Efficient Gradient Stencils for Robust Implicit Finite-Volume Solver Convergence on Distorted Grids

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1 Pepint accepted in Jounal of Computational Physics (oiginally submitted in August, 208). Download the jounal vesion at Efficient Gadient Stencils fo Robust Implicit Finite-Volume Solve Convegence on Distoted Gids Hioaki Nishikawa National Institute of Aeospace, Hampton, VA 23666, USA Mach 2, 209 Abstact Two gadient-stencil augmentation techniques ae discussed: symmetic and F-deceasing augmentations, fo efficient and obust implicit finite-volume-solve convegence on distoted unstuctued gids. The fome augments a face-neighbo stencil with exta cells to incease the symmety of the stencil as much as possible, and the latte adds futhe exta cells to decease the ecipocal of the Fobenius nom of a scaled leastsquaes matix to minimize the lowe bound of the magnitude of the gadient. These techniques ae poposed as efficient ways of ovecoming a known stability issue with a face-neighbo stencil. It is demonstated that the F-deceasing augmentation in combination with the symmetic augmentation yields obust and efficient gadient stencils on highly distoted quadilateal and tiangula gids. Intoduction Stability of implicit finite-volume solves widely used in pactical unstuctued-gid codes [, 2, 3, 4, 5, 6] is affected by vaious algoithmic components, e.g., elaxation schemes, Jacobian constuction, pseudo time steps, limite functions, discetizations, and so on. Among othes, a least-squaes (LSQ) stencil used to compute gadients is also known to impact the stability of finite-volume solves. In Ref.[7], Haide et. al. studied stability of an explicit cell-centeed finite-volume scheme fo advection in elation to the extent of gadient stencils. They have shown that the scheme can be unstable on tetahedal and hybid gids if the gadients ae computed with face-connected neighbos, and concluded that the LSQ gadient stencil should be extended beyond the face neighbos to ensue stability. Late, Zangeneh and Gooch demonstated also that a lage gadient stencil impoved stability [8]. An extended stencil can be constucted by augmenting the face-neighbo stencil with exta cells. A popula appoach is to use face neighbos of the face neighbos (see, e.g., Refs.[6, 9]); the esulting stencil is efeed to as the face2 stencil. Anothe is to use the cells shaing the vetex of the cell of inteest (see, e.g., Ref.[0]); the esulting stencil is called hee the vetex stencil. These augmented stencils ae widely used in pactical cell-centeed finite-volume solves. Howeve, the stability comes with the additional cost of pocessing a lage numbe of neighbos in each gadient calculation, especially in the vetex stencil. The numbe of vetex neighbos is not bounded, and can be unlimitedly lage. Fo example, it can be as lage as hundeds if a vetex of a cell is at a pola singulaity, fom which hundeds of gid lines ae oiginated []. If not hundeds, the numbe of vetex neighbos is known to be aound 50 on aveage in pactical unstuctued gids [0]. It is, theefoe, of geat pactical inteest to constuct a gadient stencil as obust as the vetex and face2 stencils but with a minimal numbe of neighbos. Vey few studies, howeve, ae found in the liteatue on efficient gadient stencil constuction fo obust implicit finite-volume-solve convegence on unstuctued gids. One appoach is the smat augmentation technique in Ref.[2], whee fo each vetex of a cell, an exta cell closest to the cell centoid is chosen fom cells suounding the vetex and added to the face neighbo stencil. Although it bings some impovements, it only adds as many cells as the numbe of vetices, which may not be enough to ensue stability. Ou expeience shows that an implicit solve diveges with the smat augmentation stencil on highly distoted gids. Anothe appoach is discussed in Ref.[3], which selects neighbos to constuct a fou- o thee-point stencil that esembles a stuctued-gid stencil. Howeve, they epot that the stencil exhibits an eatic behavio fo diffeent cell types, and conclude that the Geen-Gauss method is moe eliable. Yet anothe appoach can be found in Ref.[4], which attempts to constuct body-fitted gadient stencils based on bounday infomation. Impovements in accuacy ae epoted, but the algoithm does not addess the stability issue aised in Ref.[7]. Associate Reseach Fellow (hio@nianet.og), 00 Exploation Way, Hampton, VA USA,

2 n jk j m x jm x km k Figue : A tiangula gid fo a cell-centeed finite-volume discetization. The gids used in Ref.[4] ae not as distoted as those with which we ae concened, and thus emains to be demonstated fo pactical unstuctued gids in thee dimensions. This pape is intended to shed light on efficient gadient stencil constuction fo obust implicit solve convegence on unstuctued gids. Focusing on the stencil augmentation, we discuss two novel appoaches: symmetic and F-deceasing augmentations. Symmetic augmentation is motivated by the fact that a cell-centeed finitevolume scheme with a face-neighbo stencil is unstable on tetahedal gids but stable on hexahedal gids [7]. It is conjectued that a stencil having a symmetic stuctue, (i.e., consists of pais of neighbo cells located symmetically with espect to the cell of inteest), impoves the stability of implicit solves. The F-deceasing augmentation is based on anothe obsevation that an implicit finite-volume solve is stable with zeo gadients (e.g., fist-ode scheme o with a limite function). It implies that a gadient stencil minimizing the magnitude of the gadient impoves the stability of implicit solves. The F-deceasing augmentation attempts to ceate such a stencil by deceasing the ecipocal of the Fobenius nom of a scaled LSQ matix that detemines the lowe bound of the magnitude of the gadient. These augmentations ae demonstated numeically fo inviscid poblems with highly distoted gids. This pape focuses on implicit solves, which ae employed in many pactical unstuctued-gid codes [, 2, 3, 4, 5, 6]. Howeve, it is equally elevant to explicit solves because the main focus is on the spatial discetization and the gadient stencil is known to impact the stability of explicit solves also [7]. Hee, we focus on the effect of gadient stencils on iteative convegence athe than gadient accuacy because the ability to obtain solutions is moe citically impotant in pactical computations. Accuacy is not elevant unless a solution can be obtained. Nevetheless, we conside LSQ methods that ae exact fo linea functions (i.e., fist-ode accuate gadients) on abitay gids, and do not conside the Geen-Gauss gadient, which is, as is well known, zeo-th ode accuate on iegula gids. In contast to Ref.[8], whee stencils ae identified that cause instability by an eigenvalue analysis of a global esidual Jacobian and augment only such stencils, we seek simple stencil augmentation methods that can be eadily applied to existing codes without foming the exact second-ode esidual Jacobian and computing its eigenvalues. 2 Implicit Finite-Volume Solve Conside the steady Eule equations: x f + y g = 0, () 2

