On the Symmetrization in POD-Galerkin Model for Linearized Compressible Flows

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1 AIAA ScTech 4-8 January 6, San Dego, Calforna, USA 54th AIAA Aerospace Scences Meetng AIAA 6-6 On the Symmetrzaton n POD-Galerkn Model for Lnearzed Compressble Flows Mehd Tabandeh, Mngun We New Mexco State Unversty, Las Cruces, NM 883 James P. Collns Army Research Laboratory, Aberdeen Provng Grands, MD 5 Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 U φ x N L N N 3 H L a A Reduced-order models ) based on POD-Galerkn proecton have shown success n many problems snce the approach was ntroduced approxmately two decades ago. Tradtonally, the nner product used n computaton s L type to represent knetc energy. In ths paper, our work focuses on the comparson of L nner product and the symmetry nner product ntroduced by Barone and co-authors 8). The numercal smulaton and analyss are based on a lnear acoustc problem controlled by the lnearzed Euler equaton, whch, wthout vscosty, s more senstve to nstablty than the Naver-Stokes equaton. In our study, besdes the stablty advantage notced by Barone s group, much better accuracy and convergency are also shown n the usng symmetry nner product. In the test case, symmetry nner product allows to use only 8 modes for the model results to match the exact soluton, whle L nner product requres 6 modes for smlar convergency. The dynamc behavor descrbed by phase portrats of mode coeffcents also gves a cleaner pcture when symmetry nner product s used. Nomenclature Instantaneous snapshots or state vector POD bass Space coordnate Governng equaton operator Lnear part of governng equaton operator Quadratc nonlnear part of governng equaton operator Cubc nonlnear part of governng equaton operator Symmetrzed matrx Lnear coeffcent matrx temporal coefcents Coeffcent matrx for lnearzed Euler equaton I. Introducton Hgh-fdelty numercal smulatons of complex systems are often expensve despte the breathtakng advances n computers n recent years. For many applcatons, reduced-order model ) becomes attractve for ts fast computaton wth reasonable accuracy. The development and applcaton of have been seen n many dfferent research areas wth dfferent approaches [ 3]. Model order reducton, as a way to construct s, s to reduce the complexty of a system from ts orgnal number of degree of freedom e.g. large number of mesh ponts n drect numercal smulaton) to a much lower order e.g. a couple of base functons/modes). POD-Galerkn proecton, snce ts ntroducton Research Assstant, Department of Mechancal and Aerospace Engneerng Assocate Professor, Department of Mechancal and Aerospace Engneerng, Assocate Fellow AIAA of Amercan Insttute of Aeronautcs and Astronautcs Ths materal s declared a work of the U.S. Government and s not subect to copyrght protecton n the Unted States.

