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Journal of Computational Physics 230 (2011) 7775–7795

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Journal of Computational Physics

journal homepage: www.elsevier .com/locate / jcp

The multi-dimensional limiters for solving hyperbolic conservation lawson unstructured grids

Wanai Li a, Yu-Xin Ren a,⇑, Guodong Lei a, Hong Luo b

a Department of Engineering Mechanics, Tsinghua University, Beijing 100084, Chinab Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, United States

a r t i c l e i n f o

Article history:Received 31 October 2010Received in revised form 16 June 2011Accepted 17 June 2011Available online 1 July 2011

Keywords:Weighted biased averaging procedureShock capturingUnstructured grids

0021-9991/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.jcp.2011.06.018

⇑ Corresponding author.E-mail address: [email protected] (Y.-X. Ren)

a b s t r a c t

Novel limiters based on the weighted average procedure are developed for finite volumemethods solving multi-dimensional hyperbolic conservation laws on unstructured grids.The development of these limiters is inspired by the biased averaging procedure of Choiand Liu [10]. The remarkable features of the present limiters are the new biased functionsand the weighted average procedure, which enable the present limiter to capture strongshock waves and achieve excellent convergence for steady state computations. The mech-anism of the developed limiters for eliminating spurious oscillations in the vicinity of dis-continuities is revealed by studying the asymptotic behavior of the limiters. Numericalexperiments for a variety of test cases are presented to demonstrate the superior perfor-mance of the proposed limiters.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

One of the key elements in high resolution Godunov type schemes for solving the hyperbolic conservation laws is thenonlinear limiting procedure that is capable of suppressing non-physical oscillations near discontinuities while achievinghigh order accuracy in smooth regions of the solution. In the present paper, our effort is focused on the development ofthe limiters for the numerical methods on unstructured grids because of their industrial importance in offering the geomet-ric flexibility in the simulation of complex flow around realistic multi-dimensional configurations.

Remarkable progresses have been made in designing the limiting procedures for solving conservation laws. One of themost successful techniques within the framework of the finite volume methods is the MUSCL approach [32] in which a slopelimiter is adopted to control the numerical oscillations near the discontinuities. The limiters of the MUSCL schemes are usu-ally designed to satisfy the TVD conditions [14,31]. The TVD-MUSCL approach performs very well in one dimensional case.However, the TVD conditions have been proved to be too restrictive for two and three-dimensional problems and lead toschemes to be only first order accurate [13]. Nevertheless, the TVD-MUSCL schemes can be formally extended to multi-dimensional structured grid cases by applying the limiter dimension by dimension. This approach is widely used in manyflow solvers on structured grids, e.g. [28]. It is well known that a TVD discretization inevitably degrades to the first orderat local extrema [14]. To overcome this deficiency, many numerical schemes have been developed. Among them the ENO/WENO schemes [15,22,19] which posses the TVB property have been extensively studied. Many other methods which aredesigned by relaxing the TVD conditions include the monotonicity-preserving schemes [30], the multi-dimensional limitingprocess [26], the double-logarithmic reconstruction [1,9] and the high order accurate maximum-principle (MP) satisfyingschemes [37].

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7776 W. Li et al. / Journal of Computational Physics 230 (2011) 7775–7795

The limiting procedures for unstructured grids are more difficult than those for structured grids. Many of the above-men-tioned methods cannot be applied directly on unstructured grids. There are a few exceptions. For example, the ENO/WENOschemes can be extended to unstructured grids [2,12,17], but the implementation is very complicated due to the needs ofidentifying several candidate stencils and performing a reconstruction on each stencil. Currently many of the limiting pro-cedures on unstructured grids are based on certain MPs. Among them the limiters of Barth and Jespersen [5] and Venkata-krishnan [33] are widely used. Some recent developments in this line can be found in [18,8,24,26]. Other popular limitersthat are designed for the high order finite volume schemes using the k-exact reconstructions [4], the DG schemes [11]and the SV/SD schemes [34] include the TVB limiters [11], the Hermite WENO-based limiter [23] and the moment limiters[11,7,36]. The design of efficient, effective, and robust limiters is still one of the bottlenecks for high order numerical meth-ods on unstructured grids.

In this paper, a novel limiting procedure for the finite volume schemes on unstructured grids is developed. The proposedlimiting procedure is inspired by the biased averaging procedure (BAP) of Choi and Liu [10]. BAP has a number of attractiveproperties: very simple, efficient, parameter free, differentiable and applicable on unstructured grids. However, this limitingprocedure has not been widely used since it was introduced. This is mainly because its mechanism to suppress the numericaloscillations in the vicinity of the discontinuities is not well understood. Furthermore, this procedure was used in conjunctionwith the reconstruction of the upwind fluxes obtained by some flux splitting techniques. It was found that this procedurewas sometimes not sufficient to control the oscillations near the discontinuities when it was used in the MUSCL type finitevolume schemes [38]. Therefore, a further understanding of and improvements to this procedure are needed.

To improve the BAP, a weighted biased averaging procedure (WBAP) is proposed. There are several fundamental differ-ences between WBAP and BAP. Firstly, a new biased function with a free parameter e is introduced. This parameter can beused to control dissipation property of the biased functions. Secondly, the nonlinear weight functions are added in the biasedaveraging procedure to enhance its robustness in capturing strong shock waves. Thirdly, the WBAP is applied to the scaleddifferences or gradients of the flow variables, which makes the procedure preserve self-similarity and behave exactly as theslope limiter in the MUSCL approach.

The WBAP keeps the merits of BAP while being more robust in capturing strong shock waves. Furthermore, for one-dimensional case, it is proved that two variants of the WBAP limiters satisfy the TVD condition or the MP respectively whenthe free parameter e ? 0, which explains why the WBAP limiters are capable of suppressing the numerical oscillations nearthe discontinuities. The main purposes of this paper are to study the behavior of the WBAP limiters and to implement thelimiters in the finite volume schemes on the unstructured grids. For brevity, only the second order finite volume scheme isconsidered in the present paper. The application of the WBAP limiters in high order finite volume, DG and SV schemes will bediscussed in future papers.

