An optimizing finite difference scheme based on proper orthogonal decomposition for CVD equations

Transcription

1 An optiizing finite difference schee based on proper orthogonal decoposition for CVD equations Juan Du, Jiang Zhu, Zhendong Luo 3,, and I M Navon 4 School of Science, Beijing Jiaotong University, Beijing 00044, PR China Institute of Atospheric Physics, Chinese Acadey of Sciences, Beijing 0009, PR China 3 School of Matheatics and Physics, North China Electric Power University, Beijing 006, China 4 Departent of Scientific Coputing, Florida State University, Dirac Sci Lib Bldg, #483, allahassee, FL , USA SUMMARY In this article, an optiizing reduced finite difference schee (FDS based on singular value decoposition (SVD and proper orthogonal decoposition (POD for the cheical vapor deposit (CVD equations is presented Error estiates between the usual finite difference solution and the reduced POD solution of optiizing FDS are then derived Finally, soe exaples of nuerical siulation are provided to deonstrate the consistency of the nuerical and theoretical results It is shown that the optiizing reduced FDS based on POD ethod is both feasible and highly efficient Received 5 January 009; Revised July 009; Accepted August 009 KEY WORDS: cheical vapor deposit equations; finite difference schee; singular value decoposition; proper orthogonal decoposition; error estiate; nuerical siulation INRODUCION he cheical vapor deposit (CVD reaction processes, which have extensive applications, can be classified as a atheatical odel defined by the following governing nonlinear partial differential equations consisting of velocity vector, teperature field, pressure field and gas ass field Proble (I Find u, p,, C, and σ such that, u = 0, (x, y, t Ω (0,, u t + c u u = p + σ + gj, (x, y, t Ω (0,, σ = µ ε(u trσi, (x, y, t Ω (0,, t + c u = λ (, (x, y, t Ω (0,, C t + u C = D ( C, (x, y, t Ω (0,, u(x, y, 0 = v(x, y, 0 = (x, y, 0 = C(x, y, 0 = 0, (x, y Ω, u(x, y, t = v(x, y, t = 0, (x, y, t = 0, C(x, y, = C 0, (x, y, t Ω (0, Correspondence to: Zhendong Luo, School of Matheatics and Physics, North China Electric Power University, Beijing 006, China E-ail: zhdluo@63co Contract/grant sponsor: he National Science Foundation of China; contract/grant nuber: 0870 (

2 AN OPIMIZING FDS BASED ON POD FOR CVD EQUAIONS where u = (u, v represents the velocity vector, p the pressure, the teperature, C the ass fraction of MGa, g the acceleration of gravity, µ the carrier gas viscosity, λ the theral conductivity of the carrier gas and D the diffusion coefficient of MGa in the carrier gas σ the viscous stress tensor, σ = (σ ij ε(u = (ɛ ij, ɛ ij = ( u i + u j, in which u i = u, u j = v, x i = x, x j = y c x j x i represents the density of the carrier gas And c = c p c where c p represents the heat capacity of the carrier gas he CVD equations ( are a syste of iportant physical equations with a broad range of applications Although the finite difference schee (FDS is one of the ost effective approaches to achieve nuerical solution for any nonlinear partial equations [ 3], the usual FDS solutions usually involves any degrees of freedo So, how to siplify the coputational load and save CPU tie becoes a key proble considering intensive calculations and resource deands in the actual coputational process to guarantee sufficient accuracy of nuerical solutions Proper orthogonal decoposition (POD is a technique which provides a useful tool for efficiently approxiating a large aount of data and representing fluid flows with reduced nuber of degrees of freedo, ie, with lower diensional odels allowing to alleviate the coputational load and eory requireents and has been successfully applied in different fields including signal analysis and pattern recognition [4 5], fluid dynaics and coherent structures [6 ], and optial flow control probles [ 4] More recently, soe reduced order finite difference odels and MFE forulations and error estiates for the upper tropical Pacific ocean odel based on POD were presented [5 8], and a finite difference schee based on proper orthogonal decoposition for the non-stationary Navier-Stokes equations was established but its error analysis has not been derived [9] Kunisch and Volkwein have presented soe Galerkin POD ethods for parabolic probles [0] and a general equation in fluid dynaics [] he singular value decoposition approach cobined with POD technique is used to treat the Burgers equation [] and the cavity flow proble [3] hough CVD equations are also dealt with POD in [3], to the best of our knowledge, there have been no published results addressing how POD is used to reduce the FDS for CVD equations and to obtain the error estiates between the usual FDS solutions and the reduced FDS solutions based on POD In this paper, POD is used to reduce the FDS for CVD equations and the error estiates between the usual FDS solutions and the reduced FDS solutions are derived Finally, an exaple of nuerical siulation is provided to deonstrate that the errors between the reduced FDS solutions and the usual FDS solutions are consistent with derived theoretical results Moreover, it is also shown that the reduced FDS based on POD is both feasible and coputationally efficient when perforing nuerical siulations for CVD equations FDS FOR PROBLEM (I AND SNAPSHOS GENERAION Let x and y be the spatial step increent in x-direction and y-direction, respectively, and t be the tie step increent, u n j+,k and vn denote function values of u and v at point (x j,k+ j+, y k, t n and (x j, y k+, t n (0 j J, 0 k K, 0 n N = / t, respectively In the following, we apply staggered grid (see Figure FDS to discretize Proble (I ( Discretizing the continuous equation u = 0 ( yields [ u j+,k u j,k + v j,k+ v j,k ] n = 0 ( x y

