A Hgh-Order Accurate Unstructured GMRES Algorthm for Invscd Compressble Flows A. ejat * and C. Ollver-Gooch Department of Mechancal Engneerng, The Unversty of Brtsh Columba, 054-650 Appled Scence Lane, Vancouver, BC V6T Z4, Canada A precondtoned matrx-free GMRES method s presented for unstructured hgherorder computaton of D nvscd compressble flows. Lower-Upper Symmetrc Gauss-Sedel (LU-SGS has been used as the precondtonng strategy, and an approxmate frst order Jacoban as the precondtonng matrx. The numercal algorthm demonstrates promsng convergence performance for both the second and the thrd order dscretzaton methods n supersonc flow. I. Introducton RECET research on structured mesh flow solver for aerodynamc problems shows that for practcal levels of accuracy usng a hgher order accurate method can reduce computaton tme and memory usage and mprove qualty of the soluton compared to a second-order scheme., To take advantage of the hgh flexblty n mesh generaton and adaptaton for unstructured meshes, we want to apply hgher-order accurate methods for unstructured meshes. Ths approach combnes benefts of hgher-order methods and unstructured meshes. Although hgh-order accurate methods for unstructured meshes are reasonably well establshed, -6 applcaton of these methods for physcally complcated flows s stll a challenge due to very slow convergence. Ths elmnates the effcency benefts of hgher-order unstructured dscretzaton and lmts ther applcaton for practcal purposes. Consequently, convergence acceleraton becomes the key ssue for the practcal usage of hgher-order unstructured solvers. ewton-gmres famly solvers 7- are a commonly used method due to ther property of fast convergence. Snce GMRES algorthm only needs matrx vector products and theses products can be computed by matrx free approach, matrx-free GMRES 9 s a very attractve technque for dealng wth the complcated Jacoban matrces. Ths approach saves memory usage consderably and, removes the problem of explct formng of the Jacoban matrx. Recent results of an unstructured mesh solver for Posson's equaton clearly showed the possblty of reducng computatonal cost requred for a gven level of soluton accuracy usng hgher-order methods and matrx free GMRES as a convergence acceleraton technque. As Posson's equaton s smple and has a lnear flux, the resultant Jacoban matrx s well-condtoned, and the convergence rate for GMRES technque s good. In case of more complex problems, ncludng CFD problems, the Jacoban s typcally ll-condtoned and effectve precondtonng s necessary for good GMRES convergence. Several authors have studed the effect of varous precondtong methods on convergence of matrx-free GMRES both for structured and unstructured meshes. 7-9,,,4 Ther research shows ncomplete lower-upper (ILU factorzaton of the approxmate Jacoban s a very effcent precondtonng strategy for CFD problems. Delanaye et al. 5 presented an ILU precondtoned matrx-free GMRES solver for Euler and aver-stokes equatons on unstructured adaptve grds usng quadratc reconstructon. A totally matrx-free mplct method was ntroduced by Luo et al. 6 for D compressble flows usng GMRES-LUSGS (Lower-Upper Symmetrc Gauss-Sedel. They completely elmnated the storage of the precondtonng Jacoban matrx by approxmatng the Jacoban wth numercal fluxes. Our objectve n ths research s to develop an effcent and accurate unstructured solver for nvscd compressble flud flows. Ths paper presents prelmnary results of usng Lower-Upper Symmetrc Gauss-Sedel (LU-SGS for precondtonng of the matrx-free GMRES to compute the hgher-order soluton of Euler equatons. * PhD Canddate, nejat@mech.ubc.ca, Student Member AIAA. Assocate Professor, cfog@mech.ubc.ca, Member AIAA. Amercan Insttute of Aeronautcs and Astronautcs
