A refined r-factor algorithm for TVD schemes on arbitrary unstructured meshes

Transcription

1 A efined -facto algoithm fo TVD schemes on abitay unstuctued meshes Di Zhang 1,, Chunbo Jiang 1,*, Liang Cheng 2, 3, Dongfang Liang 4 1 State Key Laboatoy of Hydoscience and Engineeing, Tsinghua nivesity, Beijing , China 2 School of Civil, Envionmental and Mining Engineeing, The nivesity of Westen Austalia, 35 Stiling Highway, Cawley, WA 6009, Austalia 3 State Key Laboatoy of Coastal and Offshoe Engineeing, Dalian nivesity of Technology, Dalian, , China 4 Depatment of Engineeing, nivesity of Cambidge, Tumpington Steet, Cambidge CB2 1PZ, K SMMARY A efined -facto algoithm fo implementing TVD schemes on abitay unstuctued meshes, efeed to hencefoth as FFISAM (a Face-pependicula Fa-upwind Intepolation Scheme fo Abitay Meshes), is poposed based on an extensive eview of the existing -facto algoithms available in the liteatue. The design pinciples, as well as the espective advantages and disadvantages, of the existing algoithms ae fist systematically analyzed befoe pesenting the FFISAM. The FFISAM is designed to combine the meits of vaious existing -facto algoithms. The pefomance of the FFISAM, implemented in ten classical TVD schemes, is evaluated against fou two-dimensional pue-advection benchmak test cases whee analytical solutions ae available. The numeical esults clealy show that the FFISAM leads to a bette oveall pefomance than the existing algoithms in tems of accuacy and convegence on abitay unstuctued meshes fo the ten classical TVD schemes. Keywods: total vaiation diminishing (TVD); high esolution scheme (HRS); nomalized vaiable diagam (NVD); -facto algoithm; convection discetization; unstuctued meshes Coesponding authos: State Key Laboatoy of Hydoscience and Engineeing, Tsinghua nivesity, The Hydaulic Building, Beijing , China. Phone: addess: zhangdi10@mails.tsinghua.edu.cn (D. Zhang), jcb@mail.tsinghua.edu.cn (C. Jiang). 1

2 1. Intoduction Numeical simulation of convection-dominated flows emains one of the most challenging poblems in computational fluid dynamics [1-4]. It is well known that conventional low-ode (LO) schemes, such as the fist-ode upwind (FO), Hybid and Powe-Law schemes, although highly stable and unconditionally bounded, suffe fom excessive numeical diffusion [5 7]. Howeve, taditional high-ode (HO) schemes, such as the cental diffeencing (CD), second-ode upwind (SO), cubic-upwind intepolation (CI) and uadatic-upwind intepolation (QICK) schemes, although being able to impove the accuacy in smooth egions, often geneate unphysical oscillations in discontinuous o steep-gadient egions because of thei unbounded natue [8-10]. The endowment of the boundedness popety to the HO schemes esults in the so-called high-esolution schemes (HRS), which ae able to povide good esolution in steep-gadient egions without intoducing spuious oscillations, while at the same time ae of at least second-ode accuacy in smooth egions [11-15]. In the past decades, vaious types of HRS, such as flux-coected tanspot (FCT) techniues, total vaiation diminishing (TVD) schemes, nomalized vaiable diagam (NVD) schemes, and essentially non-oscillatoy (ENO) schemes, have been developed [16-20]. In this pape, the widely used high-esolution TVD schemes ae consideed in the context of the finite volume method (FVM). TVD schemes, oiginally developed by Haten [21], ae a goup of popula high-esolution schemes fo solving hypebolic consevation laws. Depending on some citical conditions, a TVD flux-limite often switches fom a high-ode scheme to a low-ode diffusive/compessive scheme (o vice vesa) fo the pupose of cicumventing the afoementioned numeical dilemma [22-24]. Paticulaly, in Ref. [11], a lage numbe of TVD schemes poposed in the liteatue have been gouped into thee boad categoies: OT-TVD (One-step Time-space-coupled nsteady TVD), MT-TVD (Multi-step Time-space-sepaated nsteady TVD) and SS-TVD (Semi-discete Steady-state TVD). Thei design pinciples ae also pesented with a view of choosing diffeent flux-limite foms fo vaious types of discetization methods (steady o unsteady, time-space-coupled o time-space-sepaated) in ode to achieve bette accuacy, convegence and efficiency. As is well known, TVD schemes, on the basis of a igoous mathematical foundation, possess seveal attactive featues, such as monotonicity pesevation, 2

3 computational simplicity and efficiency, and high-ode accuacy [25-28]. Fo stuctued gids, a numbe of TVD schemes have been developed in the past decades, such as Albada [11, 13, 16, 24, 29], CBISTA [11, 13, 30], Hamonic [11-13, 22, 24, 31-34], Koen [11, 13, 23, 24, 34-37], Minmod [11-13, 15, 22, 31, 33, 38-39], MSCL [11-13, 15-16, 22, 24, 31-32], OSPRE [11, 13, 24], Supebee [11-13, 15, 22, 24, 31, 33, 38], TCDF [11], MIST [11, 13, 33, 40], and WACEB (also known as the Bounded QICK limite) [11, 13, 40-42]. When implementing high-ode TVD schemes, the nodal vaiable values in two upsteam cells ae euied. This poses a difficulty when TVD schemes ae extended to abitay unstuctued gids, because the fa-upwind node is not eadily available hee. It is not staightfowad to extend TVD schemes into abitay unstuctued gids due to this eason, which is especially challenging fo thee-dimensional poblems. Although a lage numbe of TVD schemes have been poposed and analyzed, little attention has been paid to implementing these schemes on abitay unstuctued gids, patly because of the complexity of locating the fa-upsteam nodes and detemining the vaiable values thee [1-3, 6, 33, 38-39, 43-46]. In this aticle, seveal existing -facto (fa-upwind econstuction) algoithms ae eviewed, and thei advantages and disadvantages ae evealed and analyzed. A efined -facto algoithm (viz. FFISAM), being suitable fo abitay unstuctued gids, is poposed and benchmaked in seveal two-dimensional advection test poblems. Its pefomance is compaed with seveal existing algoithms, such as Dawish s and Hou s algoithms [1-3, 6, 33, 38-39, 43-46]. 2. TVD Schemes Most TVD schemes available in the liteatue can be gouped into thee boad categoies, viz. OT-TVD, MT-TVD and SS-TVD [11]. Diffeent flux-limite foms (CFL-dependent o CFL-independent, and diffeent limiting conditions) fo vaious types of discetization methods (steady o unsteady, time-space-coupled o time-space-sepaated) lead to diffeent pefomance in tems of accuacy and convegence. The extension of these schemes to unstuctued meshes can be accomplished by employing a -facto (fa-upwind econstuction) algoithm. The focus of this pape is on the extension of TVD schemes to abitay unstuctued meshes with the aid of the so-called -facto algoithm, but not the development of TVD schemes themselves. Fo this pupose, vaious SS-TVD 3