2 Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 to flud mechancs [4, 5], has become one of the most popular approaches for model order reducton. Proper orthogonal decomposton POD) provdes orthogonal bases whch represent the gven ensemble of data n a well-defned least-square optmum. Galerkn proecton s then used to proect the orgnal system to a subspace wth lmted number of these optmal bases to construct a low-order approxmate system. Usually, the same nner product, whch defnes a Hlbert space, s used n both POD and Galerkn proecton processes, though there have been exceptons []. An L norm representng knetc energy provdes a common choce for nner product. However, t s not always the best choce [9, 6]. Snce symmetrzaton has shown the advantage to make ncompletely parabolc equatons well-posed and provde stable solutons [7], t s promsng to apply the same prncple on model order reducton to mprove s. Barone et. al [8] adopted the dea of symmetrzaton along wth boundary treatments to construct a symmetry system whch shows better stablty n comparson to a usng tradtonal L type of nner product. In the current work, we appled the symmetrzaton on the same lnearzed Euler equaton, but extended the study further to understand ts effect not only on stablty but also on convergency wth the number of modes) and accuracy. The remander of ths paper s organzed as follows. In secton II, bascs of POD and Galerkn proecton are brefly revewed. In secton III, there s detaled dervaton of s for the lnearzed Euler equatons. Both L and symmetry nner product are used n dervaton. The results from s derved wth dfferent nner products are shown n secton IV, where analyses on stablty, convergency, and accuracy are also taken. Fnally, the concluson s n secton V. II.A. POD II. Bascs on POD and Galerkn Proecton An nner product space s a vector space whch assocates each par of vectors n the space wth a scalar known as the nner product. These spaces provde the means of defnng the orthogonalty between vectors wth zero nner product. An nner product generates{ a norm, thus ts space} s a normed vector space. Consder an ensemble of nstantaneous snapshots U k x) k =,..., m of real vector soluton felds on the doman x Ω. The feld can be a set of analytcal, expermental, or numercal smulaton data. The U s are assumed to belong to a Hlbert space HΩ) wth assocated nner product V, W, where V and W are the arbtary vectors. The goal s to fnd the optmal and orthogonal lnear bases to descrbe the data by ts lnear combnaton n U x, t) = a t)φ x), ) = wth φ s beng the bases. The optmalty here can be defned by the mnmzaton of the average error between the orgnal snapshot data and the reconstructon from the subspace wth lmted number of bases. POD modes are defned by the fact that the averaged proecton of the ensemble U k onto φ should be a maxmum. Ths optmzaton problem leads to an egenvalue problem as follows Rφ = λφ, ) where R s the spatal autocorrelaton tensor for the flow feld. The method of snapshots [9] allows the correlaton to be changed to between snapshots n tme sequence and usually brngs down the computatonal cost to compute POD from numercal smulaton data. II.B. Galerkn Proecton In the process of Gelerkn proecton, the above lnear combnaton wth POD bases n ) s substtuted nto the orgnal governng equaton denoted by an arbtrary lnear or nonlnear operator N, and the equaton s then proected on the subspace spanned by the same bases, The operator N n a generc nonlnear equaton can be assumed a form, N [Ux, t)], φ =. 3) N [Ux, t)] = U t LU N U, U) N 3 U, U, U) =, 4) of Amercan Insttute of Aeronautcs and Astronautcs

3 where L s the lnear operator, N and N 3 are the quadratc and cubc nonlnear operators respectvely. The proecton of 4) leads to U t, φ = LU, φ + N U, U), φ + N 3 U, U, U), φ =. 5) Expandng U by ts POD modes n 5), wth the smplfcaton by orthogonalty, the orgnal PDE becomes an ODE of mode coeffcents: N ȧ = a φ, Lφ ) N N + a a k N φ, φ k ), φ + = + N N = k= l= = k= 6) N a a k a l N3 φ, φ k, φ l ), φ. The truncaton of number of modes n N leads to a model wth lower order at N, when N s much smaller than the orgnal degree of freedom e.g. number of mesh ponts). Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 III. Reduced Order Model for Lnearzed Euler Equatons In ths work, we consdered the same one-dmensonal lnearzed Euler equaton as t was used by Barone et. al [8]: U t + A U =, 7) where U s the tme-varyng state vector, U = u ξ p, 8) whch fluctuats around a steady unform mean flow Ū = [ ū ξ p ] T, and ū ξ A = ξ ū, 9) ū wth u, ξ, and p beng the flow velocty, specfc volume, and pressure, ū, ξ, and p beng the mean values, and γ the heat capacty rato. A one-dmensonal acoustc pulse defned below as the ntal condton, u = exp x x ) ), p = ρ cu, ) ρ p ρ = γ. ) p Instead of usng numercal soluton [8], we resort to get an analytcal soluton and develop the from snapshots of the analytcal solutons. The exact soluton used here avods possble numercal perturbaton and further assure the accuracy of stablty analyss, whch was suggested to be subtle n the same work by Barone et. al [8]. The exact solutons are ux, t) = exp [ x x ū + c)t) ], ) px, t) = ρ c exp [ x x ū + c)t) ], ) ξx, t) = ξ { [ exp x x ūt) ] exp [ x x ū + c)t) ]} c [ γ c exp x x ūt) )] γ + ξ [ + γ c exp x x ūt) )], 3) γ 3 of Amercan Insttute of Aeronautcs and Astronautcs

