A Galerkin Method for a Gaseous SQU Journal for Science, 17(2) (2012) 224-231 © 2012 Sultan Qaboos University 224 A Galerkin Method for a Gaseous Ignition Model M. Salman* and J. Kim** *Department of Mathematics, Statistics and Physics, University of Qatar, Doha, Qatar, Email: msalman@qu.edu.qa. **Department of Mathematics, Tuskegee University, Tuskegee, Alabama, USA, Email: jtkim@tuskegee.edu. ABSTRACT: We consider a Galerkin procedure to solve a parabolic integrodifferential equation that arises in a gas combustion model. This model has been proposed by Kassoy and Poland, and subsequently analyzed by Bebernes, Eberly and Bressan. The problem is formulated in the variational form. In order to estimate the error, some intermediate projection has been employed. Under certain conditions on the given data, the 2 L error estimate has been obtained. A fully descretized version by using an extrapolated Crank-Nicolson method has been applied and the order of convergence derived. KEYWORDS: Crank-Nicolson, Error estimate, Galerkin method, Gaseous ignition model. ركين لنموذج إشعال غازياالطريقة ج كيم ياجنت محمد سلمان و نموذج احتراق الغاز. لقد تم اقتراح هذا عنلحل المعادلة التفاضلية المكافئة التي تنشأ ركيناالج طريقةنفترض :خصمل المشكلة في شكل طرحت وقد .تم تحليله بواسطة بيبيرنيس وابيرلي و ريسانفيما بعد ند والالنموذج من قبل كاسي وبو متنوع. وبغية تقدير الخطأ، فقد استخدم إسقاط متوسط. تم الحصول على تقدير الخطأ 2 L تحت شروط مناسبة على بيانات نيكلسون وإيجاد درجة التقارب.-تم تطبيق أدلة األبعاد التامة باستخدام طريقة كرانكومعطية. 1. Introduction assoy and Poland (1983) developed an ignition model for a reactive gas in a bounded container to describe the induction period. During this period, the spacially uniform pressure increases and causes heating effects in the system. The pressure of the gas can be expressed in terms of a space integral term in the induction model that governs the temperature perturbation ( , ).u x t This model is described by the set of equations (Bebernes and Bressan, 1988) 1 1 = ( , ) , ( , ) (0, ) , | | u t tu u e u x t dx x t           (1) K A GALERKIN METHOD FOR A GASEOUS IGNITION MODEL 225 ( , 0) = ( ), ,u x g x x  (2) ( , ) = 0, ( , ) (0, ) ,u x t x t   (3) where  is a bounded domain in n with a smooth boundary , and volume | |, > 1. The model has been subsequently studied by Bebernes and Bressan (1982), Bebernes and Bressan (1988), and Bebernes et al. (1989). Bebernes and Bressan (1982) analyzed this model and proved that for any positive value of the Frak-Kamenetski parameter  and any value of the gas constant 1,  equations (1) have a unique classical solution ( , )u x t on [0, ),T where  is a bounded domain and T can be infinite. When T is finite, the solution blows up as .t T For a critical value crit (see (Kassoy and Poland, 1983)), and crit> ,  the solution blows up in a finite time. Bebernes and Eberly (1989) used the semigroup analysis to show the existence and uniqueness of a nonextended solution. Additional comparison results have been provided in the case of a spherically symmetric domain. Blowup occurs at a time < T where T is the blowup time of the solid fuel ignition model. The location of the blowup has been also discussed. Depending on the nonlinearity of f, blowup can take place everywhere or at a single point (Bebernes and Eberly, 1989). In this paper, we study a finite element approximation to the solution of the