3 JDU, J ZHU, Z LUO AND I M NAVON 3 ( he oentu equations u t + c u u = p + σ + gj, σ + trσi = µ ε(u can be rewritten into the following equations u t + c (u u x + v u y = p x + µ ( 4 u 3 x + u y + v 3 x y, v t + c (v v y + u v x = p y + µ ( 4 v 3 y + v x + u 3 x y + g Discretizing the oentu equation (3 (4 u t + c (u u x + v u y = p x + µ ( 4 3 on x direction at point (x j+, y k, t n yields where u x + u y + v 3 x y (5 u n+ = F n t j+,k j+,k x (pn j+,k p n j,k, (6 F n j+,k = un j+,k c t x un j+,k(un j+,k u n t j,k c y vn j+,k(un j+,k+ u n j+,k + tµ ( u j+,k u j+,k + u j+,k+ y v j+,k+ v j,k+ v j+,k + v j,k n 3 x y Discretizing the oentu equation u j,k u j+,k + u j+ 3,k x + (7 v t + c (v v y + u v x = p y + µ ( 4 3 on y direction at point (x j, y k+, t n yields where G n j,k+ = v n j,k+ v y + v x + u 3 x y + g (8 v n+ = G n j,k+ j,k+ t y (pn j,k+ pj,k, n (9 t c x un j,k+ (v n j+,k+ v n t j,k+ c y vn j,k+ (v j,k+ n vj,k n + tµ ( v j,k+ v j,k+ + v j+,k+ x u j+,k+ u j,k+ u j+,k + u j,k 3 x y v j,k v j,k+ + v j,k+ 3 y n + tg n j,k (0 he stress tensor equation σ = µ ε(u trσi (

4 4 AN OPIMIZING FDS BASED ON POD FOR CVD EQUAIONS Figure Staggered esh graphics can be rewritten as such that ( σ σ σ σ 3 = µ u x ( u y + v x at point (x j+, y k, t n yield σ,j,k n = [ 4 3 µ σ = 3 (4µ ( u y + v x v y u x µ v y, σ = µ ( u y + v x, σ 3 = 3 (4µ v y µ u x ( σ + σ σ + σ 3 u j+,k u j,k x 3 µ v j,k+ v j,k ] n, y u σ,j,k n j,k+ = [µ u j,k y σ n 3,j,k = [ 4 3 µ (3 Expanding the energy equation at point (x j, y k, t n in aylor series yields n+ j,k v j,k+ v j,k y v j+ + µ,k v j x 3 µ,k ] n, u j+,k u j x,k ] n ( (3 (4 t + c u = λ ( (5 = n j,k c t x un j,k( j+,k j,k n c t y vn j,k( j,k+ j,k n +λ t( j,k j,k + j+,k x + j,k j,k + j,k+ y n (6