4 where c s the mean speed of sound. These solutons cannot be readly separated nto temporal and spatal bass functons. Therefore, we frst computed the soluton at dfferent tme moments e.g. snapshots) from the exact soluton, then used the method of snapshots to compute the POD modes, last constructed the Galerkn system wth the POD modes []. Note that dfferent boundary condtons can be appled for a doman at x L x. The soluton was performed over a non-dmensonal total tme T tot wth a unform tme nterval dt. Let an ensemble of M snapshots be gven at the dscrete tmes t m, U m) x) = U x, t m ). 4) The dscrete form of the correlaton functon s a matrx C, whch n ndex notaton reads C mn := M U x, t m), U x, t n ) Ω, 5) where m, n =,, M. The followng egen problem needs to be solved Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 Ca [] = λ a []. 6) ) Then the egenvectors of the correlaton matrx are the temporal coeffcents a [] = a [],, a[] M. The symmetry of C ndcates non-negatve egenvalues and the orthogonalty of the egenvectors. Based on the orthogonalty, the POD modes are computed by U = Mλ M m= a [] mu m). 7) The second step for constructng the reduced order model s to proect the governng PDE onto the POD bases. The lnear PDE reads U = LU, 8) t where L s the lnear operator, A ), n aforementoned lnearzed Euler equaton 7). The Galerkn proecton of 8) onto each POD mode φ s U t, φ = LU, φ. 9) Substtutng the POD decomposton of U nto 9), and applyng the algebrac rules of nner products along wth orthogonalty of the POD bass yelds an ODE, ȧ = a φ, Lφ ), ) whch s the reduced order model of 8). Ths tme-dependent system of ODE s has the order of the number of retaned POD modes M, wth =,, M. The nner products n ) are functonals of the POD modes φx), whch are tme-ndependent. These nner products make the coeffcent matrx and may be precomputed before the ntegraton of the. To calculate the nner product, l = φ, Lφ ) n ), one needs to specfy the defnton of nner product. Dfferent nner product, L or symmetry nner products, leads to dfferent dervatons and equatons from ths pont: L nner product For two arbtrary vectors, U ) and U ), the L nner product s defned for the current varables as U ), U ) [ = u ) u ) + ξ ) ξ ) + p ) p )] dω. ) L For ths nner product, and wth a modal bass U M = M a t)φ x) wth Ω φ = φ ) φ ) φ 3) =, ) 4 of Amercan Insttute of Aeronautcs and Astronautcs

5 the coeffcent matrx, L, s defned by ts entres l, Symmetry nner product [ l = Ω ūφ ) +ūφ ) φ ) φ ) ) φ 3) + ξφ + γ φ ) pφ3) ) φ ) ξφ + ūφ3) φ 3) ] dω. 3) Followng the dervaton of Gustafsson [7] for symmetrzaton of the lnearzed Euler equatons, the defnton of symmetry nner product for two arbtrary vectors, U ) and U ), s U ), U ) [ = ρ u ) u )) + α ρ ξ ) ξ ) 4) H Ω + + α p) p ) + α ρ ξ ) p ) + p ) ξ ))] dω. Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 The symmetrzng matrx H s used to defne the symmetry nner product: ρ H = α ρ ρα, 5) ρα +α where α s an arbtrary real nonzero parameter. The coeffcent matrx, L, s accordngly defned by [ l = ρūφ ) φ ) Ω + φ 3) φ) + φ ) φ3) + α ρūφ ) φ ) +α ρū φ ) φ 3) + φ3) φ ) ) + + α ūφ 3) φ 3) ] dω. 6) The egenvalues of these matrces can predct the stablty of the system. For the symmetry nner product case, and wth applyng the ntegral-by-parts rule, 6) becomes l = [ ρūφ ) φ ) Ω + φ 3) φ) + φ ) φ3) + α ρūφ ) φ ) ) ] +α ρū φ ) φ 3) + φ ) φ3) + + α ūφ 3) φ 3) dω [ + ρūφ ) φ ) φ ) φ 3) φ 3) φ ) α ρūφ ) φ ) leadng to: α ρū φ ) φ 3) + φ 3) φ ) So, the matrx L can be wrtten n matrx form l l 3 l M l l 3 l M L = l 3 l 3 l 3M l M l M l 3M ) + α ūφ 3) φ 3) ] LX, 7) l = l + b. 8) b b b b 3 b 3 b b M b M b 3M b MM. 9) 5 of Amercan Insttute of Aeronautcs and Astronautcs

