ALTERNATING EVOLUTION (AE) SCHEMES FOR HYPERBOLIC CONSERVATION

32
ALTERNATING EVOLUTION (AE) SCHEMES FOR HYPERBOLIC CONSERVATIONLAWS HASEENA AHMED AND HAILIANG LIU Abstract. The alternating evolution (AE) system of Liu [27] t u + xf (v)= 1 (v - u), t v + xf (u)= 1 (u - v), serves as a refined description of systems of hyperbolic conservation laws t φ + xf (φ)=0, φ(x, 0) = φ 0 (x). The solution of conservation laws is precisely captured when two components take the same initial value as φ 0 , or approached by two components exponentially fast when 0 if two initial states are sufficiently close. This nice property enables us to construct novel shock capturing schemes by sampling the AE system on alternating grids. In this paper we develop a class of local Alternating Evolution (AE) schemes by taking advantage of the AE system. Our approach is based on an average of the AE system over a hypercube centered at x with vertices at x ± Δx. The numerical scheme is then constructed by sampling the averaged system over alternating grids. Higher order accuracy is achieved by a combination of high-order non- oscillatory polynomial reconstruction from the obtained averages and a matching Runge-Kutta solver in time discretization. Local AE schemes are made possible by letting the scale parameter reflect the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation and implementation, and efficient computation of the solution. For the first and second order local AE schemes applied to one-dimensional scalar conservation laws, we prove the numerical stability in the sense of satisfying the maximum principle and total variation diminishing (TVD) property. The formulation procedure of AE schemes in multi-dimension is given, followed by both the first and second order AE schemes for two-dimensional conservation laws. Numerical experiments for both scalar conservation laws and compressible Euler equations are presented to demonstrate the high order accuracy and capacity of these AE schemes. Contents 1. Introduction 2 2. Scheme formulation 4 2.1. Global AE schemes 4 2.2. Local AE schemes 6 3. Stability analysis 9 3.1. Stability of first order AE schemes 9 3.2. Stability of second order AE schemes 10 4. Extension to Multi-dimensions 12 4.1. First order scheme 13 4.2. Second Order Scheme 13 5. Numerical tests 14 5.1. Scalar conservation laws 14 5.2. Euler equations of gas dynamics 18 6. Concluding remarks 20 Acknowledgments 20 References 21 Date : August, 2008; revision May 2010, April 2011. 1991 Mathematics Subject Classification. 35L65, 65M06. Key words and phrases. Conservation laws, Alternating evolution schemes, Shock capturing methods, TVD stability. 1

Transcript of ALTERNATING EVOLUTION (AE) SCHEMES FOR HYPERBOLIC CONSERVATION

ALTERNATING EVOLUTION (AE) SCHEMES FOR HYPERBOLIC

CONSERVATION LAWS

HASEENA AHMED AND HAILIANG LIU

Abstract. The alternating evolution (AE) system of Liu [27]

∂tu+ ∂xf(v) =1

ε(v − u), ∂tv + ∂xf(u) =

1

ε(u− v),

serves as a refined description of systems of hyperbolic conservation laws

∂tφ+ ∂xf(φ) = 0, φ(x, 0) = φ0(x).

The solution of conservation laws is precisely captured when two components take the same initialvalue as φ0, or approached by two components exponentially fast when ε ↓ 0 if two initial states

are sufficiently close. This nice property enables us to construct novel shock capturing schemes by

sampling the AE system on alternating grids.In this paper we develop a class of local Alternating Evolution (AE) schemes by taking advantage

of the AE system. Our approach is based on an average of the AE system over a hypercube centered

at x with vertices at x ± ∆x. The numerical scheme is then constructed by sampling the averagedsystem over alternating grids. Higher order accuracy is achieved by a combination of high-order non-

oscillatory polynomial reconstruction from the obtained averages and a matching Runge-Kutta solverin time discretization. Local AE schemes are made possible by letting the scale parameter ε reflect

the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation

and implementation, and efficient computation of the solution. For the first and second order localAE schemes applied to one-dimensional scalar conservation laws, we prove the numerical stability

in the sense of satisfying the maximum principle and total variation diminishing (TVD) property.

The formulation procedure of AE schemes in multi-dimension is given, followed by both the firstand second order AE schemes for two-dimensional conservation laws. Numerical experiments for both

scalar conservation laws and compressible Euler equations are presented to demonstrate the high order

accuracy and capacity of these AE schemes.

Contents

1. Introduction 22. Scheme formulation 42.1. Global AE schemes 42.2. Local AE schemes 63. Stability analysis 93.1. Stability of first order AE schemes 93.2. Stability of second order AE schemes 104. Extension to Multi-dimensions 124.1. First order scheme 134.2. Second Order Scheme 135. Numerical tests 145.1. Scalar conservation laws 145.2. Euler equations of gas dynamics 186. Concluding remarks 20Acknowledgments 20References 21

Date: August, 2008; revision May 2010, April 2011.

1991 Mathematics Subject Classification. 35L65, 65M06.Key words and phrases. Conservation laws, Alternating evolution schemes, Shock capturing methods, TVD stability.

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2 HASEENA AHMED AND HAILIANG LIU

1. Introduction

A multi-D hyperbolic conservation law (CL) has the form:

(1.1) φt +∇x · f(φ) = 0, x ∈ Rd, t > 0,

where φ ∈ Rm denotes a vector of conserved quantities, and f : Rm → Rd is a nonlinear convectionflux. The compressible Euler equation in gas dynamics is a canonical example. These equations areof great practical importance since they model a variety of physical phenomena that appear in fluidmechanics, astrophysics, groundwater flow, traffic flow, semiconductor device simulation, and magneto-hydrodynamics, among many others.

The need for devising accurate and efficient numerical methods for nonlinear hyperbolic conservationlaws and related models has prompted and sustained the abundant research in this area, see, for example,[25, 39, 38, 6]. The notorious difficulty encountered for the satisfactory approximation of the exactsolutions of these systems lies in the presence of discontinuities in the solution. A well known recipe toachieve both high-order accuracy and convergence to the entropy solution is the so called high-resolutionschemes. The success has been due to two factors: the local enforcement of nonlinear conservation lawsand the non-oscillatory piecewise polynomial reconstruction from evolved local moments (cell averages).

In this work we present new alternating evolution (AE) schemes to (1.1), together with stabilityanalysis and performance test through numerical experiments. These schemes can be viewed as mod-ifications of the global AE scheme proposed in [27]. In the formulation of these schemes we borrowtechniques of both the local enforcement and the high-order polynomial reconstruction from the litera-ture, but apply them to a novel approximation system

ut +∇x · f(v) =1

ε(v − u),(1.2)

vt +∇x · f(u) =1

ε(u− v).(1.3)

Here ε > 0 is a scale parameter of user’s choice. One distinguished feature of this system is the followinginequality:

‖u(·, t)− v(·, t)‖L1 ≤ ‖u0(· − v0(·)‖e−2t/ε.

This implies that exact solution for nonlinear conservation laws is captured by this system if u(x, 0) =v(x, 0) = φ0(x). When ‖u0(·) − v0(·)‖L1 is sufficiently small, both components approach each otherexponentially fast as ε ↓ 0, see [27]. Such a feature allows a sampling of the system over alternatinggrids when performing spatial discretization. On the other hand if the system is rewritten as

u+ εut = v − ε∇x · f(v), v + εvt = u− ε∇x · f(u),

one can see for ε = O(∆t) the system is able to transfer spatial changes of one component into temporalchanges of another component. The communication between two components is thus realized by thechoice of parameter ε. A word of caution: the AE system may well lose hyperbolicity in generalsetting, which by itself is an interesting issue. But this can be remedied by recasting the originalconservation laws on a fast moving reference. Numerically such a remedy is unnecessary since thedissipation produced from the averaging operation has provided an ideal fix.