5 (4 Expanding the ass equation JDU, J ZHU, Z LUO AND I M NAVON 5 at point (x j, y k, t n yields C t + u C = D ( C (7 C n+ j,k = C n j,k t x un j,k(c j+,k C j,k n t y vn j,k(c j,k+ C j,k n +D t( C j,k C j,k + C j+,k x + C j,k C j,k + C j,k+ y n (8 Inserting (6 and (9 into ( results in a Poisson equation for p as follows [ p j,k p j,k + p j+,k x + p j,k p j,k + p j,k+ y ] n = RHS, (9 where RHS = t x (F j+,k F j,k n + t y (G j,k+ G j,k n Using the sae approaches as the proof of the convergence and stability of finite difference equations of the non-stationary Navier Stokes equation in [] or [], it is not difficult to prove the convergence and stability of usual FDS (, (6, (9, (4, (6 and (8 for CVD equations, if 05( u + v t µ, µ t x, and tµ 05 y And we can conclude the following result using aylor expansion heore Usual FDS (, (6, (9, (4, (6 and (8 for CVD equations has the following error estiates E n (u n j+,k, vn, p n j,k+ j,k, j,k n, Cn j,k = (u(x j+, y k, t n, v(x j, y k+, t n, p(x j, y k, t n, (x j, y k, t n, C(x j, y k, t n (u n j+,k, vn, p n j,k+ j,k, j,k n, Cn j,k = O( t, x, y, n N, where denotes usual nor of vector (0 E n (σ n j+,k = (σ(x j+, y k, t n σ n j+,k = O( x, y, n N, ( where denotes usual nor of atrix he procedure to achieve FDS nuerical solutions for proble (I is first to solve (9, then (6 and (9, and then (4, (6 and (8, fro which we obtain a set of discrete nuerical solutions hus, if the carrier gas viscosity µ, the theral conductivity of the carrier gas λ, the diffusion coefficient D, tie step increent t, and spacial step increent x and y in x-direction and y-direction are given, by solving (, (6, (9, (4, (6 and (8 one could obtain u n j+,k, v n, p n j,k+ j,k, j,k n, Cn j,k, and σn (0 j J, 0 k K, n N j+,k Write u n i = un j+,k, vn i = v n, p n j,k+ i = pn j,k, i n = j,k n, and Cn i = Cj,k n (i = k(j ++j+, = JK, i, 0 j J, 0 k K, n N L group of values consisting of the enseble {u n l i, vn l i, p n l i, n l i, C n l i } L l= ( i (usually L N, known as snapshots, are chosen fro N group of {u n i, vn i, pn i, i n, Cn i }N n=( i Reark When one coputes real-life probles, one ay obtain the enseble of snapshots fro physical syste

6 6 AN OPIMIZING FDS BASED ON POD FOR CVD EQUAIONS trajectories via drawing saples of experients and interpolation (or date assiilation For exaple if one considers nuerical weather prediction, one can use the previous weather results to build the enseble of snapshots, then restructure the optial basis for the enseble of snapshots by the following SVD and POD, and finally cobine with a POD projection to derive a reduced optiizing dynaical syste hus, the situation of future weather change can be quickly siulated and the future weather change can be forecast, which is of ajor iportance for actual real-life applications 3 POD REDUCED MODEL FOR CVD EQUAIONS In the following, we derive POD basis fro the snapshots generated above with singular value decoposition (SVD, and then use the POD basis to develop a reduced optiizing FDS for CVD equations 3 Singular value decoposition and POD basis he enseble of snapshots {u n l i, vn l i, p n l i, n l i, C n l i } L l= ( i (usually L N can be expressed as the following L atrices A u, A v, A p, A and A C,respectively A u = u n u n u n L u n u n u n L u n u n u n L A =, A v = v n v n v n L v n v n v n L v n n n n L n n n L n n n L v n v n L, A C =, A p = p n p n p n L p n p n p n L p n C n C n C n L C n C n C n L C n C n C n L p n p n L (3 In order to obtain optial representation for A u (A v, A p, A, and A C are siilar, we eploy SVD to research FDS for CVD equations, which serves as an iportant tool to construct optial basis of optiization approxiation For atrix A u R L, there exists the SVD A u = U u ( Su 0 0 0, V u, (3 where U u R and V u R L L are all orthogonal atrices, S u =diag{σ u, σ u,, σ ul } R l l is the diagonal atrix corresponding to A u, and σ ui (i =,,, l are the positive singular values he atrices U u = (φ u, φ u,, φ u R and V u = (ϕ u, ϕ u,, ϕ ul R L L contain the orthogonal eigenvectors to the A u A u and A u A u, respectively he coluns of these eigenvector atrices are organized such that corresponding to the singular values σ ui are coprised in S u in a non-increasing order And the singular values of the decoposition are connected to the eigenvalues of the atrices A u A u and A u A u in the anner such that λ ui = σui (i =,,, l Since the nuber of esh points is by far larger than that of transient oent points, ie, L, that is also that the order for atrix A u A u is far larger than the order L for atrix A u A u, however, their null eigenvalues are identical, therefore, we ay first solve the eigen equation corresponding to atrix A u A u to find the eigenvectors ϕ uj (j =,,, L, and then by using φ uj = σ uj A u ϕ uj, j =,,, l, (33