The remainder of this paper is organized as follows. The WBAP limiters are introduced in Section 2 and its mechanism forcontrolling the numerical oscillations is presented in Section 3. In Section 4, the WBAP is generalized to multi-dimensionalunstructured grids. In Section 5 we provide extensive numerical examples to demonstrate the performance of the new lim-iter. Finally, conclusion remarks are given in Section 6.

2. The limiters based on WBAP

In this section, the limiters based on the WBAP will be presented. For the purpose of comparison, the limiting proceduresof TVD-MUSCL and BAP will also be briefly introduced. For simplicity, we consider the initial value problem of the scalar con-servation law,

ut ¼ �f ðuÞxuðx; t ¼ 0Þ ¼ u0ðxÞ

ð1Þ

where u0(x) is either a piecewise smooth function with compact support or a periodic function. On a uniform grid, integrat-

ing Eq. (1) over the control volume Ci ¼ xi�12; xiþ1

2

h i, we obtain the standard semi-discrete finite volume scheme

ddt

�ui ¼1Dx

F u�i�1=2;uþi�1=2

� �� F u�iþ1=2; u

þiþ1=2

� �h ið2Þ

where �ui is the cell average of u over Ci; F u�i�1=2;uþi�1=2

� �is the numerical flux computed by a certain Riemann solver or flux

splitting technique, and the left and right states in Eq. (2) are evaluated by applying the reconstruction technique. Here weonly consider the spatially second order finite volume scheme for which the linear reconstruction on Ci yields

uþi�1=2 ¼ �ui �Dx2

rLi

u�iþ1=2 ¼ �ui þDx2

rRi

ð3Þ

where rLi and rR

i are the reconstructed gradients or slopes. In order to suppress the numerical oscillations near the discon-tinuities, a limiting procedure has to be applied on the slopes.

W. Li et al. / Journal of Computational Physics 230 (2011) 7775–7795 7777

One of the most successful limiting procedures within the framework of the finite volume methods is the TVD-MUSCLapproach, in which the limited slope is computed by

rLi ¼

di�1=2

DxuðhiÞ

rRi ¼

diþ1=2

Dxu

1hi

� � ð4Þ

where diþ1=2 ¼ �uiþ1 � �ui; hi ¼diþ1=2

di�1=2, and u (hi) is the limiter of the TVD scheme. Sweby [31] defined a class of limiters for the

second order TVD schemes satisfying

maxð0;minðhi;1ÞÞ 6 uðhiÞ 6 maxð0;minð2hi;1Þ;minðhi;2ÞÞ: ð5Þ

Note that u(hi) is usually asked to have a symmetry of

uðhiÞhi¼ u

1hi

� �

so that the slope is uniquely determined independent of the direction of upwinding, i.e.

rLi ¼ rR

i :

The TVD-MUSCL discretization captures the discontinuities in very high resolution but is difficult to be applied directly onunstructured grids.

The BAP of Choi and Liu [10] is a limiting procedure that can suppress the oscillation near the discontinuity and can beapplied on unstructured grids. Using the BAP, the limited slopes in (3) can be written as

rLi ¼ rR

i ¼1Dx

Lðdiþ1=2; di�1=2Þ; ð6Þ

where L is defined by

Lða0; a1Þ ¼ B�1 Bða0Þ þ Bða1Þ2

� �: ð7Þ

The function B is called the biased function. There are many possible biased functions, and one example is

BðxÞ ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2p ; B�1ðxÞ ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� x2p :

In [10], this procedure performed very well when used in conjunction with the reconstruction of the upwind fluxes ob-tained by some flux splitting techniques. However, the numerical test cases in [38] show that it is sometimes not sufficient tocontrol the oscillations near the discontinuities when used in the MUSCL type finite volume schemes. Moreover, its mech-anism to suppress the numerical oscillations is not well understood.

To overcome these deficiencies, a new limiting procedure, namely the WBAP, will be proposed in this section. In the con-text of one-dimensional finite volume scheme, we present two versions of the WBAP:

Version 1:

rL ¼ Ldi�1=2

Dx;diþ1=2

Dx

� �

rR ¼ Ldiþ1=2

Dx;di�1=2

Dx

� �;

ð8Þ

Version 2:

rL ¼ rR ¼ Lð�ri; �riþ1; �ri�1Þ ð9Þ

where �ri ¼ 12Dx ðdiþ1=2 þ di�1=2Þ.

In Eqs. (8) and (9), L is the limiting function of WBAP. Before detailing the form of L, the difference between Version 1 andVersion 2 will be discussed first. Version 1 is similar in form to the TVD-MUSCL approach (Eq. (4)) and the BAP (Eq. (6)) inwhich the left and right gradients (or slopes) are computed and used as the input of the limiting function. Using this approach,it is possible to study the behavior of L by comparing with the limiters of the TVD-MUSCL approach. However, Version 1 isdifficult to be applied directly on unstructured grids and does not satisfy the condition of symmetry. In Version 2, the gradi-ents are computed by central difference which is identical to the gradients computed by the least-squares method on unstruc-tured grids. Moreover, the arguments of L in Eq. (9) are the gradients of current cell and its immediate neighbors, which makeit possible to generalize Version 2 WBAP to unstructured grids as long as the function L accepts multiple arguments.

In the present paper, L is in the following form

Lða0; a1; a2; . . . ; aJÞ ¼ a0 �W 1;a1

a0;a2

a0; . . . ;

aJ

a0

� �ð10Þ


where a0 is the first argument of L. The function W is called the WBAP limiter which is functionally identical to the limiters ofthe TVD-MUSCL schemes and is defined by

Wðh0; h1; h2; . . . ; hJÞ ¼ B�1XJ

j¼0

xjBðhjÞ" #

ð11Þ

where hj = aj/a0 (h0 = 1), B is the biased function and xj, j = 1, . . . , J are the weights. In the present paper, we consider the fol-lowing biased function and weight function which are respectively

BðxÞ ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 þ x2p ; B�1ðxÞ ¼ exffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� x2p ð12Þ

and

xj ¼ajPJj¼0aj

; aj ¼kj

BðhjÞ2 þ d: ð13Þ

In Eq. (12), e is a free parameter whose value range will be discussed in the next section. In Eq. (13), d is a small positivenumber to avoid possible division by zero, which is chosen as d = 10�10, and kj is determined by

kj ¼n; if j ¼ 01; else

If n = 1, the weights in Eq. (13) are solely determined by the smoothness indicator. If n P 1, the contribution of the centralcell is emphasized. At present time, it is worthwhile to introduce some simple properties of WBAP in terms of the followingremarks.