The numerical schemes, presented in [27] and this work, take advantage of these remarkable featuresof the AE system. There are three steps involved in the discretization procedure: (i) the AE systemis locally enforced by taking the spatial averaging over a hypercube centered at x with vertices beingits neighboring grids; (ii) the averaged system is then sampled over alternating grids for its two com-ponents; (iii) the piecewise polynomial reconstruction from averages of two components is carried out,respectively, followed by an ODE solver with matching accuracy for time discretization of the resultingsemi-discrete system. The strong stability preserving method [11] is particularly suitable. Extension ofthis procedure to multi-dimensional problems is straightforward. The scheme obtained in this manneris thus called the alternating evolution (AE) scheme. The result is a simple and high-resolution finitedifference scheme, whose derivation involves no numerical fluxes and avoids dimensional splitting.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 3

To put our study in the proper perspective we recall a few of the references from the consider-able amount of literature available on numerical methods for hyperbolic conservation laws. The finitedifference/volume solution techniques for nonlinear conservation laws fall under two main categoriesaccording to their way of sampling [33]: upwind and central schemes. The forerunners for these twolarge classes of high resolution schemes for nonlinear conservation laws are the first-order Godunov [10]and Lax-Friedrichs (LxF) schemes [9, 24], respectively.

In [10] Godunov proposed to evolve a piecewise cell average representation of the solution and eval-uate the fluxes at cell interfaces. Godunov type upwind schemes are then obtained by using an exactor approximate Riemann solver to distribute the non-linear waves between two neighboring compu-tational cells. Various higher-order extensions of the Godunov scheme have been rapidly developedsince 1970’s, employing higher-order reconstruction of piece-wise polynomials from the cell averages,including MUSCL, TVD, PPM, ENO and WENO schemes [40, 41, 12, 7, 13, 35, 36, 28].

In contrast, Godunov type central schemes are more diffusive, yet easy to formulate and implementsince no Riemann solvers are required. In the one-dimensional case, examples of such schemes forconservation laws are the second-order Nessyahu-Tadmor scheme [33] and the higher-order schemesin [29, 1, 26]. Second-order multi-D central schemes were introduced in [16], and their higher-orderextensions were developed in [21]. For non-staggered central schemes consult [15]

The two categories of schemes are somehow interlaced during their independent developments; theupwind scheme becomes Riemann solver-free when a local numerical flux is used to replace the exactRiemann solver, see Shu and Osher [35, 36], and the central scheme becomes less diffusive when variablecontrol volumes are used in deriving the scheme, see Kurganov and Tadmor [22]. The upwind featurecan be further enforced [14, 20] in central-upwind schemes, see [19] for a recent derivation of such ascheme. The relaxation scheme of Jin and Xin [17] provides another approach for solving nonlinearconservation laws.

More closely related is the recent work of Liu [30] who generalizes the NT scheme [33] by evolvingtwo pieces of information over redundant overlapping cells, therefore allows easy formulation of semi-discrete schemes. The numerical solution is represented as a convex combination of overlapping cellaverages evolved by the NT scheme. The technique has recently been applied in the development of acentral-type discontinuous Galerkin method [31].

The scheme that we design here is different in that it is based on alternatively sampling the AE system[27] with spatial accuracy enhanced by interlaced local reconstructions. The procedure discussed in thispaper opens a new way to derive robust symmetric finite difference schemes for hyperbolic conservationlaws. Also the semi-discrete scheme thus obtained offers an economic approach for achieving a matchingaccuracy in time. Another attractive feature of the AE scheme is the amount of leverage in the choiceof ε. Indeed different choices of the scale parameter in such a procedure yield different AE schemes suchas local AE schemes presented in this paper.

The AE schemes like other symmetric finite difference schemes are very easy to implement andefficient, but they can in some cases not match some more compact high order methods such as theDG method [34]. We also note that some schemes in [30] when rescaling the mesh size ∆x to 2∆x aresame as the global AE schemes presented in [27]. Hence, in terms of accuracy, the global AE schemewith N grid points is comparable only with overlapping cell schemes [30] with N/2 grid points. Butthe AE scheme is more simple to implement. Actually such simplicity in implementation is even morepronounced in the 2D case.

The AE system also distinguishes itself from the relaxation approximation [17, 18, 32] to nonlinearconservation laws. In relaxation systems the solution will deviate from the entropy solution with adistance of order O(

√ε) (here ε is the relaxation parameter); see e.g. [18], even initial data is at

equilibrium. One thus has to resolve such a parameter with a proper choice of the mesh size. Incontrast, the solution of the AE system when taking the same initial data for two components becomesthe exact solution of conservation laws (independent of ε), the choice of ε thus becomes a gift whichcan be wisely selected to stabilize the time discretization.

4 HASEENA AHMED AND HAILIANG LIU

We now conclude this section by outlining the rest of the paper. In Section 2 we give the first andsecond order formulation of the numerical method for both global and local schemes. A general algo-rithm is presented for constructing AE schemes of any desired order. Under some CFL type conditions,both the TVD property and the maximum principle are proved in Section 3 for first and second orderlocal AE schemes, respectively. Extension to multi-dimensions is discussed in Section 4, where bothfirst and second order schemes are explicitly given. Numerical tests for both scalar conservation lawsand compressible Euler equations for up to 3rd order schemes are presented in Section 5. Our numericalresults show that the AE schemes can capture shock waves as well as other existing high resolutioncentral schemes.

2. Scheme formulation

The detailed formulation and stability analysis of the first and second order global AE scheme isgiven in [27]. We present the idea here again, and give a general algorithm for constructing higher orderAE schemes.

2.1. Global AE schemes. Let u(x, t) denote the sliding average of u(·, t) over interval Ix = {ξ| x −∆x ≤ ξ ≤ x+ ∆x}:

u(x, t) =1

2∆x

∫ x+∆x

x−∆x

u(ξ, t)dξ.

Integration of the 1-D AE system (1.2)-(1.3) over Ix gives

d

dtu(x, t) = − u(x, t)

ε+

1

εL[v](x, t),(2.1)

d

dtv(x, t) = − v(x, t)

ε+

1

εL[u](x, t),(2.2)

where x serves as a moving parameter and

(2.3) L[Φ](x, t) = Φ(x, t)− ε

2∆x(f(Φ(x+ ∆x, t))− f(Φ(x−∆x, t))) .

For simplicity we take uniform distributed grids xj , j ∈ Z in one-dimensional setting. The numericalscheme is constructed by sampling both formulations (2.1) and (2.2) over alternating grids for u andv, respectively, with proper approximation of L[Φ] defined in (2.3). For example, if we sample (2.1) atx2j and (2.2) at x2j+1, respectively, then we have

d

dtu2j(t) =

1

ε[−u2j(t) + L2j [v](t)],(2.4)

d

dtv2j+1(t) =

1

ε[−v2j+1(t) + L2j+1[u](t)].(2.5)

Here we define Lk[Φ](t) := L[Φ](xk, t). In general, the numerical solution at time t = tn can berepresented as

(2.6) Φnk =

{unk k = 2j,vnk k = 2j + 1,

so that the semi-discrete scheme takes a compact form:

(2.7)d

dtΦk =

1

ε[−Φk + Lk[Φ]],

where Lk[Φ] is an approximation of L[Φ] using non-oscillatory reconstructions based on averages Φ. Letp[Φ] be a piecewise polynomial reconstruction on Ik = [xk−1, xk+1], being conservative,

(2.8) Φk =1

2∆x

∫Ik

p[Φ](x) dx,

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 5

and accurate of a desired order. With such a reconstructed p[Φ], we proceed to evaluate L[Φ] as follows:at any node xk, using piecewise polynomials p[Φ] constructed on Ik+1 and Ik−1, instead of the one onIk, we obtain

(2.9) Lk[Φ] =1

2∆x

[∫Ik∩Ik+1

+

∫Ik∩Ik−1

]p[Φ] dx− ε

2∆x[f(p[Φ]k+1)− f(p[Φ]k−1)] .

This leads to a general algorithm:

• Initialization: For Ik := [xk−1, xk+1], we define Φ0 = {u0, v0} via

Φ0k =

1

2∆x

∫Ik

φ0(ξ) dξ, k ∈ Z.

• Reconstruction: Given Φk(t), we reconstruct piecewise non-oscillatory polynomials p[Φ(t)](x)over intervals Ik for k ∈ Z, such that (2.8) holds for all k ∈ Z.

• Evolution: Given Φn = {un, vn} we obtain Φn+1, by applying a stable Runge-Kutta method tothe ODE system (2.7) over [tn, tn+1].

Here the spatial accuracy of the scheme is ensured by the order of the reconstruction p[Φ], then onemay select a proper ODE solver with matching accuracy in time to solve the semi-discrete scheme (2.7)and (2.9).