7 JDU, J ZHU, Z LUO AND I M NAVON 7 we ay obtain l (l L eigenvectors corresponding to the non-zero eigenvalues for atrix A u A u Define atrix nor α,β as A u α,β = sup Ax α / x β (where α and β are the nor x 0 of vector Let A Mu = i= σ uiφ ui ϕ ui, φ ui (i =,,, and ϕ uj (j =,,, are first colun vectors of atrices U u and V u, respectively hen, by the properties of the relationship between spectral radius and, for atrix, if < r =ranka u (r l L, then we obtain the following equation in A u B, = A u A Mu, = σ u(mu+, (34 rank(b which shows that A Mu is an optial representation of A u, ie, A Mu is an optial approxiation of A u and the error is σ u(mu + = λ u(mu + Denote the L colun vectors of atrix A u by a l u = (u n l, un l,, un l (l =,,, L, and ε l (l =,,, L by unit colun vectors except that lth coponent is, while the other coponents are 0 hen by the copatibility of the nor for atrices and vectors, we obtain that a l u P Mu (a l u = (A u A Mu ε l A u A Mu, ε l = λ u(mu +, (35 where P Mu (a l u = j= (φ uj, a l uφ uj, (φ uj, a l u are the canonical inner products for vector φ uj and vector a l u Inequality (35 shows that P Mu (a l u are the optial approxiations to a l u, whose errors are all λ u(mu + hus, a group of optial basis is found in the construction of A Mu By the property of eigenvectors, it is well known that Φ u = (φ u, φ u,, φ Mu ( L is an orthonoral atrix and {φ uj } j= is a group of POD optial bases By the sae approach as the above (35, if a l v = (v n l, vn l,, vn l, a l p = (p n l, pn l,, pn l a l = ( n l, n l,, n l, and a l C = (Cn l, Cn l,, Cn l, (l =,,, L are the L colun vectors j= (φ pj, a l pφ pj, of atrices A v, A p, A, and A C, then P Mv (a l v = M v j= (φ vj, a l vφ vj, P Mp (a l p = M p P M (a l = M j= (φ j, a l φ j, and P MC (a l C = j= (φ Cj, a l C φ Cj are the optial approxiations to a l v, a l p, a l, and al C, whose errors are a l v P Mv (a l v a l p P Mp (a l p a l P M (a l a l C P MC (a l C λ v(mv +, (36 λ p(mp +, (37 λ (M +, (38 λ C(MC +, (39 where λ v(mv +, λ p(mp +, λ (M +, and λ C(MC + are (M v + th eigenvalue for A v A v, (M p + th eigenvalue for A p A p, (M + th eigenvalue for A A, and ( + th eigenvalue for A C A C, and Φ v = (φ v, φ v,, φ vmv, Φ p = (φ p, φ p,, φ pmp, Φ = (φ, φ,, φ M, and Φ C = (φ C, φ C,, φ CMC are orthonoral atrices and {φ vj } M v j=, {φ pj} M p j=, {φ j} M j=, and {φ Cj} j= are three groups of optial basis corresponding to A v, A p, A, and A C, respectively 3 Reduced optiizing FDS based on POD for CVD equations In the following, we use POD basis to develop a reduced optiizing FDS for CVD equations

