. Introducton. The system of unsteady compressble Naver-Stokes (N.-S.) equatons s a fundamental system n ud dynamcs. To be able to solve the system qu

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1 VARIABLE DEGREE SCHWARZ METHODS FOR THE IMPLICIT SOLUTION OF UNSTEADY COMPRESSIBLE NAVIER-STOKES EQUATIONS ON TWO-DIMENSIONAL UNSTRUCTURED MESHES Xao-Chuan Ca y Department of Computer Scence Unversty of Colorado Boulder, CO 839 ca@cs.colorado.edu Charbel Farhat Department of Aerospace Engneerng Unversty of Colorado Boulder, CO 839 charbel@alexandra.colorado.edu Marcus Sarks z Department of Computer Scence Unversty of Colorado Boulder, CO 839 msarks@cs.colorado.edu Abstract We report our experences on usng a new varant of the Schwarz precondtoned GM- RES methods n the mplct soluton of the unsteady compressble Naver-Stokes equatons dscretzed on two-dmensonal unstructured meshes. We rst partton the global mesh wth the recursve spectral bsecton method nto submeshes, and then we ntroduce a famly of Schwarz methods, referred to as the Varable Degree Schwarz methods (VDS) on the overlappng submeshes. In VDS, the subdoman precondtoner s constructed by usng a polynomal n two matrx varables, namely the matrx, n ts un-factorzed form, of the current tme step k and another matrx, n ts factorzed form, obtaned at a prevous tme step j. The degree of the matrx polynomal n each subdoman s determned automatcally so that extra precondtonng s performed only n subdomans whose assocated local matrces have large condton numbers. The extra precondtonng occurs often near the body of the arfol. We show numercally that VDS s very eectve. Unlke the well-known ellptc theory, we observe that the convergence rate of VDS precondtoned GMRES degenerates very mldly wthout a coarse space for reasonably large number of subdomans. We also study the eects of the overlappng sze, the number of subdomans and the level of nexactness of the subdoman solvers. The other purpose of the study s to understand the robustness of the Schwarz methods wth respect to ow parameters, such as the CFL, the free stream Mach number and the Reynolds number. Numercal results for both subsonc and transonc problems are reported. The work was supported n part by the NSF Grand Challenges Applcatons Group grant ASC and the NASA HPCC Group grant NAG y The work was supported n part by the NSF grant ASC , and by NASA Contract No. NAS- 948 whle the author was n resdence at the Insttute for Computer Applcatons n Scence and Engneerng (ICASE), NASA Langley Research Center, Hampton, VA z The work was supported n part by the NSF grant ASC

2 . Introducton. The system of unsteady compressble Naver-Stokes (N.-S.) equatons s a fundamental system n ud dynamcs. To be able to solve the system quckly and accurately n complex geometry s one of the ultmate goals of computatonal ud dynamcs [2]. To acheve the goal, several mportant technques have been developed n the past few years, such as unstructured grd generatons for complex geometry, stable, conservatve dscretzatons of the N.-S. equatons, unstructured grd parttonngs, as well as the powerful, robust mplct precondtoned teratve solvers dscussed below. Most of the technques are developed for steady state calculatons, see e.g., [, 5, 6] and references theren. In ths paper, we put all the technques together and ntroduce a robust doman decomposton based fast precondtoned teratve method for the tme accurate soluton of unsteady problems. We study mplct methods for solvng unsteady N.-S. equatons dscretzed on twodmensonal unstructured meshes wth a combned nte element/nte volume scheme for the spatal varables and a smple backward Euler scheme for the temporal varable. It s well-known that the man advantage of mplct methods s that they allow the tme steps to be determned solely based on the physcs of the ud ow, not on the stablty property of the tme dscretzaton scheme, [4, 3, 3]. However, to advance n tme, a large, sparse, nonsymmetrc lnear system of equatons must be constructed and solved at each tme step. Dependng of the sze of the tme step, and several other ow parameters, the condtonng of the matrx may change from well-condtoned to mldly ll-condtoned. And due to the complexty of the ow pattern, at a gven tme step, the matrx may be llcondtoned n certan subregons near the arfol and relatvely well-condtoned elsewhere. Detals about the local condtonng of the matrx wll be dscussed later. To solve these systems teratvely, t s necessary to have a famly of precondtoners whose strength can be adjusted locally n each subdoman accordng to the ow condton. Overlappng Schwarz methods (OSM) s a famly of precondtoners for solvng large sparse lnear systems arsng from the dscretzaton of partal derental equatons, see e.g. [6, 8,,27]. It was orgnally desgned for scalar lnear ellptc problems. We nd OSM to be very attractve for our purpose because () they are more parallelzable than the popularly used global ILU precondtoners; (2) they are ecent for nonsymmetrc and ndente problems; (3) they have mesh ndependent convergence rates, at least for ellptc nte element problems; (4) they have adjustable strength controlled by usng the nexact soluton technques for solvng local problems. We shall further explore the exblty of OSM at the subdoman level and ntroduce a new varant below. It s well-known that when constructng a precondtoner for solvng a sngle system of lnear equatons, Au = f, all the nformaton needs to be from the matrx A. However, the ssue for tme dependent problems s derent. A sequence of nter-related systems A (k) u = f (k) has to be solved. If the matrx, and especally n ts (often nexactly) factorzed form, obtaned at a prevous tme step can be properly used, then the precondtoner at the current tme step can be obtaned cheaply. More precsely speakng, at each tme step, we solve the global lnear system by a precondtoned GMRES method ([26]) and n the precondtonng stage, followng the general overlappng Schwarz framework, we solve the local subdoman problems by another precondtoned GMRES method, wth derent precondtoners and stoppng condtons. In each subdoman the precondtoner s bult by usng a polynomal n two matrx varables, namely the matrx, n ts un-factorzed form, of the current tme step k and another matrx, n ts factorzed form, obtaned at a prevous tme step j. The degree of the matrx polynomal reects the condtonng of the subdoman matrx. Note that classcal Schwarz methods correspond to the case where the degree of