Here we recall both the first and second order schemes derived in [27]. In global AE schemes, ε ischosen such that the stability condition

(2.10) ∆t < ε ≤ Q ∆x

max|f ′|is satisfied. In system case max |f ′| is replaced by the largest eigenvalue of the Jacobian ∂uf . Thechoice of Q depends on the order of the scheme.

If the reconstructed polynomial is piecewise constant, then the first order scheme becomes

(2.11) Φn+1k = (1− κ)Φnk + κ

[Φnk+1 + Φnk−1

2− ε

2∆x

(f(Φnk+1)− f(Φnk−1)

)],

where κ = ∆tε . This when Q = 1 yields a class of monotone schemes, with the celebrated Lax-Friedrichs

scheme being a special case of κ = 1.Using a linear polynomial reconstruction of the form

(2.12) p[Φn](x) =∑k

(Φnk + snk (x− xk)) χIk(x),

where χIk is the characteristic function which takes value one on interval Ik, we arrive at the secondorder AE scheme (2.7) with

(2.13) Lk[Φ] :=

[Φk+1 + Φk−1

2+

∆x

4(sk−1 − sk+1)− ε

2∆x(f(Φk+1)− f(Φk−1))

].

The non-oscillatory property requires that snk be chosen with certain limiters. In our numerical testswe use the basic minmod limiter, see, e.g. [33, 29]:

(2.14) sk = minmod

{Φk+1 − Φk

∆x,

Φk − Φk−1

∆x

},

where

(2.15) minmod {a, b} =1

2(sgn(a) + sgn(b)) min{|a|, |b|}.

For the 3rd order AE scheme used in our numerical tests we follow the ENO reconstruction for achiev-ing the 3rd order spatial accuracy and the 3rd order TVD Runge-Kutta method for time discretization,see [35, 36]. Considering the third order SSP RK solver has better stability property, so one may use iteven for the second order (in space) scheme in the numerical simulation.

6 HASEENA AHMED AND HAILIANG LIU

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

Solu

tion u

(a) ε = 0.1∆x and ∆t = 0.5ε.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

Solu

tion u

(b) ε = 0.01∆x and ∆t = 0.5ε.

Figure 1. Plot for Burgers’ equation at discontinuity on [0, 2], T = 0.7, ∆x = 1200 ,

2nd order AE scheme to show the effect of change in ε.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

Solu

tion u

(a) ε = 0.1∆x and ∆t = 0.5ε.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

Solu

tion u

(b) ε = 0.01∆x and ∆t = 0.5ε.

Figure 2. Plot for Burgers’ equation at discontinuity on [0, 2], T = 0.7, ∆x = 1200 ,

3rd order AE scheme to show the effect of change in ε.

Finally we note that ε plays an important role as a dissipation factor. Fig. 1 and 2 show plots forthe solution of Burgers’ equation when the solution is discontinuous. From the graphs we can see thatas ε decreases, the dissipation increases and the solution is more smeared for a fixed time step ∆t. Inorder to reduce the dissipation we introduce local AE schemes which are presented in the next section.

2.2. Local AE schemes. In local schemes, we embed more local information into ε than in the globalAE schemes. Instead of a single global parameter ε, we also use local parameters to increase thenumerical accuracy of the scheme.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 7

2.2.1. First order local AE schemes. By using local parameters and integrating the 1-D AE system(1.2)-(1.3) over Ix gives

d

dtu(x, t) = − 1

2∆x

∫ x+∆x

x−∆x

∂xf(v) dx− 1

2∆x

[1

εx−∆x/2

∫ x

x−∆x

+1

εx+∆x/2

∫ x+∆x

x

](u− v) dx,(2.16)

d

dtv(x, t) = − 1

2∆x

∫ x+∆x

x−∆x

∂xf(u) dx− 1

2∆x

[1

εx−∆x/2

∫ x

x−∆x

+1

εx+∆x/2

∫ x+∆x

x

](v − u) dx.(2.17)

Note that εx−∆x/2 uses the information to the left of x and εx+∆x/2 uses information to the rightof x. Sampling equation (2.16) at x2j and (2.17) at x2j+1, respectively, and using piecewise constantpolynomial reconstruction we obtain,

Φn+1k = Φnk −

λ

2(f(Φnk+1)− f(Φnk−1))−

κk−1/2

2(Φnk − Φnk−1)−

κk+1/2

2(Φnk − Φnk+1),(2.18)

where

κk+1/2 =∆t

εk+1/2.

Use Mk+1/2 to denote the range [min(Φk,Φk+1),max(Φk,Φk+1)]. For the system case it denotes a curvein phase space connecting Φk and Φk+1 via the Riemann fan. Depending on the way εk+1/2 is defined,we can formulate the following local AE scheme.

Local AE1 scheme. We choose εk such that

(2.19) εk+1/2 ≤ Q∆x

maxs∈Mk+1/2

|f ′(s)|, ∆t ≤ min

kεk+1/2.

Q is a factor that is dependent on the order of the scheme and the stability conditions presented insection 3 provide the range of values Q can take.

Remark 2.1. For the scalar equation, we may adopt a more accurate version of the local scheme, calledlocal AE2 scheme. We choose ε such that

(2.20) εk+1/2 ≤ Q∆x

1|Mk+1/2|

∫Mk+1/2

|f ′(s)| ds, ∆t ≤ min

kεk+1/2.

This choice is motivated by the Enguist-Osher flux [8, 39]. This scheme suggests that even in the AEdiscretization, there are still different ways to incorporate local wave propagation info into the scheme.

In section 3 we will derive conditions on εk+1/2 in order that the local AE scheme (2.18)-(2.20) isnumerically stable.

2.2.2. Second order local AE schemes. If we sample (2.16) at x2j , and use piecewise linear polynomialreconstruction on [x2j−1, x2j+1], forward Euler in time with time step ∆t, we obtain a prediction for u

8 HASEENA AHMED AND HAILIANG LIU

as u∗2j which can be given as

u∗2j = un2j −∆t

2∆x(f(vn2j+1)− f(vn2j−1))

− ∆t

2∆x

1

ε2j−1/2

∫ x2j

x2j−1

[(un2j + sn2j(x− x2j))− (vn2j−1 + sn2j−1(x− x2j−1))

]dx

− ∆t

2∆x

1

ε2j+1/2

∫ x2j+1

x2j

[(un2j + sn2j(x− x2j))− (vn2j+1 + sn2j+1(x− x2j+1))

]dx,

= un2j −∆t

2∆x(f(vn2j+1)− f(vn2j−1))− ∆t

2ε2j−1/2

(un2j − vn2j−1 −

∆x

2(sn2j + sn2j−1)

)− ∆t

2ε2j+1/2

(un2j − vn2j+1 +

∆x

2(sn2j + sn2j+1)

)= un2j −∆tF2j [Φ

n],

where

Fk[Φ] =1

2∆x(f(Φk+1)− f(Φk−1)) +

1

2εk−1/2

(Φk − Φk−1 −

∆x

2(sk + sk−1)

)(2.21)

+1

2εk+1/2

(Φk − Φk+1 +

∆x

2(sk + sk+1)

).

Similarly we can sample (2.17) at x2j+1 and use piecewise linear polynomial reconstruction to obtain aprediction for v∗2j+1. Thus the second order semi-discrete AE scheme is

d

dtΦk = Fk[Φ].

Second order local AE1 scheme is designed as in the case of local first order AE1 scheme.We shall prove stability of the second order scheme,

Φ∗k = Φnk −∆tFk[Φn],(2.22)

Φn+1k =

1

2Φnk +

1

2Φ∗k −

∆t

2Fk[Φ∗],(2.23)

where Fk[Φ] is defined in (2.21).Again this procedure can be easily implemented to obtain 3rd order local AE schemes following the

algorithm outlined for global AE schemes in Section 2.1.Next we make some remarks regarding the properties of the scheme.

Remark 2.2. Conservative form: Both global and local AE schemes are conservative. i.e., theysatisfy, ∑

k

Φn+1k =

∑k

Φnk .

This is ensured by the conservative form of Fk[Φ]:∑k

Fk[Φ] = 0.

According to the celebrated Lax-Wendroff theorem [23], the conservative form is necessary for underlyingschemes to correctly approximate discontinuous weak solutions to the conservation law. It remains toshow convergence of the underlying schemes, i.e. compactness of the numerical solution. This ismeasured in terms of the total-variation-diminishing (TVD) property [12], to be analyzed in the nextsection.