II. Governng Equatons The unsteady (D Euler equatons whch model compressble nvscd flud flows, are conservaton equatons for mass, momentum, and energy. The fnte-volume formulaton of Euler equatons for an arbtrary control volume can be wrtten n the followng form of a volume and a surface ntegral: d dt cv Udv + FdA = 0 cs ( where U ρ ρu = ρv E, F ρun ρuun + Pnˆ = ρvun + Pnˆ ρ( E + P u x y n ( T In (, u = u nˆ + vnˆ and ( ρ ρu ρv E are the denstes of mass, x-momentum, y-momentum, and n x y energy, respectvely. The energy s related to the pressure by the perfect gas equaton of state: E = P /( γ + ρ(u + v /, wthγ the rato of specfc heats for the gas. III. Algorthm Descrpton The ntegral form of Eq. ( (for control volume can be wrtten n the form of Eq. (- where R (.e. resdual of the governng equatons represents the spatal dscretzaton operator. Lnearzaton n tme and applyng mplct tme ntegraton leads to mplct tme advance formula (Eq. (-: du dt + R = 0 (- I R n+ n ( + δu = R t, δ U = U U (- R where n Eq. (- s the Jacoban matrx resultng from the resdual lnearzaton. Eq. (- s a large lnear system of equatons whch should be solved at each tme step to obtan an update for the vector of unknowns. As we are only nterested n steady state soluton, the tme marchng process contnues tll the resdual of the lnear system practcally converges to zero. A. Spatal Dscretzaton For spatal dscretzaton, a hgher-order accurate least-square reconstructon scheme 6 has been used n the nteror of the doman to compute flow quanttes at flux ntegraton ponts to the desred accuracy. Fluxes at control volume boundares are calculated by Roe s flux dfference-splttng formula 7 usng reconstructed flow quanttes. Havng computed the fluxes, we use Gauss quadrature to ntegrate the fluxes to the same order of accuracy as the reconstructon. Hgh-order accurate boundary condtons are also used. 6 B. Lnear System Solver Drect methods for solvng the large lnear system of Eq.(- resultng from the dscretzaton of governng equatons at each tme step are not computatonally tractable. Therefore teratve technques such as Krylov famly methods are beng used for that purpose. 9 The lnear system s asymmetrc both n fll and values makng GMRES 8 Amercan Insttute of Aeronautcs and Astronautcs
the obvous frst choce among teratve solvers. The lnear system arsng from a hgh-order dscretzaton has four to fve tmes as many non-zero entres as a second-order scheme. Because of the sze of these matrces and the dffculty n computng ther entres analytcally (even for the second order, we use a matrx-free mplementaton of GMRES 9. In the case of Euler equatons wth nonlnear flux functon and possble dscontnutes n the soluton, usng a hgh-order dscretzaton makes the Jacoban matrx even more off-dagonally domnant and qute llcondtoned. Ths degrades or stalls the convergence of GMRES, whch s hghly dependent on the condton number of the Jacoban matrx. Therefore usng a precondtoner for GMRES becomes necessary for practcal purposes. C. Precondtoned GMRES As prevously mentoned, n order to enhance the convergence performance of GMRES solver for complcated lnear systems, t s necessary to apply precondtonng. In prncpal precondtonng s to solve a modfed lnear system whch s relatvely better condtoned than the orgnal system. Equaton (4 shows the modfed system usng rght-precondtonng. AM ( MX = b (4 M n (4 s an approxmaton to matrx A whch has smpler structure and better condton number and consequently s easer to nvert. If M s a good approxmaton to A, AM becomes close to dentty matrx, ncreasng the 9 performance of the lnear solver. We have chosen to use a matrx-free mplementaton of flexble GMRES (FGMRES. FGMRES satsfes the resdual norm mnmzaton property over the precondtoned Krylov subspace lke standard GMRES, but allows varaton n the precondtoner! As a precondtoner, we use LU-SGS, an teratve lnear solver that has recently receved a far amount of attenton both as a precondtoner and as a solver 6,0, and has been shown to be more effcent than ILU n some cases. 