonnegative,however, for a = 0 the flow speed changes sign. From the data we can find out that theconvergence order depends on the numerical flux. The optimal order is observed for the LLFflux and the LFORCE flux, since case 2(a) in the theorem is satisfied. In contrast, the half-order reduction is clearly observed for the other three numerical fluxes when a = 0, even onthe uniform mesh. Generally speaking, the convergence orders when a = 1 are better thanthose when a = 0. As shown by the conclusion in case 2(b) of Theorem 2.1, the optimal orderon the uniform mesh is shown for the GLF flux and the GFORCE flux when k is odd, andfor the Richtmyer flux when k is even.

For the Godunov flux and the GLLF flux with different parameters, the optimal order is also observedno matter whether sonic points appear or not. Limited by the length of this paper, we do not showthem here. Summing up, Theorem 2.1 is supported by the numerical results.

6. Concluding remarks. In this paper we establish the a priori L2-norm error estimate tosufficiently smooth solution of the fourth order RKDG method for one-dimensional scalar nonlinearconservation laws. The order in time is optimal and the order in space strongly depends on the ratioof the flow speed over the strength of the numerical viscosity, especially at the sonic point position.To achieve the above goal, we first extend the matrix transferring process from the linear case to thenonlinear case, and then propose a good energy equation to carry out the error estimate. Although thestability mechanism is explicitly shown by the numerical viscosity hidden in two-step time-marching,the strength of the numerical viscosity for the nonlinear problem may change everywhere and cancause a serious incompatibility with the flow speed at the sonic point. In order to obtain the optimalorder in space regardless of the sonic point, we define a new GGR projection depending on the relativeupwind effect to match the error at the element boundaries as perfectly as possible. In future work,we will extend the above studies to the multi-dimensional problem and/or the system of conservationlaws.

7. Appendix. In this section we prove Lemma 3.2 for the GGR projection when K 6= ∅. IfK = ∅, the proof is similar and so omitted.

Since the approximation property of the L2-projection is well known, we only need to put oureyesight on the difference between the two projections in each element

Gh(t)w(x)− Phw(x) = θj(t)Lj,k(x), x ∈ Ij , t ∈ [0, T ],

where θj(t) is undetermined. In this paper Lj,l(x) denotes the scaling Legendre polynomial in Ij ofdegree l ≥ 0, satisfying Lj,l(xj±1/2) = (±1)l and ‖Lj,l‖L∞(Ij) ≤ 1.

19

Table 1The errors and convergence orders for the GLF fluxes: T = 0.1 and CGLF = a + 1.

a = 0 a = 1 a = 2

L2-norm L∞-norm L2-norm L∞-norm L2-norm L∞-norm

k N nonuniform mesh

160 1.33E-04 1.20E-03 1.35E-04 1.44E-03 1.41E-04 1.34E-03

320 3.37E-05(1.99) 3.28E-04(1.87) 3.58E-05(1.92) 4.23E-04(1.77) 3.66E-05(1.95) 3.85E-04(1.80)

1 640 8.46E-06(1.99) 9.53E-05(1.78) 8.57E-06(2.06) 9.67E-05(2.13) 8.98E-06(2.03) 9.94E-05(1.95)

1280 2.13E-06(1.99) 2.38E-05(2.00) 2.19E-06(1.97) 2.45E-05(1.98) 2.26E-06(1.99) 2.69E-05(1.89)

2560 5.31E-07(2.01) 6.10E-06(1.97) 5.64E-07(1.96) 6.61E-06(1.89) 5.62E-07(2.01) 6.81E-06(1.98)

160 3.73E-06 4.94E-05 4.04E-06 7.38E-05 3.48E-06 6.47E-05

320 6.35E-07(2.55) 1.00E-05(2.30) 5.94E-07(2.77) 1.10E-05(2.74) 4.70E-07(2.89) 1.18E-05(2.46)

2 640 1.07E-07(2.57) 1.71E-06(2.55) 8.59E-08(2.79) 1.48E-06(2.89) 5.80E-08(3.02) 1.48E-06(2.99)

1280 1.78E-08(2.59) 3.14E-07(2.45) 1.22E-08(2.81) 1.82E-07(3.03) 7.26E-09(3.00) 1.72E-07(3.11)

2560 2.89E-09(2.62) 5.69E-08(2.46) 1.75E-09(2.80) 2.39E-08(2.93) 9.18E-10(2.98) 2.25E-08(2.93)

160 4.64E-08 1.02E-06 5.45E-08 1.51E-06 5.55E-08 1.59E-06

320 2.62E-09(4.14) 6.18E-08(4.04) 3.01E-09(4.18) 7.86E-08(4.26) 3.04E-09(4.19) 9.38E-08(4.08)

3 640 1.88E-10(3.80) 5.76E-09(3.42) 1.94E-10(3.96) 6.62E-09(3.57) 1.97E-10(3.94) 6.65E-09(3.82)

1280 1.21E-11(3.95) 3.71E-10(3.95) 1.25E-11(3.96) 4.41E-10(3.91) 1.26E-11(3.97) 4.38E-10(3.93)

2560 7.86E-13(3.94) 2.30E-11(4.01) 7.42E-13(4.07) 2.54E-11(4.12) 7.86E-13(4.00) 2.92E-11(3.91)

160 1.58E-09 2.99E-08 2.21E-09 7.75E-08 1.74E-09 5.72E-08

320 7.26E-11(4.44) 1.54E-09(4.28) 7.52E-11(4.88) 2.11E-09(5.20) 5.76E-11(4.91) 1.71E-09(5.06)

4 640 3.05E-12(4.57) 7.68E-11(4.32) 2.44E-12(4.95) 7.45E-11(4.82) 1.85E-12(4.96) 6.87E-11(4.64)

1280 1.26E-13(4.59) 3.77E-12(4.35) 7.89E-14(4.95) 2.50E-12(4.90) 6.18E-14(4.91) 2.77E-12(4.63)