Remark 1. Because of the specific form of L in Eq. (10), the WBAP is self-similar, i.e. it is invariant when the spatial and timevariables are scaled by the same factor. Furthermore, the function W acts exactly as the limiters of the TVD-MUSCL approach.

Remark 2. Before applying the inverse of the biased function in Eq. (11), the B(aj) are weighted averaged. The biased func-tions of WBAP are also different with those of BAP. By the introduction of the parameter e, the asymptotic behavior of WBAPcan be studied to reveal its mechanism for suppressing the numerical oscillations, which will be discussed in the nextsection.

Remark 3. The WBAP is continuously differentiable. The non-differentiable limiters are sensitive to small perturbations andmay cause slow convergence, while the differentiable one enables a smooth transition from the high order solution insmooth region to the first order solution in the vicinity of discontinuity.

Remark 4. When the solution is smooth, it is easy to prove that

u�iþ1=2 ¼ uexactiþ1=2 þ eðxiþ1=2ÞDx2 þ OðDx3Þ ð14Þ

and e(x) is Lipschitz continuous. Following the analysis of Harten et al. [15], Eq. (14) guarantees the second order accuracy ofthe scheme. At the local extrema e(x) is not Lipschitz continuous and the scheme will reduce to first order in term of L1norm. We note that a simple technique presented in [9] can be used to overcome this drawback. Furthermore, when the lim-iter based on WBAP is used in rth order reconstruction, the resulting scheme will be rth order except near the roots of higherorder derivative of u where the scheme may locally reduce to (r � 1)th order accurate in the L1 norm. This property will befurther studied in a future paper where high order reconstruction will be used.

3. The mechanism for controlling the numerical oscillations

In this section, the mechanism of WBAP for suppressing the numerical oscillations is discussed. We first consider the Ver-sion 1 of WBAP. In this case, W in Eq. (8) behaves exactly the same as the limiter in the TVD-MUSCL approach.

In the WBAP limiter, eis used to adjust the dissipation of the limiter. We note that as e ? 0, the limit of W(1,hi) is

lime!0

Wð1; hiÞ ¼max 0;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ nÞ

phiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ nh2i

q0B@

1CA ð15Þ

for the biased function presented in Eq. (12). It is easy to prove that when n 6 3, lime?0 W(1,hi) lies in the TVD region ofSweby with the superbee and minmod limiters as its upper and lower bounds (Eq. (5)). Therefore, for some values of n,W(1,hi) is a limiter with asymptotically TVD property. This behavior is shown in Fig. 1. In this figure, it is also shown that

Fig. 1. The behavior of the WBAP.


even if e is chosen as a finite value, for example e = 1, W(1,hi) is also inside or almost inside the TVD region when hi P 0.Furthermore, for a finite value of e, W(1,hi) is always smooth. The asymptotically TVD behavior of W(1,hi) explains whythe WBAP is capable of removing the numerical oscillations near the discontinuities at least for the one dimensional case.

For the Version 2 of WBAP, one of its remarkable advantages is that it can be straightforwardly applied to multi-dimen-sional unstructured grid cases. Therefore, its shock capturing mechanism will be studied in more detail. We note that theasymptotic behavior of the Version 2 WBAP can be also studied. To be general, we consider Eq. (11) with arbitrary numberof arguments. When e ? 0, we have

WL1ð1; h1; . . . ; hJÞ ¼ lime!0

Wð1; h1; . . . ; hJÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþJ

nþPJ

k¼11=h2

k

r; if h1; . . . ; hJ > 0

0; otherwise:

8<: ð16Þ

Eq. (16) is a new limiter denoted as WBAP-L1 hereafter which can be used to replace W in Eq. (10). Comparing with Eq. (15),WBAP-L1 can be considered as a logical generalization of the TVD limiter to multiple arguments cases. The Version 2 WBAPprovides new TVD-like limiters that can be applied to multi-dimensional problems. And furthermore, in what follows, wewill prove that for one-dimensional case, WBAP-L1 satisfies a certain MP.

Lemma 1. Assume WBAP-L1 limiter is used in the limiting procedure of Eqs. (9) and (10) . If n 6 14, then

uþi�1=2; u�iþ1=2 2 ½m;M�

where m and M are the minimum and maximum values of �uni�2; �u

ni�1; �u

ni ; �u

niþ1; �u

niþ2

�.

Proof. The use of WL1 in Eqs. (9) and (10) yields the following limited linear reconstruction

uiðxÞ ¼ �ui þWL1 1;�ri�1

�ri;�riþ1

�ri

� �� ri � ðx� xiÞ ¼ �ui þWL1 1;

di�1

di;diþ1

di

� �� di

2Dx� ðx� xiÞ

where di ¼ 2Dx � �ri ¼ �uiþ1 � �ui�1. And the left and right states are given by

u�iþ1=2 ¼ �ui þWL1 1;di�1

di;diþ1

di

� �� di

4

uþi�1=2 ¼ �ui �WL1 1;di�1

di;diþ1

di

� �� di

4

ð17Þ

According to Eq. (16), the specific form of WL1 is

WL1 1;di�1

di;diþ1

di

� �� di

4¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnþ2Þ=16

nd2

i

þ 1d2

i�1

þ 1d2

iþ1

r; if di; diþ1 and di�1 > 0

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnþ2Þ=16

nd2

i

þ 1d2

i�1

þ 1d2iþ1

r; if di; diþ1 and di�1 < 0

0; otherwise

8>>>>><>>>>>:

