The Finite Volume Method for Solving Systems. of Non-linear Initial-Boundary. Value Problems for PDE's

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1 Appled Matematcal Sceces, Vol. 7, 13, o. 35, HIKARI Ltd, Te Fte Volme Metod for Solvg Systems of No-lear Ital-Bodary Vale Problems for PDE's 1 Ema Al Hssa ad Zaab Moammed Alwa 1 Departmet of Matematcs, College of Scece Uversty of AL- Mstasrya, Iraq dr_emaslta@yaoo.com Departmet of Matematcs, College of Scece Uversty of AL- Mstasrya, Iraq lowgt_9@yaoo.com Copyrgt 13 Ema Al Hssa ad Zaab Moammed Alwa. Ts s a ope access artcle dstrbted der te Creatve Commos Attrbto Lcese, wc permts restrcted se, dstrbto, ad reprodcto ay medm, provded te orgal wor s properly cted. Abstract Ts paper cocered te fte volme metod tat appled to solve some ds of systems of o-lear bodary vale problems (ellptc, parabolc ad yperbolc) for PDE's. Keywords: Fte Volme Metod, Cotrol Volme, System, Bodary Vale Problems 1. Itrodcto Oe of te most mportat sorces appled matematcs s te bodary vale problems, sc as matematcal models, bology (te rate of growt of mcroorgasm), [1], cemcal egeerg (a exotermc cemcal reacto, eat codcto assocated wt radato effects, deformato of sells), [1], eat ad mass trasfer or petrolem egeerg, tools for scetfc comptg, [7]. Geerally speag, a bodary vale problem cossts of a eqato wt bodary codtos.

2 1738 E. A. Hssa ad Z. M. Alwa Te matematcal metods are dvded to two ds, te frst d s aalytc, wc s sed to fd te exact solto of te problem, ad te oter d wc ormally gves a approxmate soltos sc as [6], [8], [9], [11]. Egeerg scece ad may braces of appled matematcs ( fld dyamcs, bodary layer teory, eat trasfer), all of tem gve may partal dfferetal eqatos. Oly a few of tem ca be solved by aalytcal metods, so most cases, oe ca go for mercal metods of approxmate solto. Tere are may metods to solve partal dfferetal eqatos, sc as metod of les []. I ts paper, te fte volme metod s devoted to stdy o-lear system of bodary vale problems [7].. Formlato of Fte Volme Metod of Lear System of Mltdmeso of Ellptc Problems Ts secto s cocered wt te dscretzato of lear system of twodmeso ellptc problem by ( fte volme metod) FVM o Ω = (,a) (,a) wt rectaglar mesed ad let Ω be ope boded polygoal sbset of R d, d = or 3 ad admssble fte volme mes of Ω deoted by T s gve by a famly of cotrol volmes wc are ope polygoal covex sbset of Ω. A famly of sbset of Ω cotaed yper-plaes of R d s deoted by ε, [5]. Let s, for stace, cosder te case d =, let Ω = (,a) (,a) ad f 1, f C (Ω, R). Let T = (, ) =1,,, N1; =1,,, N ; be a admssble mes of (,a) (,a) tat satsfyg te followg addtoal assmpto [3]. 1,,, N1 >, 1,,, N >, sc tat Let, = [ x 1, x + 1 ] [ y 1, y + 1 ]. Let (x ) =,N1+1 ad (y ) =,N+1, sc tat: N 1 = 1, = 1 N = 1. = 1 ad let x, = (x, y ), for = 1,,, N 1 ; = 1,,, N xx (x,y) + yy (x,y) F 1 (,v) = f 1 (x, y), x a, y a...(1) v xx (x,y) + v yy (x,y) F (,v) = f (x, y), x a, y a...() were: F 1 (, v) = F 1 (x, y, (x, y)) or F 1 (, v) = F 1 (x,y,v(x, y))...(3) F (, v) = F (x, y, (x, y)) or F (, v) = F (x,y,v(x, y))...(4)

