Hybrid WENO Schemes with Lax-Wendroff Type Time...

J. Math. Studydoi: 10.4208/jms.v50n3.17.03

Vol. 50, No. 3, pp. 242-267September 2017

Hybrid WENO Schemes with Lax-Wendroff Type TimeDiscretization

Buyue Huang and Jianxian Qiu ∗

School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathemat-ical Modeling and High-Performance Scientific Computation, Xiamen University,Xiamen 361005, Fujian, P.R. China.

Received February 24, 2017; Accepted June 5, 2017

Abstract. In this paper, we investigate the performance of a class of the hybrid weightedessentially non-oscillatory (WENO) schemes with Lax-Wendroff time discretizationprocedure using different indicators for hyperbolic conservation laws. The main ideaof the scheme is to use some efficient and reliable indicators to identify discontinuityof solution, then reconstruct numerical flux by WENO approximation in discontin-uous regions and up-wind linear approximation in smooth regions, hence reducingcomputational cost but still maintaining non-oscillatory properties for problems withstrong shocks. Numerical results show that the efficiency and robustness of the hybridWENO-LW schemes.

AMS subject classifications: 65M60, 35L65

Key words: Time discretization methods, WENO approximation, troubled-cell indicator, Hyper-bolic conservation laws, Hybrid schemes.

1 Introduction

In this paper, we investigate the performance of a class of hybrid weighted essentiallynon-oscillatory (WENO) schemes with Lax-Wendroff time discretization, termed as hy-brid WENO-LW schemes, with different discontinuity indicators for solving hyperbolicconservation laws:

{ut+∇· f (u)=0,u(x,0)=u0(x).

(1.1)

∗Corresponding author. Email addresses: [email protected] (B. Huang), [email protected] (J. Qiu)

http://www.global-sci.org/jms 242 c©2017 Global-Science Press

B. Huang and J. Qiu / J. Math. Study, 50 (2017), pp. 242-267 243

The first finite volume WENO scheme was constructed by Liu et al. [16], and the thirdand fifth-order finite difference WENO schemes in multi-space dimensions were pre-sented by Jiang and Shu [12], in which they setup a framework to compute the smooth-ness indicators and nonlinear weights which is the key of WENO schemes in the com-bination of lower order flux to obtain a higher order approximation. Further Balsaraand Shu [1] and Gerolymos et al. [8] extended the WENO schemes to higher order. Theweights for combination of lower order flux is important for WENO approximation. Forthe case of system, WENO schemes use local characteristic decompositions and flux split-ting to avoid or reduce spurious oscillation. But the calculations of nonlinear weightsand local characteristic decomposition are expensive. To overcome these drawbacks,Jiang and Shu [12] computed the nonlinear weights from pressure or entropy insteadof the characteristic values for Euler equations. Pirozzoli [17] developed an efficient hy-brid compact-WENO scheme, which used compact up-wind schemes to treat smoothregions of the flow field and WENO schemes to handle discontinuities regions. Hilland Pullin [11] developed a hybrid scheme which combines the tuned center-differenceschemes with WENO schemes, hence achieving automatically the nonlinear weights forWENO schemes in regions of smooth flow away from shocks. But a switch was still nec-essary for the schemes. Li and Qiu [15] developed hybrid WENO schemes with Runge-Kutta time discretization which combine pure WENO schemes with simple upwind lin-ear schemes, in which they investigated using the different troubled-cell indicators whichare borrowed from discontinuous Galerkin (DG) schemes as switches to identify whereWENO approximation or upwind linear approximation is applied.

The main idea of hybrid WENO schemes is using WENO approximation in discon-tinuity and other efficient approximation such as up-wind linear one in smooth regionof solution to reduce computational cost. The troubled-cell indicators which can identifywhere is discontinuity of the solution are key components of hybrid WENO schemes.In [15], Li and Qiu had investigated using the different troubled-cell indicators to iden-tify discontinuity of the solution. There are many troubled-cell indicators based on lim-iters of DG schemes which are listed by Qiu and Shu [20]. Among them, the total vari-ation bounded (TVB) limiter [4–7] borrowed from the finite volume methodology is aslope limiter based on minmod function. Biswas, Devine and Flaherty (BDF) investi-gated the moment-based limiter [2] and Burbeau, Sagaut and Bruneau (BSB) investi-gated an improved moment limiter [3]. Krivodonova et al. (KXRCF) designed a lim-iter [13] to detect discontinuities for DG methods based on super convergent propertyat the outflow boundary in smooth regions. There are also many other troubled-cellindicators borrowed from finite volume and finite difference methodology, such as themonotonicity-preserving (MP) limiter [25], and modifications of MP (MMP) limiter [22].Qiu and Shu [21] used some limiters as troubled-cell indicators for Runge-Kutta discon-tinuous Galerkin (RKDG) methods with WENO limiters to compare their performance.And Zhu and Qiu [28] used these troubled-cell indicators for adaptive RKDG methods.Li and Qiu [15] applied them in hybrid WENO with Runge-Kutta (RK) time discretiza-tion schemes to identify where WENO approximation or upwind linear approximation

244 B. Huang and J. Qiu / J. Math. Study, 50 (2017), pp. 242-267

is applied.Hybrid WENO procedure is used to discrete the spatial derivative term. The time

derivative term there must also be discretized. There are mainly two different approachesto approximate the time derivative. The first approach is to use an ODE solver, such as aRunge-Kutta or a multistep method [23,24], to solve the method of lines ODE obtained af-ter spatial discretization. The second approach is a Lax-Wendroff type time discretizationprocedure based on the idea of the classical Lax-Wendroff scheme [14]. In Lax-Wendroffapproach, all the time derivatives are converted into spatial derivatives by using tem-poral Taylor expansion and differentiated versions of PDE, then the spatial derivativesare discretized by, e.g., the hybrid WENO approximations. Formulation and code ofLax-Wendroff type time discretization are more complex than those of TVD Runge-Kuttatime discretization [23, 24]. But less CPU time cost can rise competitive of the methods.Qiu and Shu [19] have explored a class of WENO schemes with a Lax-Wendroff timediscretization procedure. They applied local characteristic decomposition and WENOapproximation to reconstruct the first order time derivative, then used central differenceapproximations to reconstruct higher order time derivatives in order to reduce computa-tion cost but still maintain good properties of PDE and high order accuracy.

