A Fictitious Time Integration Method for a One-Dimensional Hyperbolic Boundary Value Problem

Transcription

1 Journa o mathematics and computer science 14 (15) A Fictitious ime Integration Method or a One-Dimensiona Hyperboic Boundary Vaue Probem Mir Saad Hashemi 1,*, Maryam Sariri 1 1 Department o Mathematics, Basic Science Facuty, University o Bonab, Bonab 55517, Iran * hashemi_math396@yahoo.com, hashemi@bonabu.ac.ir Artice history: Received Juy 14 Accepted August 14 Avaiabe onine November 14 Abstract his paper gives an estimate or the initia-boundary vaue probem o wave equations by using the Fictitious ime Integration Method (FIM) previousy deveoped by Liu and Aturi [1]. Given exampes conirm that FIM is highy eicient approach to ind the true soutions. It is interesting that the FIM can easiy treat the boundary vaue probems without any iteration and has high eiciency and high accuracy. Keywords: Wave Equation, method o ine, Fictitious ime Integration Method, Group preserving scheme, Lie group, Geometric numerica integration. 1. Introduction Partia dierentia equations (PDEs) are irst divided into two categories: non-evoutionary and evoutionary. hen, the atter is urther cassiied into as paraboic and hyperboic types according to the number o rea characteristic ines. A non-evoutionary PDE is usuay named eiptic type PDE because it exists no rea characteristic ine. Our task is to deveop a non-iterative agorithm having the advantages o easy to numerica impementation, and a great exibiity appying to the most paraboic type boundary vaue probems (BVPs) without resorting on specia treatments. his paper is motivated by using the evoutionary property o paraboic type PDE and the accurcay o numerica time integrators, and proposes a natura and mathematica equivaent approach to transorm the underying equation into a paraboic PDE without destroying the origina structure. In this paper, irsty by transorming the dependent variabe u( into a new one by v( y, : (1 u( such that the origina equation is naturay and mathematicay equivaenty written as a quasiinear paraboic equation, we use the method o ine or semi-discretization o wave 87

2 equation and then we appy the GPS, which irsty derived by Liu []. GPS uses the Cayey transormation and the Pade approximations in the augmented space, namey Minkowski. Ater tha Liu [3] appied the GPS or backward heat conduction probems, which is an i-posed probem and considered its stabiity and showed that obtained resuts is better than the Gaerkin soutions. Lee et a. [4] or soving the initia vaue probems o sti ordinary dierentia equations have proposed a modiied GPS. Abbasbandy and Hashemi have deveoped a combination o method o ine and GPS to obtain the soution o a severey i-posed Lapace equation [5]. Aso a combination o GPS and Lie symmetries are introduced by Hashemi et.a in [6]. he dynamic behavior o a singe-mass, two degree o reedom biinear osciator has considered by Liu [7], by using the GPS. Estimation o the temperature-dependent therma conductivity o a one-dimensiona inverse heat conduction probem has studied by one-step GPS in [8]. One o the most imortant hyperboic second-order equations is the wave equation u c =, (1) tt u xx where x signiies the spatia variabe, t the time variabe, u = u( the unknown unction and c is a given positive constant. he wave equation describes vibrations o a string. Physicay u ( represents the vaue o the norma dispacement o a partice at position x and time t. Our task is to deveop a non-iterative agorithm having the advantages o easy to numerica impementation, and a great exibiity appying to the most hyperboic type BVPs without resorting on specia treatments. Let us begin with a discussion o the oowing quasiinear hyperboic equation: u( = F( u, u ut, ), (, () u ( = H(, (, (3) where : c, is the boundary o the probem domain, and F and H are given t x unctions. It is known that or the evoutionary type PDEs a semi-discretization o the spatia coordinates together with numerica time integrators or initia vaue probems (IVPs) can hep us to ind numerica soutions eectivey.. A ictitious time integration approach First we propose the oowing transormation: and introduce a viscosity damping coeicient in Eq. (1): Mutipying the above equation by v( ) = (1 ) u(, (4) = u F( u, u x, u t, ). (5) 1 and using Eq. (5) we have 88