4 schemes, designed fo obtaining the steady-state solution of one-dimensional (1D) advection euation [11, 13], ae eviewed in this section. The 1D advection euation can be witten as: whee ( xt, ) t ( a) = (1) x = denotes the tanspoted vaiable and a is the advection velocity. Without loss of geneality, we assume that the velocity is a positive constant ( a > 0 ). The opposite case ( a < 0 ) can be handled by symmety at each cell inteface. A pseudo-time stepping techniue is employed to get the steady-state solution, and we define t as the time step and x as the size of the contol volume. The flux tem in E. (1) is delibeately placed on the ight-hand side of the euation to emphasize the semi-discete fom. It is obvious that finding the steady-state solution of E. (1) is essentially euivalent to solving the steady euation ( a ) = 0. x We would like to emphasize that, although in this pape vaious -facto algoithms ae discussed and evaluated only based on the SS-TVD schemes, these algoithms ae actually eually applicable to othe types of TVD schemes (e.g., OT- TVD, MT-TVD and othe miscellaneous TVD schemes) Semi-discete Steady-state Linea Schemes Integating E. (1) ove a contol volume Ci = xi x 2, x i+ x 2, one can obtain the geneal semi-discete flux-consevative fom: ( ) d i a = dt x i+ 1/2 i 1/2 whee the inteface value, i + 12, can be obtained by using the well-known k-schemes, fistly intoduced by Van Lee [11, 13, 36, 47]: Fo unifom gids, 1 = + ψ 2 ( )( ) i+ 1/2 i i+ 1/2 i i 1 (2) whee ( ) 1 k 1 k = + +, 2 2 ψ i + 1/2 i + 1/2 i+ 1/2 ( i+ 1 i) ( ) = (3) i i 1 Fo non-unifom meshes, xi i+ 1/2 = i+ ψ ( i+ 1/2 ) 2 x i 1/2 4

5 whee ( ) 1 + k 1 k = +, i+ 1/2 = 2 2 x x ψ i + 1/2 i + 1/2 i+ 1/2 i 1/2 i+ 1 i = x x x i+ 1/2 i+ 1 i i i 1, =, xi xi+ 1/2 xi 1/2 x i-1/2 xi xi 1 = (4) whee i + 1/2 denotes the atio of two consecutive gadients: one acoss the cell face / x i + in uestion ( ) 1/2 ( / x) i -1/2 and the othe acoss the immediate upwind inteface, which is essentially a local measue of smoothness, and k is a vaiable epesenting diffeent schemes. Actually, the k-schemes can be deemed as the sum of a diffusive FO tem and an anti-diffusive tem. The anti-diffusive pat is multiplied ψ i+ 1/2 by the flux limite function ( ), which is a linea function of i 1/2. As illustated in Figue 1, some classical linea high-ode advection schemes can be deived by taking diffeent values of k, e.g. SO (k = -1): ( ) ψ + = i 1/2 1 Fomm (k = 0): ( ) = ( + ) ψ i + + 1/2 i 1/2 1 2 CI (k = 1/3): ( ) = ( + ) ψ i + 1/2 i + 1/ QICK (k = 1/2): ( ) = ( + ) ψ i + 1/2 i + 1/2 CDS (k = 1): ( ) ψ i + 1/2 = i + 1/ (5) It is well known that these linea high-ode schemes ae vulneable to unphysical spatial oscillations when they ae applied to captue shocks o steep vaiations of the tanspoted vaiable, pimaily due to thei unbounded natue [8-10]. It is noted that the semi-discete flux-limited fom can also be fomulated in a slightly diffeent fom fom the one given by E. (3) o E. (4), as those outlined in [15, 20, 26, 38, 42]. Though simple algebaic manipulations, it is possible to convet those diffeent foms and thei coesponding flux-limites into the usual semidiscete flux-limited fom (E. (3) o E. (4)), which is the only fom consideed in this pape fo the pupose of consistency Semi-discete Steady-state TVD Citeion In ode to achieve the desied boundedness fo the usual semi-discete flux-limited fom (E. (3) o E. (4)), seveal boundedness citeia have been developed in the liteatue, such as Sweby s TVD citeion [13, 22], Gaskell s CBC citeion [48, 49] 5

6 and Spekeijse s positivity citeion [16]. Sweby s TVD citeion, which has been boadly applied to the constuction of SS-TVD flux-limites [11, 13, 15-16, 22, 26, 33-39, 40, 44], is adopted hee to constuct nonlinea monotonicity-peseving SS-TVD schemes that ae capable of suppessing spuious oscillations in the vicinity of discontinuities o steep gadients. As shown in Figue 1, Sweby s TVD citeion is defined as: ψ ( ) ( ) ( ) 0 i+ 1/2 min 2 i 1/2, 2 when + i+ 1/2 > 0 ψ i+ 1/2 = 0 when i+ 1/2 0 Obviously, Sweby s TVD citeion (E. (6)) is independent fom the CFL numbe. This is an attactive attibute to have fo the steady-state solution of the semi-discete advection euation (E. (2)) based on a pseudo-time stepping techniue. Note that, although most flux-limites poposed so fa satisfy the above citeion, fo negative values of i + 1/2, the violation of this citeion does not cause poblems in pactice, as evidenced in [11, 13-14, 16, 24, 29, 35]. (6) 2.3. SS-TVD Flux-limites In the past decades, a lage numbe of SS-TVD flux-limites have been poposed and analyzed. Some of those ae outlined below fo the convenience of discussion (efe to Figue 2 ). 1). Piecewise-linea SS-TVD Flux-limites: Minmod limite [11-13, 15, 22, 31, 33, 38-39, 46]: ( ) max 0, min (, 1) = (7) ψ i + 1/2 i + 1/2 Supebee limite [11-13, 15, 22, 24, 31, 33, 38, 46]: ( ) max 0, min( 2, 1 ), min (, 2) = (8) ψ i + 1/2 i + 1/2 i + 1/2 MSCL limite [11-13, 15-16, 22, 24, 31-32, 46]: + 1 = max 0, min 2,, 2 2 (9) i 1/2 ( ) + ψ i+ 1/2 i+ 1/2 Koen limite [11, 13, 23-24, 34-37]: = max 0, min 2,, 2 3 (10) i 1/2 ( ) + ψ i+ 1/2 i+ 1/2 WACEB limite [13, 41] (also known as the Bounded QICK limite [40, 42]): = max 0, min 2,, 2 4 (11) i 1/2 ( ) + ψ i+ 1/2 i+ 1/2 6

7 MIST limite [33, 40] (also known as the SPL-1/2 limite [11, 13]): = max 0, min 2,,, (12) i 1/2 i 1/2 ( ) + + ψ i+ 1/2 i+ 1/2 2). Gadually-switching Smooth SS-TVD Flux-limites: Hamonic limite [11-13, 22, 24, 31-34, 46]: i+ 1/2 + i+ 1/2 ψ ( i+ 1/2 ) = + i+ 1/2 1 OSPRE limite [11, 13, 24]: 3 i 1/2 ( + i+ 1/2 + 1) ψ ( i+ 1/2 ) = 2 2 ( i+ 1/2 ) + i+ 1/2 + 1 Albada limite [11, 13, 16, 24, 29] (also known as the GVA-0 limite [13]): i 1/2 ( + i+ 1/2 + 1) ψ ( i+ 1/2 ) = TCDF limite [11]: ( ) i+ 1/2 2 ( ) i 1/2 ( + i+ 1/2 + 1) ( i+ 1/2 ) + 1 when i+ 1/2 < ( i+ 1/2 ) 2( i+ 1/2 ) + 2 i+ 1/2 when 0 i+ 1/2 < 0.5 ψ ( i+ 1/2 ) = 0.75 i+ 1/ when 0.5 i+ 1/2 < ( i+ 1/2 ) i+ 1/ /2 ( 1/2 ) when 1.5 i + < + i + (13) (14) (15) (16) Geneally speaking, a piecewise-linea flux-limite (e.g. Es. (7-12)) acts simply as switches between linea schemes and offes the advantage of geat flexibility. The oveall accuacy of the piecewise-linea flux-limites can be impoved by enlaging the egion of a specified high-ode scheme (e.g. QICK o CI). Howeve, the flux-limites of this type suffe fom an advese effect on convegence behavio unde some cicumstances because of thei discontinuous natue. In contast, the smooth flux-limites (e.g. Es. (13-16)) often exhibit bette convegence than the piecewiselinea ones at a pice of accuacy [11, 13]. In summay, this section povides a bief eview of vaious SS-TVD schemes designed fo obtaining the steady-state solution of the 1D advection euation. The key elements of the SS-TVD schemes include the geneal semi-discete fluxconsevative fom (viz. E. (2)) to discetize the advection tem and the usual semi- 7