In this paper, follow [15], we explore fifth order finite difference hybrid WENO schemeswith third order Lax-Wendroff type time discretization with different troubled-cell indi-cators. The emphasis of the paper is also on comparison of the performance of hybridschemes using different indicators, with an objective of obtaining efficient and reliable in-dicators to obtain better performance of hybrid schemes to save computational cost. Thecomparison between hybrid WENO-LW schemes with hybrid WENO-RK schemes [15]is also addressed. The arrangement of this paper is as follows. In Section 2 we givethe description of hybrid WENO-LW schemes with high order up-wind linear schemesand some different troubled-cell indicators. Then we provide detailed numerical stud-ies of hybrid WENO-LW schemes in one- and two-dimensional cases, and compare withhybrid WENO-RK schemes [15] to address the issues of efficiency and non-oscillatoryproperty in Section 3. Concluding remarks are given in Section 4.

2 Description of hybrid WENO schemes with Lax-Wendroff typetime discretization

In this section, we describe in detail implementation of hybrid schemes of WENOschemes with Lax-Wendroff type time procedure. We start with the description in theone-dimensional scalar case.

2.1 Lax-Wendroff type time discretization

In this subsection, we review the Lax-Wendroff type time discretization method [19].Consider the one-dimensional scalar conservation laws:


{ut+( f (u))x =0,u(x,0)=u0(x). (2.1)

For simplicity, we denoted the cell by Ii =[xi−1/2,xi+1/2], the grid points by xi =(xi−1/2+xi+1/2)/2 which are uniformly distributed, spatial step ∆x=xi+1/2−xi−1/2 and time stepby ∆t= tn+1−tn. Let uni be the approximation of the point values u(xi,tn), and the firstthree time derivatives of u by u′, u′′, and u′′′. By a temporal Taylor expansion for u(x,t+∆t) at t, we obtain:

u(x,t+∆t)=u(x,t)+∆tu′+∆t2

2u′′+

∆t3

6u′′′+··· (2.2)

We can also take a higher order Taylor expansion to get higher order accuracy in time. Inthis paper, we expand up to third order in time, although the procedure can be naturallyextended to any higher order. The first three time derivatives of u will be converted tospatial derivatives by using differentiated versions of PDE (2.1), then the spatial deriva-tives are discretized by following procedures.

Step 1.1: The first time derivative u′=− fx(u) will be discretized by hybrid WENOwhich will be described in subsection 2.2.

Step 1.2: For the second time derivative u′′=−( f ′(u)u′)x, and the third time derivative

u′′′=−( f ′(u)u′′+ f ′′(u)(u′)2)x

will be approximated by conservative central difference formula, respectively. Due to theextra ∆t factor, if we would like to obtain (2r+1)-th order accuracy in spatial discretiza-tion, we need only an approximation of order 2r for u′′, and (2r−1) for u′′′, respectively.In this paper, we consider r=2, and would like a fourth order central difference formulato approximate u′′ and u′′′ at the point (xi,tn) as follow:

− 112∆x

(gi−2−8gi−1+8gi+1−gi+2). (2.3)

For u′′, gi= f ′(ui)u′i, with ui, u′i denoted by the point values of u and u

′ at (xi,tn), u′i whichare computed in Step 1.1. For u′′′,

gi = f ′(ui)u′′i + f′′(ui)(u′i)

2.

For system cases, the approximation of the first time derivative u′=− f (u)x shouldbe performed in the local characteristic directions to avoid spurious oscillations. See [12]and [18] for details. For the second and higher time derivatives, we convert them to spa-tial derivatives as before. It is enough to ensure the ENO property by using simple centralapproximations, and we need neither local characteristic decomposition nor WENO ap-proximation. But we should notice that f ′(u) is a matrix (the Jacobian) and f ′′(u) is athree-dimensional matrix (tensor), etc.. So it is still very complicated of the code of Lax-Wendroff time discretizations methods. However, we will see in the next section that themethods would save CPU time for certain problems.


2.2 Hybrid WENO reconstruction procedure

The spatial derivative f (u)x at (xi,tn) can be approximated by a high order conserva-tive finite difference scheme:

f (u)x|x=xi≈1

∆x( f̂i+ 12− f̂i− 12 ), (2.4)

where f̂i+ 12 are the numerical fluxes. We define an implicit function v as:

1∆x

∫ x+ ∆x2x− ∆x2

v(ξ)dξ= f (u), (2.5)

then take the derivative of equation (2.6), we have:

f (u)x|x=xi =1

∆x(v(xi+ 12 )−v(xi− 12 ). (2.6)

If the numerical flux f̂i+1/2 is taken to be the (2r+1)th order approximation to vi+1/2=v(xi+1/2), then 1∆x ( f̂i+1/2− f̂i−1/2) is the (2r+1)th order approximation to fx(u) at x= xi.For the purpose of keeping the stability, we need consider the upwind quality of theschemes. We will split flux f (u) into two parts:

f (u)= f+(u)+ f−(u) withd f+(u)

du≥0 and d f

−(u)du

≤0.

