A Galerkin Method for a Gaseous Ignition Model

Transcription

1 SQU Journal for Science, 17() (01) Sultan Qaboo Univerity A Galerkin Method for a Gaeou Ignition Model M Salan* and J Ki** *Departent of Matheatic, Statitic and Phyic, Univerity of Qatar, Doha, Qatar, Eail: alan@queduqa **Departent of Matheatic, Tukegee Univerity, Tukegee, Alabaa, USA, Eail: jtki@tukegeeedu ABSTRACT: We conider a Galerkin procedure to olve a parabolic integrodifferential equation that arie in a ga cobution odel Thi odel ha been propoed by Kaoy and Poland, and ubequently analyzed by Beberne, Eberly and Brean The proble i forulated in the variational for In order to etiate the error, oe interediate projection ha been eployed Under certain condition on the given data, the L error etiate ha been obtained A fully decretized verion by uing an extrapolated Crank-Nicolon ethod ha been applied and the order of convergence derived KEYWORDS: Crank-Nicolon, Error etiate, Galerkin ethod, Gaeou ignition odel طريقة جاالركين لنموذج إشعال غازي محمد سلمان و جنتاي كيم ملخص: نفترض طريقة جاالركين لحل المعادلة التفاضلية المكافئة التي تنشأ عن نموذج احتراق الغاز لقد تم اقتراح هذا النموذج من قبل كاسي وبوالند وفيما بعد تم تحليله بواسطة بيبيرنيس وابيرلي و ريسان وقد طرحت المشكلة في شكل متنوع وبغية تقدير الخطأ فقد استخدم إسقاط متوسط تم الحصول على تقدير الخطأ L تحت شروط مناسبة على بيانات معطية وتم تطبيق أدلة األبعاد التامة باستخدام طريقة كرانك-نيكلسون وإيجاد درجة التقارب 1 Introduction K aoy and Poland (1983) developed an ignition odel for a reactive ga in a bounded container to decribe the induction period During thi period, the pacially unifor preure increae and caue heating effect in the yte The preure of the ga can be expreed in ter of a pace integral ter in the induction odel that govern the teperature perturbation u( x, t ) Thi odel i decribed by the et of equation (Beberne and Brean, 1988) u ut u = e ut( x, t ) dx, ( x, t ) (0, ), (1) 4

2 A GALERKIN METHOD FOR A GASEOUS IGNITION MODEL u( x,0) = g ( x ), x, () u( x, t ) = 0, ( x, t ) (0, ), (3) where i a bounded doain in n with a ooth boundary, and volue, >1 The odel ha been ubequently tudied by Beberne and Brean (198), Beberne and Brean (1988), and Beberne et al (1989) Beberne and Brean (198) analyzed thi odel and proved that for any poitive value of the Frak-Kaenetki paraeter and any value of the ga contant 1, equation (1) have a unique claical olution u( x, t ) on [0, T ), where i a bounded doain and T can be infinite When T i finite, the olution blow up a t T For a critical value crit (ee (Kaoy and Poland, 1983)), and > crit, the olution blow up in a finite tie Beberne and Eberly (1989) ued the eigroup analyi to how the exitence and uniquene of a nonextended olution Additional coparion reult have been provided in the cae of a pherically yetric doain Blowup occur at a tie <T where T i the blowup tie of the olid fuel ignition odel The location of the blowup ha been alo dicued Depending on the nonlinearity of f, blowup can take place everywhere or at a ingle point (Beberne and Eberly, 1989) In thi paper, we tudy a finite eleent approxiation to the olution of the ga cobution odel that i decribed by the partial differential equation (Beberne and Eberly, 1989) ut u = f ( u) ut( x, t ) dx, ( x, t ) (0, ), (4) u( x,0) = g ( x ), x, (5) u( x, t ) = 0, ( x, t ) (0, ) (6) We aue f i a Lipchitz function uch that f ( u) > 0, f ( u) 0, and f ( u) 0 In thi work we develop etiate for error when a Galerkin ethod i applied The error i optial in the ene of the L nor Thi work i otivated by that of Cannon and Lin (1990a, 1990b) An extenive tudy of the finite eleent ethod for parabolic equation can be found in a book by Thoée (006) Forulation of the variational proble and Galerkin approxiation Let S h 1 be a finite dienional ubpace of the Sobolev pace H 0 ( ) uch that 1 inf ( w v h ( w v ) ) Ch v, v H ( ) H 0( ), (7) ws h where 1, i the L nor, and i the Sobolev nor defined on H ( ) 1 Proble (4) i equivalent to finding a u H ( ) H 0( ) uch that 1 ( ut, v ) ( u, v ) = ( f ( u), v ) utdx v dx, for all v H 0( ), where (, ) i the inner product on L ( ) defined a ( u, v ) = uv dx i defined a a olution to The continuou Galerkin approxiation U :[0, T ] Sh where Gx ( ) i the ( Ut, ) ( U, ) = ( f ( U ), ) U tdx dx, S h U ( x,0) = G ( x ), (9) L projection of ( ) g x into S h, ie, (8) 5