Remark 2.3. Numerical viscosity: The local AE2 scheme has the smallest numerical viscosity sinceit uses the average of the local speed over a cell, followed by the local AE1 scheme which uses themaximum of the local speed over a cell, followed by the global AE scheme which takes the maximumof the speed over all cells.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 9

3. Stability analysis

We show that both first and second order AE schemes are stable in the sense of satisfying themaximum principle, as well as having bounded total variation. Let Φn := {un, vn} be a computedsolution. We use the following notations:

(3.1) |Φn|∞ = maxk|Φnk |, TV [Φn] =

∑k

|Φnk+1 − Φnk |.

3.1. Stability of first order AE schemes.

Theorem 3.1. Let Φ := {un, vn} be computed from the first order local AE scheme (2.18) for scalarhyperbolic conservation laws. If

(a) for local AE1,

(3.2) εk+1/2 ≤∆x

maxs∈Mk+1/2

|f ′(s)|, ∆t ≤ 1

2minkεk+1/2,

(b) or for local AE2,

(3.3) εk+1/2 ≤∆x

1|Mk+1/2|

∫Mk+1/2

|f ′(s)| ds, ∆t ≤ 1

2minkεk+1/2

holds, then

(3.4) |Φn+1|∞ ≤ |Φn|∞, n ∈ N.

Moreover, we have

(3.5) TV [Φn+1] ≤ TV [Φn], n ∈ N.

Proof. We can rewrite equation (2.18) as

Φn+1k = Φnk −

λ

2f ′k(Φnk+1 − Φnk )−

κk−1/2

2(Φnk − Φnk−1)−

κk+1/2

2(Φnk − Φnk+1).

where λ = ∆t∆x and f ′k =

∫ 1

0

f ′(Φk + η(Φnk+1 − Φnk )) dη. Re-arranging, we obtain,

Φn+1k =

(1−

κk−1/2

2−κk+1/2

2

)Φnk +

(−λ

2f ′k+1/2 +

κk+1/2

2

)Φnk+1

+

2f ′k−1/2 +

κk−1/2

2

)Φnk−1.(3.6)

When (3.2), or (3.3) holds, the coefficients in the expression for Φn+1k in (3.6) are non-negative and we

have a convex combination of the grid point values. We can write,

|Φn+1k | ≤

(1−

κk−1/2

2−κk+1/2

2

)maxk|Φnk |+

(−λ

2f ′k+1/2 +

κk+1/2

2

)maxk|Φnk+1|

+

2f ′k−1/2 +

κk−1/2

2

)maxk|Φnk−1|,

≤ |Φn|∞.

Thus the maximum principle (3.4) follows.The TVD property follows from [37, Lemma 2.1] when the scheme is recast into a standard viscosity

form. The proof of asserted properties is thus complete.

10 HASEENA AHMED AND HAILIANG LIU

3.2. Stability of second order AE schemes. For the second order AE schemes, the slope limiter istypically chosen to ensure that the total variation does not increase under the operation of reconstruc-tion:

TV [p[un](x)] ≤ TV [un], TV [p[vn](x)] ≤ TV [vn].

This property is satisfied by the minmod limiters defined in (2.14). With such a reconstruction weprove the following:

Theorem 3.2. Let Φn be computed from the second order AE scheme (2.22)-(2.23) for scalar hyperbolicconservation laws. Let the slope sk be defined in (2.14). If

(a) for local AE1,

(3.7) εk+1/2 ≤1

4

∆x

maxs∈Mk+1/2

|f ′(s)|,

1

3≤ κk+1/2 ≤

1

2,

(b) or for local AE2,

(3.8) εk+1/2 ≤1

4

∆x

1|Mk+1/2|

∫Mk+1/2

|f ′(s)| ds,

1

3≤ κk+1/2 ≤

1

2.

holds, then

(3.9) |Φn+1|∞ ≤ |Φn|∞, n ∈ N.

Also, we have that

(3.10) TV [Φn+1] ≤ TV [Φn], n ∈ N.

Remark 3.3. The specific choice of bounds given in (3.7) and (3.8) are for convenience only. The proofas below remains valid if we take θ1 ≤ κk+1/2 ≤ θ2 with Q = 1 − θ2

2θ1for any positive numbers θi

satisfying

θ2 ≤1

2, and

θ2

2< θ1 < θ2.

Proof. We first want to show that the maximum principle (3.9) holds. We will show that

(3.11) |Φ∗|∞ = |Φn −∆t F [Φn]|∞ ≤ |Φn|∞.

Then, from equation (2.23) we have

|Φn+1|∞ ≤ 1

2|Φn|∞ +

1

2|Φ∗ −∆t F [Φ∗]|∞.

Using (3.11) twice,

|Φn+1|∞ ≤1

2|Φn|∞ +

1

2|Φ∗|∞ ≤

1

2|Φn|∞ +

1

2|Φn|∞ = |Φn|∞.

which is the desired property (3.9).We now prove (3.11) as follows:

Φ∗k = Φnk −λ

2(fk+1 − fk−1)−

κk−1/2

2

(Φnk − Φnk−1 −

∆x

2(sk + sk−1)

)−κk+1/2

2

(Φnk − Φnk+1 +

∆x

2(sk + sk+1)

),(3.12)

where fk is used to denote f(Φnk ). Define the modified flux as

f+k := fk +

∆x

2λκk+1/2 sk, f−k := fk +

∆x

2λκk−1/2 sk,

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 11

and rewrite (fk+1 − fk−1) as (fk+1 − fk + fk − fk−1). We can now re-write equation (3.12) in terms ofthe modified flux as

Φ∗k = Φnk −λ

2

(f−k+1 − f

−k + f+

k − f+k−1

)−κk−1/2

2

(Φnk − Φnk−1

)−κk+1/2

2

(Φnk − Φnk+1

).

Also set

(3.13) β±k+1/2 :=

f±k+1 − f

±k

Φnk+1 − Φnkif Φnk+1 6= Φnk ,

0 if Φnk = Φnk+1.

Using this definition, we have

Φ∗k = Φnk −λ

2β−k+1/2(Φnk+1 − Φnk )− λ

2β+k−1/2(Φnk − Φnk−1)

−κk−1/2

2

(Φnk − Φnk−1

)−κk+1/2

2

(Φnk − Φnk+1

).(3.14)

Re-arranging,

Φ∗k =

(1−

κk−1/2

2−κk+1/2

2− λ

2β+k−1/2 +

λ

2β−k+1/2

)Φnk +

(κk−1/2

2+λ

2β+k−1/2

)Φnk−1

+

(κk+1/2

2− λ

2β−k+1/2

)Φnk+1.(3.15)

When (3.7) or (3.8) holds, all the coefficients in the expression for Φ∗k are non-negative. We will showthis by looking at the coefficient for Φnk+1 and the coefficients for the other terms follow.

λ

κk+1/2|β−k+1/2| ≤

λ

κk+1/2

|fk+1 − fk||Φnk+1 − Φnk |

+∆x

2κk+1/2

|κk+1/2 sk+1 − κk−1/2 sk||Φnk+1 − Φnk |

≤ λ

κk+1/2|f ′|+ 1

2max

{1,κk−1/2

κk+1/2

}.

From (2.14) it follows that the maximum value∆x|sk+1||Φnk+1 − Φnk |

or∆x|sk|

|Φnk+1 − Φnk |can take is 1. Note that

the slopes sk+1 and sk have the same signs since minmod limiters are used. Now if (3.7) holds, we have∣∣∣∣κk−1/2

κk+1/2

∣∣∣∣ ≤ 3/2, and

λ

κk+1/2|β−k+1/2| ≤

1

4+

3

4= 1.

This implies that (κk+1/2

2− λ

2β−k+1/2

)≥ 0.

On similar lines and κk+1/2 ≤ 1/2 again we can show that(1−

κk−1/2

2−κk+1/2

2− λ

2β+k−1/2 +

λ

2β−k+1/2

)≥ 0, and

(κk−1/2

2+λ

2β+k−1/2

)≥ 0.

We can take the maximum norm on (3.15) to obtain,

|Φ∗k| ≤(

1−κk−1/2

2−κk+1/2

2− λ

2β+k−1/2 +

λ

2β−k+1/2

)maxk|Φnk |

+

(κk−1/2

2+λ

2β+k−1/2

)maxk|Φnk−1|+

(κk+1/2

2− λ

2β−k+1/2

)maxk|Φnk+1|,

which is a convex combination and hence we have

|Φ∗|∞ ≤ |Φn|∞.