3 whee f = f(u) and g = g(u) ae the fluxes, and u is the vecto of consevative vaiables, which is discetized at a cell j (see Figue ) by a cell-centeed finite-volume discetization: Res j = Φ jk A jk = 0, (2) k {k j} whee {k j } is a set of face neighbos of the cell j, A jk is the magnitude of the scaled face-nomal vecto n jk acoss j and k: A jk = n jk. The numeical flux Φ jk is defined by the Roe flux [5]: Φ jk = 2 [f n(u L ) + f n (u R )] 2 A n (u R u L ), (3) whee f n = f ˆn x + gˆn y, ˆn jk = n jk /A jk = (ˆn x, ˆn y ), A n = f n / u evaluated by the Roe-aveages [5], and the subscipts L and R denote linealy econstucted solutions fom within the cell j and the neighbo cell k, espectively: u L = u j + u j x jm, u R = u k + u k x km, (4) whee u j and u k ae the solution values stoed at the centoids of the cell j and k, espectively, and x jm and x km ae the vectos pointing fom the centoids of j and k, espectively, to the face midpoint. To pefom the econstuction, the gadients, e.g., u j and u k, need to be computed at all cells fom the cuent solution values stoed at the cell centoids. The LSQ method is widely used fo the gadient calculation on unstuctued gids; the main subject of this pape is the impact of the LSQ stencil on the implicit finite-volume solve. The esulting global system of esidual equations, 0 = Res(U h ), (5) whee U h denotes the global solution vecto, is solved by an implicit defect-coection solve, which is epesentative of pactical unstuctued-gid solves [, 2, 3, 4, 5, 6].: U n+ h = U n h + U h, (6) Res U U h = Res(U n h), (7) whee n is the iteation counte, and the Jacobian Res/ U is the exact diffeentiation of the fist-ode vesion of the esidual Res (i.e., zeo LSQ gadients). The linea system (7) is elaxed by the Gauss-Seidel elaxation scheme to a specified toleance o a fixed numbe of elaxations. Note that the implicit solve is elevant not only to steady poblems but also to unsteady poblems, whee a global system of esidual equations needs to be solved at each time step fo implicit time-stepping schemes. The stability of the implicit solve elies on vaious algoithmic components and paametes as mentioned ealie. Ou focus is on the effect of the LSQ gadient stencil. Theefoe, we fix all othe paametes: 50 Gauss-Seidel elaxations pe iteation, no limites, the Roe flux and its exact Jacobian. A pseudo-time tem is added to the Jacobian in a testbed code used hee, but the CFL numbe fo a pseudo time is set to be 0 6 by default. The CFL numbe will be adjusted based on the esidual convegence behavio (e.g., educed by an ode of magnitude if the esidual inceases); it is typically applied only duing the initial tansient and the CFL numbe quickly eachess Least-Squaes Gadient Method The LSQ gadient method is based on a polynomial fit ove a set of neaby cells. Fo second-ode finitevolume schemes, the gadients need to be at least fist-ode accuate on geneal unstuctued gids; and thus it suffices to fit a linea polynomial. Suppose we wish to compute the gadient of a solution vaiable u at a cell j, and have a set {g j } of N( 2) neaby cells (i.e., gadient stencil) available fo fitting the linea polynomial: u k = u j + x u j (x k x j ) + y u j (y k y j ), (8) whee k {g j }, (x j, y j ) and (x k, y k ) denote the centoid coodinates of the cell j and the neighbo k, espectively, and x u j and y u j ae the deivatives we wish to compute. As the numbe of neighbos often exceeds two on unstuctued gids, the polynomial fit typically leads to an ovedetemined poblem: Ax = b, (9) 3

4 whee A = w (x x j ) w (y y j ).... w k (x k x j ) w k (y k y j ).. w N (x N x j ) w N (y N y j ) [ x u j, x = y u j ], b = w (u u j ).. w k (u k u j ). w N (u N u j ), (0) and w k is the weight applied to the equation coesponding to the neighbo cell k. The following invese-distance weight is widely used in finite-volume methods: w k = d p, d k = (x k x j ) 2 + (y k y j ) 2, () k whee p is a paamete, e.g., zeo (unweighted LSQ), one (fully weighted LSQ), o any othe eal value. The ovedetemined LSQ system can be solved in vaious ways. One is the nomal equation appoach, whee we fom a 2 2 system by multiplying Equation (9) by the tanspose of A, denoted by A T, fom the left: and invet it (e.g., via Cholesky decompostion) to obtain the gadient: A T Ax = A T b, (2) x = ( A T A ) A T b. (3) Anothe is the QR factoization via Householde tansfomation [6], which diectly solves the ovedetemined system (9) as x = R Q T b, (4) whee Q is the othonomal matix and R is the uppe tiangula matix geneated fom A by the QR factoization [6]. In eithe appoach, the solution can be expessed in the following fom: [ x u j y u j ] = k {g j} [ c x jk c y jk ] (u k u j ), (5) whee c x jk and cy jk ae the LSQ coefficients to be computed and stoed at all cells once fo a given stationay gid. It is clea that the cost of the gadient calculation is diectly popotional to the numbe of neighbos involved in the gadient stencil. Note that the LSQ gadient calculation can incopoate bounday solutions o gadients if available fom a physical bounday condition [7]. Howeve, we do not conside using bounday values since such is not always possible (e.g., not all solution vaiables ae known at an outflow bounday). As mentioned in the Intoduction, the set of neighbos {g j } affects the stability of finite-volume solves. In fact, it has been known by pactitiones that adding moe neighbos tends to stabilize a finite-volume solve. Ref.[7] has fomally shown that a finite-volume scheme is unstable with the face-neighbo gadient stencil on tetahedal and hybid gids, and adding exta cells to the stencil cues the instability. Late, Ref.[8] also showed that a lage stencil size led to stability. In this egad, the most obust stencil would be the vetex stencil on tiangula/tetahedal gids and the face2 stencil on quadilateal/hexahedal gids [6], but these stencils substantially incease the cost of the gadient computation and equie a lage memoy fo stoing the LSQ coefficients, especially in thee dimensions [0]. Ou objective is to constuct a gadient stencil that achieves simila obustness with a fewe numbe of neighbos. Two augmentations ae poposed, which, in combination, would esult in a obust and efficient gadient stencil. 4