gas combustion model that is described by the partial differential equation (Bebernes and Eberly, 1989) 1 1 = ( ) ( , ) , ( , ) (0, ) , | | t tu u f u u x t dx x t          (4) ( , 0) = ( ), ,u x g x x  (5) ( , ) = 0, ( , ) (0, ) .u x t x t   (6) We assume f is a Lipschitz function such that ( ) > 0, ( ) 0,f u f u  and ( ) 0.f u  In this work we develop estimates for error when a Galerkin method is applied. The error is optimal in the sense of the 2 L norm. This work is motivated by that of Cannon and Lin (1990a, 1990b). An extensive study of the finite element method for parabolic equations can be found in a book by Thomée (2006). 2. Formulation of the variational problem and Galerkin approximation Let hS be a finite dimensional subspace of the Sobolev space 1 0 ( )H  such that 1 0( ( ) ) , ( ) ( ) ,inf s s s w S h w v h w v Ch v v H H           (7) where 1,s   is the 2 L norm, and s  is the Sobolev norm defined on ( ). s H  Problem (4) is equivalent to finding a 1 0( ) ( ) s u H H    such that 1 0 1 1 ( , ) ( , ) = ( ( ), ) , for all ( ) , | | t tu v u v f u v u dx v dx v H              (8) where ( , )  is the inner product on 2 ( )L  defined as ( , ) = .u v uv dx  The continuous Galerkin approximation : [0, ] hU T S is defined as a solution to 1 1 ( , ) ( , ) = ( ( ), ) , | | t t hU U f U U dx dx S                  ( , 0) = ( ) ,U x G x (9) where ( )G x is the 2 L projection of ( )g x into ,hS i.e., M. SALMAN and J. KIM 226 ( , ) = 0 for .hG g S   Given a basis =1{ } M i i for ,hS U can be written as =1 ( , ) = ( ) ( ) . M i i i U x t t x  Then the variational equation can be written as the nonlinear initial value problem ( ) ( ) = ( ( )) , (0) = ,B t A t F t C a a a a g (10) where A, B, and C are the matrices = ( , ) , 1 1 = ( ) = ( , ) , | | = ( , ) , i j ij i j i i i j A B b dx dx C                    for , = 1, 2, ,i j M , and the vectors a, F, and g are defined by =1 ( ) = ( ( )) , ( ) = ( ( ), ) , = ( , ) . i M i i j i i t t f g       a F a g The matrix B is positive definite, since =1 = M T ij i j i B b  a a 2 2 =1 =1 1 1 = ( ) | | M M i i i i i i dx            2 21 1 = ( ) | | U U dx        21 > 0 for 0 ,U U    where we used the Schwarz inequality 22 2 2 ( ) 1 =| | .U dx U dx dx U        With the assumption that ( )f u is uniformly Lipschitz, then it follows from the theory of ordinary differential equations that the initial-value problem (10) has a unique solution for > 0.t 3. Projection of the solution Let : [0, ] hW T S such that ( ( ), ) = 0 for all .hu W S     (11) Then W is the elliptic projection of 1 0( ) ( ) s u H H    into hS that satisfies the following properties (Thomée, 2006) , s s u W Ch u  (12) 1 , s s u W Ch u     (13) . s t t s u W Ch u  (14) A GALERKIN METHOD FOR A GASEOUS IGNITION MODEL 227 4. Error estimates Let = = ,u U u W W U       where = u W  and = .W U  From (8), (9) and (11), we get ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 1 1 ( , ) ( , ) ( ( ), ) | | 1 1 ( , ) ( ( ) ( ), ) ( ) , | | t t t t t t t t t t t W U W U W W U U W u f U U dx dx W u f u f U u U dx dx                                                              i.e., 1 1 1 1 ( , ) ( , ) = ( , ) ( ( ) ( ), ) . | | | | t t t tf u f U dx dx dx dx                                  (15) We choose =  and rewrite the equation 2 221 1 1 1 1 1 ( ) = ( , ) ( ( ) ( ), ) . 