3 the matrx polynomals always equals to one. In our method, the degree of the polynomal vares from subdoman to subdoman dependng the ow condtons, and therefore we refer to the methods as varable degree Schwarz methods (VDS). In ths paper, we also nvestgate the derence between the Schwarz famly of precondtoners and other methods such as the smple Jacob teratve method and the global ILU precondtoned teratve method. Wthn the Schwarz precondtoners, we try to understand the role of the overlappng sze between subdomans, the eect of the number of subdomans, and the eect of the nexact subdoman solvers. Snce the constructon of the precondtoner s expensve, we explore the possblty of re-usng the precondtoner for several tme steps. We restrct our attenton to sequental computers, and sngle level Schwarz algorthms. For steady state problems, some studes can be found n [8]. For other recent developments n unsteady calculatons, we refer the readers to [2, 3, 3]. Our focus s on the Schwarz algorthms, therefore only the smplest tme dscretzaton, namely the rst order backward Euler scheme, s consdered n ths paper. Hgher order schemes and ther nuence on the Schwarz precondtoners wll be dscussed n a forthcomng paper. The physcal model we choose to test our algorthms contans a sngle element NACA2 arfol at a rather large angle of attack wth a modest Reynolds number. Both subsonc and transonc cases are studed n the paper. The paper s organzed as follows. In x2, we dscuss the unsteady compressble N.-S. equatons n the conservatve form, the boundary condtons and a dscretzaton scheme. In x3, we study a precondtoned teratve method and ntroduce a new varant of the overlappng Schwarz precondtoners. Numercal experments for both subsonc and transonc ows are reported n x4. x5 ncludes a few nal remarks. 2. The two-dmensonal unsteady N.-S. equatons. In ths secton, we descrbe the two-dmensonal unsteady compressble N.-S. equatons n ts conservatve form. We also dscuss the spatal and temporal dscretzatons of the equatons on unstructured meshes. Followng the notons of Farhat, Fezou and Lanter [2, 3], and Fezou and Stouet [5], for the spatal varables, we use a combned nte element/nte volume scheme and for the temporal varable we use a smple backward Euler method. The scheme s of second order n space and rst order n tme. 2.. Governng equatons. Let < 2 be the ow doman and ts boundary, as shown n Fg.. The conservatve form of the N.-S. equatons s gven + r ~x F(W(~x;t)) = Re r ~x R(W(~x; t)); where ~x and t denote the spatal and temporal varables, and W = u v E C A ; r ~x @y C A ; F(W)= F (W) F 2 (W) Here the functons F and F 2 denote the convectve uxes F (W )= u u 2 + p uv v(e + p) C A ; F 2(W )= 2! ; R(W)= R (W) R 2 (W) v uv v 2 + p v(e + p) C A! :