5 v x v x j θ j x k x j k Figue 2: Symmetic stencil constuction: a cell is chosen fo a face k, fom those between the dotted lines defined by the angle θ, whose centoid is located acoss fom the neighbo k and closest to the (dash) line passing though j and k. The dotted line on the ight is a eflection of the left one with espect to the dash line. (a) Vetex stencil (20 cells). (b) Face neighbo stencil (3 cells). (c) Face2 stencil (9 cells). (d) Symmetic stencil (6 cells). Figue 3: Gadient stencils. Stencil membes ae coloed within the vetex stencil, and so all cells ae coloed in the vetex stencil. 5

6 4 Appoaches to LSQ Stencil Augmentation 4. Symmetic Augmentation We begin with the face neighbo stencil {k j } fo a cell j, and add to it cells fom the union of the vetex and face2 neighbos {v j } that will symmetize the stencil as much as possible. The symmetic augmentation begins with one of the face neighbos, k {k j }, and seach fo a cell located symmetically with espect to the centoid of the cell j fom the face neighbo k. It is pefomed as follows. Fist, define the initial symmetic stencil, denoted by {sym j }, as the face neighbo stencil: Then, fo each k {k j }, define a unit vecto pointing fom j to k: {sym j } {k j }. (6) e jk = x k x j x k x j, (7) whee x j is the centoid coodinate vecto of the cell j, and x k is the centoid coodinate vecto of the face neighbo k. See Figue 2. Fo a bounday face, we define x k by the face-midpoint. Then, fo all v {v j } except those aleady in the cuent stencil, compute the diection cosine: d v = x v x j x v x j e jk, (8) whee d v, and x v is the centoid coodinate vecto of the cell v. Define a subset {v j } of {v j} satisfying the following condition: d v < cos(θ), θ = 3π 4, (9) whee θ = 3π/4 has been chosen, instead of π/2, to exclude those located fa fom the line passing though x j and x k. The subset {v j } contains, theefoe, a goup of cells in the egion between the two dotted lines in Figue 2. Note that d v < 0 fo all v in {v j }. Finally, select the one with the minimum d v, and add it to the stencil {sym j }. Pefoming the same fo all faces {k j }, we aive at the final symmetic augmentation stencil {sym j }. The algoithm as descibed is diectly applicable to abitay gids in two and thee dimensions. (a) Face neighbo stencil. (b) Symmetic stencil. Figue 4: Gadient stencils fo the cone cell. The face neighbo stencil has only one neighbo, leading to the ill-conditioning of the LSQ matix (left). The symmetic augmentation adds two exta cells, and esolves the ill-conditioning poblem (ight). Hee, the symmetic stencil tuns out to be identical to the vetex and face2 stencils. The algoithm will find one exta cell pe face, but it may fail nea a bounday with an empty set {v j }, indicating that no cells exist on the othe side. In such a case, we do not add any cell and poceed to the next face. If a symmetic cell is found fo all faces, the esulting stencil will have a symmetic configuation, i.e., the stencil consists of pais of neighbo cells located symmetically with espect to the cell of inteest, simila to a face-neighbo stencil on a egula quadilateal gid that leads to stability [7]. An example is shown and compaed with othe stencils in Figue 3, whee all stencils ae indicated by coloed tiangles ove the vetex stencil fo the cell of inteest (white). Fo this iegula gid (extacted fom the gid in Figue 6), the vetex stencil has 20 neighbos as shown in Figue 3(a). On the othe hand, the numbe of cells in the face-neighbo and face2 stencils ae limited by the numbe of faces: thee and nine. The symmetic stencil is shown in Figue 6