8 8 AN OPIMIZING FDS BASED ON POD FOR CVD EQUAIONS Write u (t = (u (t, u (t,, u (t, v (t = (v (t, v (t,, v (t, p (t = (p (t, p (t,, p (t, (t = ( (t, (t,, (t, C (t = (C (t, C (t,, C (t, (30 where u i = u j+,k, v i = v j,k+, p i = p j,k, i = j,k and C i = C j,k ( i, i = k(j ++j +, = KJ, 0 j J, 0 k K hus, (, (6, (9, (, and (4 are rewritten as the following vector forulation ( u n+, v n+, p n+, n+, C n+ = (u n, v, n p n,, n C n + t F (u n, v, n p n,, n C, n 0 n N, (3 where F (u n, v n, p n, n, C n = ( F (u n, v n, p n, n, C n,, F 5 (u n, v n, p n, n, C n, is the vector function obtained fro (, (6, (9, (, and (4 Setting (u n, v n, p n, n, C n = ( Φ u α u,n, Φ v α v,n M v, Φ p α p,n M p, Φ α,n M, Φ C α C,n, (3 where u n = (u n, u n,, u n, v n = (v n, v n,, v n and p n,, n C n likewise Inserting (3 into (3 and noting that Φ u, Φ v, Φ p, Φ, Φ C are orthogonal atrices, we ay obtain a reduced optiizing odel which has + M v + M p + M + (, M v, M p, M, L unknown values: α u,n+ α v,n+ M v α p,n+ M p α,n+ M α C,n+ = α u,n α v,n M v α p,n M p α,n M α C,n + t Φ u F (Φ u α u,n,, Φ C α C,n Φ F v (Φ v α u,n,, Φ C α C,n Φ F p 3 (Φ p α u,n,, Φ C α C,n Φ F 4 (Φ α u,n,, Φ C α C,n Φ F C 5 (Φ C α u,n,, Φ C α C,n, (33 where n = 0,,,, N, and the initial values are α u,0 = Φ u u 0, α v,0 M v = Φ v v 0 and α p,0 M p, α,0 respectively α C,0 After one has obtained α u,n, α v,n M v, α p,n M p, α,n M, and α C,n fro (33, one obtains the POD optial solutions which are written as u n i = Φ u α u,n, vi n = Φ v α v,n M v, p n i = Φ p α p,n M p, i n = Φ α,n M, and Ci n = Φ C α C,n for Proble (I by (3 hus, we obtain the optial nuerical solutions which are written as (u n j+,k, v n j,k+, p n j,k, j,k, n Cj,k n (0 j J, 0 k K, 0 n N for Proble (I, where u n = j+ u n,k i, v n = v j,k+ i n, p n j,k = p n i, j,k n = i n, and Cj,k n = C n i (j = i k(j + 0, i = KJ, 0 k K, 0 n N,respectively Reark Forula (33 with (3 and (45 is the reduced optiizing FDS based on SVD and POD for Proble (I, because it only involves ( +M v +M p +M + N (, M v, M p, M, L degrees of freedo while usual FDS (, (6, (9, (, and (4 involves 5 N degrees of freedo When it coes to real-life probles, the POD basis can be structured by the known enseble of snapshots, and then we cobine it with POD projection to derive a reduced optiizing FDS, ie, one needs only to solve the above forula (33 with (3 and (45 which has uch fewer degrees of freedo instead of the usual FDS (, (6, (9, (, and (4 hus, a sizably large econoy in coputational load and eory requireents can be attained M,