2560 5.58E-15(4.50) 1.70E-13(4.47) 2.53E-15(4.96) 8.80E-14(4.83) 1.96E-15(4.98) 1.04E-13(4.74)

k N uniform mesh

160 1.27E-04 9.60E-04 1.26E-04 1.12E-03 1.36E-04 1.17E-03

320 3.19E-05(1.99) 2.43E-04(1.98) 3.16E-05(2.00) 2.86E-04(1.97) 3.40E-05(2.00) 3.00E-04(1.96)

1 640 8.00E-06(2.00) 6.14E-05(1.99) 7.92E-06(2.00) 7.21E-05(1.99) 8.51E-06(2.00) 7.59E-05(1.98)

1280 2.00E-06(2.00) 1.54E-05(1.99) 1.98E-06(2.00) 1.81E-05(1.99) 2.13E-06(2.00) 1.91E-05(1.99)

2560 5.01E-07(2.00) 3.86E-06(2.00) 4.96E-07(2.00) 4.54E-06(2.00) 5.32E-07(2.00) 4.79E-06(2.00)

160 3.68E-06 4.36E-05 3.98E-06 6.95E-05 3.36E-06 6.24E-05

320 6.16E-07(2.58) 8.54E-06(2.35) 5.80E-07(2.78) 9.59E-06(2.86) 4.37E-07(2.94) 8.19E-06(2.93)

2 640 1.04E-07(2.56) 1.61E-06(2.41) 8.30E-08(2.80) 1.26E-06(2.93) 5.55E-08(2.97) 1.04E-06(2.97)

1280 1.74E-08(2.58) 2.95E-07(2.45) 1.18E-08(2.81) 1.60E-07(2.97) 7.00E-09(2.99) 1.31E-07(2.99)

2560 2.84E-09(2.61) 5.27E-08(2.48) 1.70E-09(2.80) 2.01E-08(2.99) 8.78E-10(2.99) 1.65E-08(3.00)

160 3.73E-08 6.37E-07 4.12E-08 8.48E-07 4.46E-08 9.25E-07

320 2.32E-09(4.01) 4.01E-08(3.99) 2.55E-09(4.01) 5.30E-08(4.00) 2.77E-09(4.01) 5.76E-08(4.00)

3 640 1.45E-10(4.00) 2.56E-09(3.97) 1.59E-10(4.00) 3.35E-09(3.99) 1.73E-10(4.00) 3.63E-09(3.99)

1280 9.10E-12(4.00) 1.62E-10(3.99) 9.97E-12(4.00) 2.10E-10(3.99) 1.08E-11(4.00) 2.28E-10(3.99)

2560 5.70E-13(4.00) 1.02E-11(3.99) 6.23E-13(4.00) 1.32E-11(4.00) 6.75E-13(4.00) 1.43E-11(4.00)

160 1.50E-09 2.32E-08 1.79E-09 4.89E-08 1.57E-09 4.59E-08

320 6.33E-11(4.56) 1.21E-09(4.26) 6.59E-11(4.76) 1.82E-09(4.75) 5.27E-11(4.90) 1.59E-09(4.85)

4 640 2.78E-12(4.51) 6.18E-11(4.30) 2.24E-12(4.88) 6.09E-11(4.90) 1.69E-12(4.96) 5.13E-11(4.95)

1280 1.22E-13(4.51) 3.05E-12(4.34) 7.30E-14(4.94) 1.96E-12(4.96) 5.35E-14(4.98) 1.62E-12(4.98)

2560 5.28E-15(4.54) 1.45E-13(4.39) 2.33E-15(4.97) 6.18E-14(4.99) 1.68E-15(4.99) 5.09E-14(4.99)

Since θj(t) ≡ 0 if Ij ⊂ Kg, it is sufficient to prove this lemma by showing one by one

(7.1) ‖θg(t)‖∞ + ‖∂tθg(t)‖∞ ≤ Chk+1, g = 1, 2, . . . , NG,

where θg(t) = θj(t) : Gming ≤ j ≤ Gmax

g is a column vector.Without losing generality, we assume f ′(u(x, t)) keeps positive on Gg × [0, T ]. Associated with

the relative upwind effect and the exact solution, we define

(7.2) p(x, t) = (−1)k+1q(x, t) = (−1)k+1β(u(x, t)) ∈ [−1, 1].

The definition of the GGR projection at the element boundaries implies for Gming ≤ j ≤ Gmax

g that

θj(t)− pj+ 12(t)θj+1(t) = dj(t) ≡ η−j+ 1

2

+ qj+ 12(t)η+

j+ 12

,

20

Table 2The errors and convergence orders for the LLF fluxes: T = 0.1.

a = 0 a = 1 a = 2


k N nonuniform mesh

160 1.65E-04 1.51E-03 1.62E-04 1.30E-03 1.69E-04 1.58E-03

320 4.22E-05(1.97) 4.25E-04(1.83) 4.22E-05(1.94) 4.28E-04(1.60) 4.30E-05(1.98) 4.57E-04(1.79)

1 640 1.07E-05(1.98) 1.06E-04(2.01) 1.06E-05(1.99) 1.09E-04(1.97) 1.08E-05(1.99) 1.20E-04(1.92)

1280 2.68E-06(1.99) 2.89E-05(1.87) 2.69E-06(1.98) 2.72E-05(2.00) 2.71E-06(1.99) 2.88E-05(2.06)

2560 6.73E-07(2.00) 6.92E-06(2.06) 6.73E-07(2.00) 7.38E-06(1.88) 6.75E-07(2.00) 7.34E-06(1.97)

160 2.17E-06 3.95E-05 2.65E-06 5.26E-05 2.84E-06 7.57E-05

320 3.04E-07(2.83) 5.38E-06(2.88) 3.39E-07(2.97) 7.99E-06(2.72) 3.39E-07(3.06) 8.18E-06(3.21)

2 640 3.94E-08(2.95) 8.66E-07(2.64) 4.28E-08(2.98) 1.20E-06(2.73) 4.24E-08(3.00) 1.07E-06(2.93)

1280 5.03E-09(2.97) 1.31E-07(2.72) 5.32E-09(3.01) 1.40E-07(3.10) 5.44E-09(2.96) 1.48E-07(2.86)