6 The frst part n 3) s a skew-symmetrc matrx, whch always has egenvalues beng pure magnary numbers.e. zero n ts real part). The second matrx s a lower trangular matrx, whch has egenvalues on ts dagonal entres: b, b, b 33,..., b MM. If they are all non-postve real numbers, the ordnary dfferental equatons for the reduced order model s stable b ). In general, b = [ ρū φ ) + α +α ρū ) ) φ φ 3) α ρū ) ] ū φ 3) φ ) φ 3) at L x) ) + + α [ + ρū ū φ 3) φ ) φ ) ) ] ) ) α ρū φ ) φ 3) ) ) + φ φ 3) + α ρū at ) φ ) ). 3) The above dervaton suggests that the stablty of usng symmetry nner product s general stable but the stablty can be altered by dfferent boundary condtons. Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 IV. Numercal Results and Analyss Two smlar cases are consdered here. In the frst case, we had M =.5, L x =, and T tot = and used snapshots equally spaced n tme for POD-Galerkn proecton process. In the second case, wth other parameters beng the same, we ran for a longer tme T tot =, whch represents exactly one perod when perodc condton s appled), and snapshots are used. Two types of boundary condtons, perodc and non-reflectve, are consdered for both cases. More mportant, we appled dfferent nner products, L and symmetry nner products, n the modelng process to compare the performance n stablty, convergency, and accuracy. IV.A. Stablty For a lnear problem, the stablty of s s determned by ts coeffcent matrx, of whch all real parts of the egenvalues need to be non-postve to be stable. s wth dfferent number of modes to 3) are examned for stablty [8]. IV.A.. T tot = When the total tme T tot = s shorter than a perod to represent a more transtonal case), fgure compared the stablty of s wth dfferent boundary condtons wth dfferent number of modes. Maxmum real part of λ L Symm Maxmum real part of λ L Symm Number of Modes a) Perodc boundary condton Number of Modes b) Non-reflectve boundary condton Fgure : Maxmum real parts of egenvalues usng L and Symmetry nner products wth dfferent boundary condtonst=). 6 of Amercan Insttute of Aeronautcs and Astronautcs

7 a) Perodc boundary condton Fgure a shows that the real part of egenvalues are all zeros for symmetry case wth perodc boundary condtons. Thus, t matches wth the dscusson of 3) whch shows all zero real part for egenvalues when b s are zeros.e. perodc condton). On the other hand, for s usng L nner products, the maxmum real parts of the egenvalues are postve and ndcate nstablty. b) Non-reflectve boundary condton There are some specfc number of mode whch L nner product has postve maxmum real parts of egenvalues and s wth these number of modes are unstable fgure b). On the other hand, s usng the symmetry nner product are always stable for all the egenvalues real parts beng negatve. IV.A.. T tot = When the total tme T tot = s exactly one perod, fgure compared the stablty of s wth dfferent boundary condtons and dfferent number of modes. Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 Maxmum real part of λ L Symm Number of Modes a) Perodc boundary condton Maxmum real part of λ L Symm Number of Modes b) Non-reflectve boundary condton Fgure : Maxmum real parts of egenvalues usng L and Symmetry nner products wth dfferent boundary condtonst=). a) Perodc boundary condton Non-dmensonal tme of T tot = represents exactly one perod when perodc condtons are appled), and the coeffcent matrx s skew-symmetrc for both types of nner product. All the egenvalues are zeros and both s are stable for any number of modes fgure a). Because of the same stablty behavor, we wll focus on ths case for later study of convergency and accuracy. b) Non-reflectve boundary condton For ths specal case there s nether ncomng nor outgong wave for the left boundary x = ), It puts the second term n 3) zero for symmetry nner product case. For the rght boundary x = L x ), whch determnes the frst term n 3), where most of the sub-terms have negatve values whch stablze the system fgure b). For the L nner product, for smlar reasons, the system s also stable for any number of modes. IV.B. Convergency Besdes the stablty, the convergency of s wth the number of modes s another crtcal feature to consder. In the convergency study, we only consdered the cases wth perodc condton and the data wth one exact perod e.g. T tot = ). 7 of Amercan Insttute of Aeronautcs and Astronautcs