9 JDU, J ZHU, Z LUO AND I M NAVON 9 4 ERROR ANALYSIS OF POD REDUCED FDS In this part, the error estiates of reduced optiizing FDS (3, (33, and (45 for Proble (I are discussed Let X u = span{φ u, φ u,, φ umu }, X v = span{φ v, φ v,, φ vmv }, X p = span{φ p, φ p,, φ pmp }, X = span{φ, φ,, φ M }, X C = span{φ C, φ C,, φ CMC } hen, for colun vectors a l u ( l L of A u, by (35 we have a l u = u n l, and there is a P Mu (u n l = P Mu (a l u = j= (φ uj, a l uφ uj = j= (φ uj, u n l φ uj X u such that While n {n, n,, n L }, u n therefore, we obtain that u n l P Mu (u n l (4 λ u(mu +, l L (4 = P Mu (u n = j= (φ uj, u n φ uj obtaining by (3 and (33, u n u n λ u(mu +, n {n, n,, n L } (43 Using the sae approach as (43, we could obtain that v n v n λ v(mv +, n {n, n,, n L } (44 p n p n λ p(mp+, n {n, n,, n L } (45 n n C n C n λ (M +, n {n, n,, n L } (46 λ C(MC +, n {n, n,, n L } (47 When n {n, n,, n L }, we ay as well let t n (t nl, t nl+ and t n be the nearest point to t nl Coparing (3 (33 with (3, (3 (33 can be written in siilar fors as (, (6, (9, (, and (4 as follows where F n = j+ u n,k j+,k c t u n+ n t = F j+,k j+,k x (p n j+,k p n j,k, (48 x u n j+,k(u n j+,k u n t j,k c y v n j+,k(u n j+,k+ u n j+,k + tµ ( u j+,k u j+,k + u j+,k+ y v j+,k+ v j,k+ v j+,k + v j,k n ; 3 x y u j,k u j+,k + u j+ 3,k x + (49 v n+ = G n j,k+ j,k+ t y (p n j,k+ pj,k, n (40

10 0 AN OPIMIZING FDS BASED ON POD FOR CVD EQUAIONS where G n j,k+ = v n t j,k+ c x u n j,k+ (v n j+,k+ v n t j,k+ c y v n j,k+ (v j,k+ n vj,k n + tµ ( v j,k+ v j,k+ + v j+,k+ x u j+,k+ u j,k+ u j+,k + u j,k 3 x y n+ j,k v j,k v j,k+ + v j,k+ 3 y n + tg n j,k; = n j,k c t x u n j,k( j+,k j,k n c t y v n j,k( j,k+ j,k n +λ t( j,k j,k + j+,k x C n+ j,k + j,k j,k + j,k+ y n ; = C n j,k t x u n j,k(c j+,k C j,k n t y v n j,k(c j,k+ C j,k n +D t( C j,k C j,k + C j+,k x [ p j,k p j,k + p j+,k x + C j,k C j,k + C j,k+ y n ; (4 (4 (43 + p j,k p j,k + p j,k+ y ] n = RHS, (44 where RHS = (F j+,k F j,k n /( t x + (G j,k+ G j,k n /( t y, and σ,j,k n = [ 4 3 µ u j+,k u j,k x 3 µ v j,k+ v j,k ] n, y u σ,j,k n j,k+ = [µ u j,k y σ3,j,k n = [ 4 3 µ v j,k+ v j,k y v j+ + µ,k v j x 3 µ,k ] n, u j+,k u j x,k ] n (45 If u n j+,k, vn j+,k, un, v n, u n j,k+ j,k+ j+,k, v n j+,k, u n and v n are all bounded, by j,k+ j,k+ subtracting (48 (45 fro (, (6, (9, (4, (6, and (8 respectively, and using( and [ u j+,k u j,k + v j,k+ v j,k ] n = 0, (46 x y we can obtain that and u n+ u n+ + v n+ v n+ + n+ n+ + C n+ C n+ M( u n u n + v n v n + n n + C n C n, (47 p n p n M 0 ( u n u n + v n v n + n n, (48 σ n i σ n i M 0 ( u n u n + v n v n, i =,, 3 (49 where M = + M 0, M 0 = C t/ in( x, y, µ x, µ y, µ x y, C is a constant independent of t, x, and y Suing (47 fro n l, n l +,, n can yield that u n u n + v n v n + n n + C n C n u n l u n l + v n l v n l + n l n l + C n l C n l n +M 0 ( u j u j + v j v j + j j + C j C j j=n l (40