2560 6.50E-10(2.95) 1.68E-08(2.97) 6.66E-10(3.00) 2.01E-08(2.80) 6.69E-10(3.02) 1.92E-08(2.95)

160 5.83E-08 1.42E-06 6.59E-08 1.89E-06 5.38E-08 1.17E-06

320 3.45E-09(4.08) 8.90E-08(3.99) 3.56E-09(4.21) 8.12E-08(4.54) 3.66E-09(3.88) 8.73E-08(3.74)

3 640 2.24E-10(3.94) 6.45E-09(3.79) 2.45E-10(3.86) 7.88E-09(3.37) 2.31E-10(3.99) 6.91E-09(3.66)

1280 1.41E-11(3.99) 4.19E-10(3.95) 1.45E-11(4.08) 4.11E-10(4.26) 1.46E-11(3.98) 4.66E-10(3.89)

2560 9.21E-13(3.94) 2.80E-11(3.90) 9.13E-13(3.99) 2.64E-11(3.96) 9.16E-13(4.00) 2.81E-11(4.05)

160 1.30E-09 3.58E-08 1.50E-09 5.14E-08 1.48E-09 5.03E-08

320 4.01E-11(5.02) 1.07E-09(5.06) 4.96E-11(4.92) 2.45E-09(4.39) 4.42E-11(5.07) 1.68E-09(4.90)

4 640 1.27E-12(4.98) 4.81E-11(4.48) 1.51E-12(5.04) 6.95E-11(5.14) 1.41E-12(4.97) 5.61E-11(4.90)

1280 4.15E-14(4.94) 1.33E-12(5.18) 4.76E-14(4.99) 2.40E-12(4.85) 4.44E-14(4.99) 1.85E-12(4.92)

2560 1.45E-15(4.84) 7.57E-14(4.13) 1.42E-15(5.07) 6.58E-14(5.19) 1.42E-15(4.97) 7.41E-14(4.64)

k N uniform mesh

160 1.59E-04 1.27E-03 1.60E-04 1.30E-03 1.65E-04 1.30E-03

320 4.07E-05(1.96) 3.31E-04(1.94) 4.09E-05(1.97) 3.35E-04(1.96) 4.16E-05(1.99) 3.34E-04(1.96)

1 640 1.03E-05(1.98) 8.41E-05(1.97) 1.03E-05(1.98) 8.46E-05(1.98) 1.04E-05(2.00) 8.46E-05(1.98)

1280 2.60E-06(1.99) 2.12E-05(1.99) 2.60E-06(1.99) 2.13E-05(1.99) 2.61E-06(2.00) 2.13E-05(1.99)

2560 6.52E-07(1.99) 5.32E-06(1.99) 6.52E-07(2.00) 5.33E-06(2.00) 6.54E-07(2.00) 5.33E-06(2.00)

160 2.07E-06 3.06E-05 2.48E-06 4.90E-05 2.54E-06 5.04E-05

320 2.78E-07(2.90) 4.57E-06(2.74) 3.16E-07(2.97) 6.38E-06(2.94) 3.20E-07(2.99) 6.48E-06(2.96)

2 640 3.67E-08(2.92) 6.51E-07(2.81) 3.99E-08(2.99) 8.13E-07(2.97) 4.01E-08(2.99) 8.19E-07(2.98)

1280 4.75E-09(2.95) 8.86E-08(2.88) 5.01E-09(2.99) 1.02E-07(2.99) 5.03E-09(3.00) 1.03E-07(2.99)

2560 6.08E-10(2.97) 1.17E-08(2.92) 6.28E-10(3.00) 1.29E-08(2.99) 6.29E-10(3.00) 1.29E-08(3.00)

160 4.76E-08 8.63E-07 5.18E-08 1.03E-06 5.28E-08 1.05E-06

320 3.14E-09(3.92) 6.01E-08(3.84) 3.30E-09(3.97) 6.55E-08(3.97) 3.33E-09(3.99) 6.63E-08(3.98)

3 640 2.03E-10(3.95) 3.98E-09(3.92) 2.08E-10(3.99) 4.18E-09(3.97) 2.09E-10(3.99) 4.21E-09(3.98)

1280 1.29E-11(3.97) 2.57E-10(3.95) 1.31E-11(3.99) 2.63E-10(3.99) 1.31E-11(4.00) 2.64E-10(3.99)

2560 8.13E-13(3.99) 1.63E-11(3.98) 8.19E-13(4.00) 1.65E-11(4.00) 8.20E-13(4.00) 1.65E-11(4.00)

160 1.02E-09 1.96E-08 1.25E-09 3.69E-08 1.28E-09 3.81E-08

320 3.42E-11(4.90) 7.93E-10(4.63) 3.99E-11(4.97) 1.25E-09(4.89) 4.03E-11(4.98) 1.27E-09(4.91)

4 640 1.13E-12(4.92) 2.94E-11(4.76) 1.26E-12(4.99) 4.01E-11(4.96) 1.27E-12(4.99) 4.05E-11(4.97)

1280 3.69E-14(4.94) 1.03E-12(4.84) 3.95E-14(4.99) 1.27E-12(4.98) 3.96E-14(5.00) 1.27E-12(4.99)

2560 1.19E-15(4.96) 3.48E-14(4.88) 1.24E-15(5.00) 3.98E-14(4.99) 1.24E-15(5.00) 3.99E-14(5.00)

where η is the error of L2-projection and pj+1/2(t) = p(xj+1/2, t). Since θGmaxg +1(t) = 0, there holds

an upper triangle system of linear equations

Yg(t)θg(t) = dg(t),

where dg(t) = dj(t) : Gming ≤ j ≤ Gmax

g is a vector and Yg(t) = ygij(t) is a sparse matrix with thenonzeros

ygi,i(t) = 1, and ygi,i+1(t) = −pi+ 12(t).