4 and the functons R and R 2 denote the dusve uxes R (W )= xx xy u xx + v xy + k " Pr x C A ; R 2(W) = xy yy xy + v yy + k " Pr y In the above expressons, s the densty, U ~ =(u; v) T s the velocty vector, E s the total energy per unt of volume, and p s the pressure. These varables are related by the state equaton for perfect gas p =( ) E 2 k~ Uk 2 ; where denotes the rato of specc heats ( =:4, for ar). The specc nternal energy " s related to the temperature va " = c v T = E 2 k~ Uk 2 : In the dusve uxes, xx ; xy, and yy are the components of the two-dmensonal Cauchy stress tensor, k s the normalzed thermal conductvty, Pr = c p =k s the Prandtl number (Pr =:72, for ar), and Re = U ~ L = s the Reynolds number, where, U ~, L, and denote the characterstc densty, velocty, length, and dusvty, respectvely. The components of Cauchy stress tensor are related to the velocty va xx = 2 ; yy = 2 where denotes the C A ; xy Boundary condtons. We are nterested n unsteady, external ows around an arfol as pctured n Fg.. The doman boundary s = w [ and the far eld velocty s U ~. On the wall boundary w, a no-slp condton on U ~ and a Drchlet condton on the temperature T are mposed,.e., (2) ~U = ~; and T = T w : No boundary condton s speced for the densty. In the far eld, the vscous eect s assumed to be neglgble, therefore a unform free-stream velocty U ~ s mposed on (3) =; ~ U = cos sn! ; and p = ; M 2 where s the angle of attack, and M s the free-stream Mach number Dscretzaton. Let the temporal varable t be dscretzed as t k = t k + t k, where t k s the dscrete tme ncrement and t =. We consder the ncrement W k = W k W k, where W k s an approxmaton of W (;t k ). Note that when an algorthm s wrtten n the \delta" form [3, 28], the ncrement W k s the unknown varable rather than W k. Here, we use a rst-order nte derence approxmaton for the temporal varable, namely, the backward Euler scheme gven as (4) W k t k +(r W(F k )r ~x )W k Re (r W (R k 3 ) r ~x )W k

5 w ~U Fg.. The computatonal doman and ts wall and far eld boundares. = r( F k )+ Re rrk ; where F k and R k are approxmatons of F(W (;t k )) and R(W (;t k )), respectvely. We determne the tme step sze n the followng way. Let CFL be a pre-selected postve number. For each element, wth sze h, of the nte element mesh, we dene an element tme step sze by and then the global tme step s dened by t k = h CFL (C + ku k 2 )+2=(Re P r h ) (5) t k = mnft k g: Here C s the element sound speed, and U s the element velocty vector. The computatonal doman s dscretzed by a trangular grd as pctured n Fg.2. We use unstructured grds snce they provde exblty for tessellatng about complex geometres and for adaptng to ow features, such as shocks and boundary layers. We locate the varables at the vertces of the grd, whch gves rse to a cell-vertex scheme. The space of solutons s taken to be the space of pecewse lnear contnuous functons. The dscrete system s obtaned va a mxed Galerkn nte element/nte volume formulaton; see Farhat et al. [2, 3], and Fezou and Stouet [5] for detals. In short, the dscretzed system for (4) s obtaned as follows: For the tme dervatve of (4) we use a \mass-lumpng" technque, n whch we replace the mass matrx by some dagonal matrx. For the convectve terms of the left-hand sde of (4), we use a rst order scheme that s an extenson of Van Leer's MUSCL [29] scheme to the case of unstructured grds (see Fezou and Stouet [5]) wth a Roe approxmate Remann solver [24]. For the dusve terms of the left-hand sde of (4), we use a Galerkn nte element (rst-order quadrature ntegraton). For the convectve terms of the rght-hand sde of (4), we use a second-order scheme that s agan an extenson of Van Leer's MUSCL scheme to the case of unstructured grds (see Fezou and Stouet [5]) wth a Roe approxmate Remann solver [24]. We also use Van Albada's lmtng procedure to reduce numercal oscllatons of the solutons. For the dusve terms of the rght-hand sde of (4), we use a regular Galerkn nte element method (second-order quadrature ntegraton). 4

6 Fg. 2. The top left gure shows the un-decomposed nonunform shape regular nte element mesh. The top rght gure shows the decomposton of the mesh nto non-overlappng subdomans, and the bottom gure s a blow-up of the gure around the arfol. 5