8 discete flux-limited fom (viz. E. (3) o E. (4)) to appoximate the consevative ψ i+ 1/2 flux. It is the CFL-independent limite ( ) schemes and detemines the accuacy of the scheme. that distinguishes diffeent SS-TVD 3. Existing -Facto Algoithms fo nstuctued Meshes As displayed in Figue 3, fo a pescibed face of f on an unstuctued gid, only the upsteam node C and the downsteam node D ae diectly available. Since the index-based notation, which is used above in Es. (2-16) fo stuctued gids, is not suitable fo unstuctued meshes, the moe appopiate notation, shown in Figue 3, is adopted hee [38]. Appaently, by the definition of an imaginay fa-upwind node, which is an upwind node of C, the usual semi-discete flux-limited fom can be ewitten as the following expessions: Fo unifom meshes, 1 ( D C) f = C+ ψ ( f )( C ), whee f = (17) 2 Fo non-unifom meshes, xc C f = C+ ψ 2 ( f ) x x D C C whee f = xd xc xc x C ( ), xc 2 ( xf xc) C = (18) whee ψ ( f ) is the CFL-independent limite (e.g., Es. (7-16)). As can be seen fom Es. (17-18), the fa-upwind node location x and vaiable value ae euied fo the calculation of the face value f. Howeve, on abitay unstuctued gids, the fa-upwind node is not eadily available and thus it is not staightfowad to implement these SS-TVD schemes on unstuctued gids. We would like to eemphasize that, although vaious fa-upwind econstuction techniues (viz. -facto algoithms) ae pesented and discussed hee only in the context of the SS-TVD schemes, these algoithms can be eually applied to othe types of TVD schemes (e.g., OT-TVD, MT-TVD and othe miscellaneous TVD schemes), which ae not taken into account in this pape Extapolation -Facto Algoithms Bune s -Facto Algoithm 8

9 Since du =2 d on a unifom one-dimensional gid ( du is the magnitude of the f vecto between the nodes and C, and d f is that between the node C and the cente of the face f.), Bune [50] poposed the following modification to the definition of the -facto fo implementing vaious TVD schemes on an abitay unstuctued gid: D -C D -C D -C f = = ( ) d 2d ( f C) C C f whee d is the unit vecto fom C to D, and vaiable at the cente of the dono cell C. (19) ( ) C epesents the gadient of the As pointed out by Dawish and Moukalled [38], Bune s -facto algoithm is inconsistent because, when applied to one-dimensional meshes, it does not ecove to the oiginal TVD fomulation: D -C D -C f = - ( ) d 2 d C C f (20) Dawish s -Facto Algoithm It is appaent that the main difficulty in implementing TVD schemes on an abitay unstuctued mesh lies in the need fo defining a vitual fa-upwind node. Dawish and Moukalled [38] poposed the so-called exact -facto techniue, which is boadly used to esolve this issue in the liteatue [1, 2, 6, 33, 38-39, 43-46]. Dawish s -facto fomulation can be witten as: D -C D -C D -C f = = = (21) ( ) ( ) ( ) C D C D D D C whee D and C denote the vaiable values at the nodes that staddle the consideed inteface and thus ae eadily available fo abitay unstuctued gids. In ode to compute the tem involving, it was assumed that: ( ) d ( du + dd )=( ) d 2 dd = D - C C = -( ) d 2dd (22) D C whee d is the unit vecto fom C to D, du is the magnitude of the vecto between the nodes and C, dd is the magnitude of the vecto between the nodes C and D, and is the pedicted value at the vitual fa-upwind node, as shown in Figue 3. Essentially, Dawish s -facto algoithm extends vaious TVD schemes to abitay unstuctued gids by constucting a vitual fa-upwind node and futhe using the eual-distance and eual-gadient assumptions [1, 2, 6, 38]. nde these 9

10 assumptions, the distance between and C, du, is supposed to be eual to that between C and D, dd. Besides, it assumes that the vaiable unifomly changes fom to D,which means that the gadient keeps as a constant along the whole segment (D). It can be poved that E. (22) can be diectly deived fom the afoementioned two assumptions Zhang s Fa-upwind Reconstuction Techniue Zhang et al. [2, 6] debated that Dawish s fa-upwind econstuction techniue has two inheent dawbacks. Fistly, the imaginay node may be located seveal cells away fom the dono cell C, when extemely non-unifom gids ae adopted. Secondly, it is not all that appopiate to assume that the gadient stays the same along the whole segment D. In ode to ovecome the afoementioned two dawbacks, New-Techniue-2 is poposed and evaluated in thei pape, whee a pedicto-coecto pocedue is employed to detemine the value of du and a paabolic function is used to epesent the vaiable vaiation between the nodes and D on the basis of C, D and ( ) C. Ref. [6] clealy shows that this techniue esults in a bette oveall pefomance in tems of accuacy and convegence fo many NVD schemes, when compaed with Dawish s techniue (E. (22)). Since the New-Techniue-2 is oiginally designed fo NVD schemes, the implementation of this algoithm in TVD schemes has not been epoted and is beyond the scope of this study Intepolation -Facto Algoithms Li s -Facto Algoithm Li et al. [33, 44, 45] pointed out that the value of in Dawish s -facto algoithm is essentially extapolated by the vaiable value of the downsteam node D ( D ) and the gadient value at the upsteam node C ( ( ) C ), as demonstated in E. (22). With the aid of Gauss s theoem, ( ) C can be achieved by intepolating the values on all faces of the dono cell C, which is the only pocess that concens the fa-upwind infomation. Futhemoe, Li has demonstated that, if a paabolic distibution of along nodes, C and D in one-dimensional gid is assumed, can be exactly estimated by E. (22). Howeve, if the distibution of is exponential as in some applications, E. (22) will lead to an undesiable o even an eoneous estimation of 10

11 . In view of this, Li [44] poposed a scheme that estimates the value of by intepolation athe than extapolation. In Li s econstuction method [44], the same eual-distance assumption as that of Dawish s algoithm is employed, which is essentially euivalent to that the vitual fa-upwind node is detemined in such a way that it lies along the line joining nodes D and C, with C at the cente of the D segment. In addition, the gadient of the node, instead of the node C, is used to estimate the value of whee in E. (21). = + ( ) d (23) is the centoid of the cell which contains the vitual fa-upwind node, ( ) indicates the gadient of, and nodes and, as illustated in Figue 4. d stands fo the vecto between the Li et al. [44] stated that this algoithm allows moe upwind infomation to be included. Especially, when the distibutions of, C and D ae not paabolic (e.g. obeying an exponential distibution), Dawish s extapolation algoithm (E. (22)) leads to an undesiable pediction of, but Li s intepolation algoithm (E. (23)) woks easonably well. Additionally, it should be pointed out that the seach pocess fo is actually vey complex, because diffeent mathematical algoithms ae needed fo vaious kinds of polygonal gids (e.g. tiangle, tetahedon, uadangle, hexahedon, pentagon, and so on.), which conseuently pevent this techniue fom being widely used on abitay unstuctued gids Hou s -Facto Algoithm Given that the flux though the inteface depends pimaily on the nomal vaiable tanspotation, Hou et al. [33, 45] and Zhang et al. [2, 6] pointed out that, as demonstated in Figue 5, fo the inteface f between C and D, the face value should be calculated using auxiliay nodes C f and D, which ae the pojections of C and D, espectively, along the line L that uns though the cente of the inteface and pependicula to it. Moeove, in the discetization of the convection tem, the diffeences between the physical uantities at D and D (o C and C ) ae ignoed. This is essentially euivalent to assuming that the node D (o C ) has the same value and gadient as the node D (o C ), i.e., ( ) D = ( ) D and ( ) C = ( ) C. D = D, C = C, As shown in Figue 5, in Hou s fa-upwind econstuction techniue [33], the 11