Here, a simple Lax-Friedrichs splitting is applied as

f±(u)=12( f (u)±αu), (2.7)

in which α is set as maxu | f ′(u)| over the whole range of u. Let f̂+i+ 12and f̂−

i+ 12be the

numerical fluxes at xi+1/2 which are (2r+1)th order approximation of v(x) in (2.6) withthe positive and negative parts of f (u), respectively, and f̂i+1/2 is defined as f̂+i+ 12

+ f̂−i+ 12

.

Now we describe in detail for the reconstruction procedure of f̂+i+ 12

, and the recon-

struction procedure of f̂−i+ 12

is mirror symmetric with respect to xi+1/2 of that for f̂+i+ 12.

From the definition of v(x) in (2.6) for f+(u), we have

f+(ui)=1

∆x

∫ xi+1/2xi−1/2

v(ξ)dξ=vi, (2.8)

where vi is the cell average of v(x) on the Ii.Step 2.1: The troubled-cell indicator which will be described in Subsection 2.3 is ap-

plied to identify troubled cell, namely the locations of discontinuity of the numericalsolution.


Step 2.2: Choose the following big stencil: T = {Ii−r,..., Ii+r}, it is easy to obtain thereconstructed polynomial which satisfies

1∆x

∫Ij

Q(ξ)dξ=1

∆x

∫Ij

v(ξ)dξ=vj = f+(uj), j= i−r,...,i+r. (2.9)

Step 2.2.1: If there is not troubled-cell in stencil T, the final reconstruction of thenumerical flux of f+(u) at x=xi+1/2 is directly given by f̂+i+1/2=Q(xi+ 12 ), and which is alinear upwind approximation to f+(u).

Step 2.2.2: If there is at least one troubled-cell in stencil T, then the numerical fluxesf̂+i+1/2 will be reconstructed by 2r+1 order WENO approximation, the detail of 2r+1order WENO approximation we refer to [12].

2.3 Review of troubled-cell indicators

In this subsection, we review some troubled-cell indicators listed by [15,20,28]. Someof these troubled-cell indicators are designed as limiter for DG schemes. We constructa quadratic polynomial for the numerical solution on cell Ij at time step tn, which isdenoted by uh(x). The quadratic polynomial uh(x) should satisfies the following condi-tions:

uh(xk)=unk , k= j−1, j, j+1, (2.10)

where unj is still the approximation of the point values u(xj,tn). And we obtain:

uh(x)=u(0)j +u(1)j

2(x−xj)∆x

+u(2)j ·12

[3(

2(x−xj)∆x

)2−1]

, x∈ [xj− 12 ,xj+ 12 ], (2.11)

where

u(0)j =124

(unj−1+22unj +u

nj+1), u

(1)j =

14(unj+1−unj−1), u

(2)j =

112

(unj−1−2unj +unj+1).

For scalar equations, we use point values of solution to construct uh(x). But for systemequations, entropy values are used to construct uh(x).

Now we begin to describe the formula of troubled-cell indicators:

1. The average total variation (ATV) indicator [15]. Let

TV≡TV(un)=∑j|unj+1−unj |, (2.12)

where N is the number of cells. 0 < θ < 1 is a constant called the ATV parameter. Ifθ ·|unj+1−unj |≥

TVN , cells Ij and Ij+1 are declared as troubled cells. ATV indicator is based

on the average total variation of the numerical solution at every time step. So the choiceof θ is important. But it is difficult to chose accurately, because θ is problem-depended.


If the chosen of θ is too large, computational cost will increase unnecessarily; conversely,spurious oscillations will appear.

2. The minmod-based TVB limiter [6] (TVB). Let

ũj(mod)= m̃(ũj,u

(0)j+1−u

(0)j ,u

(0)j −u

(0)j−1),˜̃uj(mod)= m̃( ˜̃uj,u(0)j+1−u(0)j ,u(0)j −u(0)j−1), (2.13)

whereũj =uh(xj+ 12 )−u

(0)j , ˜̃uj =−uh(xj− 12 )+u(0)j

and the function m̃ is given by

m̃(a1,a2,··· ,an)={

a1 if |a1|≤M(∆x)2,m(a1,a2,··· ,an), otherwise,

(2.14)

where the minmod function m is defined as:

m(a1,a2,··· ,an)={

s·mina≤j≤n |aj| if sign(a1)= ···= sign(an)= s,0 otherwise,

(2.15)

and M>0 is the TVB parameter.

If ũj(mod) 6= ũj or ˜̃uj(mod) 6= ˜̃uj, cell Ij is identified as troubled cell. The value of M is

also problem-depended. For scalar equations, M is proportional to the second deriva-tive of the initial data at smooth extremum. But it is difficult to estimate M for systemof equations. If the chosen of M is too small, computational cost will increase unneces-sarily because many cells are identified as troubled-cells, conversely, there will be somespurious oscillations.

3. A limiter designed by Xu and Shu (XS) [27]. Let

φj =β j

β j+γj(2.16)

where

β j =ξ j

αj−1+

ξ j

αj+2, γj =

(umax−umin)2αj

,

with αj =(unj −unj−1)2+ε and ξ j =(unj−1−unj+1)2+ε. And umax and umin are the maximumand minimum values of u(x,t) at the time step of tn.