Here and below the row indices and the column indices are both taken between Gming and Gmax

g in anatural order. A simple manipulation yields the unique solution

θg(t) = Zg(t)dg(t),21

Table 3The errors and convergence orders for the Richtmyer fluxes: T = 0.1 and CR = 0.2.

a = 0 a = 1 a = 2


k N nonuniform mesh

160 6.29E-04 3.38E-03 5.51E-04 4.07E-03 3.69E-04 2.71E-03

320 1.95E-04(1.69) 1.19E-03(1.51) 1.92E-04(1.52) 1.43E-03(1.51) 9.77E-05(1.92) 7.57E-04(1.84)

1 640 5.84E-05(1.74) 3.92E-04(1.60) 6.39E-05(1.59) 4.41E-04(1.69) 2.51E-05(1.96) 1.98E-04(1.93)

1280 1.69E-05(1.79) 1.19E-04(1.72) 2.06E-05(1.63) 1.33E-04(1.73) 6.34E-06(1.98) 4.95E-05(2.00)

2560 4.77E-06(1.83) 3.49E-05(1.77) 6.52E-06(1.66) 4.09E-05(1.70) 1.60E-06(1.99) 1.32E-05(1.91)

160 2.03E-06 3.16E-05 2.07E-06 4.51E-05 1.97E-06 3.76E-05

320 3.08E-07(2.72) 4.90E-06(2.69) 3.33E-07(2.64) 7.65E-06(2.56) 3.00E-07(2.71) 8.99E-06(2.06)

2 640 3.80E-08(3.02) 8.46E-07(2.53) 4.01E-08(3.05) 1.10E-06(2.80) 3.26E-08(3.20) 1.05E-06(3.10)

1280 4.77E-09(3.00) 1.17E-07(2.85) 4.56E-09(3.14) 1.25E-07(3.13) 4.53E-09(2.85) 1.62E-07(2.70)

2560 6.66E-10(2.84) 2.17E-08(2.43) 6.11E-10(2.90) 2.14E-08(2.55) 5.40E-10(3.07) 1.96E-08(3.05)

160 1.45E-07 1.82E-06 1.20E-07 1.95E-06 9.37E-08 1.87E-06

320 1.37E-08(3.41) 1.76E-07(3.37) 1.06E-08(3.50) 1.95E-07(3.32) 6.76E-09(3.79) 1.45E-07(3.69)

3 640 1.26E-09(3.44) 1.85E-08(3.25) 7.97E-10(3.73) 1.47E-08(3.73) 4.37E-10(3.95) 1.05E-08(3.78)

1280 1.07E-10(3.56) 1.58E-09(3.55) 5.53E-11(3.85) 1.07E-09(3.78) 2.77E-11(3.98) 6.29E-10(4.06)

2560 8.61E-12(3.63) 1.44E-10(3.45) 3.69E-12(3.91) 7.12E-11(3.91) 1.78E-12(3.96) 4.44E-11(3.82)

160 1.18E-09 2.27E-08 1.30E-09 2.85E-08 1.27E-09 4.30E-08

320 4.16E-11(4.83) 1.15E-09(4.31) 3.39E-11(5.26) 1.18E-09(4.60) 4.08E-11(4.96) 1.74E-09(4.62)

4 640 1.28E-12(5.02) 3.55E-11(5.02) 1.39E-12(4.61) 6.18E-11(4.25) 1.37E-12(4.90) 5.54E-11(4.98)

1280 5.45E-14(4.55) 1.89E-12(4.23) 4.53E-14(4.94) 1.85E-12(5.06) 3.91E-14(5.13) 2.02E-12(4.78)

2560 1.75E-15(4.96) 8.57E-14(4.47) 1.43E-15(4.99) 8.04E-14(4.53) 1.29E-15(4.92) 7.99E-14(4.66)

k N uniform mesh

160 6.27E-04 3.29E-03 5.47E-04 4.00E-03 3.64E-04 2.58E-03

320 1.94E-04(1.70) 1.14E-03(1.53) 1.91E-04(1.52) 1.37E-03(1.55) 9.63E-05(1.92) 6.94E-04(1.90)

1 640 5.80E-05(1.74) 3.74E-04(1.61) 6.35E-05(1.59) 4.29E-04(1.67) 2.48E-05(1.96) 1.79E-04(1.95)

1280 1.68E-05(1.79) 1.15E-04(1.70) 2.05E-05(1.63) 1.30E-04(1.73) 6.27E-06(1.98) 4.54E-05(1.98)

2560 4.73E-06(1.83) 3.38E-05(1.77) 6.47E-06(1.66) 4.03E-05(1.68) 1.58E-06(1.99) 1.14E-05(1.99)

160 1.68E-06 2.63E-05 1.73E-06 3.18E-05 1.86E-06 3.68E-05

320 2.09E-07(3.01) 3.30E-06(3.00) 2.15E-07(3.01) 3.94E-06(3.01) 2.31E-07(3.01) 4.59E-06(3.00)

2 640 2.61E-08(3.00) 4.13E-07(3.00) 2.68E-08(3.00) 4.94E-07(3.00) 2.88E-08(3.00) 5.76E-07(3.00)

1280 3.26E-09(3.00) 5.16E-08(3.00) 3.35E-09(3.00) 6.19E-08(3.00) 3.60E-09(3.00) 7.22E-08(3.00)

2560 4.08E-10(3.00) 6.46E-09(3.00) 4.19E-10(3.00) 7.74E-09(3.00) 4.51E-10(3.00) 9.03E-09(3.00)

160 1.39E-07 1.62E-06 1.21E-07 2.06E-06 8.91E-08 1.60E-06

320 1.34E-08(3.38) 1.69E-07(3.25) 1.02E-08(3.57) 1.79E-07(3.53) 6.31E-09(3.82) 1.19E-07(3.75)

3 640 1.21E-09(3.46) 1.67E-08(3.34) 7.67E-10(3.73) 1.37E-08(3.71) 4.16E-10(3.92) 7.88E-09(3.91)

1280 1.03E-10(3.56) 1.52E-09(3.46) 5.32E-11(3.85) 9.50E-10(3.85) 2.66E-11(3.97) 5.04E-10(3.97)