7 3(d). It has an extended stencil with six neighbos (compae it with the face neighbo stencil), addessing the instability issue [7] with a fewe numbe of neighbos than the vetex and face2 stencils, and having pais of neighbo cells appoximately located in a symmetical configuation. Note that the use of the bounday face centoid in the symmetic augmentation has the impotant effect fo a cell having only one neighbo that it povides enough neighbos to avoid ill-conditioning of the LSQ poblem as illustated in Figue 4. Finally, it is pointed out that the symmetic augmentation adds two emote cells in each coodinate diection fo an othogonal hexahedal/quadilateal gid. Numeical expeiments show that the symmetic augmentation geatly impoves convegence on distoted gids, but it can still lead to instability on highly-cuved gids. This bings the second step: F-deceasing augmentation, which is the subject fo the next section. 4.2 F-Deceasing Augmentation To futhe augment the symmetic stencil, we intoduce anothe appoach based on the magnitude of the gadient. Conside the nomal equation: A T Ax = A T b, (20) whee A T A = wk x 2 2 k k {g j} w 2 k x k y k wk x 2 k y k k {g j} w 2 k y 2 k, x = xu j y u j, A T b = wk x 2 k u k k {g j} w 2 k y k u k, (2) k {g j} k {g j} k {g j} x k = x k x j, y k = y k y j, u k = u k u j, k =, 2,,, N. (22) Let us escale the system without changing the solution x: s A T Ax = b, (23) whee b = s A T b, s = k {g j} w 2 kd k, (24) so that b = O( u), whee u is a typical solution vaiation ove the stencil. It is easy to show that the L 2 nom 2 of the vecto s A T Ax is bounded as follows: s A T Ax 2 2 = 2 l t ix 2 i= 2 l i 2 2 x 2 2 = s A T A 2 F x 2 2, (25) i= whee l t i denotes the i-th ow of the matix s A T A, the inequality is due to the Cauchy-Bunyakovsky-Schwaz inequality applied to each i, and F denotes the Fobenius nom defined by B F = m i= n j= b i,j 2 fo an abitay m n matix B. Substituting Equation (23) into the left hand side, we deive x 2 ( b 2 b 2 s s A T = A F s A T = A F A T A F ) b 2. (26) Theefoe, the magnitude of the gadient is bounded fom below, and the lowe bound is detemined by the measue F defined by F = s A T A F. (27) fo a given nonzeo solution vaiation b 2. Note that the measue F only detemines the lowe bound, and thus it does not necessaily pedict the magnitude of the gadient accuately. This is also because the magnitude of the vecto b can also vay with the stencil although it emains of O( u). Nevetheless, the measue F can be easily and efficiently computed fo a given stencil, and has been found to seve well as a guide fo stencil 7

8 (a) Symmetic/F-deceasing stencil {symf j} (7 cells). (b) Face/F-deceasing stencil {facef j} (6 cells). Figue 5: F-deceasing augmentation applied to the symmetic and face-neighbo stencils. The F-deceasing augmentation has added one exta cell to the symmetic stencil, and thee exta cells to the face-neighbo stencil. augmentation. Ou poposal is to augment a gadient stencil with cells that will decease the measue F, i.e., potentially educing the magnitude of the gadient and thus giving an effect of stabilizing the implicit solve. In this note, the augmentation based on F is efeed to as the F-deceasing augmentation. It is an independent algoithm that can be applied to any gadient stencil. Hee, we apply it to the symmetic stencil to geneate a futhe impoved stencil {symf j }:. Constuct the symmetic stencil {sym j } as descibed in Section Compute A T A and s with {g j } = {sym j }, and then F. 3. Let F 0 = F, (A T A) 0 = A T A, s 0 = s, and {symf j } = {sym j }. 4. Let {v R j } be a subset of cells in {v j} not in {symf j }, and N R be the numbe of cells in {v R j }. 5. If N R = 0, no futhe augmentation is possible. Stop. 6. Fo m = to N R, pefom the following: () Compute A T A and s by adding the contibution of the m-th cell at (x m, y m ): A T A = (A T A) 0 + w 2 m x 2 m w 2 m x m y m, s = s 0 + w 2 md m, (28) w 2 m x m y m w 2 m y 2 m whee x m = x m x j, y m = y m y j, d m = x 2 m + ym, 2 w m = d p. (29) m (2) Compute F = s/ A T A F. (3) If F < KF 0 add the m-th cell to {symf j } and set F 0 = F, (A T A) 0 = A T A, and s 0 = s. The set {v j } is the union of the vetex and face2 stencils, but a lage set may be used if needed. Fo the facto K, we set K = 0.85 in ode to accept only significant eductions. The algoithm is efficient as it does not equie any matix invesion, and diectly applicable to abitay gids in two and thee dimensions. Note that the algoithm is dependent on the ode of {vj R } applied in Step 6, but finding the best fom all possible odeings can take a significant amount of effot and thus the effect of odeing is not studied in this wok. Figue 5(a) shows a stencil obtained by the F-deceasing augmentation applied to the symmetic stencil in the pevious example. It can be seen that one additional cell has been added to the symmetic stencil, deceasing the measue F fom 34.0 to Compae it with Figue 3(d). The total numbe of neighbos is 7, and it is still significantly smalle than that of the vetex stencil. Shown in Figue 5(b) is a stencil obtained by applying the 8

9 (a) Entie gid with the aifoil located at the cente. (b) Close view. Figue 6: Iegula tiangula gid fo the Joukowsky aifoil case. F-deceasing augmentation to the face-neighbo stencil, which is denoted by facef. In this case, the F-deceasing augmentation has added thee exta cells, deceasing the measue F fom 67.5 to Compae it with Figue 3(b). As will be shown late, thee ae cases, whee the symf stencil successfully stabilizes the implicit solve, but the facef stencil fails to do so. The F-augmentation is not intended to addess all poblems, and theefoe is best used in combination with the symmetic augmentation athe than the face-neighbo stencil. Ref.[2] states that a Fobenius nom is used as an estimate of the condition numbe of a LSQ system to futhe augment the smat-augmentation stencil fo the pupose of impoving gadient accuacy. It is not clea if they used s A T A F (because the equation is not shown in Ref.[2]), but it is inteesting that they added exta cells to educe it wheeas we add cells to incease it (and theeby decease the lowe bound of the gadient magnitude). Appaently, the fome just attempts to coect ill-conditioned LSQ systems, not to impove gadient accuacy in the geneal sense due to the lack of a igoous poof fo the elationship between the level of the gadient eo and a Fobenius nom. On the othe hand, ou objective is clea and mathematically justified: to decease the lowe bound of the gadient magnitude. It is impotant to note that Equation (26) simply states that a lage Fobenius nom will decease the lowe bound fo a given b 2. It does not guaantee that the magnitude of the gadient is actually educed, no does it indicate anything about accuacy. Typically, gadient eos incease fo a lage stencil, and such could be the case fo F-augmented stencils; howeve we focus on iteative convegence athe than accuacy fo the eason mentioned in the Intoduction. It is emphasized that we only popose it as a pactical measue to guide a stencil augmentation, i.e., a means to add exta cells to a given initial stencil. 5 Numeical Expeiments The sym and symf stencils ae compaed with the vetex, face2, facef, and face-neighbo stencils fo inviscid poblems on highly iegula gids. Ou focus is hee on iteative convegence, which is citically impotant as thee will be no point of speaking about accuacy if numeical solutions cannot be obtained. 5. Highly iegula tiangula gid Gadient stencils ae compaed fo an inviscid flow ove a Joukowsky aifoil. A tiangula gid was geneated ove a domain defined by the aifoil of a unit chod and the oute bounday located at a distance of 50. See Figue 6. Initially, a smooth quadilateal gid was geneated. It was then subdivided into tiangles by andom diagonal insetions, and then futhe andomized by diagonal swapping. The esulting gid has 4235 nodes and 8228 tiangles (2 nodes on the aifoil) with the minimum, maximum, and aveage cell aeas being.97e-06, 7.3, and 0.95, espectively. The skewness measue defined by the dot poduct of the unit face nomal vecto 9