For avoiding denominator becoming zero, we usually take the positive constant ε=10−6. XS indicator is a strong troubled-cell indicator presented in [27]. It is obvious that0≤φj≤ 1 and φj =O(∆x2) in the smooth regions. In strong discontinuous regions, φj isclose to 1 due to γj� β j. In this paper, we identify the cells Ij and Ij+1 as troubled cellswhen φj >∆x.


4. The monotonicity-preserving limiter (MP) [25]. Let

ũ−j+ 12

=median(u−j+ 12

,uminj+ 12,umaxj+ 12

) (2.17)

whereumin

j+ 12=max[min(u(0)j ,u

(0)j+1,u

MDj+ 12

), min(u(0)j ,uULj+ 12

,uLCj+ 12

)],

umaxj+ 12

=min[max(u(0)j ,u(0)j+1,u

MDj+ 12

), max(u(0)j ,uULj+ 12

,uLCj+ 12

)],(2.18)

withuMD

j+ 12= 12 (u

(0)j +u

(0)j+1−dmaxj+ 12

),

uULj+ 12

=u(0)j +α(u(0)j −u

(0)j−1),

uLCj+ 12

=u(0)j +12 (u

(0)j −u

(0)j−1)+

β3 d

maxj− 12

,

dmaxj+ 12

=m(4dj−dj+1,4dj+1−dj,dj,dj+1,dj−1,dj+2),

dj =u(0)j+1−2u

(0)j +u

(0)j−1.

(2.19)

The median function is defined by

median(x,y,z)= x=m(y−x,z−x), (2.20)

where m is the minmod function defined by formula (2.15). α and β are the parameters ofMP indicator, which are suggested to take α=2, and β=4 in [25]. If u−

j+ 126= ũ−

j+ 12or u+

j+ 12satisfies a symmetric condition, we identify the cell Ij as troubled cell.

5. Multi-resolution (MR) indicator [9]. Let

dj =uj−ũj, (2.21)

where ũj = 12 (uj−1+uj+1) is a approximation to uj using the values of u(x) at xj−1 andxj+1. So dj is the corresponding approximation error satisfies:

dj≈{

(∆x)p[u(p)], p≤2,

(∆x)2u(2), p≥2.(2.22)

MP indicator is based on the multi-resolution analysis of Harten [9]. If |dj|≥ εMR∆x, thecell Ij is identified as a troubled cell, where εMR is the multi-resolution parameter.

6. BDF limiter [2]. Let

ũ(1),modi =m(

u(1)i ,u(0)i+1−u

(0)i ,u

(0)i −u

(0)i−1

), (2.23)


where m is still the minmod function defined in (2.15). The BDF limiter is a momentlimiter designed by Biswas at al. [2]. If ũ(1),modi are different from the first argument of(2.23), we identify Ij as the troubled cell.

7. BSB limiter [3]. Let

û(1),modi =m(

u(1)i ,u(0)+i+ 12−u(0)i ,u

(0)i −u

(0)−i− 12

), (2.24)

whereu(0)+

i+ 12=u(0)i+1−u

(1)i+1, u

(0)−i− 12

=u(0)i−1−u(1)i−1. (2.25)

The BSB limiter is the modified moment limiter designed by Burbeau et al. [3]. If BDFindicator is enacted and û(1),modi is different from the first argument of (2.24), we identifyIj as the troubled cell.

8. Modifications of MP (MMP) limiter [22]. Let

φj =min(

1,∆ûminj∆minuj

)(2.26)

where∆ûminj =u

(0)j −min

(u(0)j−1,u

(0)j ,u

(0)j+1

),

∆minuj =u(0)j −min

(uh(xj+ 12 ),u

h(xj− 12 ))

.(2.27)

MMP limiter is a modification of the MP limiter. If φj 6= 1, we identify the cells Ij as thetroubled cell.

9. KXRCF indicator [13]. Let

φj =

∫∂I−j|uh|Ij−uh|Inj |ds

∆xk+1

2 |∂I−j |‖uh|Ij‖, (2.28)

where ∆x is the diameter of the circumscribed circle in the element Ij, k is the degreeof the polynomial uh approximating to u(x), Inj is the neighbor of Ij on the side of ∂I

−j

and the norm is L1 norm in one-dimension cases and L∞ norm in two-dimension cases.KXRCF indicator is a shock-detection technique by Krivodonova et al. [13]. If φj >1, weidentify the cell Ij as troubled cell.

From the description of troubled-cell indicators, we can find that ATV, TVB, XS, MPand MR indicators with problem depended parameters, but BDF, BSB, MMP and KXRCFindicators are parameter free.

2.4 Two-dimensional cases

Consider the two-dimensional scalar conservation laws:


{ut+( f (u))x+(g(u))y =0,

u(x,y,0)=u0(x,y).(2.29)

As the one-dimensional cases, we make a temporal Taylor expansion for u(x,t+∆t)at t, and then obtain

u(x,y,t+∆t)=u(x,y,t)+∆tu′+∆t2

2u′′+

∆t3

6u′′′+··· (2.30)

If we need a third order accurate solution in time, we would like to reconstruct up to thethird time derivatives: u′, u′′, u′′′. We again use spatial derivatives to replace time deriva-tives by the PDE (2.29). Then we apply the finite difference hybrid WENO procedure toapproximate the first time derivative u′=− f (u)x−g(u)y in a dimension-by-dimensionfashion. On the other hand, as in the one-dimensional situation, the second order timederivative u′′=−( f ′(u)u′)x−(g′(u)u′)y, the third order time derivative

u′′′=−( f ′′(u)(u′)2+ f ′(u)u′′)x−(g′′(u)(u′)2+g′(u)u′′)y,

can be approximated by simple and suitable orders central differences approximation,again in a dimension-by-dimension fashion. For system cases, we would like to use localcharacteristic decomposition to reconstruct the first time derivative in which we applyWENO procedure.