3 Fte volme metod 1739 wt bodary codtos: (x, ) = φ 1 (x), (x, a) = φ (x), x a...(5) v(x, ) = φ 3 (x,), v(x, a) = φ 4 (x) (, y) = ψ 1 (y), (a, y) = ψ (y), y a...(6) v(, y) = ψ 3 (y), v(a, y) = ψ 4 (y) Now, tegratg eqatos (1) ad () over eac cotrol volme. Te mercal sceme of te fte volme metod: + 1 ( +1,, ) (, 1, ) + 1 (,+1, ) + 1 (,, 1 ), F 1 (,v) =, f 1, (7) 1 (v +1, v, ) + 1 (v, v 1, ) + 1 (v,+1 v, ) + 1 (v, v, 1 ), F (,v) =, f, (8) 1 Te mercal sceme: +1, 4, + 1, +,+1 +, 1, f 1 (,v) =, f 1, (9) v +1, 4v, + v 1, + v,+1 + v, 1, f (,v) =, f, (1) Example (1): Cosder te followg system of bodary vale problems: xx (x,y)+ yy (x,y) v(x,y) = 4 x 3 y, x 1, y 1 (11) v xx (x,y) + v yy (x,y) (x,y) = 6xy + x 3 x y, x 1, y 1 (1) wt te bodary codtos: (, y) = y, (1, y) =1 + y, y 1 (13) v(, y) =, v(1, y) = y 3 (x, ) = x, (x, 1) = x + 1, x 1 (14) v(x, ) =, v(x, 1) = x 3 Te exact solto of te problem (x, y) = x + y ; v(x, y) = x 3 y.

4 174 E. A. Hssa ad Z. M. Alwa Solvg te problem by sg te FVM wt =., =., ad tegratg eqatos (11) ad (1) over eac cotrol volme, cosderg te fte volme sceme s gve as: ( +1,, ) + 1 (, 1, ) + 1 (,+1, ) + 1 (,, 1 ).4v, = (15) 1 (v +1, v, ) + 1 (v, v 1, ) + 1 (v,+1 v, ) + 1 (v, v, 1 ).4, =.58 (16) 1 ad after some comptatos, oe may ave: +1, 4, + 1, +,+1 +, 1.4v, = (17) v +1, 4v, + v 1, + v,+1 + v, 1.4, =.58 (18) Moreover:, =,,1 =.4,, =.16,,3 =.36,,4 =.64,,5 = 1 1, =.4,, =.16, 3, =.36, 4, =.64, 5, = 1 v, = v,1 = v, = v,3 = v,4 = v,5 = v 1, = v, = v 3, = v 4, = v 5, = Evalate eqato (17) ad (18) for all = 1,, 3, 4; = 1,,3,4 to get te followg: Table (1): Nmercal reslts for example (1) by sg FVM. x,1,,3,4 (x,y),,

5 Fte volme metod 1741 Table (): Comparso betwee te mercal ad exact soltos of example(1) by sg FVM of example (1) x v,1 v, v,3 v,4 v(x,y) v, v, Nmercal solto Exact solto Solto x Fg.(1): Nmercal ad exact reslts for of example (1) by sg FVM.

6 174 E. A. Hssa ad Z. M. Alwa.8 Nmercal solto Exact solto.6 Solto x Fg.(): Nmercal ad exact reslts for v of example (1) by sg FVM. 3. Te Formlato of Fte Volme Metod of Nolear System of Mltdmeso Ellptc Problem Sce FVM for olear system of two-dmesoal ellptc problem of PDE s s dffclt to be sed drectly. I ts secto, cocered Newto s metod [4] s a way to solve olear system of ellptc problem of PDE s. Te fctoal terato s: ( ) W = were: ( 1) W J ( 1) 1 ( 1) F( W )...(19) (W ) F(W) = (f 1 ( 11, 1,...,, v 11, v 1,..., v ), f ( 11, 1,...,, v 11, v 1,..., v ),..., f ( 11, 1,...,, v 11, v 1,..., v )) t ( ) ( ) ( ) W = ( 11, 1,, ( ), ( ) v 11, ( ) ( ) v 1,, v )t