3 = v (1 ) F( u, u x, u t, ). (6) v Recaing that = u( by Eq. (5), and adding it on both the sides o the above equation we obtain v = v (1 ) F( u, u ut, ) u. (7) Finay by using u = v/(1, ux = vx/(1 and ut = vt/(1, etc. we can change Eqs. (1) and () into a paraboic type PDE: v v vx vt = v (1 ) F(,,, ) (, (8) v( ) = (1 ) H(, (. (9) here is maybe another seection o Eq. (5) by using v( ) = q( ) u(, where we require that q() = 1. However, when q ( ) is more compex than 1 the resuting PDE is more compex than Eq. (9), and there seems no good reason to seect a more compex q ( ). he above idea has been proposed and used by Liu and his coworkers in [9-15], whose numerica resuts are very we and satisactory..1. Semi-discretization Let v ( ) := v( x, t, ) be a numerica vaue o v at the grid point x, t ), and at the i, i ( i ictitious variabe. Appying a semi-discrete procedure on the Eq. (8), yieds a couped system o ordinary dierentia equations (ODEs): v = [ v ] [ ] i1, vi, vi 1, v i, 1 vi, vi, ( x) ( 1 i, vi, (1 ) F( xi, t 1 vi, vi 1, vi 1,,, 1 (1 ) x vi, 1 vi, 1, (1 ) t, ), (1) where x and t are uniorm spatia grid engths in x and t directions, and m is the number o subintervas in each direction, assuming the same. In this section we have transormed the boundary vaue probem o the one-order hyperboic PDE in Eq. (1) to an evoutionary probem o a paraboic PDE in Eq. (8), and inay arrived to an initia vaue probem in the n-dimensiona ODE system (1) with dimensions n = m. he initia vaue o Eq. (1) is given through a guess because the true initia condition o v ( ) = u( is not known a priori; however, when x, t ) is ocated on the boundary, the boundary condition (9) has to be satisied. ( i 89

4 3. GPS or dierentia equations system Upon etting v = ( v1,1, v1,,, v m, m ) and denoting a vector with the i -th component being the right-hand side o Eq. (1) we can write it as a vector orm: n v = ( v, ), vr, R (11) where n is the number o tota grid points inside the domain. GPS can preserve the interna symmetry group o the considered ODE system. Athough we do not know previousy the symmetry group o dierentia equations system, Liu [] has embedded it into an augmented dierentia system, which concerns with not ony the evoution o state variabes themseves but aso the evoution o the magnitude o the state variabes vector. We note that v = v v = v. v, (1) where the superscript signiies the transpose, and the dot between two n-dimensiona vectors denotes their inner product. aking the derivatives o both the sides o Eq. (1) with respect to we have d v ( v ) v =. dt v v (13) hen, by using Eqs. (11) and (13) we can derive d v v =. dt v (14) It is interesting that Eqs. (11) and (14) can be combined together into a simpe matrix equation: ( v, ) n n d v v v =. (, ) d v v v v (15) It is obvious that the irst row in Eq. (15) is the same as the origina equation (11), but the incusion o the second row in Eq. (15) gives us a Minkowskian structure o the augmented state variabes o X ) := ( v, v, which satisies the cone condition: X gx =, (16) where I n 1 g = n. 1 1 (17) n 9

5 is a Minkowski metric, and I n is the identity matrix o order n. In terms o ( v, v ), Eq. (16) becomes X gx = v. v v = v v =. (18) It oows rom the deinition given in Eq. (1), and thus Eq. (16) is a natura resut. Consequenty, we have an n 1 dimensiona augmented system: with a constraint (17), where X = AX (19) n n A:= t ( v, ) v ( v, ) v () satisying A g ga =, (1) is a Lie agebra so (n,1) o the proper orthochronous Lorentz group SOo (n,1). his act prompts us to devise the group-preserving scheme (GPS), whose discretized mapping G must exacty preserve the oowing properties: i: G gg = g, ii: detg = 1, iii: G >, where G is the -th component o G. Athough the dimension o the new system is raised one more, it has been shown that the new system permits a GPS given as oows: X = G( ) X, () 1 where X denotes the numerica vaue o X at, and G ( ) SO ( n,1) is the group vaue o G at. he Lie group can be generated rom A so(n,1) by an exponentia mapping, ( 1) In [ ( )] = exp A, (3) 91

6 where a := cosh( v b := sinh( v ), ). (4) (5) Substituting Eq. (3) or G () into Eq. (), we obtain where v 1 = v, (6) ( 1). v = v. 4. Numerica procedure Starting rom an initia vaue o v, which can be guessed in a rather ree way, we empoy the i above GPS to integrate Eq. (11) rom = to a seected ina time. In the numerica integration process we can check the convergence o v, at the and 1-steps by i m i, =1 1 [ v v ], (8) i, where is a seected convergent criterion. I at a time the above criterion is satisied, then the soution o u is given by In practice, i a suitabe i, vi, ( ) u i, =. (9) 1 is seected we ind that the numerica soution is aso approached very we to the true soution, even the above convergent criterion is not satisied. he viscosity coeicient introduced in Eq. (5) can increase the stabiity o numerica integration. In particuar we woud emphasize that the present method is very stabe and eectivey without needing o any iteration. Beow we give numerica exampes to dispay some advantages o the present FIM. 5. Numerica exampes In this section, we consider two exampes o wave equation to exhibit the eiciency and power o FIM. 9