3 Numerical results

In this section, we perform extensive numerical experiments on 1D and 2D Eulerequations to present the performances of hybrid WENO schemes with Lax-Wendroff typetime discretization using some different troubled-cell indicators and compare them withhybrid WENO schemes with Runge-Kutta time [15] . We denote TVB-1 and TVB-2 by theminmod-based TVB indicator with the TVB parameters M=0.01 and M=10. For chosenof parameter, we take the ATV parameter θ = 0.3 to the shock density wave interactionproblem, and take θ = 0.7 to other problems. We take the multi-resolution parameter ofMR indicator εMR =0.5. In all numerical tests of this paper, the CFL number is 0.5.

3.1 One-dimensional case

We first consider one-dimensional Euler equations of gas dynamics with four dif-ferent initial conditions. At the same time we compare the performance with hybridWENO-RK schemes. The PDEs are:


Table 1: Lax problem. The total CPU time for N=100×2n,(n=0,1,··· ,4) cells, the ratios of the total CPUtime by hybrid WENO-LW schemes over the same order hybrid WENO-RK scheme using the same indicators.

Scheme CPU of hybrid CPU of hybrid Ratioor indicators WENO-LW schemes WENO-RK schemes

ATV 1.2720 2.2734 0.56TVB-1 1.1811 2.2710 0.52TVB-2 1.0909 2.0872 0.52

XS 1.4131 2.3970 0.59MP 1.7415 2.9156 0.60MR 1.3348 2.2932 0.58BDF 1.6622 3.0157 0.55BSB 1.6665 2.9997 0.56

MMP 1.5263 2.7160 0.56KXRCF 1.3095 2.3205 0.56

ρ

ρv

E

t

+

ρv

ρv2+p

v(E+p)

x

=0, (3.1)

where ρ is density, v is velocity, p is pressure and E is energy which can be obtained byequation E= pγ−1 +

12 ρv

2 with γ=1.4.

Example 3.1. The Lax problem. The initial condition is

(ρ,v,p)=

{(0.445,0.698,3.538) if x≤0,(0.5,0,0.571) if x>0.

(3.2)

The computational domain is [−0.5,0.5], and we compute this problem till t= 0.16 withinflow and outflow boundary condition, respectively.

We compare the performances of hybrid WENO-LW schemes with different troubled-cell indicators. We show the density and time history of reconstruction of fluxes byWENO approximation of hybrid WENO-LW schemes in Figure 1. In the density fig-ure, the solid line represents the exact solution, ”�” represent the numerical solution ofhybrid WENO-LW schemes. In the time history figure, ”�” represents the troubled-cell.

Then we present CPU-L1-error and cell-percentage curves of the two different hybridschemes with all indicators using 100×2n(n=0,··· ,4) uniform cells in Figure 2. The CPUtime for the hybrid WENO-LW schemes and hybrid WENO-RK schemes are shown inTable 1.


−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(a) ATV

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(b) TVB-1

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(c) TVB-2

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(d) XS

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(e) MP

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(f) MR

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(g) BDF

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(h) BSB

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(i) MMP

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

density

−0.5 0 0.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

t

(j) KXRCF

Figure 1: Lax problem by hybrid WENO5-LW3 scheme with different indicators, 400 cells, t = 0.16.Density and time history of reconstruction of fluxes by WENO approximation.


10−2 1000

0.005

0.01

0.015

0.02

0.025

0.03

CPU time

Erro

r

ATVTVB1TVB2XSMPMRBDFBSBMMPKXECF

0 1 2 3 4

10

20

304050

n

Perc

enta

ge


(a) Lax-Wendroff type time discretization

10−2 1000

0.005

0.01

0.015

0.02

0.025

0.03

CPU time

Erro

r


0 1 2 3 4

10

20

304050

n

Perc

enta

ge


(b) Runge-Kutta time discretization

Figure 2: Lax problem by the hybrid WENO scheme with two different time discretizations. CPU-L1-error (left)and cell-percentage (right).

In order to save space, in this paper we only show the results by fifth order in spatialdiscretization and third order in time discretization, because the performances of otherorders schemes are similar. We can observe that the numerical results for all cases keepsharp transition and mostly free oscillation. But hybrid WENO-LW schemes are moreefficient than hybrid WENO-RK schemes even save nearly half of the CPU time. Thisproperty is more obvious with mesh refinement. In this case, for hybrid WENO-LWschemes, the best troubled-cell indicators (lead to less CPU time and smaller percentageof reconstruction of fluxes) are ATV, MR ,XS ,TVB and KXRCF indicators.

Example 3.2. The shock density wave interaction problem. The initial condition is

(ρ,v,p)=

{(3.857143,2.629369,10.333333) if x


−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(a) ATV

−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(b) TVB-1

−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(c) TVB-2

−5 0 50

1

2

3

4

5

xdensity

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(d) XS

−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(e) MP

−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(f) MR

−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(g) BDF

−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(h) BSB

−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(i) MMP

−5 0 50

1

2

3

4

5

x

density

−5 0 5

0.20.40.60.81

1.21.41.6

x

t

(j) KXRCF

Figure 3: Shock density wave problem by hybrid WENO5-LW3 scheme with different indicators, 400cells, t=1.8. Density and time history of reconstruction of fluxes by WENO approximation.