12 imaginay fa-upwind node is defined as the intesection of the line L and the line l, which passes though (the closest centoid of the futhe upwind cells) and is pependicula to the nomal of the inteface. Hee, is selected by compaing the distances between the line L and the centoids of all cells adjacent to the opposite vetex P 2 except fo the dono cell C. Evidently, the vaiable values at the nodes, C intepolation fom the values at the cell-centes, C and D can be obtained by and D, espectively. But, fo simplicity and efficiency, Hou et al. [33] suggests to diectly eplace the fome by the latte. Conseuently, E. (18) can be efomulated as: whee x C C f = C+ ψ 2 ( f ) x x D C C f =, xc =2( xf xc ) xd xc xc x C (24) Numeical esults pesented in [33, 45] confim that Hou s -facto algoithm (E. (24)) pefoms bette than Li s (E. (23)) and Dawish s (E. (22)) algoithms because it esults in highe accuacy and less oscillation in steep gadient egions, especially fo low uality two-dimensional (2D) unstuctued meshes. Since Hou's algoithm was deived only based on 2D unstuctued tiangula gids, it needs to be futhe impoved to be applicable to abitay unstuctued gids. 4. A Refined -Facto Algoithm fo Abitay nstuctued Meshes This section pesents a futhe analysis of the afoementioned existing -facto algoithms, and theoetically demonstates the advantages and disadvantages of these algoithms. Futhemoe, based on the analysis, a efined -facto algoithm, efeed to as FFISAM, is poposed and evaluated in ode to moe efficiently extend TVD schemes to abitay unstuctued meshes Futhe Analysis of Existing -Facto Algoithms Dawish s -Facto Algoithm Despite the well-known Dawish s -facto algoithm is widely used in the liteatue [1, 2, 6, 33, 38-39, 43-45], a futhe analysis of this fa-upwind econstuction techniue eveals the following issues that need to be esolved: 12

13 1). As afoementioned, it is appaent that the flux though the inteface depends pimaily on the nomal vaiable tanspotation. As a conseuence, the vaiables in the effective advection diection (nomal to the inteface) should have been taken into account in calculating the flux. In fact, as shown in Figue 6, in Dawish s algoithm (E. 22), the flux is solely detemined by the vaiables along the line joining the centoids of the upwind and downwind cells (C and D). This is not physically appopiate fo non-othogonal gids and may be a souce of numeical eos. 2). To obtain the fa-upwind node vaiable value, this techniue assumes that the gadient does not change along the whole segment (D). Conseuently, the value at the cente of the upwind cell ( C ) can be witten as: = -( ) CD (25) C D C Obviously, this is contadictoy to the eal physical situation, because C is an unknown fixed value befoe the new pseudo-time step. 3). Although E. (22) is able to accuately simulate paabolic distibutions of along the cells, C and D, when the exponential distibution is consideed, E. (22) will esult in undesiable o even eoneous estimations fo the value of [33, 44-45]. Clealy, impoved econstuction techniues ae euied, eithe by extapolation o intepolation Li s -Facto Algoithm In Li s -facto algoithm [44], the gadient of node (the cente of the cell containing the vitual node ) instead of node C is used to estimate the fa-upwind node vaiable value. This suggests that this techniue obtains the value of by intepolation athe than extapolation, which is an impovement ove Dawish s algoithm. Howeve, the following aspects could possibly be futhe impoved: 1). Although Li s algoithm ovecomes the afoementioned second and thid dawbacks of Dawish s algoithm to a cetain degee, the fist dawback of the latte still exist in the fome. 2). As discussed in Section 4.2, since the cell containing the vitual node is not guaanteed to be the closest centoid of the fa-upwind cells. The way of defining the node could potentially be futhe impoved fo this algoithm. Fo instance, as shown in Figue 6, the cell H is close to the vitual node than the cell I containing the node. The use of H could lead to a bette appoximation of the value of. 13

14 3). Li s algoithm was oiginally developed fo inne cells, and theefoe a special teatment o fomulation may be euied fo bounday cells. Othewise, the vitual node may be located outside of the computational domain, and thus the cell containing the node cannot be identified, viz. the node situation, as displayed in Figue 7. does not exist in this 4). In addition, although it is convenient to implement Li s algoithm on tiangula o uadilateal gids, its application to abitay unstuctued gids, consisting of hexahedons, pentagons and hexagons, may be complicated and time-consuming, as diffeent mathematical algoithms ae euied fo vaious types of polygonal unstuctued gids Hou s -Facto Algoithm Hou s -facto algoithm [33, 45] ovecomes all the afoementioned thee dawbacks of Dawish s algoithm, but it is developed specifically fo 2D tiangula gids. The extension of this algoithm to abitay unstuctued gids has not been attempted yet A Refined -Facto Algoithm The -facto algoithm poposed in this pape (viz. FFISAM) is essentially an extension of Li s and Hou s intepolation algoithms that etains the attactive attibutes of the both algoithms and ovecomes the issues identified above. In FFISAM, the following euation is employed: xc C f = C + ψ 2 ( f ), = + d ( ) x x C D C C whee f =, D = D, C = C, xc = 2( xf xc )(26) xd xc xc x whee ψ ( f ) is the CFL-independent flux limite (e.g., Es. (7-16)). is defined as the closest effective fa-upwind cell, which is detemined in a diffeent way fom that in Li s (E. (23)) and Hou s (E. (24)) algoithms. ( ) epesents the gadient of and can be obtained with the aid of Gauss s theoem. is the vitual fa-upwind node, and d denotes the vecto between the centoid of and the node. All the above definitions will be descibed in detail in the following paagaphs with the help of Figue 6 and Figue 7. 14

15 Fo the Inne Domain As demonstated in Figue 6, fo the inteface ab between C and D on an abitay unstuctued mesh, consideing the fact that the flux though the inteface depends pincipally on the nomal vaiable tanspotation, the face value calculated using auxiliay nodes C f should be and D, which ae the pojections of C and D, espectively, along the line L that cuts though the cente of the consideed inteface (ab) pependiculaly. As afoementioned in Section 3.2.2, fo the discetization of the convection tem, the diffeences between the physical uantities at D and D (o C and C ) can be neglected, which means that the node D (o C ) has the same vaiable value as the node D (o C ), viz. D = D and as shown in E. (26). C = C, Futhemoe, simila to Dawish s and Li s algoithms, the eual-distance assumption is employed in FFISAM in ode to detemine the location of the vitual fa-upwind node. Howeve, as illustated in Figue 6, hee the vitual node is detemined in such a way that it lies along the line joining nodes D and C, with C at the cente of the D segment (viz. du=dd ), which is slightly diffeent fom the way in Dawish s and Li s algoithms. Additionally, to obtain the value of, all the upwind-cell s nodes ae taken into account except fo the two end nodes of the chosen face (a, b), viz. the nodes c, d, e and f in Figue 6. Then, the closest effective fa-upwind cell is identified by compaing the distances between the vitual node and the centoids of all cells adjacent to the afoementioned consideed nodes (except fo the upwind cell C itself ), such as B, E, F, G, H, I and J in Figue 6. Afte identifying the cell be used to calculate the value of It should be emphasized that the cell the cell and eventually the face value, E. (26) can f. of FFISAM is not always the same as of Li s algoithm, which contains the vitual fa-upwind node. As evidenced in Figue 6, the cell H is selected as the cell I is chosen as the fo the fome, howeve, the fo the latte. It is speculated that FFISAM is moe pefeable in this situation, because the cell H is close to the vitual node than the cell I, and the use of H can lead to a bette appoximation of the value of. Fo the Bounday Domain When it comes to the bounday domain, some mino coections to the above innedomain s fa-upwind econstuction pocedue ae necessay to obtain a easonable estimation of the value of. 15