2560 8.22E-12(3.65) 1.27E-10(3.57) 3.53E-12(3.91) 6.23E-11(3.93) 1.68E-12(3.99) 3.18E-11(3.99)

160 9.10E-10 2.01E-08 9.35E-10 2.62E-08 9.92E-10 2.99E-08

320 2.76E-11(5.05) 6.50E-10(4.95) 2.82E-11(5.05) 7.90E-10(5.05) 3.01E-11(5.04) 9.17E-10(5.03)

4 640 8.55E-13(5.01) 2.04E-11(4.99) 8.76E-13(5.01) 2.45E-11(5.01) 9.35E-13(5.01) 2.86E-11(5.00)

1280 2.67E-14(5.00) 6.40E-13(5.00) 2.73E-14(5.00) 7.66E-13(5.00) 2.92E-14(5.00) 8.94E-13(5.00)

2560 8.34E-16(5.00) 2.00E-14(5.00) 8.54E-16(5.00) 2.40E-14(5.00) 9.12E-16(5.00) 2.80E-14(5.00)

where Zg(t) = zgij(t) is the upper triangle matrix with the non-zero entries

(7.3) zgij(t) =

j−1∏m=i

pm+ 12(t), j ≥ i.

Note that∏m2

m=m1= 1 if m2 < m1 in this paper; hence zgii(t) = 1. Below we are going to prove (7.1)

for two cases stated in Theorem 2.1.Case 2(a): It is easy to know from the approximation property of the local L2-projection (or

Lemma 3.2) that

(7.4) ‖dg(t)‖∞ ≤ Chk+1, ‖∂tdg(t)‖∞ ≤ Chk+1.

22

Table 4The errors and convergence orders for the GFORCE fluxes: T = 0.1; CGLF = a + 1 and CR = 0.2.

a = 0 a = 1 a = 2


k N nonuniform mesh

160 1.73E-04 1.44E-03 1.62E-04 1.48E-03 1.70E-04 1.72E-03

320 4.38E-05(1.99) 3.76E-04(1.93) 4.09E-05(1.98) 4.50E-04(1.72) 4.31E-05(1.98) 4.41E-04(1.97)

1 640 1.12E-05(1.97) 1.02E-04(1.88) 1.03E-05(1.99) 1.16E-04(1.96) 1.08E-05(2.00) 1.13E-04(1.97)

1280 2.78E-06(2.01) 2.52E-05(2.02) 2.60E-06(1.99) 2.75E-05(2.07) 2.70E-06(1.99) 3.09E-05(1.87)

2560 6.97E-07(2.00) 6.77E-06(1.90) 6.53E-07(1.99) 7.25E-06(1.92) 6.74E-07(2.00) 7.30E-06(2.08)

160 3.21E-06 4.60E-05 3.11E-06 6.66E-05 2.81E-06 6.00E-05

320 4.85E-07(2.73) 7.82E-06(2.56) 4.15E-07(2.91) 7.87E-06(3.08) 3.44E-07(3.03) 9.30E-06(2.69)

2 640 7.66E-08(2.66) 1.48E-06(2.40) 5.73E-08(2.86) 1.42E-06(2.47) 4.24E-08(3.02) 1.18E-06(2.98)

1280 1.21E-08(2.66) 2.51E-07(2.56) 7.88E-09(2.86) 1.51E-07(3.24) 5.51E-09(2.94) 1.74E-07(2.77)

2560 1.91E-09(2.66) 4.45E-08(2.50) 1.10E-09(2.84) 1.84E-08(3.03) 6.66E-10(3.05) 2.05E-08(3.08)

160 4.33E-08 9.24E-07 5.65E-08 1.41E-06 6.00E-08 1.22E-06

320 2.95E-09(3.88) 8.04E-08(3.52) 3.39E-09(4.06) 8.40E-08(4.06) 3.88E-09(3.95) 1.25E-07(3.28)

3 640 1.88E-10(3.97) 5.66E-09(3.83) 2.27E-10(3.90) 7.48E-09(3.49) 2.31E-10(4.07) 6.64E-09(4.24)

1280 1.22E-11(3.95) 3.73E-10(3.92) 1.38E-11(4.04) 4.04E-10(4.21) 1.47E-11(3.97) 4.32E-10(3.94)

2560 7.57E-13(4.00) 2.40E-11(3.96) 8.86E-13(3.96) 3.09E-11(3.71) 9.12E-13(4.01) 2.72E-11(3.99)

160 1.46E-09 2.71E-08 1.66E-09 5.12E-08 1.40E-09 4.17E-08

320 5.24E-11(4.80) 1.27E-09(4.42) 5.43E-11(4.93) 2.31E-09(4.47) 4.54E-11(4.94) 1.37E-09(4.93)

4 640 2.26E-12(4.54) 6.24E-11(4.35) 1.72E-12(4.98) 7.27E-11(4.99) 1.60E-12(4.83) 8.46E-11(4.02)

1280 9.94E-14(4.50) 3.39E-12(4.20) 5.08E-14(5.08) 2.04E-12(5.16) 4.85E-14(5.04) 2.47E-12(5.10)

2560 3.80E-15(4.71) 1.31E-13(4.69) 1.64E-15(4.96) 7.66E-14(4.73) 1.44E-15(5.08) 6.77E-14(5.19)

k N uniform mesh

160 1.69E-04 1.19E-03 1.56E-04 1.39E-03 1.65E-04 1.37E-03

320 4.28E-05(1.98) 3.09E-04(1.95) 3.93E-05(1.99) 3.59E-04(1.95) 4.16E-05(1.99) 3.55E-04(1.95)

1 640 1.08E-05(1.99) 7.88E-05(1.97) 9.86E-06(1.99) 9.12E-05(1.98) 1.04E-05(2.00) 9.03E-05(1.98)

1280 2.70E-06(1.99) 1.99E-05(1.99) 2.47E-06(2.00) 2.29E-05(1.99) 2.61E-06(2.00) 2.27E-05(1.99)