ð18Þ

We firstly consider the case di, di+1 and di�1 > 0, which is corresponding to the first branch of Eq. (18). In this case it istrivial to show that

u�iþ1=2 P �ui P m; uþi�1=2 6�ui 6 M: ð19Þ


Furthermore, if n 6 14, we have

WL1 1;di�1

di;diþ1

di

� �� di

46

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

nd2

iþ 1

d2i�1þ 1

d2iþ1

vuut 6 diþ1 ð20Þ

and

WL1 1;di�1

di;diþ1

di

� �� di

46

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

nd2

iþ 1

d2i�1þ 1

d2iþ1

vuut 6 di�1 ð21Þ

In terms of Eq. (17),

u�iþ1=2 ¼ �ui þWL1 1;di�1

di;diþ1

di

� �� di

46 �ui þ diþ1 ¼ �uiþ2 6 M ð22Þ

and

uþi�1=2 ¼ �ui �WL1 1;di�1

di;diþ1

di

� �� di

4P �ui � di�1 ¼ �ui�2 P m ð23Þ

According to Eqs. (19), (22) and (23), we have

uþi�1=2; u�iþ1=2 2 ½m;M�

For the case di, di+1 and di�1 < 0 which is corresponding to the second branch of Eq. (18), the proof can be done in a similarmanner. If di, di+1 and di�1 are not of the same sign, it is straightforward to show that m 6 uþi�1=2 ¼ u�iþ1=2 ¼ �ui 6 M. h

Theorem 1. For the discrete finite volume scheme Eq. (2) with a monotone Lipschitz continuous flux function, if Eq. (16) is used inthe limiting procedure of Eqs. (9) and (10) and n 6 14, then the scheme ensures the maximum principle, i.e.

�unþ1i 2 ½m;M�

under the CFL condition, where m and M are the minimum and maximum values of �uni�3; �u

ni�2; �u

ni�1; �u

ni ; �u

niþ1; �u

niþ2; �u

niþ3

� .

Proof. Using Lemma 1, one can obtain

u�i�1=2; uþi�1=2; u�iþ1=2; uþiþ1=2 2 ½m;M�

The application of the Theorem 2.2 of [37] to the case of linear reconstruction leads to �unþ1i 2 ½m;M� under the CFL condi-

tion. h

The theorem of the present paper shows that the WBAP limiter asymptotically satisfies the MP when e ? 0. This propertyensures the shock capturing capability of the present limiter. Similar to the MLP [26], the MP satisfied by the WBAP-L1 in-volve a larger stencil than the MP corresponding to the Barth limiter [5]. Therefore, the present limiter is less restrictive inthe admissible range of the limited gradient. Numerical results in Section 5 show that the present limiter can capture themulti-dimensional flow features better than Barth limiter and is less sensitive to the computational grids. We note thatthe WBAP-L1 limiter is considerably simpler than the full WBAP and can be used as an independent limiter. Its performancewill be further studied in Section 5.

Next, the influence of e in Eq. (12) will be studied. We have already shown that, when e ? 0, the resulting WBAP-L1 lim-iter satisfies the MP. It is interesting to note that when e ?1, the WBAP also has a limit in the following form,

WL2ð1; h1; . . . ; hJÞ ¼ lime!þ1

Wð1; h1; . . . ; hJÞ ¼nþ

PJk¼11=hk

nþPJ

k¼11=h2k

ð24Þ

which is termed as the WBAP-L2 limiter. Although it has not been proved that WL2 satisfies the MP, extensive numerical testsin Section 5 show that WL2 can also produce essentially oscillation-free numerical results. In fact, if the WBAP-L2 limiter ismodified as

WL2ð1; h1; . . . ; hJÞ ¼nþPJ

k¼11=hk

nþPJ

k¼11=h2

k

; if h1; . . . ; hJ > 0

0; otherwise

8><>:

the modified WBAP-L2 limiter is also MP-satisfying. Furthermore, for the case of h1, . . . ,hJ P 0, we can prove that

WL1ð1; h1; . . . ; hJÞP WL2ð1; h1; . . . ; hJÞ ð25Þ

Fig. 2. The influence of e in WBAP, n = 5.


by using the Cauchy–Schwarz inequality, which means that WL2 is in fact more dissipative than WL1 for the monotonenumerical solutions. We consider a special case h = h1 = � � � = hJ, the influence of e can be shown clearly in Fig. 2 in termsof the W � h curves. According to this figure, the WBAP-L1 limiter is the least dissipative while the WBAP-L2 limiter isthe most dissipative for h P 0. For other e, the dissipation of WBAP lies basically in-between WBAP-L1 and WBAP-L2. Theresults suggest that e can be chosen arbitrary from infinite small to infinite large, and a larger value of e will result in a moredissipative WBAP limiter. Because the two limiters, WBAP-L1 and WBAP-L2 are computationally more efficient than WBAP,they will be applied in most of the test cases of the present paper.

Remark 5. The numerical experiments in Section 5 indicate that in all test cases the WBAP-L1 and WBAP-L2 perform verywell and there is no need to use other WBAP limiters in practice. However, the WBAP provides a general method to designlimiter for unstructured grid simulation. In fact, we can choose biased functions and weights which are different with thoseof the present paper. It is possible that these WBAP limiters have better performance, or can be asymptotically reduced tonew simpler limiters.

4. The WBAP limiter on 2D unstructured grids

In this section, the WBAP for unstructured grids will be presented. Without loss of generality, we consider the implemen-tation of WBAP on the triangular grids. The application of WBAP on other types of grids, e.g. the quadrilateral grids, is similar.

To be specific, we consider the 2D Euler equations. Before applying the WBAP, we first reconstruct a linear function oneach cell from the cell-averaged values of the conservative variables. On cell i, the reconstruction polynomial is:

ui ~x�~xið Þ ¼ �ui þrui � ~x�~xið Þ ð26Þ

where u is any one of the components of the state variable vector U ¼ ðq;qu;qv ;qEÞ; �ui is the cell average of the conserva-tive variable u on cell i and~xi is the centroid of cell i. The gradient rui in Eq. (26) are unknown parameters which are com-puted using the least-squares method [4,25]. For the linear reconstruction, it is sufficient to choose cell i and its immediateneighbors labeled as Si = {j1, j2, j3} as the reconstruction stencil which is shown in Fig. 3. It is expected that Eq. (26) conservesthe cell average value on every cell of the stencils, which yields

Fig. 3. Reconstruction and limiting stencils for cell i. Si = {j1, j2, j3}.