16 As illustated in Figue 7, fo the inteface ab between C and D, the same eual- distance assumption as the inne-domain case is adopted to detemine the location of the vitual node. If the above inne-domain s econstuction pocedue was followed, cell A would have been selected as the closest effective fa-upwind cell because only the cell A is adjacent to the node c, which is the upwind-cell s uniue node except fo the two end nodes of the consideed face (a, b). Howeve, unde this cicumstance, it is not so advisable to estimate the value of, by using the vaiable value and gadient of A, because the centoid of A is fa away fom the vitual node. The adoption of the centoid of the upwind-cell s bounday face bc, viz. the node d, may lead to a bette appoximation of. On this basis, it is suggested that should be chosen fom the centoid of the closest effective fa-upwind cell (following the pocedue in the inne-domain case) and the centoids of the upwind-cell s bounday faces by compaing the distances between these centoids and the vitual node. Moeove, if the centoid of the upwind-cell s bounday face is selected as, to ende FFISAM moe efficient, the value of is simply estimated as the vaiable value of this centoid (e.g., = d in Figue 7.) It deseves ou attention that, if Hou s algoithm was used in Figue 7 (following the pocedue in Section 3.2.2), the cell A would have been chosen as the cell and the node A would have been defined as the vitual fa-upwind node. Appaently, this would have been contadictoy to the definition of the node, in view of the fact that the node A is actually in the downsteam of the centoid of the upwind-cell C. Besides, none of the cells contain the node in this case, because it lies outside of the computational domain as displayed in Figue 7. Hence, it can be concluded that Li s and Hou s algoithms also euie diffeent fomulations fo bounday cells in ode to be applicable in all situations. To ensue the physical boundedness of,, all the neighboing cells and bounday faces of the up-wind cell C ae consideed except fo the down-wind cell D, such as B, E, F, G, H, I and J in Figue 6 and the bounday face bc in Figue 7. The following euation is employed to estict the value of whee min C and min max { { } }. = min max,, (27) C C max C espectively epesent the maximum and minimum values of the afoementioned consideed neighboing cells, bounday faces and the cell C itself. With the aid of FFISAM (E. (26) and E. (27)), the implementation of TVD schemes on abitay unstuctued meshes is now simple and staightfowad even fo 16

17 thee-dimensional poblems, because the value of f can be calculated in the same way fo all the faces iespective of the dimensionality of contol volumes and the topological stuctue of unstuctued meshes. The attactive attibutes of FFISAM ae summaized. Fist of all, the face position can be taken into consideation fo non-unifom unstuctued gids as shown by E. (26). Secondly, simila to Li s (E. (23)) and Hou s (E. (24)) algoithms, FFISAM obtains the value of the imaginay fa-upwind node ( ) by intepolation athe than extapolation. This allows moe upwind infomation to be included elative to Bune s (E. (19)) and Dawish s (E. (22)) extapolation algoithms. Thidly, the use of FFISAM facilitates the extension of TVD schemes to abitay unstuctued meshes without having to deal with some issues as in Li s and Hou s algoithms discussed ealie on. The last but not the least, the implementation of FFISAM is simple and staightfowad, and does not involve a complex and time-consuming seach pocedue fo. Based on these, it is easonable fo us to believe that FFISAM offes impovements ove the existing -facto algoithms in tems of numeical popeties. A numbe of numeical tests will be pesented in the next section to validate FFISAM. 5. Numeical Test Cases This section pesents seveal classic numeical examples to evaluate the elative pefomance of the above -facto algoithms fo ten classical SS-TVD flux-limites, including Albada, Hamonic, Koen, Minmod, MSCL, OSPRE, Supebee, MIST, WACEB and TCDF. Fo abitay unstuctued meshes, only Dawish s algoithm (E. (21) and E. (22)) and the newly-poposed FFISAM (E. (26) and E. (27)) ae compaed, because Hou s and Li s algoithms ae not suitable fo abitay unstuctued meshes. As afoementioned, since Hou s algoithm was oiginally designed fo tiangula o tetahedal gids, it is included in compaison in Test 4 whee both an abitay unstuctued mesh and an unstuctued tiangula gid ae employed. As fo Li s algoithm, since the wok concening the seach of (the centoid of the cell containing the vitual node ) is pohibitively complex and time-consuming fo abitay polygonal unstuctued gids (e.g. tiangle, tetahedon, uadangle, hexahedon, pentagon, and so on.), it is not included in the compaison. Pescibed velocity fields ae used and the accuacy is uantitatively measued by the aveage L2 nom of the diffeence between the exact and numeical solutions: 17

18 whee 1 E = St (28) N n a 2 ( ( i i )) N i= 1 n i is the calculated solution afte n time steps, a i the exact analytical solution and N the numbe of contol volumes [1, 2, 6, 11] Test 1: Pue convection of a step pofile In this poblem we conside the advection of a passive scala in an obliue unifom velocity field of (u, v)=(1, 1). The govening consevation euation of this poblem is: ( u) ( v) + + = 0 t x y whee is the tanspoted vaiable, and u=1 and v=1 ae the Catesian components of the given velocity vecto. As shown in Figue 8, the inlet bounday conditions ae pescibed as: ( 0, ) 1 y = fo 0 y 1 (,0) 0 (29) x = fo 0< x 1 (30) The step pofile povides the most stingent gadient vaiation, with the pupose of evaluating the method s ability to esolve a shap font, with minimum numeical diffusion and without oscillations [6, 11, 30]. An abitay unstuctued mesh composed of 4057 cells is employed in this test, as displayed in Figue 9. To each the steady state, a pseudo-time stepping appoach is adopted and computations ae pefomed with fou diffeent time steps, yielding maximum Couant numbes (Cu) of 0.5, 1.0, 1.5 and 2.0, espectively. Moeove, the fist-ode implicit Eule method is used fo the time discetization because it allows lage time steps to be taken, which is especially useful fo solving steady flow poblems. Specifically, the FO value ( C ) in E. (18) o E. (26) is implicitly discetized and the high-ode coection tem is explicitly teated as the souce tem. Table 1 summaizes the eos of Dawish s algoithm and FFISAM fo the afoementioned ten classical SS-TVD flux-limites along the line x=0.8 when the steadystate solution is achieved. Figues compae the esulting pofiles of along x=0.8 and the decay of the esiduals at a Couant numbe of 1.0 fo each consideed TVD scheme and fo diffeent -facto algoithms. It should be pointed out that, fo the above fou diffeent Couant numbes, almost the same steady-state solution and convegence conclusions can be eached, theefoe, only the numeical esults at a 18