8 In the comparson, we consdered the energy captured by each model as an ndcaton for ts convergency. Snce dfferent defnton of nner products also provde dfferent defnton of energy as shown n ) and 4). We need to consder the energy separately by ther own defnton. To analyze the qualty of the s usng dfferent types of nner products or n the other words dfferent types of energy norms, the total energy captured wth the s compared wth the POD and exact soluton s total energy. Fgure 3a shows that for L nner product the captures less energy than the energy of the reconstructon of the same number of POD modes. On the other hand, fgure 3b shows that the symmetry nner product model s more energy effcent and captures even more energy than the reconstructon of POD modes Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 Total Energy IV.C. Total Energy POD No. of mode a) L nner product Total Energy Total Energy POD No. of mode b) Symmetry nner product Fgure 3: Energy calculaton for dfferent number of modes usng dfferent nner products. Mode Coeffcents Accuracy The fnal comparson s the accuracy of mode coeffcents computed by s bult from the same number of modes but dfferent nner products, L and symmetry, are used n the process. In fgures 4 and 5, we compared the temporal coeffcents of modes, n forms of phase portrat, for the s bult wth respectvely 8 and 6 modes. In both fgures, we use the phase portrats of a versus a, and a versus a 4 to show two dfferent types of attractors. Here, we notced that the usng symmetry nner product only needs 8 modes to match perfectly wth the exact soluton, whle the model usng tradtonal L nner product needs 6 modes to match well wth the exact soluton. The phase portrats from symmetrcal also depct a cleaner pcture of dynamcs for attractors, whch by tself s usually an ndcaton of better representaton of underlne dynamcs. 8 of Amercan Insttute of Aeronautcs and Astronautcs

9 a a a a Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 a a) L nner product a c) L nner product a b) Symmetry nner product a d) Symmetry nner product Fgure 4: Phase portrats of mode coeffcents computed by 8-mode s usng L and Symmetry nner products. 9 of Amercan Insttute of Aeronautcs and Astronautcs

10 a a a a Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 a a) L nner product a c) L nner product a b) Symmetry nner product a d) Symmetry nner product Fgure 5: Phase portrats of mode coeffcents computed by 6-mode s usng L and Symmetry nner products. V. Concluson and Dscusson We used two dfferent defnton of nner products, the tradtonal L nner product and a specally desgned symmetry nner product, to derved POD-Galerkn models for a lnearzed Euler equaton. A smple dervaton of coeffcents matrx for the resulted usng symmetry nner product suggests a unversally stable soluton for perodc boundary condtons, thus s not the case for the usng L nner product. Further analyss of general boundary condtons besde the perodc one suggests that the stablty may stay n most of cases for a symmetrcal. In our test case usng an exact soluton for a travelng wave, we notced that the L based, not surprsngly, captures less energy than the theoretcal upper bound, whch s the energy n the reconstructed flow by the same number of POD modes; however, the symmetrcal can even capture more energy surpassng such theoretcal upper bound. Usng the same test case, we also compared the accuracy of s by a comparson of phase portrats of mode coeffcents computed drectly from these s. A drect proecton of exact solutons to the same POD mode provdes an exact soluton to compare to. In ths study, we found that the usng symmetry nner product shows superor accuracy and the results can match perfectly wth the exact soluton for a 8-mode. On the other hand, the usng L nner product needs at least 6 modes to to reach of Amercan Insttute of Aeronautcs and Astronautcs