6,9 D. Jacoban Matrx Approxmaton Unlke the orgnal lnear system matrx A, the precondtoner matrx, M, must be bult explctly. We compute an approxmate form of the frst-order Jacoban matrx, reducng the sze and complexty of the precondtoner matrx M. To buld the Jacoban (precondtoner matrx we frst defne the resdual for a typcal cell n terms of flux functons at the control volume faces. For the cell wth the drect neghbors of, and (Fg., the Fgure. A typcal control volume wth ts frst neghbors resdual can be recast n the form of Eq. (5, where length. nˆ s an outward normal vector for each face and l s the face R = + (5 F nˆ ds = F(U,U ( nˆ l + F(U,U ( nˆl F(U,U ( nˆ l faces,,, Amercan Insttute of Aeronautcs and Astronautcs
The next step s takng dervatve of the resdual functon respect to the soluton vector of U at control volume and ts neghbors. Eq. (6- through Eq. (6-4 represent the row entres of the Jacoban matrx. Here, we only consder the frst neghbors as the Jacoban matrx s beng computed to the frst order of accuracy. J(, R = = ( nˆl F(U,U, (6- J(, R F(U = = ( nˆ l, (6-,U J(, R F(U,U = = ( nˆ l, (6- J(, R = F(U =,U F(U,U F(U,U ( nˆl, + ( nˆ l, + U ( nˆl, (6-4 The fluxes at control volume faces are calculated based on Roe s flux formula; 7 for example, for the face between cell and ts neghbor, cell, the flux functon can be expressed by Eq. (7. F(U,U = ( F(U + F(U A ~ (U (, U (7 Therefore the flux functon dervatve terms n Eq. (6- through Eq. (6-4 smply can be computed by gnorng changes n A ~ matrx. Equaton (8- and Eq. (8- show examples of such dervatves. F(U,U = ( F ( A ~ (, (8- F(U,U F = ( ( + U (, A ~ (8- E. Start up process and ewton-gmres teraton As convergence performance and stablty of the ewton-gmres technque, especally for compressble flows, are qute senstve to the start up, we have to do several mplct teratons before swtchng to ewton-gmres teraton. Equaton (- s used for mplct pre-teratons. For the Left hand sde we use the approxmate frst order Jacoban, whle the resduals n the rght hand sde are beng computed to the rght order of accuracy. The LU-SGS algorthm s used as a lner solver for these pre-teratons. Then we swtch to ewton-gmres teraton; at ths stage I the term n Eq. (- s removed (.e. takng nfnte tme step. GMRES s used to solve the resultant lnear t system at each ewton teraton. However completely solvng the lnear system at each ewton teraton does not necessarly accelerate overall convergence. Because of the non-lnearty of the equatons, ewton teraton would not accurately represent the soluton and solvng the system completely would result n excessve but not helpful computatons. Wasted computaton can be reduced by usng Inexact-ewton method,. Multple GMRES nner teratons at each ewton outer teraton are appled to decrease the resdual of the lnear system by some factor 4 Amercan Insttute of Aeronautcs and Astronautcs
usng restart. Our experence (at least wth the presented test cases shows that normally GMRES teraton wthout restart would provde us enough drop n resdual of the lnear system. Therefore we perform one GMRES teraton per ewton teraton. Wth ths approach we do not acheve the quadratc convergence rate of ewton method but we do reach convergence n less CPU-Tme. IV. Results To nvestgate the convergence performance and robustness of the purposed GMRES solver wth a hgher order unstructured dscretzaton, dfferent supersonc flow cases have been studed. Here for brevty, we present two of them whch show the general convergence behavor of our numercal experments. The frst test case (Fg. s a supersonc duct wth a.5 (deg ramp. The ramp starts at x=0.04m and ends at x=0.m. The total length of the duct s.m and has a heght of 0.m at the nlet secton. The nflow Mach number s equal to. We have used a coarse mesh of 578 control volumes for ths case wth some refnement at the ramp s start and end ponts to capture the