Fig. 4. Reconstruction and limiting stencils for boundary cells. Reconstruction stencil: {j1, j2,k1,k2,k3,k4,k5}; limiting stencil: {j1, j2}.


1Ci

ZCi

uið~x�~xiÞdxdy ¼ �ui ð27Þ

1Cj

ZCj

uið~x�~xiÞdxdy ¼ �uj; j 2 Si ð28Þ

where Ci stands for the area of cell i. It can be seen the Eq. (26) satisfies Eq. (27) automatically and Eq. (28) constitutes a set ofover-determined linear equation systems which can be solved by the least-squares method. In practice, the geometricweights wij are used to specify the relative importance of the control volumes in the stencil, and Eq. (28) is modified as

wij

Cj

ZCj

ui ~x�~xið Þdxdy ¼ wij�uj; j 2 Si

where the weights are chosen as wij ¼ 1= ~xj �~xi

�� .For the cells sharing one side with the boundary, the use of only the immediate neighbors is not sufficient to overcome

the stiffness in the reconstruction. Therefore, for such cells, we choose all cells that share at least one vertex with the presentcell as the reconstruction stencil, which is shown in Fig. 4.

Next, the Version 2 WBAP is applied to the reconstructed gradients using a component by component approach. ru canbe written as

ru ¼ ðu1;u2Þ ¼ @u@x;@u@y

� �

and the WBAP is applied to each component uk, k = 1,2 separately. The implementation of WBAP on unstructured grids isstraightforward. One can directly use Eq. (10) to compute the limited value of uk on cell i and the result is

~uki ¼ L uk

i ;ukj ; . . .

� �; j 2 Si; k ¼ 1;2 ð29Þ

We would like to emphasize that only the gradients of the current cell and its immediate neighbors are included in the lim-iting procedure even for the boundary cells. We also note that WBAP limiter in Eq. (10) can also be replaced by the WBAP-L1limiter in Eq. (16) or the WBAP-L2 limiter in Eq. (24).

After the limiting process, the reconstruction polynomial can be modified as

uið~x�~xiÞ ¼ �ui þr~ui � ~x�~xið Þ

where r~ui ¼ ~u1i ; ~u

2i

� �is the limited gradient.

We now give some remarks concerning the properties and implementations of the WBAP for the reconstruction onunstructured grids.

Remark 6. The WBAP limiter can be applied directly to the reconstruction in terms of the conservative variables. Numericalexperiments in Section 5 indicate that this approach produces nearly oscillation-free numerical solutions without relying onthe characteristic variables. This is a very favorable feature which makes the present approach being more efficient than thecharacteristic-wise limiting procedures. Besides the conservative and characteristic variables, the primitive variables are alsofrequently used in the literature [3]. As being remarked in [3], the use of the primitive variables enables a better control ofthe positivity of density and pressure. However, to work directly on the conservative variables is beneficial to achieve formalorder of accuracy when high order reconstruction (e.g. quadratic reconstruction) is adopted.

Remark 7. The WBAP limiter in the present paper is applied to the components of the gradients, and is therefore not rota-tionally invariant. We note that the Barth limiter [5] and the MLP limiter [26] satisfy the rotational invariance property forscalar quantities but do not for the vector valued quantities. As a matter of fact, it is extremely difficult to design a finitevolume scheme which is fully rotationally invariant if the governing equations are in terms of the Cartesian velocity com-ponents. Therefore, being not rotational invariant is not a particular drawback of the present limiting procedure. Further-more, the present approach can be also applied directly to higher order derivatives, which makes the present methodbeing able to be implemented straightforwardly in higher order reconstructions.


Remark 8. According to Eq. (29), only the gradients on cell i and its immediate neighbors are used in the limiting procedure.Therefore, the present procedure is very compact. Of course, during the reconstruction of the gradients, more cells must beinvolved, which makes the actual stencil of the limiting procedure much larger. However, since the gradients are computedand stored at a previous stage of the simulation, it will not affect the compactness of the limiting procedure. The compact-ness of the present procedure permits an optimum usage of the cache memory of the computer.

Remark 9. In one-dimensional case, we have shown that n 6 14 is sufficient to ensure the WBAP-L1 limiter to satisfy theMP. For the multidimensional cases, there has not been theoretical basis for choosing the value of n. But as has been men-tioned in the previous section, a larger n will emphasize more the contribution of the central cell (when n ?1, the limitedgradients will be the same as the unlimited ones). Numerical experiments in Section 5 indicate that n 6 5 is sufficient to cap-ture the discontinuities involved in the test cases.

Some standard approaches are adopted to implement the cell centered finite volume scheme for solving the 2D Eulerequations on unstructured grids. These approaches are well-known and mentioned briefly here for completeness.

(1) The reconstructed polynomials are used to compute left and right states of the conservative variables at the mid-pointof every cell interface.

(2) The numerical fluxes are computed using a certain Riemann solver or flux splitting technique. In this paper, if it is notmentioned explicitly, a modified Roe’s Riemann solver is adopted, which is the standard Riemann solver [27] using theLax–Friedrich flux as the entropy fix. Specifically, the absolute value of each eigenvalue appeared in the formulation ofRoe’s Riemann solver is modified as:

j~kij ¼max

4

i¼1ðjkijÞ; if 8jkij < bc; i ¼ 1; . . . ;4

ki; otherwise

8<:

where c is the local speed of sound and b is a free parameter chosen usually between 0.01 and 0.15.
(3) A third-order explicit TVD Runge–Kutta scheme [29] is used to advance the solution of the governing equations in
time for the spatially second order scheme.