7 Fte volme metod 1743 were: f1 f1 f1 f1 f1 f1 (x,y) (x,y) L (x,y) (x,y) (x,y) L (x,y) 11 1 v11 v1 v f f f f f f (x,y) (x,y) (x,y) (x,y) (x,y) (x,y) J(W) = L L 11 1 v11 v1 v M M O M M M O M f f f f f f (x,y) (x,y) L (x,y) (x,y) (x,y) L (x,y) 11 1 v11 v1 v Example (): Cosder te bodary vale problem of two-dmesoal ellptc problem: xx (x,y)+ yy (x,y) (x,y)v(x,y) = x y (x +y )+4, x (,1), y (,1)...() v xx (x,y)+v yy (x,y) (x,y) = (x + y ) +(x +y ), x (,1),y (,1)...(1) wt bodary codtos: (, y) = y, (1, y) = 1 + y, y 1...() v(, y) =, v(1, y) = y (x, ) = x, (x, 1) = 1 + x, x 1...(3) v(x, ) =, v(x, 1) = x Te exact solto of te problem s (x, y) = x + y, v(x, y) = x y. Wt =., =., we ave: Now, tegratg eqatos () ad (1) over eac cotrol volme. Te reslts of te above tegral s gve by: [ +1,, ] + 1 [, 1, ] + 1 [,+1, ] + 1 [,, 1 ].4, v, = (4) 1 [v +1, v, ] + 1 [v, v 1, ] + 1 [v,+1 v, ] + 1 [v, v, 1 ] 1.4, =.15...(5) After some comptatos, oe may ave: +1, 4, + 1, +,+1 +, 1.4, v, = (6)

8 1744 E. A. Hssa ad Z. M. Alwa v +1, 4v, + v 1, + v,+1 + v, 1.4, =.15...(7) Moreover:, =,,1 =.4,, =.16,,3 =.36,,4 =.64,,5 = 1 1, =, = 3, = 4, = 5, = v, = v 1, = v, = v 3, = v 4, = v 5, =, v,1 = v, = v,3 = v,4 = v,5 = ad te followg reslts are obtaed: Table (3): Nmercal reslts for example () by sg FVM we =1. x (1),1 (1), (1),3 (1),4 Exact solto Exact - (1), Table (4): Comparso betwee te mercal ad exact soltos of example() by sg FVM we =1of example () Exact - (1) (1) (1) (1) Exact x v,1 v, v,3 v,4 (1) solto v,

9 Fte volme metod 1745 Table (5): Nmercal reslts for example () by sg FVM we =. I x (),1 (), (),3 (),4 Exact solto Exact - (), Table (6): Comparso betwee te mercal ad exact soltos of example() bysg FVM we = of example () I x () v,1 () v, () v,3 () v,4 Exact solto Exact - v (),

10 1746 E. A. Hssa ad Z. M. Alwa Nmercal solto, = 1 Nmercal solto, = Exact solto 1. Solto x Fg.(3) : Nmercal ad exact reslts for, = 1, of example () by sg FVM..7 Nmercal solto, = 1.6 Nmercal solto, = Exact solto.5 Solto x Fg.(4): Nmercal ad exact reslts for v, = 1, of example () by sg FVM.

11 Fte volme metod Formlato of Fte Volme Metod of Nolear System of Parabolc Problem Ts secto dscssed dscretzato te system of parabolc problem o a ope polygoal sbset Ω by FVM of te form: t (x, t) xx (x, t)g 1 (, v) x (x, t) = f 1 (x, t)...(8) v t (x, t) v xx (x, t)g (, v) v x (x, t) = f (x, t)...(9) were: g 1 (, v) = g 1 (x, t, (x, t)) or g 1 (, v) = g 1 (x, t, v(x, t))...(3) g (, v) = g (x, t, (x, t)) or g (, v) = g (x, t, v(x, t))...(31) wt tal codtos of te form: (x, ) = 1 (x), x a...(3) v(x, ) = (x) ad te bodary codtos of te form: (, t) = φ 1 (t), (a, t) = φ (t), t R +...(33) v(, t) = ψ 1 (t), v(a, t) = ψ (t) were φ 1 (t), φ (t), ψ 1 (t), ψ (t), 1 (x) ad (x) are cotos fctos. Now, tegratg eqatos (8), (9) over te cotrol volme K ad wt tme depedet (, ( + 1) ). Te mercal sceme s gve by: ( 1 + ) m()f 1 (x,t) ad; ( 1 v + v ) m()f (x,t) v v v v g 1 (,v) ( + 1 ) = g (,v) ( v+ 1 v ) = (34) (35)