7 Although both approxmatons we use for the left-hand sde of (4) are spatally rstorder, they operate on the ncrement W k and as a consequence the resultng scheme s spatally second-order for any xed t. We assume W k satses strongly the wall boundary condtons on w and the far eld boundary condtons at nnty. For the ntal values, W satses strongly the wall boundary condtons on w and the far eld boundary condtons at. For the ntal values and at the nteror nodes of, W takes the free-stream boundary condton. Puttng peces of the dscretzaton together, at each tme step, we obtan a large, sparse, generally nonsymmetrc lnear system of equatons of the form (6)! D (k) + B(k) t k u k = f k ; where D (k) s a dagonal, lumped mass matrx, B (k) s the sum of the convectve and dusve terms on the left-hand sde of (4) dscretzed by the nte volume and nte element methods, respectvely, f k s the dscretzed rght-hand sde of (4), and u k s the approxmate nodal value of W k at the nte element mesh ponts. We solve (6) by a precondtoned Krylov teratve method wth a precondtoner M (k) to a certan lnear tolerance,.e., (7) " M (k)! D (k) + B(k) t k u k f k 2 km (k) f k k 2 : To smplfy the dscusson, we shall use A (k) = D(k) + t k B (k) n the rest of the paper. The precondtoner M (k) wll be ntroduced n the next secton. 3. Soluton methods and varable degree Schwarz precondtoners. In ths secton, we rst brey recall the classcal addtve and multplcatve Schwarz algorthms. Then we ntroduce a varable degree verson of Schwarz algorthms that s more sutable for solvng tme dependent problems. Unlke steady state problems, f certan nformaton or matrx factorzaton obtaned at the prevous tme steps can be used a great deal of calculatons can be saved. Let us revew the Schwarz algorthms. Suppose that, at tme step k, we need to solve a lnear system of equatons A (k) u k = f k ; where A (k) s an explctly constructed, nonsymmetrc and sparse matrx wth symmetrc non-zero pattern. Snce our man nterest s on the multcomponent N.-S. equatons n twodmensonal spaces, each element ofa (k) can be consdered as a small 44 matrx, and each unknown of the vector u (k) s a `4-sze' vector. Thus, there s a bjecton between unknowns and vertces. We denote the set of vertces (or nodes) by N = f; ;ng, where n represents the total number of nodes (or unknowns). To dene algebrac Schwarz algorthms, see e.g. [6], we rst partton the set N nto n nonoverlappng subsets fn g whose unon s N.We use the TOP/DOMDEC mesh parttonng package of Farhat et al. [4] to obtan sets N. We use the recursve spectral bsecton method ([22]) wth certan optmzaton to obtan roughly the same number of nteror and boundary nodes n each N, and also to obtan good aspect rato on the subgrds. To generate an overlappng parttonng wth overlap 6

8 ovlp, we further expand each subgrd N by ovlp number of neghborng nodes, denoted as ~N. We denote by L the vector space spanned by the set ~N. For each subspace L,we dene an orthogonal projecton operator I as follows: I s a n n block dagonal matrx whose elements are 4 4 dentty matrces f the correspondng nodes belong to ~N and to 4 4 zero matrces otherwse. Wth ths we dene A (k) = I A (k) I ; whch s an extenson to the whole subspace, of the restrcton of A (k) to L. Note that although A (k) s not nvertble n the full space, ts restrcton to the subspace spanned by ~N s nonsngular, and we dene ts nverse by (A (k) ) I (A (k) ) jl I : The classcal addtve and multplcatve Schwarz algorthms can now be smply descrbed as follows: Solve the lnear system by a Krylov subspace method, where MA (k) u = Mf (k) (8) M =(A (k) ) ++(A (k) n ) ; for the addtve Schwarz algorthm, and (9) MA (k) =I I (A (k) ) A (k) I (A (k) n ) A (k) for the multplcatve Schwarz algorthm. It has been shown that the above mentoned algorthms are very successful for steady state CFD problems, see e.g., [5, 2] and also [9]. There are three major steps n the constructon of the Schwarz precondtoners, namely ) the constructon of the matrx A (k) ; 2) the constructon of the matrces A (k) ; and 3) the ncomplete factorzaton of the matrces A (k). In fact Step ) s not necessary snce the matrces constructed n Step 2) can be used to calculate the matrx-vector multplcatons. Snce we are nterested n mplct methods, Step 2) has to be done at every tme step no matter how expensve t s. One expensve step n the constructon of the precondtons as formulated above for tme dependent problems One way toavod the frequent factorzaton of A (k) s to smply use some calculated at tme step j, where j<k. However, ths method may not be very eectve fjand k are too far apart. More dscussons on usng frozen precondtoners can be found later n the paper. Another problem wth the Schwarz precondtoners (8) and (9) s that all subdomans are treated equally n terms of the level of precondtonng n the sense that the number of s Step 3). old factorzed matrx A (j) applcatons of (A (k) ), or ts nexact verson, s the same on all subdomans, regardless of the fact that the subdoman matrces A (k) have vary derent condton numbers. Physcally speakng, the behavor of the partal derental equatons n subdomans near the body of the arfol, or near the shocks s very derent from the regons that are far from the subdomans where the real actons take place. 7