12 HASEENA AHMED AND HAILIANG LIU

We next want to show that the scheme satisfies the total variation diminishing property. It sufficesto show that

TV [Φ∗] ≤ TV [Φn].

Using equation (3.14), we can write an expression for Φ∗k − Φ∗k−1 as

Φ∗k − Φ∗k−1 =(1− κk−1/2

)(Φnk − Φnk−1) +

1

2κk+1/2(Φnk+1 − Φnk ) +

1

2κk−3/2(Φnk−1 − Φnk−2)

−λ2β−k+1/2(Φnk+1 − Φnk )− λ

2β+k−1/2(Φnk − Φnk−1)

2β−k−1/2(Φnk − Φnk−1) +

λ

2β+k−3/2(Φnk−1 − Φnk−2).

Re-arranging,

Φ∗k − Φ∗k−1 =

(1− κk−1/2 −

λ

2β+k−1/2 +

λ

2β−k−1/2

)(Φnk − Φnk−1)

+

(κk+1/2

2− λ

2β−k+1/2

)(Φnk+1 − Φnk ) +

(κk−3/2

2+λ

2β+k−3/2

)(Φnk−1 − Φnk−2).

When (3.7) or (3.8) holds, all the coefficients in the expression for Φ∗k−Φ∗k−1 are non-negative. We cannow take the absolute value and sum over k ∈ Z to obtain,∑

k

|Φ∗k − Φ∗k−1| ≤∑k

(1− κk−1/2 −

λ

2β+k−1/2 +

λ

2β−k−1/2

)|Φnk − Φnk−1|

+∑k

(κk+1/2 −

λ

2β−k+1/2 +

λ

2β+k+1/2

)|Φnk+1 − Φnk |.

Hence

TV [Φ∗] ≤∑k

|Φnk − Φnk−1| = TV [Φn].

The proof is thus complete.

4. Extension to Multi-dimensions

Consider the system of conservation laws

(4.1) ut +∇x · f(u) = 0, (x, t) ∈ Rd × (0, T ),

where u = (u1, . . . , um)T . For simplicity, we take uniform distributed grids at xα with multi-indexα = (α1, . . . , αd). Let Iα be a rectangle with vertices at {xα+β , |β| = 1}, labeled as xα±1, the numberof which amounts to 2d. We take the average of the AE equation,

ut +∇x · f(v) =1

ε(v − u),

over Iα to obtaind

dtuα +

1

εuα =

1

εLα[v](t),

where uα = 1|Iα|

∫Iαu(x, t)dx and

Lα[v] =1

|Iα|

∫Iα

v dx− ε

|Iα|

∫∂Iα

f(v) · dS.

Here, |Iα| denotes the volume of Iα, and ∂Iα indicates the boundary of Iα. Let Φ ∼ uα denote thenumerical solution, and reconstruct non-oscillatory polynomials p[Φ] over Iα from available averagesΦα, then we obtain a semi-discrete scheme

(4.2)d

dtΦα +

1

εΦα =

1

εLα[Φ],

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 13

where p[Φ]’s constructed over Iα±1 are to be used to evaluate

(4.3) Lα[Φ] =∑ 1

|Iα|

∫Iα∩Iα±1

p[Φ](x) dx−∑ 1

|Iα|

∫∂Iα∩Iα±1

f(p[Φ](x)) · ds.

Finally to obtain the same order accuracy in time, the semi-discrete (4.2)-(4.3) is to be complementedwith an ODE solver with matching accuracy for time discretization. The evolution parameter is chosensuch that

(4.4) ε

d∑j=1

max |f ′j(·)|∆xj

≤ Q, ∆t < ε,

where Q ≤ 1 depends on the order of the scheme, see (4.8) or (4.16). For system case, f ′j need to bereplaced by the dominant eigenvalues over a Riemann curve.

For the 2D setting we present the corresponding global AE scheme for

(4.5) φt + f(φ)x + g(φ)y = 0.

4.1. First order scheme. If the reconstructed polynomial is piecewise constant, then we obtain thefirst order scheme as

Φn+1k,l = (1− κ)Φnk,l + κLk,l[Φ

n],(4.6)

where Lk,l[Φn] can be evaluated as

Lk,l[Φn] =

1

4

(Φnk−1,l−1 + Φnk+1,l−1 + Φnk+1,l+1 + Φnk−1,l+1

)− ε

4∆x

(f(Φnk+1,l−1)− f(Φnk−1,l−1) + f(Φnk+1,l+1)− f(Φnk−1,l+1)

)− ε

4∆y

(g(Φnk−1,l+1)− g(Φnk−1,l−1) + g(Φnk+1,l+1)− g(Φnk+1,l−1)

),(4.7)

with κ =∆t

ε. The parameter ε is chosen so that

ε

(max |f ′|

∆x+

max |g′|∆y

)≤ 1 and ∆t < ε,(4.8)

which ensures the scalar maximum principle

|Φn+1|∞ ≤ |Φn|∞, n ∈ N.(4.9)

Again, for system case, both |f ′| and |g′| in the stability requirement (4.8) needs to be replaced bydominant eigenvalues of the corresponding Jacobian matrices.

4.2. Second Order Scheme. The second order scheme requires linear polynomial reconstruction forits formulation which on the rectangle Ikl = [xk −∆x, xk + ∆x]× [yk −∆y, yk + ∆y] has the form

pk,l[Φn](x, y) =

∑k,l

(Φnk,l + s′k,l(x− xk) + s8k,l(y − yl)

)χIkl(x, y),(4.10)

where χIkl is the characteristic function which takes value one on the rectangle Ik,l, s

′ and s8 arethe numerical derivatives corresponding to Φx and Φy. Now, using the midpoint quadrature rule inevaluation of (4.3), we obtain the second order AE scheme

d

dtΦk,l = Lk,l[Φ]

14 HASEENA AHMED AND HAILIANG LIU

with

Lk,l[Φ] =1

4(Φk−1,l−1 + Φk+1,l−1 + Φk+1,l+1 + Φk−1,l+1)

+∆x

8

(s′k−1,l−1 − s′k+1,l−1 − s′k+1,l+1 + s′k−1,l+1

)+

∆y

8

(s8k−1,l−1 + s8k+1,l−1 − s8k+1,l+1 − s8k−1,l+1

)− ε

4∆x

[f

(Φk+1,l−1 +

∆y

2s8k+1,l−1)

)− f

(Φk−1,l−1 +

∆y

2s8k−1,l−1)

)+f

(Φk+1,l+1 −

∆y

2s8k+1,l+1)

)− f

(Φk−1,l+1 −

∆y

2s8k−1,l+1)

)]− ε

4∆y

[g

(Φk−1,l+1 +

∆x

2s′k−1,l+1)

)− g

(Φk−1,l−1 +

∆x

2s′k−1,l−1)

)+g

(Φk+1,l+1 −

∆x

2s′k+1,l+1)

)− g

(Φk+1,l−1 −

∆x

2s′k+1,l−1)

)].

The non-oscillatory property requires that we choose s′k,l and s8k,l with certain limiters as in the one-dimensional polynomial reconstruction. In our proofs we use the basic minmod limiter defined as below:

s′k,l = minmod

{Φnk+2,l − Φnk,l

2∆x,

Φnk,l − Φnk−2,l

2∆x

},(4.11)

s8k,l = minmod

{Φnk,l+2 − Φnk,l

2∆y,

Φnk,l − Φnk,l−2

2∆y

},(4.12)

with

minmod {a1, a2, . . .} =

mini{ai} if ai > 0 ∀i,

maxi{ai} if ai < 0 ∀i,

0 otherwise.

(4.13)

When the second order Runge-Kutta discretization in time is used, the scheme becomes

Φ∗k,l = (1− κ)Φnk,l + κLk,l[Φn],(4.14)

Φn+1k,l =

1

2Φnk,l +

1

2(1− κ)Φ∗k,l +

κ

2Lk,l[Φ

∗],(4.15)

Indeed a careful verification shows that such a choice again yields the scalar maximum principle (4.9),provided

ε

(max |f ′|

∆x+

max |g′|∆y

)≤ 1

4and ∆t < ε.(4.16)

5. Numerical tests

5.1. Scalar conservation laws. In this section we use some model problems to numerically test thefirst, second and third order global and local AE schemes. If φ is the exact solution and Φ is thecomputed solution, then the numerical errors are calculated as:

L1 error =∑k

|φk − Φk|∆x, L∞ error = maxk|φk − Φk|.