11 JDU, J ZHU, Z LUO AND I M NAVON If t = O( x, y, (µ t, by discrete Gronwall Lea (see [4], we get that u n u n + v n v n + n n + C n C n ( u n l u n l + v n l v n l + n l n l + C n l C n l exp[c t (n nl ] (4 If t l ( l L are uniforly chosen fro t n ( l N, then (n n l N/(L If L = O(N, we obtain fro (4 and (43 (47 that u n u n + v n v n + n n + C n C n C 0 ( λ u(mu + + λ v(mv + + λ (M + + λ C(MC + (4 in which C 0 represents a constant hen the error estiates including u, v, p,, and C can be expressed as follows heore Let (u n, v, n p n,, n C n (n =,,, N be vectors constituted with solutions of usual FDS (, (6, (9, (4, (6 and (8, (u n, v n, p n, n, n be the vectors of the reduced optiizing FDS (3, (33, and (45, if n {n, n,, n L }, then the following error estiates exist u n u n λ u(mu +, v n v n λ v(mv +, p n p n λ p(mp +, n n λ (M +, C n C n λ C(MC +, σi n σi n C( λ u(mu + + λ v(mv +, i =,, 3 (43 Moreover, if n {n, n,, n L }, t = O( x, y, (µ t, u n j+,k, vn, p n j,k+ j,k, j,k n, Cj,k n, u n j+,k, v n j,k+, p n j,k n, j,k, C n j,k are all bounded, snapshots {un l j+,k, vn l j+,k, pn l j,k, n l j,k, Cn l j,k }L l= are uniforly chosen fro {u n j+,k, vn j+,k, pn j,k, n j,k, Cn j,k }N n=, L = O(N, then the following error estiates hold u n u n + v n v n + p n p n + n n + C n C n C( λ u(mu+ + λ v(mv+ + λ p(mp+ + λ (M + + λ C(MC +, σi n σi n C( λ u(mu+ + λ v(mv+ + λ p(mp+ + λ (M + + λ C(MC +, i =,, 3, (44 (45 where N = / t and L is the nuber of snapshots Given that the absolute value of each coponent of a vector cannot exceed value of any of its nors, the following result can be obtained by cobining heore and heore heore 3 Under the assuptions of heore, the following error estiate holds u(x j+, y k, t n u n j+,k + v(x j, y k+, t n v n j,k+ + p(x j, y k, t n p n j,k + (x j, y k, t n n j,k + C(x j, y k, t n C n j,k + σ (x j+, y k, t n σ n j+,k + σ (x j+, y k, t n σ n j+,k + σ 3(x j+, y k, t n σ n 3j+,k O( λ u(mu + + λ v(mv+ + λ p(mp+ + λ (M + + λ C(MC +, t, x, y, n N (46

12 AN OPIMIZING FDS BASED ON POD FOR CVD EQUAIONS Figure Physics odel of the transport gas: t = 0, ie, n = 0 initial values on boundary Reark 3 he condition L = O(N in heore exhibits the relation between the nuber L of snapshots and the nuber N of all tie instances herefore, it is not necessary to take the total transient solutions at all tie instances t n as snapshots (see [0-] heores and 3 have presented the error estiate between the solution of the reduced optiizing FDS (3, (33, and (45 and the solution of usual FDS (, (6, (9, (4, (6, and (8, and the error estiate between the solution of the reduced FDS and Proble (I respectively Because our ethod eploys the coputed FDS solutions (u n j+,k, vn, p n j,k+ j,k, j,k n, Cn j,k (n =,,, N for Proble (I as initial data for the reduced odel, the error estiates in heore 3 are correlated to the grid scales x and y, and the tie step size t However, when it coes to actual probles, the enseble of snapshots could be obtained fro physical syste trajectories by drawing saples fro experients and interpolation (or data assiilation hus, the assistant data (u n j+,k, vn, p n j,k+ j,k, j,k n, Cn j,k (n =,,, N could be replaced by the either interpolation functions of experiental or previously obtained results 5 NUMERICAL SIMULAIONS In this section, soe nuerical exaples of the physical odel of the cavity flows with the POD reduced FDS (3, (33, and (45 are presented to deonstrate the feasibility and efficiency of the POD ethod Let the side length of the cavity be (see Figure We take the spatial step increents as x = y = 3 and tie step increent as t = 0000 Except that = 4y( y and p y Ω = µ ( 4 3 v y Ω on the right boundary, other initial and boundary values are all taken as 0 ake the constants to be c = c = 005, the viscosity µ = 00, the theral conductivity λ = and the diffusion coefficient D = 05 We obtain 0 discrete values (ie, snapshots at ties t = 0, 0, 30,, 00,respectively by solving usual FDS (, (6, (9, (4, (6, and (8 And it is shown by coputing the eigenvalues that λ u6 + λ v6 + λ p8 + λ 8 + λ C8 = O(0 3 When t = 00, we obtain the solutions of the POD reduced FDS (3, (33, and (45 depicted in Figure 3 and Figure 4 on the right-hand side used = M v = 5, M p = M = = 7 optial POD bases respectively, and the solutions of usual FDS, ie, (, (6, (9, (4, (6, and (8 are depicted in Figure 3 and Figure 4 on the left-hand side (Because these figures are equal to