9 Here we propose a method that places derent level of precondtonng n derent subdomans and wll also show by numercal experments that the methods reman eectve even f j and k are far apart from each other. The dea s smple. We replace the requred matrx-vector multply n (8) or (9) () w =(A (k) ) v by another teratve procedure wth (B (j) ) as the precondtoner. Here B (j) s an ncomplete factorzaton of A (j) wth certan levels of ll-ns. More precsely speakng, to obtan w for a gven v, we run several steps of GMRES n the subspace L to drve the resdual () (B (j) ) (v A (k) ~w) 2 (B (j) We then set w := ~w. Here s pre-selected small value. Examples can be found n x4 of ths paper. In the matrx language, we replace the matrx (A (k) ) n () by a matrx polynomal poly (B (j) ) A (k) of a certan degree, whch depends of the number of GMRES teratons needed n the subspace L.To put them nto a sngle form, the addtve Schwarz precondtoner becomes ) v : 2 M = poly (B (j) ) A (k) + +poly n (B (j) n ) A (k) n : Note that ths precondtoner does not contan (A (k) ), but t contans certan spectral nformaton of (A (k) ). Ths makes t very eectve. In fact, M s a truncated seres representaton of (A (k) ) based on a splttng of A (k) nto the sum of B (j) and A (k) B (j). A dscusson on a related polynomal precondtonng method can be found n [7]. We note that n a gven subdoman, the number of GMRES teratons, or the degree of the polynomal, s determned by the condtonng of the local stness matrx. The multplcatve verson can be constructed n a smlar way. The extenson to the local multplcatve Schwarz method ([7]) s also straghtforward. We remark that snce the precondtoner changes n the GMRES loop due to the stoppng condton determned by, t s generally more approprate to use the so-called exble GMRES [25], whch s slghtly more expensve than the regular one. We do not use the exble GMRES snce the regular GMRES presents no problem for our test cases. The mplementaton of the methods s rather complcated because of the use of mult-layered Krylov teratons as a global, or outer, and also local solvers. PETSc makes our numercal tests possble. More detals regardng to the mplementaton wll be dscussed n the next secton. 4. Numercal results. The goal of ths secton s to demonstrate the usefulness of the famly of VDS precondtoners n the mplct soluton of compressble ow problems, and to compare the eectveness of the methods wth varous other methods, such as the pontwse Jacob teratve method and the global ILU() precondtoned GMRES method, for both subsonc and transonc ows. The experments were performed on a DEC Alpha workstaton (275MHz, 52MB memory), and the software was wrtten by usng the newly developed package PETSc [9] of the Argonne Natonal Laboratory. All arthmetc operatons are n 8

10 double precson. The system BLAS lbrary (dxml []) was used. Only sequental results are reported here. A parallel verson of the code s beng developed, and the results wll be reported n the future. Here we apply our computatonal algorthms to the smulaton of two-dmensonal low Reynolds number chaotc ows past a NACA2 arfol at hgh angle of attack and two derent Mach numbers. It was shown n Pullam [23] that such ows can be resolved wth a reasonable number of grd ponts. The accuracy of the computed solutons are substantated by successve mesh renements and comparsons wth the results were reported n [23]. No steady state solutons exsts for both test cases descrbed below. Test : The subsonc case wth free stream Mach number M =: and Re = 8:. We use a pre-generated shape regular trangular mesh wth 228 nodes; see Fg.2 for example. The Mach surfaces of the computed soluton at varous tme steps are gven n Fg.5. Test 2: The transonc case wth free stream Machnumber M =:84 and Re = 6:. We use a mesh wth nodes obtaned by unformly renng the mesh used n Test. The Mach surfaces of the computed soluton at varous tme steps are gven n Fg.6. Table Total CPU hours and tme steps spent for calculatng the lft curves usng explct and mplct methods wth derent CFL numbers. The total non-dmensonalzed tme nterval s (, 25) for the M =:case and (, ) for the M =:84 case. Expl. Impl. Impl. Impl. CFL=.8 CFL=25 CFL=5 CFL= M =: CPU(hours) Mesh2k Tme steps M =:84 CPU(hours) Mesh48k Tme steps In the followng dscussons, we shall refer to these two meshes as \Mesh2k" and \Mesh48k", respectvely. In the mplementaton of Schwarz precondtoners, we partton the mesh by usng the TOP/DOMDEC package [4], whch mplements the recursve spectral bsecton method. We requre that all subdomans have more or less the same number of mesh ponts. An eort s made to reduce the number of mesh ponts along the nterfaces of subdomans, whch may be needed later n our parallel code to reduce the communcaton cost. The mesh generaton and parttonng are consdered as pre-processng steps, and therefore not counted toward the CPU tme reported n Table. The sparse matrx dened by (4) s constructed at every tme step, and stored n the Compressed Sparse Row format. The subdoman matrces are obtaned by takng elements, accordng to a pre-selected ndex set, from the global matrx. A symbolc ILU() factorzaton of the subdoman matrx s performed at the very rst tme step, and re-used at all the later tme steps. Ths s possble due to the fact that the matrces, constructed at every tme step, share the same non-zero pattern. We also tested the ILU(k) (k>) precondtoners, whch are not compettve wth ILU() n terms of the CPU tme n our mplementaton for both test cases. We remark that f ILU wth drop tolerance s used then the non-zero pattern of the matrces may change and therefore the symbolc factorzaton may not be very useful. We note that at the begnnng of the ow movement,.e., when the non-dmensonalzed tme t :, the ow changes so drastcally that the use of any t n that makes the correspondng CFL number larger than : would result n the loss of tme accuracy for the 9

