8454 XMa,N.Zabaras/Joumal of Computational Physics 7(0)8448-8471 Since the input uncertainty is taken as a Gaussian random field.we use Hermite polynomials to represent: z..-...) xto-xΨ⑤.t,o-4xΨ（⑤ (37) Substitution of Eqs.(28).(31)-(33)and(37)into the stabilized second-order projection method formulation Eqs.(7)and (10)-(13).results in the following: 2n-+会会会42wW-g网m 品2会-w+容92名w-g (38) w+加玄以"男=A三+会+会 (39 "-g (40) 立名w1-9%+玄-名防=0 2w-明+22四-呀m-云g。 (42 (0dtheroonity of thepoly -0,得言若名e听- -高含名-名2a网-+n-名名 -prRaeneu (43) 0-0 (At+t)p"AtV+ 4) 暖=Vp吸l (45) 心1-+暖1-哦=0， (46) -用+客呀时g (47) wheree0.1...P.This results in (P+1)(3d+)decoupled deterministic equations. Remark 2.In Eq.(43)it is time consuming to evaluate the fourth-order product term ()directly using the method discussed in Section 4.1.To simplify this calculation,we introduce an auxiliary random variable as follows:Since the input uncertainty is taken as a Gaussian random field, we use Hermite polynomials to represent the solution: vðx;t;xÞ ¼ XP i¼0 viðx;tÞWiðnÞ; pðx;t;xÞ ¼ XP i¼0 piðx;tÞWiðnÞ; hðx;t;xÞ ¼ XP i¼0 hiðx;tÞWiðnÞ; pðx;t;xÞ ¼ XP i¼0 piðx;tÞWiðnÞ: ð37Þ Substitution of Eqs. (28), (31)–(33) and (37) into the stabilized second-order projection method formulation Eqs. (7) and (10)–(13), results in the following: 1 Dt XP i¼0 ðvnþ1=2 i  vn i ÞWi þXP i¼0 XP j¼0 XP l¼0 ^lð2vn i rvn j  vn1 i rvn1 j ÞWiWjWl ¼  Pr Da XP i¼0 XP j¼0 ið2vn j  vn1 j ÞWiWj þ PrXP i¼0 r2 ðvnþ1=2 i ÞWi  1:75kvnk ð150DaÞ 1=2 XP i¼0 XP j¼0 ~ið2vn j  vn1 j ÞWiWj XP i¼0 XP j¼0 irpn j WiWj  PrRaXP i¼0 XP j¼0 ihn j WiWjeg; ð38Þ ðDt þ sÞr2XP i¼0 pnþ1 i Wi ¼ Dt XP i¼0 r2 pn i Wi þXP i¼0 r vnþ1=2 i Wi þ s XP i¼0 r pn i Wi; ð39Þ XP i¼0 pnþ1 i Wi ¼ XP i¼0 rpnþ1 i Wi; ð40Þ 1 Dt XP i¼0 ðvnþ1 i  vnþ1=2 i ÞWi þXP i¼0 rpnþ1 i Wi XP i¼0 rpn i Wi ¼ 0; ð41Þ 1 Dt XP i¼0 ðhnþ1 i  hn i ÞWi þXP i¼0 XP j¼0 ð2vn i rhn j  vn1 i rhn1 j ÞWiWj ¼ XP i¼0 r2 hnþ1 i Wi: ð42Þ Then performing a Galerkin projection of each equation by h; Wki [8,10], and using the orthogonality of the polynomial basis Eq. (19), we obtain 1 Dt ðvnþ1=2 k  vn k Þ þ hWiWjWlWki hW2 k i XP i¼0 XP j¼0 XP l¼0 ^lð2vn i rvn j  vn1 i rvn1 j Þ ¼  Pr Da XP i¼0 XP j¼0 ið2vn j  vn1 j Þeijk  1:75kvnk ð150DaÞ 1=2 XP i¼0 XP j¼0 ~ið2vn j  vn1 j Þeijk þ Prr2 vnþ1=2 k XP i¼0 XP j¼0 irpn j eijk  PrRaXP i¼0 XP j¼0 ihn j eijkeg; ð43Þ ðDt þ skÞr2 pnþ1 k ¼ Dtr2 pn k þ r vnþ1=2 k þ skr pn k ; ð44Þ pnþ1 k ¼ rpnþ1 k ; ð45Þ 1 Dt ðvnþ1 k  vnþ1=2 k Þ þ rpnþ1 k  rpn k ¼ 0; ð46Þ 1 Dt ðhnþ1 k  hn k Þ þXP i¼0 XP j¼0 ð2vn i rhn j  vn1 i rhn1 j Þeijk ¼ r2 hnþ1 k ; ð47Þ where eijk ¼ hWiWjWki hW2 k i , k ¼ 0; 1; ... ; P. This results in ðP þ 1Þð3d þ 2Þ decoupled deterministic equations. Remark 2. In Eq. (43), it is time consuming to evaluate the fourth-order product term 2 hW2 k i PP i¼0 PP j¼0 PP l¼0^lvn i rvn j hWiWjWlWki directly using the method discussed in Section 4.1. To simplify this calculation, we introduce an auxiliary random variable as follows: 8454 X. Ma, N. Zabaras / Journal of Computational Physics 227 (2008) 8448–8471