13 JDU, J ZHU, Z LUO AND I M NAVON 3 Figure 3 Velocity (u, v strea line figure for usual FDS solution (on left-hand side figure and when = M v = 5, solutions of the reduced FDS based on POD (on right-hand side figure solutions obtained with 0 bases, they are also referred to as the figures of full basis solution Figure 5 shows the errors between solutions obtained with different nubers of optiizing POD bases and solutions with full basis Coparing the usual FDS with the POD reduced FDS by ipleenting 3000 ties nuerical siulation coputations, it is shown that the CPU tie with usual FDS, ie, (, (6, (9, (4, (6, and (8 is alost a factor of 00 larger than that required with the POD reduced FDS (3, (33, and (45 with POD optial basis and the errors between the solutions are not ore than Although what we have done here is kind of recoputing what we have already coputed by usual FDS, when it coes to actual probles, we ay structure the snapshots and POD basis with interpolation or data assiilation by drawing saples fro experients, then solve directly the reduced optiizing FDS (33 with (3 and (45 hus, the CPU tie and eory requireents in the coputational process will be greatly reduced It is also shown that the POD approach in solving the CVD equations is very effective and the consistency of the nuerical and theoretical results is deonstrated 6 CONCLUSIONS In this paper, the SVD and POD ethods are applied to derive a reduced optiizing FDS for CVD equations First, ensebles of data are copiled fro transient solutions obtained with usual FDS, a process that can be oitted in actual applications where the enseble of snapshots can be obtained fro physical syste trajectories by drawing saples fro experients and interpolation (or data assiilation hen we eploy SVD to derive POD basis fro the ensebles of data and substitute the unknowns of usual FDS with the linear cobinations of POD basis to develop the reduced optiizing

14 4 AN OPIMIZING FDS BASED ON POD FOR CVD EQUAIONS Figure 4 eperature figure, ass figure and pressure figure for usual FDS solution (on left-hand side figure and when M p = M = = 7 solutions of the reduced FDS based on POD (on right-hand side figure FDS for CVD equations, in which a uch saller basis renders the POD reduced FDS optial in a sense Finally, we have analyzed the errors between the POD reduced FDS approxiate solutions and the usual finite difference solutions finding that they are consistent with the theoretical results of error estiate and validating the feasibility and efficiency of our reduced optiizing FDS at the sae tie he theoretical and nuerical results in this article also deonstrate that the POD ethod has extensive applications for coplicated nonlinear PDEs, which akes it ore feasible to be applied in other coplicated PDEs and for real-life probles such as the air quality forecast syste, the ocean fluid dynaics forecast syste,etc REFERENCES Chung Coputational Fluid Dynaics Cabridge University Press, Cabridge, UK, 00 Xin X, Liu R,Jiang B Coputational Fluid Dynaics National Defense Science echnology Press, Changsha, 989 (in Chinese 3 Quarteroni A, Valli A Nuerical Approxiation of Partial Differential Equations Springer-Verlag, Berlin

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