12 1748 E. A. Hssa ad Z. M. Alwa Example (3): Cosder te followg system of tal-bodary vale problems: t (x,t) xx (x,t)v (x,t) x (x,t) = 3xe 3t e 3t =, x (,1), t R + v t (x,t) v xx (x,t)(x,t) v x (x,t) = xe 3t e t, x (,1), t R + (36) (37) wt tal codtos: (x, ) = v(x, ) = x, x (, 1) ad bodary codtos: (, t) =, (1, t) = e 3t, t R + (38) v(, t) =, v(1, t) = e t Te exact solto of te problem s (x, t) = xe 3t, v(x, t) = xe t. Solvg te problem by sg te FVM wt =.1, =.1, ad tegratg eqatos (36), (37) over te cotrol volme wt tme depedet leads to te followg mercal sceme gve by: [ ]( v ) =.5(e 3(+1) e 3 ).3(e 3(+1) e 3 ) (39) 1 v v.1[ v + 1 v + v 1].1 1 v + =.5(e 3(+1) e 3 ).5(e 3(+1) e 3 ) (4) = 1 = 1 =.1,.8, v = 9 =.9, 1 v = = 3 = =., 1 = 1 v = 3 v = 4 = 3 =.3, 4 v = 5 = 6 = 4 =, 5 v = 6 v = 7 = 8 = 5 =.5, 7 v = 8 v = 9 = 6 = 6, 9 v = 1 = 7 =.7, 1 v = 8 = v 1 =.1, v =., v 3 =.3, v 4 =, v 5 =.5, v 6 =.6, v 7 =.7, v 8 =.8, v 9 =.9, v 1 = 1 Evalatg eqatos (39) ad (4) for all = 1,,, 9; =,1, to get te followg reslts:

13 Fte volme metod 1749 Table (7): Comparso betwee te mercal ad exact soltos of te example (3) by sg FVM. I x 1 3 Exact solto Exact Table (8): Nmercal reslts for example (3) by sg FVM I x v 1 v v 3 Exact solto Exact

14 175 E. A. Hssa ad Z. M. Alwa Nmercal solto Exact solto.8.7 Solto x Fg.(8): Nmercal ad exact reslts for of example (6) by sg FVM Nmercal solto Exact solto.8.7 Solto x Fg.(9): Nmercal ad exact reslts for v of example (3) by sg FVM.

15 Fte volme metod Formlato of Upwd Fte Volme Metod of Nolear System of Hyperbolc Problem I ts secto, te FVM wll be appled to solve te system of olear partal dfferetal eqatos of te form: t (x,t) + x (x, t)f 1 (,v) + v(x,t) = f 1 (x,t), x R, t R + (41) v t (x,t) + v x (x, t)f (,v) + (x,t) = f (x,t), x R, t R + (4) were: F 1 (, v) = F 1 (x, t, ) or F 1 (,v) = F 1 (x, t, v) (43) F (, v) = F (x, t, ) or F (,v) = F (x, t, v) (44) wt te tal codto of te form: (x, ) = 1 (x), v(x, ) = (x) (45) were 1 (x) ad (x) are cotos fctos. Now, tegratg eqatos (41), (4) over te cotrol volme. Te mercal sceme s gve by: 1 + ( 1)f 1 (,v) + v = f 1 (x,t) (46) 1 v v + ( v v 1)f (,v) + = f (x,t) (47) Example (4): Cosder te followg olear system of oe frst order of PDE s: t (x,t) + x (x,t)v (x,t) + v(x,t) = (x t) +x t+1, x R, t R + (48) v t (x,t) + v x (x,t)(x,t) + 3(x,t) = 4x+4t 1, x (,1), t R + (49) wt tal codtos: (x, ) = x, x R (5) v(x, ) = x Te exact solto of te problem s (x, t) = x + t, v(x, t) = x t. Solvg te problem by sg te FVM wt =.1, =.1, tegratg eqato (48),(49) over cotrol volme order to get te fte volme sceme wc s gve as:

16 175 E. A. Hssa ad Z. M. Alwa v + v + ( 1) + ( v v 1) ( v ) +. v = t.1t +.3 =.8 +.4t Te mercal sceme of (48), (49) s gve by: + 1 +[ ]( v ) +. = t.1t (51) 1 v v + [ v v 1] +.3 =.8 +.4t (5) =, 1 =.1, =., 3 =.3, 4 =.4, 5 =.5, 6 =.6, 7 =.7, 8 =.8, 9 =.9, 1 = 1 v =, v 1 =.1, v =., v 3 =.3, v 4 =.4, v 5 =.5, v 6 =.6, v 7 =.7, v 8 =.8, v 9 =.9, v 1 = 1 Now, evalate eqatos (51) ad (5) for all = 1,,, 1; =,1, to get te followg: Table (9): Comparso betwee te mercal ad te exact soltos of te example (4). I x 1 3 Exact solto Exact

17 Fte volme metod 1753 Table (1): Nmercal reslts for example (4) by sg FVM I x v 1 v v 3 Exact solto Exact - v Nmercal solto Exact solto.8.7 Solto x Fg.(1) : Nmercal ad exact reslts for of example (4) by sg FVM.

18 1754 E. A. Hssa ad Z. M. Alwa Nmercal solto Exact solto.7.6 Solto x Fg.(11) Nmercal ad exact reslts for v of example (4) by sg FVM. 6. Coclso ad Ftre Wor Te followg ca be draw from te preset stdy: Te se te FVM for solvg olear system of ellptc problem of PDE gve good reslts compared wt te exact solto ad te FVM was appled to solve some ds systems of o-lear for parabolc ad yperbolc problems PDE s. A terestg ftre researc topc s to se of te FVM to solve oter types of problems, sc as fld dyamc problems, optmal cotrol problems, fld mecac problems ad te sallow water problem wt topograpy. Refereces [1] S. Aba, A. Hmprey ad N. Mlls, Bocemcal Egeerg, Academc Press, New Yor, (1965). [] W. Ames, Nmercal Metods for Partal Dfferetal Eqato, Academc Press, New Yor, Sa Fracsco, (1977). [3] A. Alam-Idrss ad M. Atot, A Error Estmate for Fte Volme Metods for te Stoes Eqatos, J. of Ieqaltes Pre ad Appl. Mat., Vol.3, Isse 1, Artcle 17, ().

19 Fte volme metod 1755 [4] R. Brde ad J. Fares, Nmercal Aalyss, 7t ed., Broo/Cole, Astrala, Prdle,(1). [5] Y. Codér, Galloët T ad Herb R., Dscrete Sobolev Ieqaltes ad L p Error Estmates for Approxmate Fte Volme Solto of Covecto- Dffso Eqato, Sbmtted, (1998). [6] Z. Ca, J. Doglas ad M. Par, Developmet ad Aalyss of Hger Order Fte Volme Metods Over Rectagles for Ellptc Eqatos, Advaces Compt. Mat., Vol.19, pp.3-33, (3). [7] E. Eymard, T. Galloët ad R. Herb, Fte Volme Metod adboo of Nmercal Aalyss, Edtors: P. G. Carlet ad J. L. Los, Vol.7, pp.713-1, Amsterdam, Nort Hollad, (3). [8] A. Hadlovčová, Peroa-Mal Eqato Propertes of Explct Fte Volme Sceme, Kybereta, Vol.43, No.4, pp.53-53, (7). [9] L. Jaggo, M. L ad Y. X, A Adaptve Dscotos Fte Volme Metods for Ellptc Problems, J. of Compt. Ad Appl. Mat., Vol.35, pp , (11). [1] D. Kameets, Dffso ad Heat Trasfer Cemcal Ketcs, Plem Press, New Yor, (1969). [11] Zag Q.,Joase, H. ad Colellad, P., A Fort-Order Accrate Fte- Volme Metod wt Strctred Adaptve Mes Refemet for Solvg te Advecto-Dffso Eqato, SIAM J. Sc. Compt., Vol.34, pp.b179- B1, (1). Receved: Jaary, 13