shock and expanson waves properly at those locatons. The second test case s a Damond arfol (5% thckness wth 77 control volumes. We use 80 control volumes along the chord wth the proper refnement at leadng and tralng edges and also apex of the arfol (Fg.. The far feld Mach number s agan and the angle of attack s zero. For both cases, 00 mplct pre-teratons have been performed wth the startng CFL=.0 whch ncreases gradually to 00, and a fxed Krylov-subspace sze of 40 s used. F. Supersonc duct The resdual convergence hstory both n terms of CPU-Tme and resdual evaluatons has been shown n Fg. 4 and Fg. 5. GMRES-LUSGS decreases the resdual by 0 order of magntude n 6 teratons for nd order and 7 teratons for rd order. For both orders of accuracy, L-norm of the resdual s reduced by 0 orders of magntude n 000 resdual evaluatons. However as each resdual evaluaton for the rd order dscretzaton s more expensve than for nd order, the rd order CPU-tme s larger. Table shows the summary of the convergence performance for ths case. Table. Convergence summary for supersonc duct nd order rd order CPU-Tme (Sec 89 86 Resdual Evaluatons 956 997 GMRES Iteratons. 6 7 G. Damond arfol Supersonc flow feld around the damond arfol has been computed, usng second and thrd order dscretzaton. The convergence summary s shown n Table.. Ths s the case that actually usng hgher-order dscretzaton has mproved the overall performance of the computaton, and both CPU-Tme and number of resdual evaluatons for the rd order dscretzaton are less than these quanttes for the nd order dscretzaton. Fgure 6 and Fg. 7 demonstrate resdual convergence hstory n terms of CPU-Tme and resdual evaluatons respectvely. Steeper slope for the rd order scheme n Fg. 7 means the lnear system arsng from the rd order dscretzaton at each GMRES (ewton teraton s a better representatve for the orgnal non-lnear system. Fgure 8 shows convergence hstory n terms of teraton number. The frst 00 teratons are mplct pre-teratons, takng just small porton of CPU-Tme (Fg. 6. They do not decrease the resduals, but they are necessary for the start up. Fgure. 9 compares computed and the exact pressure coeffcent over the surface along the chord of the arfol. Except the overshoot (at the leadng edge and undershoot (at the apex, agreement wth the exact soluton s reasonably good. It s expected by addng lmter to the present scheme these over and under shoots would be cured consderably. Fgure 0 and Fg. dsplay Mach contours of the flow fled for the nd and rd order dscretzatons. The captured shock waves at leadng and tralng edges and the expanson fan at the apex of the arfol are clearly vsble. The qualty of the 5 Amercan Insttute of Aeronautcs and Astronautcs
captured dscontnutes and hgh gradent regons n the flow feld s notceably better for the rd order dscretzaton respect to the nd order case. Table. Convergence summary for damond arfol nd order rd order CPU-Tme (Sec 9 70 Resdual Evaluatons 86 66 GMRES Iteratons 46 6 V. Concluson A precondtoned matrx-free LUSGS-GMRES algorthm has been presented for hgher-order computaton of soluton of D Euler equatons. The results show that LUSGS-GMRES works almost as effcently for the thrd order dscretzaton as for the second order one. In fact t showed that n some cases, convergence rate can be ncreased usng hgher-order dscretzaton. We are currently workng on extenson of the proposed approach for the 4 th order dscretzaton, followed by a detaled study of soluton accuracy and resource requrements for all orders of accuracy. Acknowledgements Ths research was supported by the Canadan atural Scence and Engneerng Research Councl under Grant OPG-094467. References De Rango, S., and Zngg D. W., Aerodynamc Computatons Usng a Hgher-Order Algorthm, AIAA Conference Paper 99-067, 999. Zngg, D. W., De Rango, S., emec, M., and Pullam, T. H., Comparson of Several Spatal Dscretzatons for the aver- Stokes Equatons, Journal of Computatonal Physcs, Vol.60, 000, pp. 68-704. Barth, T. J., Fredrckson P. O., and Stuke M., Hgher-Order Soluton of the Euler Equatons on Unstructured Grds Usng Quadratc Reconstructon, AIAA Conference Paper 90-00, 990. 