10 (a) Numbe of neighbos fo vaious gadient stencils: =face-neighbo, 2=face2, 3=vetex, 4=sym, 5=symF, 6=faceF. Maximum esidual nom face neighbo face2 vetex sym symf facef Iteation (b) Residual vesus iteation. Figue 7: Results fo the Joukowsky aifoil case. and the unit vecto pointing fom a cell centoid to that of a face neighbo is 0.67 on aveage with the minimum Such a seveely-distoted gid is typically avoided in pactice, but it epesents highly iegula natue that can easily aise locally on unstuctued gids fo complex geometies in thee dimensions, and thus it is of geat pactical inteest to see if an algoithm is obust enough to deal with such sevee gid iegulaity. Figue 7(a) compaes the numbe of neighbos fo diffeent gadient stencils. As can be expected, the vetex stencil has the lagest lage numbe of neighbos. The next lagest stencil is the face2 stencil followed by the symf and facef stencils. Sample stencils ae shown in Figues 3 and 5. To investigate the impact of gadient stencils on the implicit solve convegence, we conside a subsonic flow of Mach numbe 0.3 at an angle of attack.25 degees. A slip bounday condition is applied to the aifoil, and a fee steam condition is applied to the oute bounday. These bounday conditions ae imposed weakly though the numeical flux. Fo the gadient computation, we use only neighbo cells and thus skip bounday faces. The implicit solve was used to educe the maximum L esidual nom among the equation set by fou odes of magnitude. All solve paametes ae fixed as mentioned in Section 2. Also, the powe p to the invese distance weights is fixed as p = 0. fo all cases in ode to avoid the stability poblem with p = known fo implicit inviscid flow solves [8]. The LSQ coefficients ae computed by the QR factoization, which is known to be moe stable (in solving the LSQ poblem) than the nomal equation appoach that suffes fom a lage squaed condition numbe. The convegence esults ae shown in Figue 7(b). As can be seen, the solve diveged with the face-neighbo, face2, and facef stencils. The instability was found to emege in the lage-gadient egion aound the leading edge, whee a suction peak aises. On the othe hand, the solve conveged successfully with the vetex, sym, and symf stencils with the same CFL numbe adjustment fo the vetex and symf : 0 6, 0 5, 0 4, 0 3, 0 2, 0 2, and 0 6 fo the fist seven iteations, and then 0 6 fo the est of the iteation. The sym stencil equied anothe cycle of the same adjustment. The esults indicate, as desied, that the implicit solve conveges with a significantly fewe numbe of neighbos fo the sym and symf stencils. Although it took slightly moe iteations to convege, the sym stencil is vey efficient since it uses only about a half of the vetex neighbos. The convegence is vey simila fo the vetex and symf stencils; the symf stencil also seves as an efficient stencil with a fewe numbe of neighbos than the vetex stencil. Finally, the instability obseved fo the facef stencil indicates that the F-deceasing augmentation alone is not sufficient to stabilize the solve. It diveges even with K =, i.e., all neighbos ae added that educe the measue F by any amount. No significant speed-up in computing time is, howeve, obseved by the sym and symf stencils since the gadient calculation is only a faction of the entie implicit solve algoithm that is lagely dominated by the 0

11 esidual and Jacobian calculations. Nevetheless, one saves a significant amount of memoy with the sym, and symf stencils, and each gadient calculation is indeed made cheape than with the vetex stencil. Saving in computing time and memoy fo the implicit solve would be moe significant on thee-dimensional gids, whee the sym, and symf stencils ae expected to be substantially smalle than the vetex and face2 stencils. 5.2 Cuved gids To futhe demonstate the impact of the symf stencil on the solve stability, we conside cuved high-aspectatio gids, which ae typical in high-reynolds-numbe tubulent-flow simulations and known to intoduce sevee stability issues [8, 9, 20, 2]. Cuvatue intoduces an additional stability challenge to high-aspect-atio gids as discussed in Ref.[8]; theefoe, if the solve is stable on cuved high-aspect-atio gids, it is likely to be stable on non-cuved high-aspect-atio gids. We ceate a cuved domain fom a unit squae domain (ξ, η) [0, ] [0, ] by the following mapping: x = (0.002η + ) cos(θ), y = (0.002η + ) sin(θ), θ = 2 (π + θ ext) ξθ ext, (30) whee θ ext = π/4, and solve the Eule equations with souce tems deived such that the following function is the exact solution fo all vaiables: ( sin 00π ) x 2 + y 2 + π/ (3) The domain spans ove the hoizontal axis appoximately fom x = 0.4 to x = 0.4, and has the thickness about (see Figue 8(a)). The solution vaies pedominantly in the adial diection, modeling a typical bounday laye solution. The combination of the cuvatue, the high-aspect-atio, and a lage solution vaiation in the adial diection pesents a sevee difficulty in gadient accuacy and solve stability [8, 9, 20, 2]. The bounday condition is imposed weakly though the numeical flux with the ight state specified by the exact solution. Fo the gadient computation, we use only neighbo cells and skip bounday faces unless othewise stated. As in the pevious case, the implicit solve is used again to educe the maximum esidual nom by fou odes of magnitude with the same set of paametes. Howeve, in this case, we keep CFL= 0 6 fom the fist iteation to the last iteation. Fo this type of gid, an appoximate mapping technique that maps a local stencil to a body-fitted coodinate system is known to alleviate the poblems [8]. Howeve, it equies a distance function computation and is not simple to implement; hee we investigate whethe stencil augmentations esolve convegence issues y.3 y x (a) Entie gid x (b) Close view in scaled coodinates. Figue 8: A cuved quadilateal gid. We begin with a quadilateal gid with 8 nodes and 64 cells as shown in Figue 8. The gid spacing is unifom in the adial diection, but not in the cicumfeential diection, whee the gid is gently stetched towads the vetical line at x = 0. Figue 9(a) shows the compaison of the numbe of neighbos fo diffeent