2560 6.77E-07(2.00) 4.99E-06(1.99) 6.19E-07(2.00) 5.76E-06(2.00) 6.54E-07(2.00) 5.70E-06(1.99)

160 3.00E-06 4.07E-05 2.87E-06 5.26E-05 2.54E-06 4.99E-05

320 4.67E-07(2.68) 7.28E-06(2.48) 3.94E-07(2.86) 6.94E-06(2.92) 3.20E-07(2.99) 6.43E-06(2.96)

2 640 7.42E-08(2.66) 1.30E-06(2.48) 5.44E-08(2.86) 8.82E-07(2.98) 4.02E-08(2.99) 8.12E-07(2.99)

1280 1.18E-08(2.65) 2.28E-07(2.51) 7.57E-09(2.85) 1.11E-07(2.99) 5.04E-09(3.00) 1.02E-07(3.00)

2560 1.86E-09(2.67) 3.95E-08(2.53) 1.07E-09(2.83) 1.39E-08(3.00) 6.31E-10(3.00) 1.27E-08(3.00)

160 4.07E-08 7.06E-07 4.91E-08 1.03E-06 5.33E-08 1.08E-06

320 2.57E-09(3.98) 4.66E-08(3.92) 3.10E-09(3.99) 6.63E-08(3.95) 3.37E-09(3.98) 6.89E-08(3.97)

3 640 1.62E-10(3.99) 3.01E-09(3.95) 1.95E-10(3.99) 4.24E-09(3.97) 2.11E-10(3.99) 4.38E-09(3.97)

1280 1.02E-11(3.99) 1.92E-10(3.97) 1.22E-11(4.00) 2.68E-10(3.99) 1.32E-11(4.00) 2.76E-10(3.99)

2560 6.38E-13(4.00) 1.21E-11(3.99) 7.65E-13(4.00) 1.68E-11(3.99) 8.29E-13(4.00) 1.73E-11(4.00)

160 1.31E-09 2.18E-08 1.37E-09 3.93E-08 1.27E-09 3.77E-08

320 5.16E-11(4.66) 1.11E-09(4.30) 4.51E-11(4.92) 1.34E-09(4.88) 4.01E-11(4.99) 1.25E-09(4.91)

4 640 2.12E-12(4.60) 5.33E-11(4.38) 1.45E-12(4.96) 4.34E-11(4.95) 1.26E-12(4.99) 4.01E-11(4.97)

1280 8.81E-14(4.59) 2.50E-12(4.42) 4.60E-14(4.98) 1.37E-12(4.99) 3.95E-14(5.00) 1.26E-12(4.99)

2560 3.62E-15(4.60) 1.12E-13(4.47) 1.45E-15(4.99) 4.29E-14(4.99) 1.23E-15(5.00) 3.94E-14(5.00)

To get the second conclusion we have used the simple fact that

(7.5) |∂tp(x, t)| ≤ C|∂tu(x, t)| ≤ C|ut(x, t)|+ Cτ, t ∈ [0, T ]

since β(·) is Lipschitz continuous on Gg and utt(x, t) is bounded.Let xmin

g and xmaxg be two adjoint sonic point positions, such that Gg ⊂ (xmin

g , xmaxg ). Since the

continuous function achieves its maximum and minimum in the closed interval, the condition in case2(a) implies a positive constant p?g < 1 independent of h such that |p(x, t)| ≤ p?g on [xmin

g , xmaxg ]. It

follows from (7.3) that |zgij(t)| ≤ (p?g)j−i for j ≥ i. A simple calculation yields

(7.6) ‖Zg(t)‖∞ ≤ maxGming ≤i≤Gmax

g

∑i≤j≤Gmax

g

(p?g)j−i ≤ 1

1− p?g,

23

Table 5The errors and convergence orders for the LFORCE fluxes: T = 0.1 and CR = 0.2.

a = 0 a = 1 a = 2


k N nonuniform mesh

160 2.33E-04 1.88E-03 2.25E-04 2.01E-03 2.16E-04 1.89E-03

320 6.12E-05(1.93) 5.60E-04(1.75) 5.96E-05(1.91) 5.66E-04(1.83) 5.48E-05(1.98) 4.58E-04(2.04)

1 640 1.58E-05(1.95) 1.36E-04(2.04) 1.52E-05(1.97) 1.34E-04(2.08) 1.38E-05(1.99) 1.29E-04(1.83)

1280 4.03E-06(1.97) 3.64E-05(1.90) 3.85E-06(1.98) 3.54E-05(1.92) 3.47E-06(1.99) 3.40E-05(1.92)

2560 1.01E-06(1.99) 9.16E-06(1.99) 9.72E-07(1.99) 8.98E-06(1.98) 8.66E-07(2.00) 8.09E-06(2.07)

160 1.97E-06 3.54E-05 2.26E-06 5.55E-05 2.35E-06 5.85E-05

320 2.46E-07(3.00) 5.12E-06(2.79) 2.82E-07(3.00) 8.07E-06(2.78) 2.90E-07(3.02) 6.66E-06(3.14)

2 640 3.35E-08(2.87) 9.10E-07(2.49) 3.46E-08(3.03) 9.98E-07(3.02) 3.63E-08(3.00) 9.18E-07(2.86)

1280 4.09E-09(3.03) 1.05E-07(3.12) 4.45E-09(2.96) 1.37E-07(2.87) 4.64E-09(2.97) 1.31E-07(2.80)

2560 5.38E-10(2.93) 1.68E-08(2.64) 5.69E-10(2.97) 1.74E-08(2.98) 5.86E-10(2.99) 1.89E-08(2.80)

160 6.41E-08 1.28E-06 7.14E-08 1.53E-06 7.29E-08 1.62E-06

320 4.67E-09(3.78) 1.16E-07(3.47) 5.01E-09(3.83) 1.21E-07(3.67) 4.40E-09(4.05) 9.28E-08(4.12)

3 640 3.30E-10(3.82) 9.04E-09(3.68) 3.12E-10(4.01) 8.55E-09(3.82) 3.01E-10(3.87) 8.38E-09(3.47)