7 Exampe 1. We irst consider a wave equation as oows: (3) utt ux u( ) u x(, 1, u ( ) cos( x), t u (, 1, x he domain is given by = {( x, x, t }, (31) and anaytica soution is: u( = x cos xsin t. he boundary data o H ( in FIM can be written as v, ( ) = (1 )sin t, vm 1, ( ) = (1 )( sin t v ) x. i,( = (1 ) xi, vi, m 1( ) = (1 1 We ix x = t = with m = 5,which the number o equations in Eq. (1) wi be n = We m start by an initia vaue o v = 1 and integrate Eq. (3) by using the GPS with a time stepsize i, =.1 he ina time is = 1. he vaues =.1 and =.1 are seected in this equation. Fig. 1. shows that obtained approximate soution is an accurate soution. i ), Figure 1: Potting the numerica and Exact soution o Exampe 1 or a Wave equation. 93

8 Exampe. Consider utt ux u( ), u(, 1, (3) u ( ) sin( x), t u(, 1, with anaytica soution u( = sin xsin t, In the given domain he boundary data H ( can be written as: = {( x, x, y }. v, ( ) =, v m 1, ( ) =, v =, ( ) =. i,( v i, m1 1 We ix x = t = with m = 5, and the number o equations in Eq. (1) is n = We start by m an initia vaue o v = 1 or the interior nodes and integrate Eq. (3) by using the GPS with a time i, stepsize =.1 and ina time = 1. Under a given =.3 and =.1 the convergence is perormed within the range o < = 1. Finay, accuracy and powern o FIM can be seen rom the Fig.. 94

9 Figure : Potting the numerica and Exact soution o Exampe or a Wave equation. 6. Concusion In the present paper, the origina quasiinear eiptic equation is mathematicay transormed into a paraboic type evoutionary equation by in troducing a ictitious time coordinate, and adding a viscous damping coeicient to enhance the stabiity o numerica integration o the discretized equations by using a group preserving scheme. We must stress that the resuting paraboic equation is mathematicay equivaent to the origina equation, and no approximation is made. Hence, the present FIM can work very eectivey and accuratey or the soution o boundary vaue probem o quasiinear eiptic equation. Because no iteration is required, the present method is very time saving and power o method is shown graphicay. Reerences [1] C.-S. Liu, S.N. Aturi, A nove time integration method or soving a arge system o non-inear agebraic equations, CMES 31 (8) [] C.-S. Liu, Cone o non-inear dynamica system and group preserving schemes, Int. J. Non- Linear Mech. 36 (1) [3] C.-S. Liu, Group preserving scheme or backward heat conduction probems, Int. J. Heat Mass ranser 47 (4) [4] H.-C. Lee, C.-K. Chen, C.-I. Hung, A modiied group-preserving scheme or soving the initia vaue probems o sti ordinary dierentia equations, App. Math. Comput. 133 () [5] S. Abbasbandy, M.S. Hashemi, Group preserving scheme or the Cauchy probem o the Lapace equation, Eng. Ana. Bound. Eem. 35 (11) [6] M.S. Hashemi, M.C. Nucci, S. Abbasbandy, Group anaysis o the modiied generaized Vakhnenko equation, Commun. Noninear Sci. Numer. Simuat. 18 (13)

10 [7] C.-S. Liu, wo-dimensiona biinear osciator: group-preserving scheme and steady-state motion under harmonic oading, Int. J. Non-Linear Mech. 38 (3) [8] C.-S. Liu, One-step GPS or the estimation o temperature-dependent therma conductivity, Int. J. Heat Mass ranser 49 (6) [9] C.-S. Liu, A ictitious time integration method or two-dimensiona quasiinear eiptic boundary vaue probems, CMES: Computer Modeing in Engineering & Sciences, 33 (8) [1] C.-S. Liu, A ictitious time integration method or soving the discretized inverse Sturm- Liouvie probems, or speciied eigenvaues, CMES: Computer Modeing in Engineering & Sciences, 36 (8) [11] C.-S. Liu, A ictitious time integration method or soving m-point boundary vaue probems, CMES: Computer Modeing in Engineering & Sciences, 39 (9) [1] C.-S. Liu, A ictitious time integration method or a quasiinear eiptic boundary vaue probem deined in an arbitrary pane domain, CMC: Computers, Materias & Continua, 11 (9) [13] C.-S. Liu, S. N. Aturi, A nove time integration method or soving a arge system o noninear agebraic equations, CMES: Computer Modeing in Engineering & Sciences, 31 (8) [14] C.-S. Liu, S. N. Aturi, A ictitious time integration method (FIM) or soving mixed compementarity probems with appications to non-inear optimization, CMES: Computer Modeing in Engineering & Sciences, 34 (8) [15] C.-S. Liu, S. N. Aturi, A ictitious time integration method or the numerica soution o the Fredhom integra equation and or numerica dierentiation o noisy data, and its reation to the iter theory, CMES: Computer Modeing in Engineering & Sciences, 41 (9)