2 2 | | | | t t d d dx f u f U dx dx dt dt                            (16) Assuming that f is uniformly Lipschitz with 1 2 1 2( ) ( ) .f u f u L u u   (17) Then, using Schwarz and Young's inequalities implies 22 2 2 221 1 1 1 ( ) . 2 2 | | t d d C dx C L L dt dt                       (18) With the use of Poincaré-Friedrichs' inequality (Gilbarg and Trudinger, 1983) 1/ | | , n n u u         we obtain 22 2 2 22 2/1 1 1 1 | | ( ) ( ) ( ) . 2 2 | | n t n d d C C dx L C dt dt                      (19) If the Lipschitz constant of f is small enough such that 2/| | < ( ) , n n L   (20) then we can also choose  small enough so that 2/| | ( ) . n n L C      Thus, we have 22 2 221 1 1 1 ( ) . 2 2 | | t d d dx C C C dt dt               (21) Integrating both sides from 0 to t after dropping 2 C  to get 22 2 22 0 1 1 ( ) ( , 0) ( ) . | | t tdx C d                  Then, using schwarz inequality to obtain 22 2 2 0 1 ( , 0) ( ) . | | t tC d           M. SALMAN and J. KIM 228 Here 2 ( , 0) = ( , 0) ( , 0) ( , 0) ( , 0) ( , 0) ( , 0) , s s s s W U W u u U Ch g g G Ch g                  and 22 t  can be replaced by their upper bound in (12) and (14). This implies 0 ( ) . s s t ts s s u U Ch g Ch u u d    (22) This establishes the following theorem. Theorem 1. Suppose that problem (4) possesses a solution u in 1 0( ) ( ), s H H   tu in ( ), s H  and f is uniformly Lipschitz that satisfies (17) and (20). Then, the continuous Galerkin solution U of (9) satisfies (22). Proof. The next step is to get an estimate for ( ).u W  For that purpose we put = t  in (15). This yields   22 21 1 1 1 1 = ( , ) ( ( ) ( ), ) . 2 | | | | t t t t t t t d f u f U dx dx dx dt                             (23) This implies 2 2 2 22 21 1 ( ) ( ) . 2 t t t t d C C C f u f U dt                   (24) Estimating the righthand side we get 2 2 2 22 21 1 . 2 t t t d C CL C u U dt               (25) Selecting  small so that 1 > ,C   we can drop the t terms to get 22 2 .t d C C u U dt      (26) Upon integrating from 0 to t, we get 22 2 2 0 0 ( , 0) ) , t t tC C u U d           (27) where 1 1 ( , 0) ( , 0) ( , 0) ( , 0) ( , 0) , s s u U W u g G Ch g                    (28) and  1 110 0 01 ( ( , ) ( , ) ) . t t s s s tss s u U d Ch g g G Ch u u d d                     (29) The double integration can be interchanged, a process to suppress one of the integrals, then the right hand side simplifies to 1 0 0 1 1 1 ( ) . t t s ts s s u U d C g G h g u u d                (30) In view of (28) and (30), estimate (27) may become 2 1 1 01 1 1 1 ( ) . s t s ts s s s g G Ch g C g G h g u u d                     (31) This proves the theorem. A GALERKIN METHOD FOR A GASEOUS IGNITION MODEL 229 Theorem 2. Under all the assumptions mentioned in Theorem 1, we have   1 01 1 1 1 . s t ts s s s u U Ch g u u u d            (32) Note that as G being the 2 L projection of g onto ,hS , it legitimizes the estimates , s s g G Ch g  1 1 . s s g G Ch g      5. A priori estimate on extrapolated Crank-Nicolson-Galerkin method In order to get a fully discretized version of the Galerkin method, we introduce the time mesh =mt mk for = 0, 1, , ,m M where k is a uniform time step. For the rest of this section, we denote 1 1 = ( ) , 2 m m mF F F  as an averaged value of F on the nodes mt and 1.mt  In the Crank-Nicolson method, we replace the time derivative in (9) by 1= ( ) /m m mU U U k  and U by 1= ( ) / 2.m m mU U U  This defines 1mU  as a solution to the nonlinear system 1 1 ( , ) ( , ) = ( ( ), ) , . | | m mm m hU U f U U dx dx S                    The nonlinearity due to ( )mf U can be overcome by replacing the argument of ( )mf U by an extrapolated U over the time steps m and 1,m  i.e. 1 3 1 ( ) ( ) . 