1280 2.08E-11(3.99) 5.41E-10(4.06) 2.02E-11(3.95) 5.02E-10(4.09) 1.81E-11(4.05) 4.93E-10(4.09)

2560 1.33E-12(3.96) 3.77E-11(3.84) 1.25E-12(4.01) 3.83E-11(3.71) 1.13E-12(4.00) 3.22E-11(3.94)

160 1.23E-09 3.60E-08 1.16E-09 3.91E-08 1.11E-09 3.18E-08

320 3.81E-11(5.01) 1.28E-09(4.81) 4.40E-11(4.72) 1.98E-09(4.30) 4.11E-11(4.76) 1.65E-09(4.27)

4 640 1.10E-12(5.12) 3.29E-11(5.28) 1.18E-12(5.22) 5.63E-11(5.14) 1.30E-12(4.98) 4.92E-11(5.07)

1280 3.69E-14(4.89) 1.64E-12(4.33) 4.06E-14(4.86) 2.76E-12(4.35) 4.17E-14(4.96) 1.86E-12(4.73)

2560 1.17E-15(4.98) 6.63E-14(4.63) 1.32E-15(4.94) 8.29E-14(5.06) 1.28E-15(5.03) 6.73E-14(4.79)

k N uniform mesh

160 2.28E-04 1.65E-03 2.20E-04 1.69E-03 2.10E-04 1.61E-03

320 6.01E-05(1.92) 4.48E-04(1.88) 5.79E-05(1.92) 4.51E-04(1.91) 5.35E-05(1.97) 4.19E-04(1.94)

1 640 1.55E-05(1.95) 1.17E-04(1.94) 1.49E-05(1.96) 1.16E-04(1.96) 1.35E-05(1.99) 1.07E-04(1.97)

1280 3.95E-06(1.97) 2.97E-05(1.97) 3.78E-06(1.98) 2.95E-05(1.98) 3.38E-06(1.99) 2.68E-05(1.99)

2560 9.98E-07(1.99) 7.51E-06(1.99) 9.54E-07(1.99) 7.44E-06(1.99) 8.47E-07(2.00) 6.74E-06(1.99)

160 1.80E-06 2.73E-05 2.03E-06 4.06E-05 2.16E-06 4.35E-05

320 2.31E-07(2.96) 3.96E-06(2.78) 2.54E-07(3.00) 5.18E-06(2.97) 2.70E-07(3.00) 5.54E-06(2.97)

2 640 2.95E-08(2.97) 5.30E-07(2.90) 3.18E-08(3.00) 6.54E-07(2.98) 3.37E-08(3.00) 6.98E-07(2.99)

1280 3.74E-09(2.98) 7.04E-08(2.91) 3.98E-09(3.00) 8.22E-08(2.99) 4.22E-09(3.00) 8.75E-08(2.99)

2560 4.72E-10(2.99) 9.12E-09(2.95) 4.98E-10(3.00) 1.03E-08(3.00) 5.28E-10(3.00) 1.10E-08(3.00)

160 6.23E-08 1.01E-06 6.71E-08 1.22E-06 6.36E-08 1.22E-06

320 4.41E-09(3.82) 7.80E-08(3.70) 4.50E-09(3.90) 8.52E-08(3.84) 4.13E-09(3.94) 8.10E-08(3.91)

3 640 2.98E-10(3.89) 5.46E-09(3.84) 2.90E-10(3.95) 5.57E-09(3.93) 2.62E-10(3.98) 5.18E-09(3.97)

1280 1.95E-11(3.93) 3.62E-10(3.91) 1.84E-11(3.98) 3.56E-10(3.97) 1.65E-11(3.99) 3.26E-10(3.99)

2560 1.25E-12(3.96) 2.33E-11(3.96) 1.16E-12(3.99) 2.25E-11(3.99) 1.03E-12(4.00) 2.05E-11(3.99)

160 9.28E-10 1.96E-08 1.06E-09 3.20E-08 1.12E-09 3.40E-08

320 2.96E-11(4.97) 7.06E-10(4.80) 3.28E-11(5.02) 1.03E-09(4.96) 3.46E-11(5.02) 1.10E-09(4.96)

4 640 9.44E-13(4.97) 2.50E-11(4.82) 1.02E-12(5.00) 3.24E-11(4.99) 1.08E-12(5.00) 3.45E-11(4.99)

1280 2.99E-14(4.98) 8.37E-13(4.90) 3.20E-14(5.00) 1.02E-12(4.99) 3.37E-14(5.00) 1.08E-12(4.99)

2560 9.46E-16(4.99) 2.75E-14(4.93) 9.99E-16(5.00) 3.19E-14(5.00) 1.05E-15(5.00) 3.39E-14(5.00)

which together with (7.4) yield

(7.7) ‖θg(t)‖∞ ≤ ‖Zg(t)‖∞‖dg(t)‖∞ ≤ Chk+1.

Since Yg(·) is the inverse of Zg(·), we have

(7.8) ∂tθg(t) = −Zg(t)∂tYg(t)θg(t+ τ) + Zg(t)∂tdg(t).

Since ∂tYg(t) are zeroes expect ∂tpi+1/2(t) at the (i, i+ 1) position, we have ‖∂tYg(t)‖∞ ≤ C, due to(7.5). Together with (7.4), (7.6) and (7.7), the formula (7.8) implies that

‖∂tθg(t)‖∞ ≤ ‖Zg(t)‖∞‖∂tYg(t)‖∞‖θg(t+ τ)‖∞ + ‖Zg(t)‖∞‖∂tdg(t)‖∞ ≤ Chk+1.

24

Case 2(b): Since p(x, t) = −1 is achieved at the sonic point position in this case, we only havea bad boundedness

(7.9) ‖Zg(t)‖∞ ≤ Ch−1,

instead of (7.6). Along the same line as that for case 2(a), we are not able to get the optimal order.To prove (7.1), we need a more detailed observation on dg(t) and Zg(t).