5. Numerical examples

A variety of test cases are presented in this section to assess the performance of the developed WBAP limiters. Specifically,the shock capturing capability and resolution are demonstrated using the shock tube problems, blast wave, double Machreflection and forward facing step problems. The isentropic vortex problem is solved to test the accuracy. And the steadytransonic and supersonic flows around an airfoil are computed to show the convergence to steady state of the proposed lim-iter. In the simulations, only the Version 2 WBAP limiter and its variant WBAP-L1/L2 limiters are used. For the purpose ofcomparison, some other limiters, such as Barth and Jespersen [5], Venkatakrishnan [33] and MLP [26] limiters, are also ap-plied in some test cases. The CFL number for all test cases is taken as 0.8.

5.1. Rotation of slotted cylinder

This case is the two-dimensional benchmark problem proposed by LeVeque [21] which makes it possible to asses the abil-ity of a high-resolution scheme to preserve both smooth and discontinuous profile. The problem is governed by the scalaradvection equation, i.e.

Fig. 5. Slotted cylinder rotation: initial data/exact solution.

Fig. 6. Contour comparison of the slotted cylinder at t = 2p.


ut þ~v � ru ¼ 0

with the nonuniform velocity field ~v ¼ ð0:5� y; x� 0:5Þ. This problem describes that a slotted cylinder, a sharp cone and asmooth hump undergo a counterclockwise rotation about the center of a unit square [0,1] � [0,1]. The shapes of the threebodies are depicted in Fig. 5 and further details of the initial conditions can be found in [21].

After one full revolution (t = 2p) the exact solution coincides with the initial data. The numerical solutions presented inFig. 6 are computed on non-uniform triangular mesh with grid size h = 1/160. The velocity distribution across the line x = 0.5and y = 0.75 are shown in Fig. 7, where we can observe that no numerical oscillations occur near the discontinuities for bothWBAP and MLP limiters. MLP-u1 captures the discontinuities most accurately, while the WBAP-L1 and WBAP-L2 using n = 5show a bit lower resolution than MLP-u1, but higher than MLP-u2.

5.2. Shock tube problems

This test case is chosen to test the capability of WBAP in capturing the strong shock wave and contact discontinuity. Theproblem is one-dimensional in nature, and the initial conditions are

ðq;u;pÞð~xÞ ¼ðqL; uL;pLÞ if x 6 0:5ðqR; uR; pRÞ if x > 0:5

We consider two sets of initial conditions given by

Case 1:

ðqL;uL;pLÞ ¼ ð0:445; 0:698;3:528ÞðqR;uR;pRÞ ¼ ð0:5;0; 0:571Þ

Fig. 7. The solutions along vertical and horizontal lines for the slotted cylinder at t = 2p.

Fig. 8. Comparison of density distributions in one dimension.


Case 2:

ðqL;uL;pLÞ ¼ ð1;0;1000ÞðqR;uR;pRÞ ¼ ð1;0; 0:01Þ

where case 1 is the Harten–Lax problem [20] and case 2 consists of the left and middle states of the blast wave problem ofWoodward and Collella [35]. Although the problems are one-dimensional, they are computed on both one- and two-dimen-sional grids. Three types of grids are used, the first one is the one-dimensional uniform grids, the second one is the two-dimensional uniform grids, and the third one is the two-dimensional grid with large aspect ratio.

We firstly compute these problems on one-dimensional uniform grid and the grid number is 200. The effect of e is studiedby using WBAP-L1(e ? 0), WBAP-L2(e ?1) and WBAP with e = 1. n = 14 is used in all limiters mentioned above. The prob-lems are also computed using the Barth limiter. The numerical results are shown in Fig. 8. It shows that the numerical resultsare not sensitive to e and a smaller e leads to a less diffusive result. When comparing to the results of the Barth limiter, it isevident that the present limiters are more dissipative than Barth limiter. This is understandable since the Barth limiter isvery compressive in one-dimensional cases [6].

Secondly, we compute the test cases on a two-dimensional uniform triangular grid which is created by dividing the200 � 40 square grid as shown in Fig. 9. Fig. 10 shows the density distributions, which indicates that both WBAP-L1 and

Fig. 9. Uniform grids in two dimension.

Fig. 10. Comparison of density distributions in two dimension.

Fig. 11. Comparison of stencils involved in the limiting procedure in one and two dimensional cases.


WBAP-L2 limiters can capture strong shock wave and contact discontinuity in high resolution. The performance of WBAP-L1is very close to MLP-u1, and that of WBAP-L2 is slightly inferior. Comparing between Figs. 8 and 10, it is observed that res-olution to the contact discontinuity of the Barth limiter degenerates considerably on two-dimensional grid. This phenome-non is mainly due to the grid sensitivity of the Barth limiter. Referring to Fig. 11, the upper and lower bounds of the MPcorresponding to the Barth limiter are respectively

1D : M1 ¼maxð�ui�1; �ui; �uiþ1Þ; m1 ¼ minð�ui�1; �ui; �uiþ1Þ2D : M2 ¼maxð�ui; �uj1 ; �uj2 ; �uj3 Þ; m2 ¼minð�ui; �uj1 ; �uj2 ; �uj3 Þ

for one and two-dimensional grids. Since the solution is uniform in y-direction, Fig. 11 suggests for a monotone solution,�uiþ1 > �uj3 ; �ui�1 > �uj2 , so that M1 P M2 and m1 6m2. Therefore, the allowable range of the gradient after applying the Barthlimiter in two-dimensional case is smaller than that of the one-dimensional case. As a result, the two-dimensional Barth lim-iter is more diffusive. This also explains why the MLP limiter which uses a larger stencil performs better than the Barth lim-iter in two-dimensional case. Whereas, for the WBAP limiter,

/1 ¼W 1;di�1

di;diþ1

di

� �

/2 ¼W 1;@uj1=@x@ui=@x

;@uj2=@x@ui=@x

;@uj3=@x@ui=@x

� �

Fig. 12. Uniform stretched triangle grids for two-dimensional test for large aspect ratio: R = 50 (left); R = 1/50 (right).

Fig. 13. Comparison of density distribution for Harten–Lax shock tube problem on two-dimensional stretched grids.

Fig. 14. Density distribution of blast wave at t = 0.38.