4 Barth, T. J., Recent Development n Hgh-Order K-Exact Reconstructon on Unstructured Meshes, AIAA Conference Paper 9-0668, 994. 5 Delanaye, M., and Essers, J. A., An Accurate Fnte Volume Scheme for Euler and aver-stokes Equatons on Unstructured Adaptve Grds, AIAA Conference Paper, 95-70, 995. 6 Ollver-Gooch, C., and Van Altena M., A Hgher-Order Accurate Unstructured Mesh Fnte-Volume Scheme for the Advecton-Dffuson Equaton, Journal of Computatonal Physcs, Vol. 8, 00, pp. 79-75. 7 Venkatakrshnan, V., and Mavrpls, D., Implct Solvers for Unstructured Meshes, AIAA Conference Paper 9-57, 99. 8 Orkws, P. D., Comparson of ewton s and Quas-ewton s Method Solvers for aver-stokes Equatons, AIAA Journal, Vol., 99, pp. 8-86. 9 Barth, T. J., and Lnton, S., W., An Unstructured Mesh ewton Solver for Compressble Flud Flow and Its Parallel Implementaton, AIAA Conference Paper 95-0, 995. 0 Ollver-Gooch, C., Toward Problem-Independent Multgrd Convergence Rates for Unstructured Mesh Methods, In 6 th Internatonal Symposum on Computatonal Flud Dynamcs, 995. Pueyo, A., and Zngg, D. W., Improvement to a ewton-krylov Solver for Aerodynamc Flows, AIAA Conference Paper 98-069, 998. ejat, A., and Ollver-Gooch C., A Hgh-Order Accurate Unstructured GMRES Solver for Posson's Equaton, CFD 00 Conference Proceedng, 00, pp. 44-49. Blanco, M., and Zngg, D. W., A Fast Solver for the Euler Equatons on Unstructured Grds Usng a ewton-gmres Method, AIAA Conference Paper 97-0, 997. 4 Manzano, L. M., Lassalne, J. V., Wong, P., and Zngg D. W., A ewton-krylov Algorthm for the Euler Equatons Usng Unstructured Grds, AIAA Conference Paper 00-074, 00. 5 Delanaye, M., Geuzane, Ph., Essers J. A., and Rogest, P., A Second-Order Fnte-Volume Scheme Solvng Euler and aver-stokes Equatons on Unstructured Adaptve Grds wth an Implct Acceleraton Procedure, AGARD 77 th Flud 6 Amercan Insttute of Aeronautcs and Astronautcs
Dynamcs panel Symposum on Progress and Challenges n Computatonal Flud Dynamc Methods and Algorthms, Sevlle, Span, 995. 6 Luo, H., Baum, J. D., and Lohner, R., A Fast Matrx-free mplct Method for compressble Flows on Unstructured Grds, Journal of Computatonal Physcs, Vol 46, 998, pp. 664-690. 7 Roe, P. L., Approxmate Remann Solvers, Parameter vectors, and dfference schemes, Journal of Computatonal Physcs, Vol. 4, 98, pp. 57-7. 8 Saad, Y., and Schultz M. H., A Generalzed Mnmal Resdual Algorthm for Solvng on-symmetrc Lnear Systems, SIAM J. Sc., Stat. Comp. Vol. 7, 986, pp. 856-869. 9 Saad, Y., A Flexble Inner-Outer Precondtoned GMRES Algorthm, SIAM J. Sc.,Stat. Comp. Vol. 4, 99, pp. 46-469. 0 Chen, R. F., and Wang Z. J., Fast, Block Lower-Upper Symmetrc Gauss-Sedel Scheme for Arbtrary Grds, AIAA Journal, Vol. 8, 000, pp. 8-45. 7 Amercan Insttute of Aeronautcs and Astronautcs
Fgure. Doman and mesh for the supersonc duct. Fgure. Mesh around the 5% damond arfol. 8 Amercan Insttute of Aeronautcs and Astronautcs
Fgure 4. Convergence hstory n terms of CPU-Tme, supersonc duct. Fgure 5. Convergence hstory n terms of Resdual evaluatons, supersonc duct. 9 Amercan Insttute of Aeronautcs and Astronautcs
Fgure 6. Convergence hstory n terms of CPU-Tme, damond arfol. Fgure 7. Convergence hstory n terms of Resdual evaluatons, damond arfol. 0 Amercan Insttute of Aeronautcs and Astronautcs
Fgure 8. Convergence hstory n terms of teraton number, damond arfol. Fgure 9. Pressure coeffcent along the chord, damond arfol. Amercan Insttute of Aeronautcs and Astronautcs
Fgure 0. Mach contours, nd order dscretzaton, damond arfol. Fgure. Mach contours, rd order dscretzaton, damond arfol. Amercan Insttute of Aeronautcs and Astronautcs