12 Maximum esidual nom face neighbo face2 vetex sym symf facef Iteation (a) Numbe of neighbos fo vaious gadient stencils: =face-neighbo, 2=face2, 3=vetex, 4=sym, 5=symF, 6=faceF. (b) Residual vesus iteation. Figue 9: Results fo the cuved quadilateal gid case (a) Vetex stencil (8 cells). (b) Face neighbo stencil (4 cells) (c) Face2 stencil (2 cells). (d) Symmetic stencil (8 cells) (e) symf stencil (0 cells). (f) facef (6 cells). Figue 0: Gadient stencils fo the cuved quadilateal gid shown in the pola coodinates. Stencil membes ae coloed and shown ove the face2 stencil, and so all cells ae coloed in the face2 stencil. 2

13 Figue : The tiangula gid with egula connectivity shown in the pola coodinate system. gadient stencils. Fo quadilateal gids, the face2 stencil is the lagest as is well known [6, 9]. The next lagest is the symf stencil, and then the sym, vetex, facef, and face-neighbo stencils, in ode of deceasing size. It is impotant to note that facef and symf have diffeent sets of neighbos, and thus the F-deceasing augmentation depends on the initial stencil as expected. In fact, thee is no eason that the F-deceasing augmented stencil must contain a symmetic stencil. The convegence esults ae shown in Figue 9(b). The implicit solve diveged with the vetex and symmetic stencils, and conveged with othe stencils. The fact that it conveges with the face neighbo stencil indicates that no stencil augmentation is needed fo this poblem. The esults show, nevetheless, that the F-deceasing augmentation geneated gadient stencils smalle than the lagest and leads to stable iteative convegence in this paticula case. To gain some insight, let us compae the gadient stencils. Figue 0 shows the gadient stencils taken fom a epesentative inteio cell located nea the cente of the gid, displayed in the pola coodinates (, θ). Compaing the vetex stencil and the face neighbo stencil, we obseve that fou cone cells included in the vetex stencil intoduce a lage solution vaiation and seem to have destabilized the implicit solve (face 2 and symf also include the cone cells but have an additional featue as discussed futhe below). The same is obseved fo the symmetic stencil. Note that the symmetic augmentation picked up the two cone cells instead of the left and ight most cells in the θ diection due to the cuved natue of the gid combined with the non-unifom spacing in θ. The gid is ectangula in the (, θ) space because the nodes have been geneated as such, but the cell centoids ae not necessaily mapped onto a ectangula configuation if the gid spacing in θ is not unifom. In fact, we found that a simila stencil was obtained even by applying the symmetic augmentation in the (, θ) space. The fact that the face2 stencil leads to stable convegence indicates that the poblem is ovecome by adding these left and ight most cells. This is exactly what has been done by the F-deceasing augmentation. See Figues 0(e) and 0(f). Next, we conside a tiangula gid geneated fom the quadilateal gid by inseting diagonals in the same diection (i.e., between the bottom left node and the top ight node in a quadilateal). See Figue. The esulting gid has 28 tiangles in total. Figue 2(a) shows the compaison of the numbe of neighbos fo diffeent gadient stencils. The vetex stencil is the lagest stencil, followed by the face2, the symf, sym, facef, and face-neighbo stencils, in ode of deceasing size. Fo this gid, the discetization failed with the faceneighbo stencil due to ill-conditioned LSQ matices at cells having only one neighbo cell. Thee exist two such cells at the uppe left and lowe ight cones of the domain as indicated by the ed filled cicles in Figue. To avoid the poblem, we added manually the midpoints of thei bounday faces to the gadient stencils and use the exact solution to obtain the solution values at these points. The facef stencil does not suffe as the F-deceasing augmentation added an exta cell to these face-neighbo stencils. In the symf stencil, the poblem is aleady avoided by the symmetic stencil. Iteative convegence esults ae shown in Figue 2(b). The implicit solve conveged with the sym, symf, facef, and somewhat supisingly with the face-neighbo stencil, but diveged with the vetex and face2 stencils. The esults indicate that a lage numbe of neighbos does not necessaily stabilize the implicit solve, which pesents a counte example to the studies in Refs.[7, 8]. On the othe hand, the symmetic and F-deceasing augmentations have successfully added exta cells to stabilize the implicit solve. The convegence with the face-neighbo and symmetic stencils is, howeve, vey slow compaed with othe stencils. Figue 3 shows the inteio gadient stencils taken at a location close to the point (.46,.0004) in the gid. The success with the face-neighbo stencil may be due to its symmetic configuation: the top and bottom neighbos ae aligned vetically, and the left and bottom ones ae also nealy aligned in the θ diection. Note that due to the cuvatue and the non-unifom spacing in θ, again, the symmetic stencil is not pecisely symmetic in the (, θ) space. The symf and facef ae diffeent, but both successfully stabilized the implicit solve. 3