We call Q in equations (2.19) and (2.20 ) the CFL number in our numerical tests.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 15

Table 1. The L1 and L∞ errors for Burgers’ equation, Example 5.1 using N equally

spaced cells for different 1st order numerical schemes at T =0.1

π, when the solution

is continuous.

N Scheme L1 error L1 order error ratio L∞ error L∞ order error ratio

10 LxF 7.092532E-02 1.000000 6.149400E-02 1.000000AE 4.546986E-02 0.641095 3.894449E-02 0.633305

AE1 1.812761E-02 0.255587 1.431931E-02 0.232857

20 LxF 3.686688E-02 1.01 1.000000 3.150142E-02 1.03 1.000000

AE 2.246434E-02 1.05 0.609337 1.971157E-02 1.02 0.625736

AE1 8.195435E-03 1.19 0.222298 6.868932E-03 1.10 0.218051

40 LxF 1.851668E-02 1.03 1.000000 1.590225E-02 1.02 1.000000

AE 1.124649E-02 1.02 0.607370 9.921157E-03 1.01 0.623884AE1 3.872198E-03 1.10 0.209119 3.378816E-03 1.04 0.212474

80 LxF 9.240998E-03 1.02 1.000000 7.976805E-03 1.01 1.000000AE 5.630459E-03 1.01 0.609291 4.981433E-03 1.00 0.624490

AE1 1.880982E-03 1.05 0.203547 1.671873E-03 1.02 0.209592AE2 1.834574E-03 1.02 0.198526 1.632366E-03 0.99 0.204639

160 LxF 4.630626E-03 1.01 1.000000 3.998649E-03 1.01 1.000000AE 2.818238E-03 1.00 0.608608 2.496904E-03 1.00 0.624437

AE1 9.269747E-04 1.02 0.200183 8.313492E-04 1.01 0.207908

320 LxF 2.317531E-03 1.00 1.000000 2.001063E-03 1.00 1.000000

AE 1.409864E-03 1.00 0.608347 1.250070E-03 1.00 0.624703

AE1 4.600095E-04 1.01 0.198491 4.145088E-04 1.01 0.207144

Example 5.1. The one dimensional Burgers’ equation has the form

ut +

(u2

2

)x

= 0, x ∈ [0, 2],

u(x, 0) = 1 + sinπx,

u(0, t) = u(2, t).

We use the Burgers’ equation with convex flux to make a comparison of the global and local AEschemes with some standard central schemes such as the first order Lax Friedrich (LxF), second orderKT (Kurganov-Tadmor [22]) scheme, which improves upon the original NT (Nessyahu-Tadmor [33])scheme. In order to check the numerical accuracy, we compute the solution when it is continuous. We

take the final time T =0.1

πand use CFL number 0.8 and ∆t = 0.8ε for the first order scheme, and

CFL number 0.6 and ∆t = 0.6ε for the second order scheme. We can see that both the global and localschemes give the desired order of accuracy from Tables 1-2. We want to make a note that we do notuse non-linear limiters while checking the numerical accuracy of the second order schemes when thesolution is smooth.

For first order schemes, the numerical errors, the orders of accuracy for the numerical solution, andratios of the numerical errors of the AE and AE1 schemes in comparison with the LxF scheme areshown in Table 1. All the schemes give the required numerical order of accuracy. We can see that thelocal AE1 scheme gives the smallest numerical error. From Table 2, we can see that for second orderschemes, local AE1 scheme gives the smallest numerical error compared to the AE and KT schemes.Our global AE scheme is comparable to the KT scheme: when n = 20 and n = 40, L∞ error of AE is

16 HASEENA AHMED AND HAILIANG LIU

Table 2. The L1 and L∞ errors for Burgers’ equation, example 5.1 using N equally

spaced cells for different 2nd order numerical schemes at T =0.1

π, when the solution

is continuous.

N Scheme L1 error L1 order error ratio L∞ error L∞ order error ratio

10 KT 8.060034E-03 1.000000 8.127727E-03 1.000000AE 5.198982E-03 0.645032 6.680986E-03 0.828903

AE1 4.520158E-03 0.560811 5.540215E-03 0.687369

20 KT 1.267812E-03 2.86 1.000000 1.429797E-03 2.69 1.000000

AE 1.167134E-03 2.23 0.920589 1.478923E-03 2.25 1.034359

AE1 1.083947E-03 2.13 0.854975 1.305326E-03 2.16 0.912945

40 KT 2.732028E-04 2.29 1.000000 3.289424E-04 2.20 1.000000

AE 2.739386E-04 2.13 1.002693 3.350735E-04 2.18 1.018639AE1 2.641463E-04 2.07 0.966851 3.124174E-04 2.10 0.949763

80 KT 6.602794E-05 2.09 1.000000 8.180660E-05 2.04 1.000000AE 6.663412E-05 2.06 1.009181 7.996743E-05 2.09 0.977518

AE1 6.557099E-05 2.03 0.993079 7.703799E-05 2.04 0.941709

160 KT 1.645922E-05 2.02 1.000000 2.042386E-05 2.02 1.000000

AE 1.646475E-05 2.03 1.000336 1.949339E-05 2.05 0.954442AE1 1.632925E-05 2.01 0.992104 1.912202E-05 2.02 0.936259

320 KT 4.139036E-06 2.00 1.000000 5.107226E-06 2.01 1.000000AE 4.093983E-06 2.01 0.989115 4.810093E-06 2.02 0.941821

AE1 4.077607E-06 2.01 0.985159 4.763512E-06 2.01 0.932700

higher than that of KT, and for n = 40, 80, 160, the L1 error of AE is higher than that of KT. Whileas the number of cells is increased, the AE, AE1, and KT schemes give nearly the same errors.

We also do some CPU run-time comparisons in Table 3. The comparisons are made on a HP desktopwith Intel (R) Core (TM) Duo CPU E8400 @ 3.00GHz, 2.99GHz 1.95GB RAM. As expected, therun-times for the local AE schemes are larger than that for the global AE scheme.Also note that therun-time for the second order global AE scheme is slightly smaller than the run-time for second orderKT scheme.

We also plot the solution to the Burgers’ equation when it is discontinuous, that is at time T = 0.7.We use CFL number 0.2 and ∆t = 0.8ε for both the second and third order schemes. From Fig. 3and Fig. 4, we see that there is a significant increase in the resolution of discontinuities from second tothird order schemes. We can also see that the local AE schemes perform much better than the globalAE scheme.

Example 5.2. One dimensional nonlinear Buckley-Leverett problem:

ut +

(4u2

4u2 + (1− u)2

)x

= 0, x ∈ [−1, 1].

The initial condition is given by

u(x, 0) =

{1 x ∈ [− 1

2 , 0],0 otherwise.

The flux is nonlinear and we use this example to test the scheme when the initial data is discontinuous.In order to check the numerical accuracy, we take the final time T = 0.4 and N = 80. We choose CFLnumber 0.2 and ∆t = 0.8ε for both the second and third order schemes. The numerical results forsecond and third order schemes are given in Fig. 5. As in the case of Burgers’ equation, we can see that

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 17

Table 3. Run-time comparisons for Burgers’ equation, Example 5.1 using N equally

spaced cells for different order numerical schemes at T =0.1

π, when the solution is

continuous.

N Scheme 1st order CPU time 2nd order CPU time 3rd order CPU time

20 Lax - - -KT - - -

AE - - -

AE1 - 0.03 0.03

40 Lax - - -

KT - - -AE - 0.03 0.03

AE1 0.03 0.05 0.05

80 Lax - - -

NT - - -

KT - 0.03 -AE - 0.03 0.05

AE1 0.06 0.09 0.11AE2 2.50 7.67 7.94

160 Lax - - -KT - 0.06 -

AE - 0.06 0.13

AE1 0.11 0.27 0.28

320 Lax - - -

KT - 0.23 -AE 0.05 0.20 0.38

AE1 0.36 0.86 0.98

640 Lax 0.02 - -

KT - 0.75 -AE 0.13 0.63 1.41

AE1 1.22 3.47 3.98

local AE schemes perform better than the global AE scheme. Also, the resolution of discontinuities isas expected higher for third order schemes than the second order schemes.

We now test the numerical accuracy using a two-dimensional scalar equation with continuous initialdata.