10−2 1000

0.02

0.04

0.06

0.08

0.1

0.12

CPU time

Erro

r


0 1 2 3 4

20

406080100

n

Perc

enta

ge



10−2 1000

0.02

0.04

0.06

0.08

0.1

0.12

CPU time

Erro

r


0 1 2 3 4

20

406080100

n

Perc

enta

ge



Figure 4: Shock density wave problem problem by the hybrid WENO scheme with two different time discretiza-tion. CPU-L1-error (left) and cell-percentage (right).

The computational domain is [−5,5], we compute this problem till t = 1.8 with in-flow and outflow boundary condition, respectively. We show the performance of hybridWENO-LW schemes in Figure 3, and the reference“exact” solution is computed by the5th-order finite difference WENO scheme [12] using 6400 grid points. Then we presentcomparison of hybrid WENO-LW schemes with the two different time discretization onCPU-L1-error and cell-percentage curves of all indicators using 100×2n(n=0,··· ,4) uni-form cells in Figure 4. Table 2 presents the CPU time comparison among hybrid WENO-LW schemes with hybrid WENO-RK schemes, and we can observe that TVB, MR andKXRCF indicators perform better than others.

Example 3.3. The blast wave interaction problem. The initial condition is


Table 2: Shock density wave problem. The total CPU time for N=100×2n(n=0,1,··· ,4)cells, the ratios of thetotal CPU time by hybrid WENO-LW schemes over the same order hybrid WENO-RK scheme using the sameindicators.


ATV 1.5311 2.9220 0.52TVB-1 1.4968 2.8959 0.52TVB-2 1.3258 2.4457 0.54

XS 1.7021 3.0705 0.55MP 1.7523 3.0542 0.57MR 1.5190 2.7213 0.56BDF 1.9128 3.4855 0.55BSB 1.8332 3.3283 0.55

MMP 1.7817 3.2765 0.54KXRCF 1.4496 2.6364 0.55

Table 3: Blast wave problem. The total CPU time for N=100×2n(n=0,1,··· ,4)cells, the ratios of the totalCPU time by hybrid WENO5-LW3 schemes over the same order hybrid WENO-RK scheme using the sameindicators.


ATV 2.7950 5.2616 0.53TVB-1 2.5849 5.1078 0.51TVB-2 2.3707 4.4763 0.53

XS 3.0549 5.3526 0.57MP 3.4427 6.2660 0.55MR 2.9027 5.2652 0.55BDF 3.4550 6.9515 0.50BSB 3.3518 6.7832 0.49

MMP 3.4115 6.4476 0.53KXRCF 2.9193 5.2933 0.55

(ρ,v,p)=

(1,0,1000) if 0≤ x


0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t(a) ATV

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(b) TVB-1

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(c) TVB-2

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(d) XS

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(e) MP

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(f) MR

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(g) BDF

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(h) BSB

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(i) MMP

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

de

nsity

0 0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

t

(j) KXRCF

Figure 5: Blast wave problem by hybrid WENO5-LW3 scheme with different indicators, 400 cells, t=1.8.Density and time history of reconstruction of fluxes by WENO approximation.


10−1 1000

0.05

0.1

0.15

0.2

0.25

0.3

CPU time

Erro

r


0 1 2 3 4

20

406080

100

n

Perc

enta

ge



10−1 1000

0.05

0.1

0.15

0.2

0.25

0.3

CPU time

Erro

r


0 1 2 3 4

20

406080

100

n

Perc

enta

ge



Figure 6: Blast wave problem by the hybrid WENO scheme with two different time discretization. CPU-L1-error(left) and cell-percentage (right).

3.2 Two-dimensional case

We have observed that ATV, TVB, MR and KXRCF indicators perform better thanother troubled-cell indicators for hybrid WENO-LW schemes in one-dimensional cases.Now we consider hybrid WENO-LW schemes with the above indicators for 2D Eulerequations of gas dynamics with two different initial conditions. The PDEs are:

ρ

ρu

ρv

E

t

+

ρu

ρu2+p

ρuv

u(E+p)

x

+

ρv

ρuv

ρv2+p

v(E+p)

y

=0. (3.5)


Table 4: Double Mach reflection problem. Comparison on CPU time and percentage of reconstruction of fluxesby WENO approximation among hybrid WENO schemes with the two different time discretization procedures.

Nx×Ny Scheme or indicators hybrid WENO-LW schemes hybrid WENO-RK schemesCPU Percentage CPU Percentage

480×120 ATV 82.26 6.56 95.05 7.12TVB-2 80.03 6.43 104.84 8.10

MR 71.76 3.05 91.23 3.89KXRCF 69.48 3.84 87.89 4.19WENO 129.29 100.00 273.51 100.00

960×240 ATV 761.74 4.48 987.36 4.96TVB-2 778.63 4.18 1034.83 5.98

MR 749.35 1.97 894.61 2.48KXRCF 701.14 2.72 890.73 2.84WENO 1112.56 100.00 2314.68 100.00

1920×480 ATV 6187.74 3.24 7011.19 3.54TVB-2 5785.61 3.17 7401.62 4.94

MR 5307.68 1.26 6582.81 1.63KXRCF 5290.87 1.92 6574.98 2.00WENO 9002.84 100.00 18931.58 100.00

Example 3.4. Double Mach reflection problem.

This problem is originally presented by [26]. The computational domain for this prob-lem is [0,4]×[0,1]. The reflecting wall lies at the bottom, starting from x= 16 . Initially aright-moving Mach 10 shock is positioned at x= 16 , y=0 and makes a 60

o angle with thex-axis. For the bottom boundary, the exact post-shock condition is imposed for the partfrom x = 0 to x = 16 and a reflective boundary condition is used for the rest. At the topboundary, the flow values are set to describe the exact motion of a Mach 10 shock. Wecompute the solution up to t=0.2.