11 the same accuracy. It s nterestng to notce that the attractors, represented by phase portrats, from the symmetrcal, depcts a more organzed pcture and therefore a cleaner representaton of underlne dynamcs. VI. Acknowledgment The authors gratefully acknowledge the support from Army Research Lab ARL) through Army Hgh Performance Computng Research Center AHPCRC) and Mcro Autonomous Systems and Technology MAST) CTA. The authors also acknowledge Dr. Matthew Barone for all the help and frutful dscusson. References Downloaded by Mngun We on February 5, 6 DOI:.54/6.6-6 Woon, S. and Marshall, S., Desgn of multvarable control systems usng reduced-order models, Electroncs Letters, Vol., No. 5, July 975, pp Lall, S., Marsden, J. E., and Glavašk, S., A subspace approach to balanced truncaton for model reducton of nonlnear control systems, Int. J. Robust Nonlnear Control, Vol.,, pp Bergmann, M., Corder, L., and Brancher, J.-P., Optmal rotary control of the cylnder wake usng proper orthogonal decomposton reduced-order model, Physcs of fluds, Vol. 7, No. 9, 5, pp Rowley, C. W. and Wllams, D. R., Dynamcs and control of hgh-reynolds-number Flow over open cavtes, Ann. Rev. Flud Mech., Vol. 38, 6, pp Mchopoulos, J., Tsompanopoulou, P., Housts, E., Rce, J., Farhat, C., Lesonne, M., and Lechenault, F., DDEMA: A Data Drven Envronment for Multphyscs Applcatons, Computatonal Scence ICCS 3, edted by P. Sloot, D. Abramson, A. Bogdanov, Y. Gorbachev, J. Dongarra, and A. Zomaya, Vol. 66 of Lecture Notes n Computer Scence, Sprnger Berln Hedelberg, 3, pp Cortal, J., Farhat, C., Gubas, L., and Raashekhar, M., Compressed Sensng and Tme-Parallel Reduced-Order Modelng for Structural Health Montorng Usng a DDDAS, Computatonal Scence ICCS 7, edted by Y. Sh, G. van Albada, J. Dongarra, and P. Sloot, Vol of Lecture Notes n Computer Scence, Sprnger Berln Hedelberg, 7, pp Rowley, C. W. and Marsden, J. E., Reconstructon Equatons and the Karhunen-Loève Expanson for Systems wth Symmetry, Phys. D, Vol. 4,, pp Rowley, C. W., Kevrekds, I. G., Marsden, J. E., and Lust, K., Reducton and reconstructon for self-smlar dynamcal systems, Nonlnearty, Vol. 6, 3, pp Rowley, C. W., Colonus, T., and Murray, R. M., Model reducton for compressble flow usng POD and Galerkn proecton, Phys. D, Vol. 89, No., Feb. 4, pp We, M. and Rowley, C. W., Low-dmensonal models of a temporally evolvng free shear layer, J. Flud Mech., Vol. 68, 9, pp We, M., Qawasmeh, B. R., Barone, M., van Bloemen Waanders, B. G., and Zhou, L., Low-dmensonal model of spatal shear layers, Physcs of Fluds, Vol. 4, No.,, pp. 48. Qawasmeh, B. R. and We, M., Low-dmensonal models for compressble temporally developng shear layers, Journal of Flud Mechancs, Vol. 73, 9 3, pp Noack, B. R., Afanasev, K., Morzynsk, M., Tadmor, G., and Thele, F., A herarchy of low-dmensonal models for the transent and post-transent cylnder wake, Journal of Flud Mechancs, Vol. 497, No., 3, pp Berkooz, G., Holmes, P., and Lumley, J. L., The proper orthogonal decomposton n the analyss of turbulent flows, Ann. Rev. Flud Mech., Vol. 5, 993, pp Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamcal Systems and Symmetry, Cambrdge Unversty Press, Cambrdge, Colonus, T., Rowley, C. W., Freund, J. B., and Murray, R. M., On the choce of norm for modelng compressble flow dynamcs at reduced-order usng the POD, IEEE Conference on Decson and Control, Las Vegas, NV,. 7 Gustafsson, B. and Sundstrom, A., Incompletely Parabolc Problems n Flud Dynamcs, SIAM Journal on Appled Mathematcs, Vol. 35, No., 978, pp. pp Barone, M. F., Kalashnkova, I., Segalman, D. J., and Thornqust, H. K., Stable Galerkn reduced order models for lnearzed compressble flow, Journal of Computatonal Physcs, Vol. 8, No. 6, 9, pp Srovch, L., Turbulence and the dynamcs of coherent structures. I - Coherent structures. II - Symmetres and transformatons. III - Dynamcs and scalng, Quarterly of Appled Mathematcs, Vol. 45, Oct. 987, pp D.M. Luchtenburg, B. N.. M. S., An ntroducton to the POD Galerkn method for flud flows wth analytcal examples and MATLAB source codes, Tech. rep., Berln Insttute of Technology, Germany, 9. of Amercan Insttute of Aeronautcs and Astronautcs