2 2 m m mf U f U U   We denote these extrapolated values by 1 3 1ˆ = . 2 2 m m mF F F  (33) This produces the new linearized equation in 1mU  as 1 1ˆ( , ) ( , ) = ( ( ), ) , . | | mm m m hU U f U U dx dx S                    (34) Note that this extrapolation process will result in a second order accuracy 2 1 1/2 3 1 ˆ = = ( ) , 2 2 m m m mu u u u O k   with 1/2 1/2= (., ).m mu u t  We shall estimate the error ( , )m mU u t  ( , ) = ( , ) ( , ) ( , )m m m m m mU u t U W t W t u t        = ,m m  where the estimate of m is shown in (12). We now consider m by writing ( , ) ( , ) = ( , ) ( , ) ( , ) ( , )m m mm m mU U W W                    1 1ˆ= ( ( ) ( ), ) , , | | m m m m m m u u f U f u U dx dx W t t                              (35) where M. SALMAN and J. KIM 230 1 1 ( ) = ( ( ) ( )) . 2 m m mf u f u f u  This implies 1 1 ( , ) ( , ) | | mm m dx dx                    1 1ˆ= ( ( ) ( ), ) , . | | m m m m m m m m u u f U f u u dx dx u t t                                   (36) Setting = ,m  we can get   2221 1 1 ˆ( ) ( ) , 2 2 | | m m mm m m m m m u dx C f U f u u t                              (37) where we have used the Poincaré-Friedrichs' inequality .m m   This implies   2 222 21 1 1 ˆ( ) ( ) . 2 2 | | m m m m m m m u dx C f U f u u t                           (38) The last two norms on the right hand side are of orders 2s h and 4 ,k respectively. Moreover 1/2 1/2 ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )m m m m m mf U f u f U f u f u f u      2 1/2 ˆ( )m mC U u k   (39) 2 1/2 ˆ ˆ ˆ( )m m m mC u u k       2 1( ) , s m mC h k      where ˆ ,m ˆm and ˆmu are the extrapolated representations for ,m m and ,mu respectively. On the other hand, the left hand side of (38) is bounded below by   22 21 1 1 1 . 2 2 | | 2 m m mdx              (40) Now, in view of (39) and (40), estimate (38) can be written as 2 2 2 2 2 1 1(1 ) ( ) , s m m mCk Ck Ck h k        or 2 2 2 2 2 2 1 1(1 2 )( ) ( ) . s m m m mCk Ck Ck Ck h k          (41) A repeated application of (41), with a small k, implies 22 2 2 2 1 0( ) ( ) . s m C k C h k      If 0 and 1 are both calculated with an accuracy 2 ( ) ( ), s O h O k we get the following result 2 ( ) , s m C h k   which proves the following theorem. Theorem 3. The extrapolated Crank-Nicolson solution mU of (34) satisfies 2 ( ) ,max s m m m u U C h k   where C depends on u. A GALERKIN METHOD FOR A GASEOUS IGNITION MODEL 231 6. References BEBERNES, J. and BRESSAN, A. 1982. Thermal Behaviour for a Confined Reactive gas. J. Diff. Equations, 44: 118-133. BEBERNES, J. and BRESSAN, A. 1988. Total Blowup Versus Single Point Blowup. J. Diff. Equations, 73(1): 30-44. BEBERNES, J. and EBERLY, D. 1989. Mathematical Problems from Combustion Theory. Applied Mathematics Series, vol. 83, Springer-Verlag, New York. BEBERNES, J., BRESSAN, A., KASSOY, D. and RILEY, N. 1989. The Confined Non-Diffusive Thermal Explosion with Spatially Homogeneous Pressure Variation. Comb. Sci. Tech., 63: 45-62. CANNON, J. and LIN, Y-P. 1990a. A Galerkin Procedure for Diffusion Equations with Integral Boundary Conditions. Int. J. Engng. Sci., 28(7): 579-587. CANNON, J. and LIN, Y-P. 1990b. 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