3 M SALMAN and J KIM ( G g, ) = 0 for S h Given a bai { i} i M =1 for S h, U can be written a M U ( x, t ) = i ( t ) i ( x ) i =1 Then the variational equation can be written a the nonlinear initial value proble Ba( t ) Aa( t ) = F( a( t )), Ca(0) = g, (10) where A, B, and C are the atrice A = ( i, j ), B = ( bij ) = ( i, j ) i dx i dx, C = ( i, j ), for i, j =1,,, M, and the vector a, F, and g are defined by a( t) = ( i ( t)), M Fa ( ) = ( f ( i i ), j ), i =1 g = ( g, i ) The atrix B i poitive definite, ince T a Ba = M bij i j i =1 M M = ii ( ii dx ) i=1 i=1 = U ( U dx ) 1 U > 0 for U 0, where we ued the Schwarz inequality ( U dx ) U dx 1 dx = U With the auption that f ( u ) i uniforly Lipchitz, then it follow fro the theory of ordinary differential equation that the initial-value proble (10) ha a unique olution for t > 0 3 Projection of the olution Let W :[0, T ] Sh uch that Then W i the elliptic projection of (Thoée, 006) ( ( u W ), ) = 0 for all S h (11) 1 u H ( ) H 0( ) into S h that atifie the following propertie 1 u W Ch u, (1) u W Ch u, (13) t t u W Ch u (14) 6

4 A GALERKIN METHOD FOR A GASEOUS IGNITION MODEL 4 Error etiate Let u U = u W W U =, where = u W and = W U Fro (8), (9) and (11), we get ( t, ) (, ) ( Wt Ut, ) ( W U, ) ( Wt, ) ( W, ) ( Ut, ) ( U, ) ( Wt, ) ( u, ) ( f ( U ), ) U tdx dx ( Wt ut, ) ( f ( u ) f ( U ), ) ( ut Ut ) dx dx, ie, ( t, ) (, ) = ( t, ) ( f ( u ) f ( U ), ) t dx dx t dx dx We chooe = and rewrite the equation 1 d 1 d ( dx ) = ( t, ) ( f ( u) f ( U ), ) tdx dx dt dt (16) Auing that f i uniforly Lipchitz with f ( u1) f ( u) L u1 u (17) Then, uing Schwarz and Young' inequalitie iplie 1 d 1 d C ( dx ) t C L L dt dt (18) With the ue of Poincaré-Friedrich' inequality (Gilbarg and Trudinger, 1983) u 1/ n n u, we obtain 1 d 1 d / n C C ( dx ) ( ) t ( L C) dt dt n (19) If the Lipchitz contant of f i all enough uch that / n L < ( ), n (0) / n then we can alo chooe all enough o that ( ) L C Thu, we have n 1 d 1 d ( dx ) C C C t dt dt (1) Integrating both ide fro 0 to t after dropping C to get ( ) (,0) t dx C 0 ( t ) d Then, uing chwarz inequality to obtain 1 t (,0) C0 ( t ) d (15) 7

5 M SALMAN and J KIM and Here (,0) = W (,0) U(,0) W (,0) u (,0) u (,0) U (,0) Ch g g G Ch g, can be replaced by their upper bound in (1) and (14) Thi iplie t u U Ch g Ch 0 ( u u t ) d () Thi etablihe the following theore 1 Theore 1 Suppoe that proble (4) poee a olution u in H ( ) H0( ), ut in H ( ), and f i uniforly Lipchitz that atifie (17) and (0) Then, the continuou Galerkin olution U of (9) atifie () Proof The next tep i to get an etiate for ( u W ) For that purpoe we put = t in (15) Thi yield 1 d 1 1 = (, ) ( ( ) ( ), ) 1 1 t t t f u f U t t dx t dx t dx (3) dt Thi iplie 1 d C C 1 t t C t f ( u) f ( U ) t (4) dt Etiating the righthand ide we get 1 1 d C CL t t C t u U (5) dt Selecting all o that 1 > C, we can drop the t t ter to get d C t C u U dt (6) Upon integrating fro 0 to t, we get t t (,0) C 0 t C 0 u U ) d, (7) where (,0) u (,0) U (,0) W (,0) u(,0) g G (8) 1 Ch g, 1 and t t 0 g ( (, ) t(, ) ) u d (9) The double integration can be interchanged, a proce to uppre one of the integral, then the right hand ide iplifie to t t 1 0 u U d C 0 g G h ( g u u 1 1 t ) d 1 (30) In view of (8) and (30), etiate (7) ay becoe 1 t 1 g G Ch g C 1 0 g G h ( g u u ) 1 1 t d 1 (31) Thi prove the theore 8