For Double Mach reflection problem, we show the numerical results of hybrid WENO-LW schemes and hybrid WENO-RK schemes on the most refined mesh with 1920×480uniform cells in Figure 7. The figures are showing 30 equally spaced density contoursfrom 1.5 to 22.7. Reconstructions of fluxes by WENO approximation at the last time stepwith the ATV, TVB2, MR and KXRCF indicators are shown in Figures 8. We can observethat the numerical results of all cases keep sharp transition and are mostly oscillation-free.

In Table 4 we present the CPU time and the percentages of reconstruction of fluxes byWENO approximation by the two hybrid schemes with the ATV, TVB-2, MR and KXRCFindicators. It is easy to observe that the hybrid WENO-LW schemes cost about 60% CPUtime of pure WENO-LW scheme. The hybrid WENO-LW scheme is nealy similar to hy-brid WENO-RK scheme for the percentages of reconstruction of fluxes by WENO ap-


x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(a) ATV

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(b) TVB-2

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(c) MR

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(d) KXRCF

Figure 7: Double Mach reflection problem by the hybrid WENO5-LW3 (left) scheme and hybrid WENO5-RK3 scheme (right) with 1920×480 cells, t=0.2. Thirty equally spaced density contours from 1.5 to22.7. From top to bottom: ATV, TVB-2, MR and KXRCF.

proximation, but cost less time. And the percentages are all less than 10% with the fourindicators. The smaller percentage of reconstruction of fluxes by WENO approximationis presented with the finer meshes as we expect.


x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(a) ATV

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

xy

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(b) TVB-2

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(c) MR

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(d) KXRCF

Figure 8: Double Mach reflection problem by the hybrid WENO5-LW3 (left) scheme and hybrid WENO5-RK3 scheme (right) with 1920×480 cells, t=0.2. Reconstructions of fluxes by WENO approximationat the last time step. From top to bottom: ATV, TVB-2, MR and KXRCF.

Example 3.5. A Mach 3 wind tunnel with a step problem.

This problem is also originally from [26]. The wind tunnel is 1 length unit wide and3 length units long. The step is 0.2 length units high and is located 0.6 length units fromthe left-hand end of the tunnel. The problem is initialized by a right-going Mach 3 flow.


x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(a) ATV

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(b) TVB-2

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(c) MR

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(d) KXRCF

Figure 9: Forward step problem by the hybrid WENO5-LW3 (left) scheme and hybrid WENO5-RK3scheme (right) with 1920×480 cells, t=0.2. Thirty equally spaced density contours from 1.5 to 22.7.From top to bottom: ATV, TVB-2, MR and KXRCF.

Reflective boundary conditions are applied along the wall of the tunnel and in/out flowboundary conditions are applied at the entrance/exit. The corner of the step is a singularpoint and we treat it the same way as in [26], which is based on the assumption of anearly steady flow in the region near the corner. We compute the solution up to t=4.

For forward step reflection problem, numerical results of hybrid WENO-LW schemes


x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(a) ATV

x0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(b) TVB-2

x0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(c) MR

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(d) KXRCF

Figure 10: Forward step problem by the hybrid WENO5-LW3 (left) scheme and hybrid WENO5-RK3scheme (right) with 1920×480 cells, t=0.2. Reconstructions of fluxes by WENO approximation at thelast time step. From top to bottom: ATV, TVB-2, MR and KXRCF.

and hybrid WENO-RK schemes on the most refined mesh with 960×320 uniform cellsare given in Figure 9. ALL the figures are showing 30 equally spaced density contoursfrom 0.32 to 6.15. Reconstructions of fluxes by WENO approximation at the last time stepwith the ATV, TVB-2, MR and KXRCF indicators are shown in Figures 10. We can observethat numerical results of all cases keep sharp transition and are mostly oscillation-free.


Table 5: Forward step problem. Comparison on CPU time and percentage of reconstruction of fluxes by WENOapproximation among hybrid WENO schemes with the two different time discretization procedures.

Nx×Ny Scheme or indicators hybrid WENO-LW schemes hybrid WENO-RK schemesCPU Percentage CPU Percentage

280×80 ATV 83.12 23.58 105.66 24.63TVB-2 79.59 17.48 95.13 13.74

MR 70.33 18.96 81.54 8.40KXRCF 65.20 17.69 77.49 9.04WENO 106.03 100.00 225.46 100.00

480×160 ATV 728.21 17.24 824.18 17.92TVB-2 728.74 14.59 787.98 10.79

MR 655.66 12.67 677.31 6.37KXRCF 654.24 13.38 665.95 7.19WENO 946.19 100.00 1862.37 100.00

960×320 ATV 6199.52 13.60 7789.95 14.15TVB-2 6538.66 14.19 7444.96 10.39

MR 5777.98 8.59 6469.16 4.77KXRCF 5879.61 10.37 6461.70 5.82WENO 7986.02 100.00 15585.91 100.00

Table 5 indicates that hybrid WENO-LW schemes are more efficient than hybrid WENO-RK schemes, but save less time compared with pure WENO schemes because of morepercentages of reconstruction of fluxes by WENO approximation.