Example 5.3. Linear two-dimensional advection:

φt + φx + φy = 0, −1 ≤ x, y ≤ 1,

with initial data,φ(x, y, 0) = sinπ(x+ y).

This equation causes to transport the initial data. Periodic boundary conditions are used and the errorsare calculated at final time T = 1 using exact solution φ = sin(π(x + y − 2t)). We do not use anylimiters while checking the accuracy of the second order scheme since the solutions are smooth for alltimes and the numerical derivative for the linear reconstruction is taken to be the central differenceΦi+1,j−Φi−1,j

2∆x andΦi,j+1−Φi,j−1

2∆y in the x− and y− direction respectively. The results given in Tables 4

and 5 indicate that the scheme gives the desired accuracy both in the L1 and L∞ norms.

Next we test the global and local AE schemes for the Euler system of equations.

18 HASEENA AHMED AND HAILIANG LIU

Table 4. The L1 and L∞ errors for linear advection, Example 5.3, using N equallyspaced cells for global 1st order AE scheme at T = 1, when the solution is continuous.

N L1 error L1 order L∞ error L∞ order

20 1.32E+00 - 4.67E-01 -40 6.84E-01 0.9824 2.55E-01 0.9039

80 3.54E-01 0.9679 1.35E-01 0.9288

160 1.80E-01 0.9798 7.00E-02 0.9609320 9.12E-02 0.989 3.56E-02 0.9798

640 4.58E-02 0.9948 1.79E-02 0.9902

Table 5. The L1 and L∞ errors for linear advection, Example 5.3, using N equallyspaced cells for global 2nd order AE scheme at T = 1, when the solution is continuous.

N L1 error L1 order L∞ error L∞ order

20 1.47E-01 - 5.26E-02 -

40 1.91E-02 3.0444 7.16E-03 2.9824

80 3.33E-03 2.5656 1.28E-03 2.5333160 7.24E-04 2.223 2.81E-04 2.2036

320 1.73E-04 2.0713 6.76E-05 2.0623

640 4.28E-05 2.0204 1.68E-05 2.0159

5.2. Euler equations of gas dynamics. In this section we apply our second and third order globalAE and local AE1 schemes to the Euler equations of polytropic gas.

Example 5.4. The Euler system of equations has the form ρρvE

t

+

ρvρv2 + pv(E + p)

x

= 0,

where p = (γ − 1)(E − 12ρv

2); γ = 1.4. We test the Euler equation with different initial data that arestandardly used [25, 39].

(1) Lax initial data: Here, x ∈ [0, 1] and

(ρ, ρv,E) (0) =

{(0.445, 0.311, 8.928) x ∈ [0, 0.5](0.5, 0, 1.4275) x ∈ [0.5, 1]

The final time T = 0.16.(2) Sod initial data: Here, x ∈ [0, 1] and

(ρ, ρv,E) (0) =

{(1, 0, 2.5) x ∈ [0, 0.5](0.125, 0, 0.25) x ∈ [0.5, 1]

The final time T = 0.1644.(3) Osher-Shu problem: Here, x ∈ [−5, 5] and

(ρ, ρv,E) (0) =

{(3.857143, 2.629369, 10.333333) x ∈ [−5,−4](1 + 2 sin(5x), 0, 1) x ∈ [−4, 5]

The final time T = 1.8.(4) Woodward-Colella problem: Here, x ∈ [0, 1] and

(ρ, ρv,E) (0) =

(1, 0, 2500) x ∈ [0, 0.1](1, 0, 0.025) x ∈ [0.1, 0.9](1, 0, 250) x ∈ [0.9, 1]

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 19

The final time T = 0.01, 0.03, 0.038.

We plot the density, velocity and pressure profiles for all the four types of initial data in the rest ofthe paper.

For Lax initial data, we choose CFL number 0.5 and ∆t = 0.8ε. The reference solution is calculatedusing the ENO scheme with local LxF numerical flux, see [35, 36] with 5120 grid points with ε = 0.1∆xand ∆t = 0.95ε. Fig. 6 makes a comparison between second order (density) and third order global AEand local AE1 schemes. We can see that in general, AE1 captures the corners much better than theAE scheme and the resolution of discontinuities is much sharper in third order schemes as compared tosecond order.

For the Sod, Osher-Shu and Woodward-Colella initial data, the reference solution is calculated usingthe third order ENO scheme with local LxF numerical flux and with 2560 grid points.

For Sod initial data, we choose CFL number 0.5 and ∆t = 0.8ε. Fig. 7 shows the density, velocityand pressure profiles at the final time for both second and third order AE1 schemes. As in the case ofLax initial data, the resolution of third order scheme is better than that of the second order scheme.

In the Osher-Shu problem, there are several extrema in the smooth regions which present a goodcase for examination of the accuracy of shock-capturing. In this case, we choose CFL number 0.6 and∆t = 0.8ε. The density, velocity and pressure profiles for third order AE, AE1 with 400 grid points inFig. 8. The final time T = 1.8. We note that with 400 grid points difference between local and globalschemes in third order is not significant. Fig. 9 gives plots for second and third order AE1 scheme with800 grid points. We can see that a lot of peaks are captured in the third order but not in the secondorder scheme.

The Woodward-Colella problem involves the interaction of two blast waves. The initial data givenpresents one shock at x = 0.1 and the other at x = 0.9. The boundaries at x = 0 and x = 1 aresolid walls with a reflective boundary condition. After a certain time, these two shocks collide witheach other. At the final time step of t = 0.038, the flow field involves two shocks and three contactdiscontinuities. For numerical computations with Woodward Colella initial data, we choose CFL = 0.3and ∆t = 0.8ε. Fig. 10 show the density, velocity and pressure profiles for both second and third orderlocal AE1 scheme for time T = 0.01 and Fig. 11 shows third order AE1 scheme for time T = 0.03, 0.038.

Example 5.5. Explosion problem for the Euler equation. The 2D Euler equation can be written as

φt + f(φ)x + g(φ)y = 0, φ = (ρ, ρu, ρv, E)T , p = (γ − 1)[E − ρ

2(u2 + v2)

],

f(φ) = (ρu, ρu2 + p, ρuv, u(E + p))T , g(φ) = (ρv, ρuv, ρv2 + p, v(E + p))T ,

where γ = 1.4. This example chosen from [39] consists of a high density and high pressure region insidea bubble of radius 0.4 centered at the origin and a low density and pressure region outside the bubblewhich causes the explosion. The initial data is

(p, ρ, u, v) (0) =

{(1, 1, 0, 0) x2 + y2 < 0.16,

(0.1, 0.125, 0, 0) x2 + y2 ≥ 0.16.

The solution is computed at time T = 0.25. In our numerical experiments, we use constant extensionboundary conditions on all the four walls. CFL number used is 0.4 and time step is taken as ∆t = 0.95ε.

We use the general minmod limiter [33] with θ = 1.5

s′k,l = minmod

Φk+1,l − Φk, l

∆x,

Φk+1,l − Φk−1,l

2∆x, θ

Φk,l − Φk−1,l

∆x

},

s8k,l = minmod

Φk,l+1 − Φk, l

∆x,

Φk,l+1 − Φk−1,l−1

2∆x, θ

Φk,l − Φk,l−1

∆x

}.

for the numerical derivative calculations. This θ−dependent family of limiters with θ ∈ (1, 2) is knownto be the less dissipative limiter than the case θ = 1. In this test, the solution exhibits a circular shockand circular contact discontinuity moving away from the center of the circle and circular rarefactionwave moving in the opposite direction, a complex wave pattern emerging as time evolves. We compute

20 HASEENA AHMED AND HAILIANG LIU

this 2D bubble explosion solution by second order AE scheme with mesh ∆x = ∆y = 1400 , CFL = 0.5,

and ∆t/ε = 0.95 The isolines of the density and pressure are given in Figure 12 (left).A reference solution for this problem with cylindrical symmetry can be produced by the one -

dimensional inhomogeneous system

Ut + F (U)r = S(U),

where r > 0 denotes the radial variable,

U =

ρρξE

, F =

ρξρξ2 + pξ(E + p)

, S = −1

r

ρξρξ2

ξ(E + p)

,

with p = 0.4(E − 12ρξ

2). Here ξ is the velocity in radial direction. The initial data can be rewritteninto the Sod type initial data for r ∈ [0, 1]

(p, ρξ, E) =

((1, 0, 2.5) r < 0.4,

(0.125, 0, 0.25) 0.4 < r < 1.