14 Maximum esidual nom face neighbo sym symf facef Iteation (a) Numbe of neighbos fo vaious gadient stencils: =face-neighbo, 2=face2, 3=vetex, 4=sym, 5=symF, 6=faceF. (b) Residual vesus iteation. Figue 2: Results fo the cuved tiangula gid case (egula connectivity) (a) Vetex stencil (2 cells). (b) Face neighbo stencil (3 cells) (c) Face2 stencil (9 cells). (d) Symmetic stencil (6 cells) (e) symf stencil (0 cells). (f) facef (7 cells). Figue 3: Gadient stencils fo the cuved tiangula gid having egula connectivity, shown in the pola coodinates. Stencil membes ae coloed within the vetex stencil, and so all cells ae coloed in the vetex stencil. 4

15 Figue 4: The tiangula gid with iegula connectivity shown in the pola coodinate system Maximum esidual nom face neighbo face2 vetex sym symf facef (a) Numbe of neighbos fo vaious gadient stencils: =face, 2=face2, 3=vetex, 4=sym, 5=symF, 6=faceF Iteation (b) Residual vesus iteation. Figue 5: Results fo the cuved tiangula gid case (iegula connectivity). Finally, we conside an iegula tiangula gid geneated by inseting diagonals with andom diections in the quadilateal gid (see Figue 4). The gid lines in the adial diection ae kept staight, which is a typical featue acoss a bounday laye in a pactical unstuctued gid. A stiking diffeence fom the pevious egula tiangulation is that the numbe of cells aound a node is no longe bounded. As expected, andom diagonal insetion has inceased the size of the vetex stencil significantly as shown in Figue 5(a). The next lagest is the face2 stencil, and then the symf, sym, facef, and face-neighbo stencils, in ode of deceasing size. In this gid, thee exist two cells having only one neighbo at the lowe left and lowe ight cones of the domain as indicated by the ed filled cicles in Figue 4; and we again employed the bounday face midpoints to avoid the ill-conditioning of the LSQ matix fo the face neighbo stencil. The facef stencil has no poblem since the F-deceasing augmentation added, again, an exta cell to these stencils (the exta cell is the one that appeas in Figue 6(f) but not in Figue 6(b)). Figue 5(b) shows the iteative convegence esults. The implicit solve diveged with the face, face2, facef, and sym stencils, and conveged with the vetex and symf stencils. As can be seen in Figue 6, whee the inteio gadient stencils ae taken again at a location close to the point (.46,.0004) in the gid, the face-neighbo stencil does not have a symmetic configuation because of the andom diagonal insetion. The sym stencil again does not pecisely symmetize the stencil in the (, θ) space. The esults demonstate that as intended, the F-deceasing augmentation impoved the sym stencil, and led to convegence with significantly fewe numbes of neighbos than the vetex stencil. As in the aifoil case, the facef stencil fails to stabilize the implicit solve. It confims again that the F-deceasing augmentation alone is not sufficient. Fo the conveged solutions, we compaed the discetization and gadient eos, and found that the eos wee vey simila: the L eos in the x-velocity component u ae 5.552E-02 (vetex) and 5.967E-02 (symf ), 5

16 (a) Vetex stencil (2 cells). (b) Face neighbo stencil (3 cells) (c) Face2 stencil (8 cells). (d) Symmetic stencil (6 cells) (e) symf stencil (7 cells). (f) facef (4 cells). Figue 6: Gadient stencils fo the cuved tiangula gid having iegula connectivity, shown in the pola coodinates. Stencil membes ae coloed within the vetex stencil, and so all cells ae coloed in the vetex stencil. the L eos in the gadient ae 4.057E+0 (vetex) and 3.952E+0 (symf ) fo x u, and 2.308E+02 (vetex) and 2.26E+02 (symf ) fo y u. Vey simila eo esults have been obtained fo othe vaiables and fo othe types of gids. Futhe discussion and investigation on gadient accuacy ae left as futue wok. 6 Concluding Remaks We have discussed two gadient-stencil augmentation methods fo obust implicit finite-volume-solve convegence: symmetic and F-deceasing augmentations. The symmetic augmentation constucts a gadient stencil with a symmetic popety simila to a face-neighbo stencil in a quadilateal gid. It is an attempt to develop an efficient emedy to addess the stability issue in a cell-centeed finite-volume scheme with the face-neighbo gadient stencil [7]. The F-deceasing augmentation adds exta cells to an existing stencil that decease the ecipocal of the Fobenius nom of a scaled LSQ matix, which detemines the lowe bound of the gadient magnitude. It is based on the obsevation that implicit finite-volume solves ae moe stable with smalle gadients. Numeical expeiments demonstate that the symmetic stencil indeed helps the implicit solve convege on a highly iegula tiangula gid with a significantly fewe numbe of neighbos than the vetex stencil. Fo cuved gids with vaious element types: quadilateal, egula tiangles, and andom tiangles, the symmetic stencil is not sufficient, and the F-deceasing augmentation has been found citically impotant fo stabilizing the implicit solve. Othe conventional stencils ae not entiely successful fo these cuved gids, leading to divegence. Only the symf stencil, which is constucted by the F-deceasing augmentation applied to the symmetic stencil, successfully stabilized the implicit solve fo all the test cases. It is noted, howeve, that all the conclusions ae not based on a igoous mathematical poof that guaantees stability. This pape was intended to shed light on efficient gadient stencil constuction and popose pactical guides fo a stencil augmentation. Fo a pactical unstuctued gid, the study suggests that one would employ the symmetic and F-deceasing augmentation nea viscous sufaces, and the symmetic augmentation elsewhee. Ideally, we would like to switch fom the latte to the fome when a stong cuvatue effect is detected, but developing a eliable geomety quantity to detemine a highly-cuved egion fo geneal thee-dimensional unstuctued gids emains an open 6