11 3 Mach., Re 8, Angle of Attack 3, Mesh2k 3 Mach.84, Re 6, Angle of Attack 3, Mesh48k + > Explcty scheme wth CFL.8 + > Explcty scheme wth CFL x > Implcty scheme wth CFL x > Implcty scheme wth CFL 25 o > Implcty scheme wth CFL 5 o > Implcty scheme wth CFL 5 2 * > Implcty scheme wth CFL 2 * > Implcty scheme wth CFL Lft.5 Lft Non Dmensonalzed Tme Non Dmensonalzed Tme Fg. 3. Lft curves obtaned by explct and mplct methods. The left gure shows the subsonc cases wth M =:,Re = 8: and the number of mesh ponts s 228. The rght gure s for the transonc cases wth M =:84, Re = 6: and the number of mesh ponts s entre calculaton. Ths mples that small t n have to be used when t :, and therefore, the mplct method has to be abandoned for ths ntal perod of tme. In our experments, the mplct solver s turned on at t =:. The soluton for the perod <t:s obtaned wth the explct method wth CFL= The eect of usng large CFL numbers. One of the bggest advantages of mplct methods s that large tme steps, or large CFL numbers, can be used. We rst examne ths clam by comparng the tme accuracy of two solutons obtaned by usng the mplct methods wth CFL=25, 5 and, respectvely, and the soluton obtaned by a second order (n tme and space) explct method, [2]. From Fg.3, one can easly see that no two derent CFL numbers gve dentcal solutons. However, the mportant thng for engneerng purposes s to capture correctly the \phase" and \ampltude" of the lfts. Small errors n ampltude and phases are usually admssble. Ths can often substantally shorten the turnaround tmes n the ntal aerodynamc desgn and analyss. For the mplct methods, we use the multplcatve VDS precondtoned GMRES method as the global lnear solver. The subdoman precondtoners are re-computed at every 5 tme steps. In the Schwarz methods, we use 8 subdomans, and ovlp =. Each subdoman problem s solved by an ILU() precondtoned GMRES method wth the stoppng tolerance set to =. The stoppng tolerance for the global lnear system solver s = 3. We also test several cases wth smaller, such as 4 and 5. The resultng lft curves are not dstngushable from the ones shown n Fg.3. In Table, we report the total number of tme steps and CPU tme n hours spent on the entre computaton, not ncludng the mesh generaton and parttonng. We observe from our experments that even wth a CFL number,, not much tme accuracy s lost for a certan perod of tme, see Fg.3. However, f the tme accuracy s requred for a longer perod of tme, we do recommend a smaller CFL number. We remark that our dscussons here on the eects on usng large CFL numbers s based on our rst order tme dscretzaton scheme. The results may change slghtly f hgher order schemes are used The Schwarz parameters. The number of subdomans and the sze of overlap are two mportant parameters for Schwarz methods. We here test the addtve and multplcatve VDS methods for both test cases. Instead of runnng the entre calculaton as