6 A GALERKIN METHOD FOR A GASEOUS IGNITION MODEL Theore Under all the auption entioned in Theore 1, we have u U Ch 1 t g u u 1 u t 1 d (3) Note that a G being the L projection of g onto S h,, it legitiize the etiate g G Ch g, 1 g G Ch g 1 5 A priori etiate on extrapolated Crank-Nicolon-Galerkin ethod In order to get a fully dicretized verion of the Galerkin ethod, we introduce the tie eh t = k for = 0,1,, M, where k i a unifor tie tep For the ret of thi ection, we denote 1 F = ( F F 1), a an averaged value of F on the node t and t 1 In the Crank-Nicolon ethod, we replace the tie derivative in (9) by U = ( U 1 U ) / k and U by U U U U = ( 1) / Thi define 1 a a olution to the nonlinear yte ( U, ) ( U, ) = ( f ( U ), ) U dx dx, S h The nonlinearity due to f ( U ) can be overcoe by replacing the arguent of f ( U ) by an extrapolated U over the tie tep and 1, ie 3 1 f ( U ) f ( U U 1) We denote thee extrapolated value by 3 1 Fˆ = F F 1 (33) Thi produce the new linearized equation in U 1 a ˆ ( U, ) ( U, ) = ( f ( U ), ) U dx dx, S h (34) Note that thi extrapolation proce will reult in a econd order accuracy uˆ 3 1 = u u 1 = u 1/ O ( k ), u = u(, t ) We hall etiate the error U u(, t) U u(, t ) = U W (, t ) W (, t ) u(, t ) =, with 1/ 1/ where the etiate of i hown in (1) We now conider by writing 9 (, ) (, ) = ( U, ) ( U, ) ( W, ) ( W, ) ˆ u u = ( f ( U ) f ( u ), ) U dx dx W,, (35) t t where

7 M SALMAN and J KIM 1 f ( u ) = ( f ( u ) f ( u 1)) Thi iplie (, ) dx dx (, ) ˆ u u = ( f ( U ) f ( u ), ) u dx dx u, t t Setting =, we can get 1 ( ˆ u dx C f U ) f ( u ) u, t where we have ued the Poincaré-Friedrich' inequality Thi iplie (36) (37) 1 ( ˆ u dx C f U ) f ( u ) u t 4 The lat two nor on the right hand ide are of order h and k, repectively Moreover f ( Uˆ ) ( ) ( ˆ f u f U ) f ( u 1/ ) f ( u 1/ ) f ( u ) ˆ (38) C ( U u1/ k ) (39) C ( ˆ ˆ uˆ u 1/ k ) C ( 1 h k ), where ˆ, ˆ and uˆ are the extrapolated repreentation for, and u, repectively On the other hand, the left hand ide of (38) i bounded below by dx (40) Now, in view of (39) and (40), etiate (38) can be written a 1 (1 Ck ) Ck 1 Ck ( h k ), or 1 Ck (1 Ck )( Ck 1 ) Ck ( h k ) (41) A repeated application of (41), with a all k, iplie C ( 1 k 0 ) C ( h k ) If 0 and 1 are both calculated with an accuracy C ( h k ), which prove the following theore Theore 3 The extrapolated Crank-Nicolon olution U of (34) atifie ax u U C ( h k ), where C depend on u O( h ) O( k ), we get the following reult 30

8 A GALERKIN METHOD FOR A GASEOUS IGNITION MODEL 6 Reference BEBERNES, J and BRESSAN, A 198 Theral Behaviour for a Confined Reactive ga J Diff Equation, 44: BEBERNES, J and BRESSAN, A 1988 Total Blowup Veru Single Point Blowup J Diff Equation, 73(1): BEBERNES, J and EBERLY, D 1989 Matheatical Proble fro Cobution Theory Applied Matheatic Serie, vol 83, Springer-Verlag, New York BEBERNES, J, BRESSAN, A, KASSOY, D and RILEY, N 1989 The Confined Non-Diffuive Theral Exploion with Spatially Hoogeneou Preure Variation Cob Sci Tech, 63: 45-6 CANNON, J and LIN, Y-P 1990a A Galerkin Procedure for Diffuion Equation with Integral Boundary Condition Int J Engng Sci, 8(7): CANNON, J and LIN, Y-P 1990b A Priori L error Etiate for Finite Eleent Method for Nonlinear Diffuion Equation with Meory SIAM J Nuer Anal, 7(3): GILBARG, D and TRUDINGER, NS 1983 Elliptic Partial Differential Equation of Second Order Springer- Verlag KASSOY, D and POLAND, J 1983 The Induction Period of a Theral Exploion in a Ga between Infinite Parallel Plate Cobution and Flae, 50: THOMÉE, V 006 Galerkin Finite Eleent Method for Parabolic Proble Springer Serie in Coputational Matheatic, Springer-Verlag New York, USA Received 1 February 011 Accepted 15 Noveber