17 aea of eseach. Although the poposed techniques have been shown to bing significant impovements, the esulting gadient stencils ae not necessaily optimal, i.e., not the smallest stencil that stabilizes the implicit solve. To attempt such an optimization, a technique would need to be developed that emoves cells fom a stencil without losing obust convegence. Anothe topic left fo futue wok is the investigation of gidesolution effects, i.e., whethe the stability behavio changes on a fine gid whee the gadient is appopiately esolved. The poposed augmentation techniques ae independent of the gid connectivity unlike othe popula stencils (e.g., face2 and vetex), and can be diectly applied to a geneal point cloud in two and thee dimensions. These techniques may find thei applications in high-ode econstuction-based finite-volume methods [22] and second/thid-ode node-centeed edge-based methods [23] fo geneating efficient gadient stencils. Finally, it is emphasized that the pesent study has focused on only one of many algoithmic aspects affecting implicit solve stability. Existing techniques such as an appoximate mapping method [8], mesh optimizations [8], and Jacobian-Fee Newton Kylov methods [5], continue to be useful, and they may tun out to be even moe effective when combined with these impoved gadient stencils. Acknowledgments This wok was suppoted by the Hypesonic Technology Poject, though the Hypesonic Aibeathing Populsion Banch of the NASA Langley Reseach Cente, unde Contact No. 80LARC7C0004. The autho would like to thank Jeffey A. White (NASA Langley Reseach Cente) fo valuable comments and discussions. Refeences [] FUN3D online manual. [2] Desciption of the DLR TAU Code. [3] scflow, Softwae Cadle, MSC Softwae Company. visited on 0/30/208. [4] Y. Nakashima, N. Watanabe, and H. Nishikawa. Development of an effective implicit solve fo genealpupose unstuctued CFD softwae. In The 28th Computational Fluid Dynamics Symposium, C08-, Tokyo, Japan, 204. [5] Mohagna J. Pandya, Bois Diskin, James L. Thomas, and Neal T. Fink. Impoved convegence and obustness of USM3D solutions on mixed element gids. AIAA J., 54(9): , Septembe 206. [6] Jeffey A. White, Robet Baule, Badley J. Passe, Seth C. Spiegel, and Hioaki Nishikawa. Geometically flexible and efficient flow analysis of high speed vehicles via domain decomposition, pat, unstuctuedgid solve fo high speed flows. In JANNAF 48th Combustion 36th Aibeathing Populsion, 36th Exhaust Plume and Signatues, 30th Populsion Systems Hazads, Joint Subcommittee Meeting, Pogammatic and Industial Base Meeting, Newpot News, VA, 207. [7] F. Haide, J.-P. Coisille, and B. Coubet. Stability analysis of the cell centeed finite-volume MUSCL method on unstuctued gids. Nume. Math., 3(4): , [8] Reza Zangeneh and Cal Ollivie-Gooch. Mesh optimization to impove the stability of finite-volume methods on unstuctued meshes. Comput. Fluids, 56:590 60, 207. [9] Chandan B. Sejekan and Cal F. Ollivie-Gooch. Impoving finite-volume diffusive fluxes though bette econstuction. Comput. Fluids, 39:26 232, 206. [0] Andew W. Cay, Andew J. Dogan, and Moi Mani. Towads accuate flow pedictions using unstuctued meshes. In Poc. of 39th AIAA Fluid Dynamics Confeence, AIAA Pape , San Antonio, [] Hioaki Nishikawa, Bois Diskin, James L. Thomas, and D. H. Hammond. Recent advances in agglomeated multigid. In 5st AIAA Aeospace Sciences Meeting, AIAA Pape , Gapevine, Texas, 203. [2] A. Schwöppe and B. Diskin. Accuacy of the cell-centeed gid metic in the DLR TAU-code. In A. Dillmann, G. Helle, H. P. Keplin, W. Nitsche W., and I. Peltze, editos, New Results in Numeical and Expeimental Fluid Mechanics VIII. Notes on Numeical Fluid Mechanics and Multidisciplinay Design, Volume 2, pages Spinge,

18 [3] Eme Soze, Chistoph Behm, and Cetin C. Kiis. Gadient calculation methods on abitay polyhedal unstuctued meshes fo cell-centeed cfd solves. In Poc. of 52nd AIAA Aeospace Sciences Meeting, AIAA Pape , National Habo, Mayland, 204. [4] M. Xionga, X. Denga, X. Gaoa, Y. Dongb, C. Xua, and Z. Wanga. A novel stencil selection method fo the gadient econstuction on unstuctued gid based on OpenFOAM. Comput. Fluids, 72: , 208. [5] P. L. Roe. Appoximate Riemann solves, paamete vectos, and diffeence schemes. J. Comput. Phys., 43: , 98. [6] Gilbet Stang. Linea Algeba and Its Applications. Academic Pess, second edition, 980. [7] A. Haselbache. On constained econstuction opeatos. In Poc. of 44th AIAA Aeospace Sciences Meeting and Exhibit, AIAA Pape , Reno, Nevada, [8] B. Diskin and J. L. Thomas. Compaison of node-centeed and cell-centeed unstuctued finite-volume discetizations: Inviscid fluxes. AIAA J., 49(4): , 20. [9] D. J. Maviplis. Revisiting the least-squaes pocedue fo gadient econstuction on unstuctued meshes. In Poc. of 6th AIAA Computational Fluid Dynamics Confeence, AIAA Pape , Olando, Floida, [20] B. Diskin and J. L. Thomas. Accuacy of gadient econstuction on gids with high aspect atio. NIA Repot No , [2] E. Shima, K. Kitamua, and T. Haga. Geen-Gauss/weighted-least-squaes hybid gadient econstuction fo abitay polyheda unstuctued gids. AIAA J., 5(): , 203. [22] Alieza Jalali and Cal Ollivie-Gooch. Highe-ode unstuctued finite volume RANS solution of tubulent compessible flows. Comput. Fluids, 43:32 47, 207. [23] H. Nishikawa and Y. Liu. Accuacy-peseving souce tem quadatue fo thid-ode edge-based discetization. J. Comput. Phys., 344: ,