12 we dd n the prevous secton, we run the tests for only tme steps startng at t =:. In terms of the non-dmensonal tme, ths means tme ntervals (; :86) for Test and (; 2:28) for Test 2. In the rest of the paper, we shall use MaxIt to denote the maxmum number of global GMRES teratons and TotalIt the total numbers of global GMRES teratons wthn ths lnear system solves. To measure the approxmate cost of the methods wthout ncludng any machne dependent factors, we use EMatVec to denote the equvalent number of matrx-vector multplcatons, whch ncludes the actual stness matrx-vector multplcatons and the precondtonng-matrx-vector multplcatons. Snce derent subdomans may need derent number of matrx-vector multplcatons, we take the average over all subdomans and convert t nto a multple of the equvalent global matrx-vector multplcatons. Suppose that ~n s the sze of the global matrx. Note that ~n s 4 tmes the number of mesh ponts. Then, one global matrx-vector multplcaton requres roughly 28~n ops. Our prmary teratve solver GMRES has a complexty of I(I+ 2)~n+ I(cost of a precondtoned matrx-vector multplcaton). Here I s the number of teratons. For example, the pure GMRES cost, wthout countng the cost of the matrx-vector multplcatons, for 4 teratons s about 24~n, whch s a lttle less than the cost of dong global stness matrx-vector multplcaton. Let us rst dscuss the dependence of the convergence rate on the number of subdomans. We use 5 derent decompostons of, wth both Mesh2k and Mesh48k. The number of subdomans goes from 8 to 28. We run both Test and 2, wth ovlp equals to one ne mesh cell. In Table 2, we present the maxmum number of global GMRES teratons wthn one hundred tme steps and ts correspondng EMatVec. If multplcatve VDS s used even wthout the specal subdoman colorng or orderng, MaxIt s ndependent of the number of subdomans, for reasonably large number of subdomans, such as 28. An nterestng case s shown on the top left porton of Table 2, whch ndcates that f addtve VDS s used for the subsonc problem, the number of maxmum teratons does grow, though not very fast, as the number of subdomans becomes large. In ths case, we beleve that a coarse space may be useful to reduce the dependence on the number of subdomans. However, we have not mplemented the coarse grd solver yet. For transonc problems, our tests show that the use of a coarse level grd s not necessary wth both addtve and multplcatve VDS precondtoners. Whether overlap s useful or not s a rather subtle ssue. It depends on the global lnear stoppng parameter dened n (7) and the local lnear stoppng parameter dened n (). In Table 3, we report the case for = 6 and varyng. Accordng to the results n Table 3 and a large number of other tests we dd (not beng reported here), large overlaps can reduce the number of teratons and CPU tme only f the stoppng parameter s small. In our stuaton when = 3,we nd = oers the best CPU results, and therefore we do not need large overlaps. In the rest of the tests, we shall use ths set of and, wth overlap. We next exam the VDS methods. We focus on the case wth 8 subdomans, and use GMRES/ILU() as the local subdoman solvers. The parttonngs used for Mesh2k and Mesh48k are derent. The subdomans are numbered as n Fg.4. The results obtaned for one hundreds tme steps startng at t =: are summerzed n Table 4. We have also run the test for t equals to other values and the results are more or less the same. We use the same local stoppng condton, namely = for all subdomans and for both subsonc and transonc problems. It turns out the requred degrees of local precondtonng

13 Fg. 4. The left gure shows the parttonng of Mesh2k nto 8 subdomans and the rght ones shows that for Mesh48k. polynomals are qute derent. For the subsonc case, subdomans 6 and 8 need more teratons (4 and 6 respectvely) than other subdomans. The left pcture of Fg.4 shows that these two subdomans cover the top porton of the arfol. Only two teratons are needed for subdomans that are far away from the arfol, such as, 2 and 3. The number of teratons reects the condtonng of the subdoman matrx. For the transonc case, all subdomans need ether one or two more teratons. For both test problems, Table 2 and Table 4 show that both the number of global teratons and the number of local teratons are surprsngly small, whch ndcate that the lnear system of equatons (6) s n fact not too ll-condtoned. We beleve that ths s due to the use of relatvely small tme steps (5) A comparson wth the pontwse Jacob teratve method. For comparson purpose, we solve both test problems by usng the smplest teratve method, namely the pontwse Jacob method, whch s often referred to as the Jacob precondtoned Rchardson's method. Jacob method has a few very attractve features, such as beng easy to parallelze. Note that for multcomponent test problems, a pont corresponds to a 4 4 matrx. When usng the Jacob method, a dense 4 4 matrx needs to be nverted at every mesh pont. In our experments, at each tme step before solvng the lnear system, we compute the nverse of these 4 4 matrces explctly and save them for the Jacoban teratons. As before, we run the tests for one hundred tme steps and record the maxmum number of teratons as well as the total number of teratons, see Table 5. We observe that, n terms of teraton numbers for solvng lnear systems, the transonc problem s easer to handle than the subsonc problem, whch s more of an ellptc system. Comparng the TotalIt, whch equals to the total number of EMatVec, n Table 5, and the total EMatVec numbers n Table 2, we found that Jacob s consderably more expensve than the Schwarz precondtoned GMRES methods. We beleve that the requred number of teratons would grow much faster f ner meshes are used than that of the Schwarz precondtoned GMRES methods. 2