19 Couant numbe of 1.0 ae demonstated fo conciseness of pesentation. Obviously, fo Koen, Supebee, MSCL, WACEB and MIST, when compaed with Dawish s algoithm, FFISAM does esult in significant impovement in accuacy and convegence, as evidenced in Table 1 and Figues Special attention needs to be paid to the well-known Supebee scheme, which is known as one of the most compessive diffeencing schemes and does poduce the most accuate esult in this test, as illustated in Figue 11. Moeove, the combination of Supebee with FFISAM poduces a much impoved pofile ove the widely-used Dawish s algoithm. In addition, as shown in Figues 15-19, compaisons of the numeical solutions, obtained with vaious -facto algoithms, indicate that FFISAM gives moe accuate pedictions and slightly bette convegence popeties when compaed to Dawish s algoithm fo OSPRE, Albada, Hamonic, Minmod and TCDF. In summay, the numeical esults clealy show that the newly-poposed FFISAM leads to a bette pefomance elative to the widely-used Dawish s algoithm in tems of accuacy and convegence fo all the consideed ten SS-TVD flux-limites in this test, which agees with the theoetical analysis in the pevious sections Test 2: Pue convection of a sine-suae pofile As illustated in Figue 20, the same physical domain, govening euation, time discetization scheme and velocity field as the step-pofile case ae adopted in conducting the advection test of a sine-suae pofile. The inlet bounday conditions ae given as: ( 0, y) = sin π y fo 0 y 3 10 y = fo 3 /10 < y 1 ( 0, ) 0 (,0) 0 x = fo 0< x 1 (31) The sine-suae is a elatively steep pofile that is often used to benchmak vaious flux-limites in steep gadient egions. In this test, an abitay unstuctued mesh composed of 5728 cells is employed, as depicted in Figue 21. Computations ae pefomed with fou diffeent time steps as well, yielding maximum Couant numbes of 0.5, 1.0, 1.5 and 2.0, espectively. The eos of vaious -facto algoithms fo the ten classical SS-TVD fluxlimites when the steady-state is eached ae also demonstated in Table 1. As 19

20 expected, almost the same conclusions as the step-pofile case can be deduced fom the numeical esults. Howeve, fo compactness of pesentation, only the esulting pofiles of along x=0.6 and the coesponding esidual eos (Cu=1.0) fo seven TVD schemes (Koen, Supebee, MSCL, WACEB, TCDF, OSPRE and Albada) ae shown in Figues Again, FFISAM exhibits bette oveall pefomance than Dawish s algoithm in tems of accuacy and convegence Test 3: Pue convection of a semi-ellipse pofile In the thid test, we conside the advection of a semi-ellipse pofile, and, as illustated in Figue 29, the inlet bounday conditions ae set as: and 2 ( 0, ) 1 [ /(1/6)] y = y fo y 1/6 ( 0, y ) = 0 elsewhee 2 (,0) 1 [ x/(1/6)] x = fo x 1/6 ( x,0) = 0 elsewhee (32) The same physical domain, govening euation, velocity vecto and time discetization scheme as the fist test case ae employed. Besides, an abitay unstuctued mesh of 5249 cells is used, as shown in Figue 30. Fou time steps ae adopted which coespond to the maximum Couant numbes of 0.5, 1.0, 1.5 and 2.0, espectively. Afte eaching the steady-state, the eos along x=0.8 fo diffeent flux-limites ae summaized in Table 1. The esulting pofiles of along this line and the coesponding esidual eos (Cu=1.0) ae given in Figues fo seven of the ten TVD schemes that ae compaed in this study. The pefomance of FFISAM in othe thee schemes is simila to the seven schemes shown hee. It is inteesting to obseve that the Supebee scheme does tend to compess the elatively smooth semi-ellipse pofile into a step-like pofile due to its inheent suaing effect, as evidenced in Figue 32. It appeas that FFISAM intoduces moe sevee suaing-effect than Dawish s algoithm fo this TVD flux-limite. This is likely the eason fo the smalle eo of Dawish's algoithm than FFISAM shown in Table 1 fo this case, in contast to pevious two test cases whee FFISAM povides bette pedictions than Dawish's algoithm when the Supebee scheme is used. Except fo the above diffeence, almost the same conclusions as Test 1 and Test 2 20

21 can be obtained in this test. As with pevious tests, FFISAM demonstates excellent oveall pefomance fo the ten SS-TVD flux-limites elative to Dawish s algoithm in tems of accuacy and convegence Test 4: Pue convection in a otational velocity field The fome thee tests show an oveall bette pefomance of FFISAM compaed to Dawish s algoithm when the velocity field is unifom. To evaluate the ability of FFISAM in non-unifom flow fields, the advection of a double-step pofile in a otational velocity field is simulated in Test 4. As shown in Figue 38, the inlet bounday conditions ae pescibed as: (,0) 1 x = fo 0.8 x 0.6 (,0) 0 x = fo 0.6 x 0 and 1 x 0.8 (33) The double-step pofile is tanspoted clockwise fom the inlet bounday (x<0, y=0) to the outlet bounday (x>0, y=0) by a otational velocity field defined as: = y, V = x (34) As shown in Figue 39, this poblem is fistly solved on an abitay unstuctued gid composed of 6428 cells at fou diffeent time steps, yielding maximum Couant numbes of 0.5, 1.0, 1.5 and 2.0, espectively. The same govening euation and time discetization scheme as Test 1 ae selected in this test. The accuacy is uantitatively measued by the aveage L2 nom at the outlet plane (0 x 1.0, y=0), and Table 1 summaizes the accuacy of diffeent -facto algoithms fo the ten TVD schemes. In fact, almost the same conclusions as in the pevious tests can be dawn hee fo diffeent -facto algoithms (FFISAM and Dawish s). The esulting pofiles of along the outlet plane and the coesponding esidual eos (Cu=1.0) ae demonstated in Figues fo six of the ten TVD schemes that ae compaed in this study. The pefomance of FFISAM in othe fou schemes is simila to the six schemes shown hee. The pefomance of FFISAM is delibeately tested on an unstuctued tiangula mesh in ode to compae with Hou's algoithm, because Hou's algoithm is only applicable to unstuctued tiangula o tetahedal gids. As illustated in Figue 46, in Test 4, calculations ae pefomed on an unstuctued tiangula mesh consisting of 8682 cells. Table 2 summaizes the eos of diffeent -facto algoithms (Dawish s, Hou s and FFISAM) fo all the ten SS-TVD schemes. 21

22 As expected, FFISAM povides the most accuate pedictions among all the thee -facto algoithms, and Hou s algoithm gives slightly bette pefomances than Dawish s algoithm, which is in line with the conclusions in Ref. [33]. Besides, Figues exhibit the esulting pofiles of along the outlet plane and the coesponding convegence pocesses (Cu=1.5) fo six TVD schemes (Koen, Supebee, MSCL, WACEB, TCDF and Hamonic). It is obvious that FFISAM shows the apidest decay of the esiduals among all the thee algoithms in these figues. Actually, when it comes to the othe fou SS-TVD schemes, the same conclusions can be dawn as well, although these ae not pesented fo the compactness of the pape. In conclusion, even in this moe complicated situation, whethe on abitay unstuctued gids o on tiangula gids, the numeical esults also demonstate the appaent advantages of FFISAM ove the othe -facto algoithms fo the ten SS-TVD schemes in tems of accuacy and convegence. 6. CONCLSIONS In this aticle, the details of seveal existing -facto (viz. fa-upwind econstuct- tion) algoithms ae eviewed fom the pespective of extending TVD schemes to abitay unstuctued gids. Thei espective advantages and disadvantages ae also evealed and analyzed. Based on the eview, a efined -facto algoithm, efeed to as the FFISAM, is developed, which ovecomes some of the inheent dawbacks of the existing algoithms. In the FFISAM, the vaiables in the effective advection diection (viz. nomal to the inteface) ae chosen fo calculating the flux, which is believed to be physically sound. Moeove, since the FFISAM detemines the vitual fa-upwind node value by intepolation athe than extapolation, it allows moe upwind infomation to be included and conseuently gives moe accuate pedictions than Dawish s algoithm in some cases, e.g., with an exponential distibution of the advected uantity. In addition, the FFISAM ensues a easonable vitual fa-upwind node location on abitay unstuctued gids. Fou advection tests ae conducted to evaluate the elative pefomance of vaious -facto algoithms fo ten classical SS-TVD schemes, including Albada, Hamonic, Koen, Minmod, MSCL, OSPRE, Supebee, MIST, WACEB and TCDF. The numeical esults demonstate that the FFISAM leads to a highe accuacy and faste convegence than the existing algoithms on abitay unstuctued meshes. 22