First we recall the superconvergence of the L2-projection on the uniform mesh, i.e., h = hj forever.By the Legendre expansion of w = w(x) on each element, we have

(7.10) η(x) = wj,k+1Lj,k+1(x) + rj(x), x ∈ Ij ,

where [4, 28]

wj,k+1 =(−1)k+1(2k + 3)hk+1

22k+3(k + 1)!

∫ 1

−1∂k+1x w

(xj +

hx

2

)(x2 − 1)k+1dx,

and the remainder satisfies ‖rj‖∞ ≤ Chk+2 if w ∈ W k+2,∞(I). Here xj is the middle point in Ij . Itis well known that

|wj,k+1| ≤ Chk+1, and |wj,k+1 − wj+1,k+1| ≤ Chk+2.

Substituting the above conclusions in the definition of dj(t), we have

dj(t) = [1 + pj+ 12(t)]wj,k+1 + wj(t),

where wj(t) = pj+1/2(t)[wj+1,k+1 − wj,k+1 + (−1)k+1rj+1(x+j+1/2)] + rj(x−j+1/2).

Letting wg(t) = wj,k+1 and wg(t) = wj(t), and we get the important expression

(7.11) θg(t) = Zg(t)dg(t) = Hg(t)wg + Zg(t)wg(t),

where Hg(t) = hgij(t) is the upper triangle matrix with the non-zeroes

(7.12) hgij(t) = zgij(t)[1 + pj+ 1

2(t)]

= (−1)j−i

(j−1∏m=i

[−pm+ 12(t)]−

j∏m=i

[−pm+ 12(t)]

), j ≥ i.

From the definitions, it is easy to see that

‖wg‖∞ ≤ Chk+1, ‖wg(t)‖∞ ≤ Chk+2.

Noticing (7.9), it is sufficient to prove ‖θg(t)‖∞ ≤ Chk+1 by showing the next proposition.

Proposition 7.1. Under the condition as in Theorem 2.1, ‖Hg(t)‖∞ is bounded for case 2(b).

Proof. Since u(x, t) is sufficiently smooth, it follows from assumption A2 that p(x, t) is Lipschitzcontinuous with respect to x ∈ [xmin

g , xming ], and the Lipschitz constant is independent of t. Note that

p(xming , t) = p(xmax

g , t) = −1 in this case. Therefore, there are two separating points xleftg and xrightg ,together with a positive number p?g < 1, all independent of t and h, such that −p?g ≤ p(x, t) ≤ p?g on

[xleftg , xrightg ]× [0, T ] and p(x, t) ≤ 0 otherwise. Let Gleftg and Gright

g be two integers such that

xGleftg + 1

2≤ xleftg < xGleft

g + 32, and xGright

g − 32< xrightg ≤ xGright

g − 12,

the elements in Gg are divided into three groups. For Gleftg + 1 ≤ j ≤ Gright

g − 1, we have the similarboundedness as in case 2(a), namely,

|hgij(t)| ≤ 2(p?g

)j−max(i,Gleftg +1)

.

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Their sum at the same row is bounded by 2/(1 − p?g). Otherwise, there holds a nice boundedness inthe head to tail connection by a sequence of positive numbers, namely,

|hgij(t)| =

∣∣∣∣∣i′−1∏m=i

pm+ 12(t)

∣∣∣∣∣∣∣∣∣∣j−1∏m=i′

[−pm+ 12(t)]−

j∏m=i′

[−pm+ 12(t)]

∣∣∣∣∣ ≤j−1∏m=i′

[−pm+ 12(t)]−

j∏m=i′

[−pm+ 12(t)],

with i′ = i if j ≤ Gleftg or i′ = max(i, Gright

g ) if j ≥ Grightg . Since all multipliers are not greater than 1,

the sum of |hgij(t)| at the same row is bounded by 1 for two groups with small and big j, respectively.Summing up the above conclusions, we have∑

i≤j≤Gmaxg

|hgij(t)| ≤ 2 +2

1− p?g, Gmin

g ≤ i ≤ Gmaxg ,

and complete the proof of this proposition.

To prove ‖∂tθg(t)‖∞ ≤ Chk+1 in case 2(b), we start from a new inequality due to (7.8)

(7.13) ‖∂tθg(t)‖∞ ≤ ‖Zg(t)∂tYg(t)‖∞‖θg(t+ τ)‖∞ + ‖Zg(t)∂tdg(t)‖∞.

Since |f ′(·)|, α(·) and π(·) are all bounded, it follows from

|f ′(u)|(−1)k+1β(u) + 1

=

12 |f′(u)|+ α(u), if k is odd,

12 |f′(u)|+ 1

4π(u), if k is even,

that this quantity is bounded. Since u(x, t) satisfies (1.1), we can get from (7.5) that

(7.14)∣∣∂tp(x, t)∣∣ ≤ C|f ′(u(x, t))||ux(x, t)|+ Cτ ≤ C [p(x, t) + 1] + Cτ.

This implies that the non-zero entry in the strictly upper triangle matrix Zg(t)∂tYg(t) = hgij(t) isbounded in the form

|hgi,j+1(t)| ≤ C|zgij(t)[pj+ 12(t) + 1]|+ Cτ |zgij(t)| = C|hgij(t)|+ Cτ |zgij(t)|, j ≥ i.

Together with (7.9) and Proposition 7.1, since τ ≤ h, we have

(7.15) ‖Zg(t)∂tYg(t)‖∞ ≤ C‖Hg(t)‖∞ + Cτ‖Zg(t)‖∞ ≤ C.

It follows from (7.14) that

|∂tdj(t)| = |∂tpj+ 12(t)η+

j+ 12

| ≤ C[pj+ 12(t) + 1]hk+1 + Cτhk+1.

Along the same line as for ‖Zg(t)∂tYg(t)‖∞, we have

(7.16) ‖Zg(t)∂tdg(t)‖∞ ≤ C[‖Hg(t)‖∞ + τ‖Zg(t)‖∞

]hk+1 ≤ Chk+1.

Since ‖θg(t+ τ)‖∞ ≤ Chk+1 has been just proved, we can get the expected result from (7.13), (7.15)and (7.16). We have now completed the proof of Lemma 3.2.

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