14 4.4. A comparson wth a global ILU() precondtoned GMRES method. In Table 5, we show the maxmum and total number of teratons when usng the global ILU() precondtoned GMRES method for both test cases. In terms of the number of EMatVec, the method outperforms, by a factor of 5% to 3%(Comparng Table 5 and the bottom part of Table 2), the multplcatve VDS precondtoned GMRES methods we dscussed n the paper. Unfortunately, ts parallelzaton on dstrbuted memory computers s not very easy The eect of frozen precondtoner. Fnally, we examned the eect of usng the same precondtoner, or part of the precondtoner, for several tme steps wthout dong the factorzaton at every tme step. In Table 6, we summerze the results for usng derent numbers of frozen steps, namely Froz = ; 5; :::. There s a range of optmal Froz one can choose from; smlar numbers of EMatVec were obtaned n our mplementaton for Froz rangng from 5 to 5. For the subsonc case, we can go a bt further, e.g., take Froz =. 5. Conclusons. We proposed and tested a famly of varable degree Schwarz (VDS) precondtoned GMRES methods for solvng lnear systems that arse from the dscretzaton of unsteady, compressble N.-S equatons on 2D unstructured meshes for both subsonc and transonc ows past a sngle element NACA2 arfol. We found that wth mplct methods, larger tme steps can be used and the overall smulaton tme can be reduced sgncantly comparng wth the explct method. In VDS, the level of precondtonng n each subdoman vares accordng to the local ow condton, therefore extra precondtonng s performed only when and where t s needed. For subsonc problems, we found that the condtonng of the subdoman matrces changes qute a bt from one ow regon to another, and extra local precondtonng n subdomans n whch the ow changes drastcally can sgncantly reduce the total number of global lnear teratons. Ths s somewhat less obvous for transonc ow, whch needs a nearly unformly small global and local number of teratons. When usng VDS, the best results are obtaned wth small overlap. For the multplcatve verson, the convergence rate depends very mldly on the number of subdomans (up to 28 subdomans has been tested), and for the addtve verson, a slght dependence s observed for the subsonc test problem and therefore a coarse space mght be useful. We also compared our methods wth the smple pont(means 44 block for our multcomponent problems) Jacob teratve method and the global ILU() precondtoned GMRES method. We found that Jacob s sgncantly slower than the proposed methods, especally for the subsonc case, and f sequental computers are the prmary computng platform, then the global ILU() precondtoned GMRES s a wnner over all methods we have tested. All of our tests were done on a sequental computer. Consderable eort s needed n order to obtan a well-balanced parallel mplementaton. We remark that our mesh parttonng s obtaned wthout the knowledge of the ow condton and our experences show that a soluton dependent mesh parttonng would oer a more computatonally balanced decomposton of the problems. 3

15 Fg. 5. The Mach dstrbuton at M =:, for non-dmensonalzed tme t =2;3; :::; 9. 4

16 Fg. 6. The Mach dstrbuton at M =:84, for non-dmensonalzed tme t =2;3; :::; 9. 5

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19 Table 2 Each test s for tme steps and at each tme step the ntal precondtonedresdual s reduced by a factor of = 3 by usng GMRES/VDS wth ovlp =. We use GMRES/ILU() as nexact local solvers to reduce the local precondtoned resdual by a factor of =. Here CFL=5. ASM Test Test 2 subdomans MaxIt TotalIt EMatVec MaxIt TotalIt EMatVec MSM Test Test Table 3 GMRES teraton numbers to reduce the precondtoned resdual of Test to = 6 usng GM- RES/(addtve VDS) wth 8 subdomans. We use GMRES/ILU() as nexact local solver wth derent local stoppng crtera. Here CFL=. ovlp = ovlp = ovlp = 2 ovlp = 3 teraton =5:e =3:e =:e =:e exact 5 9 Table 4 The maxmum (MaxIt) and total (TotalIt) local GMRES/ILU() teraton numbers. The global solver s GMRES/(multplcatve VDS). The parameters are = 3,=, ovlp = and the local solvers are ILU(). Here CFL= Test, MaxIt TotalIt Test 2, MaxIt TotalIt

20 Table 5 The number of equvalent EMatVec operatons needed for tme steps startng at t =:. = 3, =, CFL= Jacob Global ILU() Test, MaxIt 74 EMatVec Test 2, MaxIt 46 4 EMatVec Table 6 The number of EMatVec operatons needed for tme steps startng at t =:. In GMRES/(multplcatve VDS), = 3,=, CFL=5, number of subdomans s 8 and ovlp =. For the Froz=2 case, the numbers are taken for 2 tme steps dvded by 2. Froz= Test, EMatVec TotalIt Test 2, EMatVec TotalIt