We compute this reference solution by second order global AE scheme with 2560 points, CFL = 0.4and ∆t/ε = 0.9. Figure 12 (right) shows the comparison of density and pressure as functions of rat the output time T = 0.25. Performance of the AE scheme for this explosion problem is clearlydemonstrated.

6. Concluding remarks

In this work we have developed several AE schemes based on the alternating evolution approximationto nonlinear hyperbolic conservation laws. The AE system captures the exact solution when initiallyboth components are chosen as the given initial data for the conservation laws. Such a feature allows fora sampling of two components over alternating grids. High order accuracy is achieved by a combinationof high-order non-oscillatory polynomial reconstruction from the obtained averages and a matchingRunge-Kutta solver in time discretization. Local AE schemes are made possible by letting the scaleparameter ε reflect the local distribution of nonlinear waves. The AE schemes have the advantage ofeasier formulation and implementation, and efficient computation of the solution. For the first andsecond order local AE schemes applied to scalar conservation laws, we have proved both the maximumprinciple and total variation diminishing (TVD) property. Numerical tests for both scalar conservationlaws and compressible Euler equations presented in this work have demonstrated the high order accuracyand capacity of these AE schemes. Extension to multi-dimensional problems is straightforward, thederived first and second order AE schemes have been included and tested using both a linear transportproblem and an explosion problem for the 2D Euler equation.

We note that the development of higher order methods that could potentially yield better accuracyor reduced computational costs is a subject of ongoing research. One of the most promising approachesis the discontinuous Galerkin (DG) method, for which the theoretical basis has been provided in aseries of papers by Cockburn et al. [4, 3, 2, 5]. Our AE schemes are simple to implement and havelower computational costs. In terms of accuracy, the advantage is still for the DG methods and futurework will be dedicated to making the AE scheme more accurate by exploiting compact high orderapproximations.

Acknowledgments

The authors thank the anonymous referees who provided valuable comments resulting in improve-ments in this paper. Liu’s research was partially supported by the National Science Foundation undergrant DMS09-07963.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 21

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[J. Comput. Phys. 32 (1979), no. 1, 101–136]. J. Comput. Phys., 135(2):227–248, 1997. With an introduction by Ch.Hirsch, Commemoration of the 30th anniversary {of J. Comput. Phys.}.

Iowa State University, Mathematics Department, Ames, IA 50011

E-mail address: [email protected]; [email protected]

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 23

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(a) NT scheme.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(b) KT scheme.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(c) AE scheme.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(d) AE1 scheme.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(e) AE2 scheme.

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Exact

AE

KT

AE1

AE2

NT

(f) Comparison with all schemes.

Figure 3. Comparison of plots for Burgers’ equation at discontinuity on [0, 2], T =0.7, ∆x = 1

40 , ∆t = 0.8ε, 2nd order scheme.

24 HASEENA AHMED AND HAILIANG LIU

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(a) AE scheme.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(b) AE1 scheme.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(c) AE2 scheme.

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Exact

AE

AE1

AE2

(d) Comparison with all schemes.

Figure 4. Comparison of plots for Burgers’ equation at discontinuity on [0, 2], T =0.7, ∆x = 1

40 , ∆t = 0.8ε, 3rd order scheme.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 25

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) AE1 scheme.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) AE1 scheme.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) AE2 scheme.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d) AE2 scheme.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact

AE

AE1

AE2

(e) Comparison of global AE, local AE1 and AE2 schemes.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact

AE

AE1

AE2

(f) Comparison of global AE, local AE1 and AE2 schemes.

Figure 5. Comparison of plots for Buckley-Leverett problem 5.2 on [−1, 1], T = 0.4,∆x = 1

40 , ∆t = 0.8ε, 2nd(Left) and 3rd order (Right) scheme.

26 HASEENA AHMED AND HAILIANG LIU

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

De

nsity

(a) Density with AE scheme, 2nd order.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

De

nsity

(b) Density with AE scheme, 3rd order.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

Den

sity

(c) Density with AE1 scheme, 2nd order.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

Den

sity

(d) Density with AE1 scheme, 3rd order.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

Ve

locity

(e) Velocity with AE1 scheme, 2nd order.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

Ve

locity

(f) Velocity with AE1 scheme, 3rd order.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

3.5

4

x

Pre

ssu

re

(g) Pressure with AE1 scheme, 2nd order.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

3.5

4

x

Pre

ssu

re

(h) Pressure with AE1 scheme, 3rd order.

Figure 6. Plots for the Euler equation with Lax initial data on [0, 1], T = 0.16,∆x = 1

200 , ∆t = 0.8ε, 2nd and 3rd order scheme.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 27

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Density

(a) Density with 2nd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Density

(b) Density with 3rd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Velo

city

(c) Velocity with 2nd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Velo

city

(d) Velocity with 3rd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Pre

ssure

(e) Pressure with 2nd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Pre

ssure

(f) Pressure with 3rd order scheme.

Figure 7. Plots for the Euler equation with Sod initial data on [0, 1], T = 0.1644,∆t = 0.8ε, ∆x = 1

200 , AE1 scheme.

28 HASEENA AHMED AND HAILIANG LIU

−5 −4 −3 −2 −1 0 1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

(a) Density with AE scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

(b) Density with AE1 scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

x

Velo

city

(c) Velocity with AE scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

x

Velo

city

(d) Velocity with AE1 scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50

2

4

6

8

10

12

x

Pre

ssure

(e) Pressure with AE scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50

2

4

6

8

10

12

x

Pre

ssure

(f) Pressure with AE1 scheme.

Figure 8. Plots for the Euler equation with Osher-Shu initial data on [−5, 5], T = 1.8,∆t = 0.8ε, ∆x = 1

40 , 3rd order scheme.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 29

−5 −4 −3 −2 −1 0 1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

(a) Density with 2nd order scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

(b) Density with 3rd order scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

x

Velo

city

(c) Velocity with 2nd order scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

x

Velo

city

(d) Velocity with 3rd order scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50

2

4

6

8

10

12

x

Pre

ssure

(e) Pressure with 2nd order scheme.

−5 −4 −3 −2 −1 0 1 2 3 4 50

2

4

6

8

10

12

x

Pre

ssure

(f) Pressure with 3rd order scheme.

Figure 9. Plots for the Euler equation with Osher-Shu initial data on [−5, 5], T = 1.8,∆t = 0.8ε, ∆x = 1

80 , AE1 scheme.

30 HASEENA AHMED AND HAILIANG LIU

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

x

Density

(a) Density with 2nd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

x

Density

(b) Density with 3rd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−5

0

5

10

15

20

x

Velo

city

(c) Velocity with 2nd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−5

0

5

10

15

20

x

Velo

city

(d) Velocity with 3rd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

350

400

450

500

x

Pressure

(e) Pressure with 2nd order scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

350

400

450

500

x

Pressure

(f) Pressure with 3rd order scheme.

Figure 10. Plots for the Euler equation with Woodward Colella initial data on [0, 1],T = 0.01, ∆x = 1

800 , ∆t = 0.8ε, AE1 scheme.

AE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS 31

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

x

Density

(a) Density at T = 0.03.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

x

Density

(b) Density at T = 0.038.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−4

−2

0

2

4

6

8

10

12

14

x

Velo

city

(c) Velocity at T = 0.03.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

0

2

4

6

8

10

12

14

16

x

Velo

city

(d) Velocity at T = 0.038.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

x

Pressure

(e) Pressure at T = 0.03.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

350

400

450

x

Pressure

(f) Pressure at T = 0.038.

Figure 11. Plots for the Euler equation with Woodward Colella initial data on [0, 1],∆x = 1

800 , ∆t = 0.8ε, 3rd order AE1 scheme.

32 HASEENA AHMED AND HAILIANG LIU

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(a) Density contour

0 0.2 0.4 0.6 0.8 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Density ρ

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(c) Pressure contour

0 0.2 0.4 0.6 0.8 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d) Pressure p

Figure 12. Isolines of density and pressure to the explosion problem (left). Compar-ison between the two dimensional solutions and the one dimensional radial solutions(right). Example 5.5 for 2nd order AE scheme on [−1, 1] × [−1, 1], T = 0.25,∆t = 0.95ε, ∆x = ∆y = 1/400.