Because W is a special averaging operator, it is less sensitive to the grid. Comparing between Figs. 8 and 10, we note that thebetter resolution of WBAP achieved in two-dimensional case is mainly due to the fact that the grid number in x-direction forthe two-dimensional case is doubled comparing to the one-dimensional case.

Thirdly, we test the performance of the WBAP limiters on two-dimensional stretched grids with large aspect ratio. Twogrids used in the tests are generated by dividing two 400 � 40 rectangular grids, for which the aspect ratios are 50 and 1/50,

Fig. 15. Two sets of grids for accuracy test.

Table 1Accuracy test for isentropic vortex at t = 2.0 on regular grids, h = 10.

Case Grid L1 error Order L1 error Order

Unlimited h/20 1.66E�03 3.49E�02h/40 3.76E�04 2.14 8.09E�03 2.11h/80 8.34E�05 2.17 1.70E�03 2.25h/160 2.09E�05 1.99 4.41E�04 1.95h/320 5.27E�06 1.99 1.25E�04 1.81

Barth h/20 3.69E�03 1.11E�01h/40 1.61E�03 1.20 5.05E�02 1.14h/80 6.65E�04 1.27 2.61E�02 0.95h/160 3.32E�04 1.00 1.29E�02 1.02h/320 1.71E�04 0.96 7.51E�03 0.78

MLP-u1 h/20 1.81E�03 4.48E�02h/40 4.38E�04 2.05 1.39E�02 1.69h/80 8.53E�05 2.36 2.50E�03 2.48h/160 2.10E�05 2.02 5.26E�04 2.25h/320 5.27E�06 1.99 1.49E�04 1.82

WBAP-L1, n = 1 h/20 2.91E�03 7.93E�02h/40 6.55E�04 2.15 2.15E�02 1.88h/80 1.23E�04 2.41 4.44E�03 2.27h/160 2.48E�05 2.31 1.02E�03 2.12h/320 5.20E�06 2.26 2.79E�04 1.87





respectively. Typical cells of these two grids are shown in Fig. 12. The HLL approximate Riemann solver [16] is used to com-pute the fluxes. The WBAP and the Barth limiters can maintain the uniformity in y-directions while the MLP limiter cannot.


In terms of the density distributions depicted in Fig. 13, the resolution of the Barth limiter is lower and very sensitive to thegrid aspect ratio. The WBAP limiters are not very sensitive to the aspect ratio but slight oscillations can be observed in thenumerical results in the vicinity of the contact discontinuity, which however can be removed by using the characteristicvariables.

5.3. Blast wave problem

This example is to solve the interaction of two blast waves subjected to the initial conditions

Table 2Accurac

Case

Unlim

Barth

MLP

WBA

WBA

WBA

WBA

ðq;u;v ;pÞ ¼ð1;0;0;1000Þ; 0 6 x < 0:1ð1;0;0;0:01Þ; 0:1 6 x < 0:9ð1;0;0;100Þ; 0:9 6 x 6 1

8><>:

on the domain [0,1] � [0,0.1] with uniform triangle grids shown in Fig. 9, which has 401 vertices in the x-direction and 41vertices in the y-direction. Reflecting boundary conditions are imposed at x = 0 and x = 1. This problem was studied exten-sively in [35]. We solved the problem to t = 0.38 using the Barth limiter, MLP-u1 limiter and WBAP limiter. The density dis-tributions are depicted in Fig. 14. This is a severe numerical test for the performance of limiters. All limiters can capturestrong discontinuities. We notice that WBAP limiter can capture the flow structure much better than Barth limiter and veryclose to MLP-u1 limiter.

5.4. Isentropic vortex problem

The isentropic vortex transport problem [17,26] is used to examine the accuracy of the numerical scheme in computingmulti-dimensional flow without shock waves. The mean flow is q = 1, p = 1, and (u,v) = (0,0). We add, to mean flow, an isen-tropic vortex expressed by the following perturbations,

y test for isentropic vortex at t = 2.0 on irregular grids, h = 10.

Grid L1 error Order L1error Order

ited h/20 5.08E�03 0.104765h/40 1.21E�03 2.07 2.95E�02 1.83h/80 2.40E�04 2.34 7.87E�03 1.91h/160 5.02E�05 2.26 2.00E�03 1.98h/320 1.16E�05 2.11 5.15E�04 1.96

h/20 8.46E�03 2.32E�01h/40 3.83E�03 1.14 1.29E�01 0.85h/80 1.84E�03 1.06 6.25E�02 1.04h/160 9.18E�04 1.00 3.14E�02 1.00h/320 4.65E�04 0.98 1.58E�02 0.99

-u1 h/20 5.24E�03 0.13725h/40 1.31E�03 1.99 4.70E�02 1.54h/80 2.95E�04 2.16 1.15E�02 2.03h/160 5.67E�05 2.38 2.61E�03 2.14h/320 1.25E�05 2.18 5.19E�04 2.33

P-L1, n = 1 h/20 6.74E�03 1.87E�01h/40 1.82E�03 1.89 5.97E�02 1.65h/80 4.53E�04 2.01 2.09E�02 1.52h/160 9.88E�05 2.20 5.61E�03 1.89h/320 2.04E�05 2.28 1.49E�03 1.91

P-L1, n = 5 h/20 6.17E�03 0.174609h/40 1.55E�03 1.99 5.64E�02 1.63h/80 3.78E�04 2.04 1.82E�02 1.64h/160 7.67E�05 2.30 4.58E�03 1.99h/320 1.59E�05 2.27 1.13E�03 2.02



Fig. 16. Density distribution across the vortex center.

Table 3Compute time comparison with different limiters on the h = 1/160 non-uniform grids.

Limiter Barth MLP-u1 MLP-u2 WBAP-L1 WBAP-L2Time (s) 59.9 60.7 72.7 81.1 73.0

Fig. 17. Unstructured mesh (nelem = 10,382, npoint = 5306, nbound = 150) used for computing transonic and supersonic flow past a NACA0012 airfoil.