4 Concluding remarks

In this paper, we have investigated the hybrid WENO finite difference scheme withLax-Wendroff type time discretization using some different troubled-cell indicators, thencompared them with hybrid WENO-RK finite difference schemes. Extensive one-dimen-sional simulations on the hyperbolic systems of Euler equations indicate that althoughsome indicators pick less troubled-cell like XS, but they take more computerized timebecause of complex code. In summary ATV, TVB-2, MR and KXRCF indicators are bet-ter than other indicators. We then apply these four indicators to two-dimensional Eulerequations. All of numerical results suggest that hybrid WENO-LW scheme with these”best” indicators are more efficient than hybrid WENO-RK schemes. And the best indi-cators are MR and KXRCF indicators. These approach can also be applied to the finitevolume schemes. And the implementation of this method for structured curved meshesand three-dimensional problems are ongoing.


References

[1] D.S. Balsara, C.-W. Shu. Monotonicity preserving WENO schemes with increasingly high-order of accuracy. Journal of Computational Physics, 2000, 160: 405-452.

[2] R. Biswas, K. D. Devine, J. E. Flaherty. Adaptive finite element methods for conservationlaws. Applied Numerical Mathematics, 1994, 14: 255-283.

[3] A. Burbeau, P. Sagaut, C.H. Bruneau. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. Journal of Computational Physics, 2001, 169: 111-150.

[4] B. Cockburn, S. Hou, C.-W. Shu. The Runge-Kutta local projection discontinuous Galerkinfinite element method for conservation laws V: the multidimensional case. Mathematics ofComputation, 1990, 54: 545-581.

[5] B. Cockburn, S.-Y. Lin, C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkinfinite element for conservation laws III: one dimensional systems. Journal of ComputationalPhysics, 1989, 84: 90-113.

[6] B. Cockburn, C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finiteelement method for conservation laws II: general framework. Mathematics of Computation,1989, 52: 411-435.

[7] B. Cockburn, C.-W. Shu. The Runge-Kutta discontinuous Galerkin finite element method forconservation laws IV: multidimensional systems. Journal of Computational Physics, 1998,141: 199-224.

[8] G.A. Gerolymos, D. Snchal, I. Vallet. Very-high-order WENO schemes. Journal of Compu-tational Physics, 2009, 228: 8481-8524.

[9] A. Harten. Adaptive multiresolution schemes for shock computations. Journal of Compu-tational Physics, 1994, 115: 319-338.

[10] A. Harten, B. Engquist, S. Osher, S. Chakravathy. Uniformly high order accurate essentiallynon-oscillatory schemes III. Journal of Computational Physics, 1987, 71: 231-303.

[11] D.J. Hill, D.I. Pullin. Hybrid tuned center-difference-WENO method for large eddy simula-tions in the presence of strong shocks. Journal of Computational Physics, 2004, 194: 435-450.

[12] G. Jiang, C.-W. Shu. Efficient implementation of weighted ENO schemes. Journal of Com-putational Physics, 1996, 126: 202-228.

[13] L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, J. Flaherty. Shock detection and lim-iting with discontinuous Galerkin methods for hyperbolic conservation laws. Applied Nu-merical Mathematics, 2004, 48: 323-338.

[14] P. D. Lax, B. Wendroff. Systems of conservation laws. Communications on Pure and AppliedMathematics, 1960, 13: 217-237.

[15] G. Li, J. Qiu. Hybrid weighted essentially non-oscillatory schemes with different indicators.Journal of Computational Physics, 2010, 229: 8105-8129.

[16] X.D. Liu, S. Osher, T. Chan. Weighted essentially non-oscillatory schemes. Journal of Com-putational Physics, 1994, 115: 200-212.

[17] S. Pirozzoli. Conservative hybrid compact-WENO schemes for shock-turbulence interac-tion. Journal of Computational Physics, 2002, 178: 81-117.

[18] J. Qiu, C.-W. Shu. On the construction, comparison, and local characteristic decompositionfor high order central WENO schemes. Journal of Computational Physics, 2002, 183: 187-209.

[19] J. Qiu, C.-W. Shu. Finite difference WENO schemes with Lax-Wendroff type time discretiza-tions. SIAM Journal of Scientific Computing, 2003, 24: 2185-2198.


[20] J. Qiu, C.-W. Shu. A comparison of troubled-cell indicators for Runge-Kutta discontinu-ous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM Journal ofScientific Computing, 2005, 27: 995-1013.

[21] J. Qiu, C.-W. Shu. A comparison of troubled-cell indicators for Runge-Kutta discontinu-ous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM Journal ofScientific Computing, 2005, 27: 995-1013.

[22] W.J. Rider, L.G. Margolin. Simple modifications of monotonicity-preserving limiters. Journalof Computational Physics, 2001, 174: 473-488.

[23] C.-W. Shu. Total-variation-diminishing time discretizations. SIAM Journal on Scientific &Statistical Computing, 1988, 9: 1073-1084.

[24] C.-W. Shu, S. Osher. Efficient implementation of essentially non-oscillatory shock capturingschemes. Mathematics of Computation, 1988, 77: 439-471.

[25] A. Suresh, H.T. Huynh. Accurate monotonicity-preserving schemes with Runge-Kutta timestepping. Journal of Computational Physics, 1997, 136: 83-99.

[26] P. Woodward, P. Colella. The numerical simulation of two-dimensional fluid flow withstrong shocks. Journal of Computational Physics, 1984, 54: 115-173.

[27] Z. Xu, C.-W. Shu. Anti-diffusive flux corrections for high order finite difference WENOschemes. Journal of Computational Physics, 2005, 205: 458-485.

[28] H. Zhu, J. Qiu. Adaptive runge-kutta discontinuous galerkin methods using different indi-cators: One-dimensional case. Journal of Computational Physics, 2009, 228: 6957-6976.

Hybrid WENO Schemes with Lax-Wendroff Type Time...

Documents

Transcript of Hybrid WENO Schemes with Lax-Wendroff Type Time...