23 ACKNOWLEDGEMENTS The authos gatefully acknowledge the financial suppot povided by the National Natual Science Foundation of China (Gant No ) and the suppot fom Austalian Reseach Council though a Discovey Gant (Poject ID: DP ). 23

24 REFERENCES 1. O. bbink, R.I. Issa. A method fo captuing shap fluid intefaces on abitay meshes. Jounal of Computational Physics; 153 (1) (1999), D. Zhang, C. Jiang, D. Liang, Z. Chen, Y. Yang, Y. Shi, A efined volume-of-fluid algoithm fo captuing shap fluid intefaces on abitay meshes. Jounal of Computational Physics, 274 (2014), F. Denne, B.G.M. van Wachem. Compessive VOF method with skewness coection to captue shap intefaces on abitay meshes. Jounal of Computational Physics, 279 (2014), R. Scadovelli, S. Zaleski. Diect numeical simulation of fee-suface and intefacial flow. Annual Review of Fluid Mechanics, 31 (1) (1999), B.P. Leonad, J.E. Dummond. Why you should not use Hybid, Powe-Law o elated exponential schemes fo convective modelling-thee ae much bette altenatives. Intenational Jounal fo Numeical Methods in Fluids, 20 (6) (1995), D. Zhang, C. Jiang, C. Yang, Y. Yang. Assessment of diffeent econstuction techniues fo implementing the NVSF schemes on unstuctued meshes. Intenational Jounal fo Numeical Methods in Fluids, 74 (3) (2014), W. Shyy. A study of finite diffeence appoximations to steady-state, convectiondominated flow poblems. Jounal of Computational Physics, 57 (3) (1985), T. Hayase, J.A.C. Humphey, R. Geif. A consistently fomulated QICK scheme fo fast and stable convegence using finite-volume iteative calculation pocedues. Jounal of Computational Physics, 98 (1) (1992), D.B. Spalding. A novel finite diffeence fomulation fo diffeential expessions involving both fist and second deivatives. Intenational Jounal fo Numeical Methods in Engineeing, 4 (4) (1972), P. Tamamidis, D.N. Assanis. Evaluation of vaious high-ode-accuacy schemes with and without flux limites. Intenational Jounal fo Numeical Methods in Fluids, 16 (10) (1993), D. Zhang, C. Jiang, D. Liang, L. Cheng. A eview on TVD schemes and a efined flux-limite fo steady-state calculations. Submitted to Jounal of Computational Physics, (2015). 12. R.J. LeVeue. High-esolution consevative algoithms fo advection in incompessible flow. SIAM Jounal on Numeical Analysis, 33 (2) (1996), N.P. Wateson, H. Deconinck. Design pinciples fo bounded highe-ode convection 24

25 schemes-a unified appoach. Jounal of Computational Physics, 224(1) (2007), V. Dau, C. Tenaud. High ode one-step monotonicity-peseving schemes fo unsteady compessible flow calculations. Jounal of Computational Physics, 193 (2) (2004), Y. Wang, K. Hutte. Compaisons of numeical methods with espect to convectively dominated poblems. Intenational jounal fo numeical methods in fluids, 37 (6) (2001), V. Venkatakishnan. Convegence to steady state solutions of the Eule euations on unstuctued gids with limites. Jounal of Computational Physics, 118 (1) (1995), G.S. Jiang, C.W. Shu. Efficient implementation of weighted ENO schemes. Jounal of Computational Physics, 126 (1) (1996), J.P. Bois, D.L. Book. Flux-coected tanspot. I. SHASTA, A fluid tanspot algoithm that woks. Jounal of Computational Physics, 11 (1) (1973), S.T. Zalesak, Fully multidimensional flux-coected tanspot algoithms fo fluids. Jounal of Computational Physics, 31 (3) (1979), M.S. Dawish, F.H. Moukalled. Nomalized vaiable and space fomulation methodology fo high-esolution schemes. Numeical Heat Tansfe, 26 (1) (1994), A. Haten. High esolution schemes fo hypebolic consevation laws. Jounal of Computational Physics, 49 (3) (1983), P.K. Sweby. High esolution schemes using flux limites fo hypebolic consevation laws. SIAM Jounal on Numeical Analysis, 21 (5) (1984), W. Hundsdofe, R.A. Tompet. Method of lines and diect discetization: a compaison fo linea advection. Applied Numeical Mathematics, 13 (6) (1994), F. Kemm. A compaative study of TVD-limites well-known limites and an intoduction of new ones. Intenational Jounal fo Numeical Methods in Fluids, 67 (4) (2011), Y.N. Jeng,.J. Payne. An adaptive TVD limite. Jounal of Computational Physics, 118 (2) (1995), M.K. Kadalbajoo, R. Kuma. A high esolution total vaiation diminishing scheme fo hypebolic consevation law and elated poblems. Applied Mathematics and Computation, 175 (2) (2006), P.L. Roe. Some contibutions to the modelling of discontinuous flows. Lage-scale Computations in Fluid Mechanics, 1 (1985),

26 28. M. Aoa, P.L. Roe. A well-behaved TVD limite fo high-esolution calculations of unsteady flow. Jounal of Computational Physics, 132 (1) (1997), G.D. Van Albada, B. Van Lee, W.W. Robets J. A compaative study of computational methods in cosmic gas dynamics. Astonomy and Astophysics, 108 (1982), M.A. Alves, P.J. Oliveia, F.T. Pinho. A convegent and univesally bounded intepolation scheme fo the teatment of advection. Intenational Jounal fo Numeical Methods in Fluids, 41 (1) (2003), H.Q. Yang, A.J. Pzekwas. A compaative study of advanced shock-captuing schemes applied to Buges' euation. Jounal of Computational Physics, 102(1) (1992), B. Van Lee. Towads the ultimate consevative diffeence scheme. II. Monotonicity and consevation combined in a second-ode scheme. Jounal of Computational Physics, 14 (4) (1974), J. Hou, F. Simons, R. Hinkelmann. A new TVD method fo advection simulation on 2D unstuctued gids. Intenational Jounal fo Numeical Methods in Fluids, 71 (10) (2013), C.M. Oldenbug, K. Puess. Simulation of popagating fonts in geothemal esevois with the implicit Leonad total vaiation diminishing scheme. Geothemics, 29 (1) (2000), M. Čada, M. Toilhon. Compact thid-ode limite functions fo finite volume methods. Jounal of Computational Physics, 228 (11) (2009), W. Hundsdofe, B. Koen, M. Vanloon, J.G. Vewe. A positive finite-diffeence advection scheme. Jounal of Computational Physics, 117 (1) (1995), J. Liu, M. Delshad, G.A. Pope, K. Sepehnooi. Application of highe-ode flux-limited methods in compositional simulation. Tanspot in Poous media, 16 (1) (1994), M.S. Dawish, F. Moukalled. TVD schemes fo unstuctued gids. Intenational Jounal of Heat and Mass Tansfe, 46 (4) (2003), V. Pzulj, B. Basaa. Bounded convection schemes fo unstuctued gids. 15th AIAA Computational Fluid Dynamics Confeence, Anaheim, CA, (2001). 40. F.S. Lien, M.A. Leschzine. psteam monotonic intepolation fo scala tanspot with application to complex tubulent flows. Intenational Jounal fo Numeical Methods in Fluids, 19 (6) (1994), B. Song, G.R. Liu, K.Y. Lam, R.S. Amano. On a highe-ode bounded discetization scheme. Intenational Jounal fo Numeical Methods in Fluids, 32 (7) (2000), K.B. Kuan, C.A. Lin. Adaptive QICK-based scheme to appoximate convective 26