ðdu; dvÞ ¼ v2p

e0:5ð1�r2Þð��y; �xÞ

dT ¼ �ðc� 1Þv2 8cp2e1�r2

; dS ¼ 0

with �x; �yð Þ ¼ ðx� 5; y� 5Þ; r2 ¼ �x2 þ �y2, and the vortex strength v = 5.We first compute the solution upto t = 2.0 to test the accuracy of the limiter. Two types of grids, namely regular grids and

irregular grids shown in Fig. 15 are used for test. The numerical results are shown in Table 1 for the regular grid and Table 2for the irregular grid. In terms of the errors and convergence rate of the density, it is clear that the present limiter is moreaccurate than Barth limiter. From Tables 1 and 2, both WBAP limiters with n = 1 and n = 5 show second order convergence,and the limiters with n = 5 give smaller errors. It is interesting to notice that on regular grids the L1 error produced by WBAPcan be smaller than that produced by the unlimited scheme. One possible explanation for this phenomenon is that the WBAPcan reduce to linear weighted average procedure for sufficiently smooth flows with very small gradient. In this case, theWBAP may behave similar to the WENO scheme which can actually increase the accuracy by proper averaging procedure.Of course, the present results are largely by coincidence. However, it does suggest that it is possible to further improve

Fig. 18. Transonic flow past the NACA0012 airfoil.

Fig. 19. Convergence comparison in the supersonic flow with different limiters.


the WBAP by using a better weighted averaging procedure. Fig. 16 shows the distribution of density, which indicates that thepresent limiter is much less dissipative than the Barth limiter. To study the efficiency of the limiters, we audit the compu-tational time using different limiters on a h = 1/160 regular grid. According to Table 3, the computational time of the WBAP-L2 limiter is nearly the same as MLP-u2 limiter and a bit longer than that of Barth and MLP-u1 limiter. Due to the square rootoperation in Eq. (16), WBAP-L1 takes the longest time.

5.5. Transonic and supersonic flows around NACA 0012 airfoil

Transonic and supersonic steady flows about the NACA0012 airfoil are simulated to demonstrate the superior conver-gence property of the WBAP limiters for steady flow computation. The unstructured grid used in the simulation is shownin Fig. 17, which consists of 10,382 elements, 5306 grid points, and 150 boundary points.

The first case considered is the Mach 0.8 transonic flow at the angle of attach of 1.25�. The Mach number contours com-puted using the WBAP-L1 limiter are shown in Fig. 18(a). Fig. 18(b) compares the convergence histories of the numericalsolutions obtained using WBAP, Barth, Venkatakrishnan and MLP-u2 limiters. It is observed that the WBAP and Venkata-

Fig. 20. Supersonic flow past the NACA0012 airfoil.

Fig. 21. Comparison of density contours for double Mach reflection. Thirty equally spaced contour lines from q = 1.7 to 21.3.


krishnan limiters can converge to machine zero, while for Barth and MLP-u2 limiter, the residual stalls after a drop of aboutthree orders of magnitude. Fig. 19 shows the pressure coefficient distributions on the surface of the airfoil obtained using theBarth limiter and the present WBAP limiter. As expected, both limiters are able to capture the shock wave without numericaloscillations. However, the solution obtained by Barth limiter exhibits some small wiggles in the smooth region.

The second case is the Mach 2 supersonic flow at zero angle of attack. The Lax Riemann solver is used in this case in orderto avoid the carbuncle phenomenon in computing the supersonic flow. The Mach number contours using WBAP-L1 limiter isshown in Fig. 20(a). The history of convergence is shown Fig. 20(b), which indicates that WBAP-L2 achieves the best conver-gence followed by WBAP-L1.The converge properties of other limiters are significantly worse than the WBAP limiters.

5.6. Double Mach reflection problem

One of the popular test cases for high-resolution schemes is the double Mach reflection problem [35]. The whole compu-tational domain is [0,4] � [0,1]. The wall is located at the bottom of the computational domain beginning at x = 1/6. Initially,a right-moving shock with Ma = 10 is located at x = 1/6, y = 0, inclined 60� with respect to the x-axis. The computational iscarried out until t = 0.21.

In Figs. 21 and 22., the density contours computed using WBAP and Barth limiter on two set of grids are presented. Bothlimiters can capture the discontinuity without oscillations, while the resolution of the WBAP is higher especially for the sliplines. This observation is further verified in Fig. 23.

Fig. 22. Close-up view around the double Mach stem.

Fig. 23. Density distribution on line y = 0.3.


Fig. 24. Unstructured mesh used for computing supersonic flow in a wind tunnel with a step at M1 = 3.

Fig. 25. Comparison of density contours for the Mach 3 wind tunnel with a step. Thirty equally spaced contour lines from q = 0.5 to 6.


5.7. A Mach 3 wind tunnel with a step

This is another popular case for high resolution schemes in [35]. The computational domain is [0,3] � [0,1]. The corner ofthe step is located at (x,y) = (0.6,0.2). The initial conditions are (q,u,v,p) = (1.4,3,0,1) which stand for a Mach 3 uniform flowimpacting the step at the initial time.

The grid near the corner of wind tunnel are showed in Fig. 24. The density contours at t = 4.0 are plotted in Fig. 25. Again,both limiters exhibit very robust shock capturing performances. The resolution of the WBAP is higher than that of the Barthlimiter, which is more remarkable for the contact discontinuity.

6. Conclusions

A class of accurate and robust limiters has been developed for the second order finite volume methods on unstructuredgrids. The limiters are based on a novel procedure termed as WBAP. WBAP limiters are shown to possess the asymptoticallyTVD or MP satisfying properties and can capture discontinuities essentially free from numerical oscillations. The WBAP lim-iters are simple, smooth and applicable on multi-dimensional unstructured grids. The developed WBAP limiters have beentested for a variety of problems on unstructured grids. The superior performance of the WBAP limiters is demonstrated interms of accuracy, robustness and convergence performance in comparison with other frequently-used limiters. Anotherremarkable feature of the WBAP limiters is that they can be applied to higher order reconstructions which will be fully ex-plored in the future work.

Acknowledgments

This work was supported by project-10932005 of NSFC. The authors would like to thank the anonymous reviewers fortheir valuable suggestions and detailed comments, which significantly improved the quality of this manuscript.

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