27 tanspot. AIAA Jounal, 38 (12) (2000), H. Jasak, H.G. Welle, A.D. Gosman. High esolution NVD diffeencing scheme fo abitaily unstuctued meshes. Intenational Jounal fo Numeical Methods in Fluids, 31 (2) (1999), L.X. Li, H.S. Liao, L.J. Qi. An impoved -facto algoithm fo TVD schemes. Intenational Jounal of Heat and Mass Tansfe, 51 (3) (2008), J. Hou, F. Simons, R. Hinkelmann. Impoved total vaiation diminishing schemes fo advection simulation on abitay gids. Intenational Jounal fo Numeical Methods in Fluids, 70 (3) (2012), S. Bidadi, S.L. Rani. Quantification of numeical diffusivity due to TVD schemes in the advection euation. Jounal of Computational Physics, 261 (2014), B. Van Lee. pwind-diffeence methods fo aeodynamic poblems govened by the Eule euations. In Lage-scale Computations in Fluid Mechanics, 1 (1985), B.P. Leonad. The LTIMATE consevative diffeence scheme applied to unsteady one-dimensional advection. Compute Methods in Applied Mechanics and Engineeing, 88 (1) (1991), P.H. Gaskell, A.K.C. Lau. Cuvatue-compensated convective tanspot: SMART, A new boundedness-peseving tanspot algoithm. Intenational Jounal fo Numeical Methods in Fluids, 8 (6) (1988), C. Bune, R. Waltes, Paallelization of the Eule euations on unstuctued gids, AIAA Pape, ( ) (1997),

28 Table 1 Accuacy of vaious -facto algoithms fo ten classical SS-TVD flux-limites on abitay unstuctued gids in Test 1, 2, 3 and 4. Scheme Step Sine-Suae Semi-Ellipse Double-Step Rotation Dawish s FFISAM Dawish s FFISAM Dawish s FFISAM Dawish s FFISAM Supebee 6.88E E E E E E E E-02 Koen 8.10E E E E E E E E-02 MSCL 8.17E E E E E E E E-02 WACEB 8.07E E E E E E E E-02 TCDF 8.28E E E E E E E E-02 MIST 9.00E E E E E E E E-02 Minmod 9.77E E E E E E E E-02 OSPRE 8.87E E E E E E E E-02 Albada 9.04E E E E E E E E-02 Hamonic 8.63E E E E E E E E-02 Table 2 Accuacy of vaious -facto algoithms fo ten classical SS-TVD flux-limites on an unstuctued tiangula gid in Test 4. Algoithm Supebee Koen MSCL WACEB TCDF MIST Minmod OSPRE Albada Hamonic Dawish s 1.11E E E E E E E E E E-02 Hou s 1.08E E E E E E E E E E-02 FFISAM 1.01E E E E E E E E E E-02 28

29 Figue 1. A gaphical epesentation of the SS-TVD citeion (viz. Sweby s TVD citeion) and the k-schemes. Figue 2. A gaphical epesentation of seveal existing classical SS-TVD flux-limites. 29

30 Figue 3. The moe appopiate notation adopted fo abitay unstuctued gids and a gaphical epesentation of Dawish s -facto algoithm. Figue 4. A gaphical epesentation of Li s -facto algoithm on two-dimensional abitay unstuctued gids. Figue 5. A gaphical epesentation of Hou s -facto algoithm on two-dimensional unstuctued tiangula gids. 30

31 Figue 6. A gaphical epesentation of the newly-poposed -facto algoithm (FFISAM) on abitay unstuctued gids (fo the inne domain). Figue 7. A gaphical epesentation of the newly-poposed -facto algoithm (FFISAM) on abitay unstuctued gids (at the bounday). 31

32 Figue 8. Pue convection of a step pofile by a unifom velocity field. Figue 9. Physical domain and abitay unstuctued mesh used fo the pue convection of a step pofile. Figue 10. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (Koen / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. 32

33 Figue 11. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (Supebee / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 12. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (MSCL / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 13. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (WACEB / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. 33

34 Figue 14. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (MIST / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 15. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (OSPRE / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 16. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (Albada / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. 34

35 Figue 17. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (Hamonic / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 18. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (Minmod / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 19. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 1 (TCDF / Step): (a) scala pofiles; (b) decay of the nom of the esiduals. 35

36 Figue 20. Pue convection of a sine-suae pofile by a unifom velocity field. Figue 21. Physical domain and abitay unstuctued mesh used fo the pue convection of a sine-suae pofile. Figue 22. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 2 (Koen / Sine-suae): (a) scala pofiles; (b) decay of the nom of the esiduals. 36

37 Figue 23. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 2 (Supebee / Sine-suae): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 24. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 2 (MSCL / Sine-suae): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 25. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 2 (WACEB / Sine-suae): (a) scala pofiles; (b) decay of the nom of the esiduals. 37

38 Figue 26. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 2 (TCDF / Sine-suae): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 27. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 2 (OSPRE / Sine-suae): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 28. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 2 (Albada / Sine-suae): (a) scala pofiles; (b) decay of the nom of the esiduals. 38

39 Figue 29. Pue convection of a semi-ellipse pofile by a unifom velocity field. Figue 30. Physical domain and abitay unstuctued mesh used fo the pue convection of a semi-ellipse pofile. Figue 31. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 3 (Koen / Semi-ellipse): (a) scala pofiles; (b) decay of the nom of the esiduals. 39

40 Figue 32. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 3 (Supebee / Semi-ellipse): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 33. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 3 (MSCL / Semi-ellipse): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 34. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 3 (WACEB / Semi-ellipse): (a) scala pofiles; (b) decay of the nom of the esiduals. 40

41 Figue 35. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 3 (TCDF / Semi-ellipse): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 36. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 3 (MIST / Semi-ellipse): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 37. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 3 (Albada / Semi-ellipse): (a) scala pofiles; (b) decay of the nom of the esiduals. 41

42 Figue 38. Pue convection of a double-step pofile in a otational velocity field. Figue 39. Physical domain and abitay unstuctued mesh used fo the pue convection of a double-step pofile in a otational velocity field. Figue 40. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 4 (Koen / Double-step Rotation / Abitay unstuctued gid): (a) scala pofiles; (b) decay of the nom of the esiduals. 42

43 Figue 41. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 4 (Supebee / Double-step Rotation / Abitay unstuctued gid): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 42. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 4 (WACEB / Double-step Rotation / Abitay unstuctued gid): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 43. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 4 (TCDF / Double-step Rotation / Abitay unstuctued gid): (a) scala pofiles; (b) decay of the nom of the esiduals. 43

44 Figue 44. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 4 (OSPRE / Double-step Rotation / Abitay unstuctued gid): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 45. Compaison of accuacy and convegence fo vaious -facto algoithms (Cu=1.0) in Test 4 (Hamonic / Double-step Rotation / Abitay unstuctued gid): (a) scala pofiles; (b) decay of the nom of the esiduals. Figue 46. Physical domain and unstuctued tiangula mesh used fo the pue convection of a double-step pofile in a otational velocity field. 44