Numerical Solution of Convection Diffusion Problem Using Non- Standard Finite Difference Method and Comparison With Standard Finite Difference Methods

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1 IOSR Journal of Matematics (IOSR-JM) e-issn: , p-issn: X. Volume 12, Issue 3 Ver. IV (May. - Jun. 2016), PP Numerical Solution of Convection Diffusion Problem Using Non- Standard Finite Difference Metod and Comparison Wit Standard Finite Difference Metods GetaunTadesse 1, Parca Kalyani 2 1,2 Scool ofmatematical &Statistical Sciences, Hawassa University, Hawassa, Etiopia. Abstract: In tis article we found te numerical solution of singularly perturbed one dimensional convection diffusion equation using Non-Standard finite difference metod by following te Mickens Rules. To compare te results wit te known metods we also found solution of one dimensional convection diffusion equation using standard backward and central finite difference scemes. Te work as been illustrated troug te examples for different values of small parameter ϵ, wit different step lengts. Te approximate solution is compared wit te solution obtained by standard finite difference metods and exact solution. It as been observed tat te approximate solution is an excellent agreement wit exact solution. Low absolute error indicates tat our numerical metod is effective for solving perturbation problems. Keywords:Convection diffusion problem; Non-standard finite difference metod; Perturbation problem; Absolute error. I. Introduction Te non-standard finite difference approac was initiated almost tree decades ago bymickens [1]. An important observation fromtis pioneer researcer [2] was tat te traditional procedures in te design of finite difference scemes ave to be suitably canged by nonstandard procedures to avoid instabilityand caotic beavior. Subsequently, a remarkable effort was made to designnonstandard finite difference approac for a variety of ordinary and partial differentialequations of interest in applications [3]. One of te culminating points of tis effortwas fromte autor s point of view, te identification bymickens s five rules for te constructionof non-standard finite difference scemes as more reliable numerical metods.since te publication ofmickens s book, te nonstandard finite difference approac wasextensively been applied to differential models originating problems from Engineering, Pysics, Biology, Cemistry, etc. In all tese contributions of different areas ofapplication, te non-standard finite difference sceme ave sown a great potential inreplicating te essential pysical properties of te exact solutions of te involved differentialmodels.despite te success of te new approac, Mickens s imself acknowledgetat te general rules for constructing te nonstandard finite difference sceme are notprecisely known at present time. Consequently, tere exists a certain level of ambiguityin te practical implementation of non-standard procedures to te formulation offinite difference scemes for differential equations. Singularly perturbed differential equations is one of te area of increasing interest in teapplied matematics and engineering since recent years.in tis type of problems, tereare regions were te solution varies very rapidly known as boundary layers and te regionwere te solution varies uniformly known as te outer region. Standard finite difference or finite element metods are applied on te singularly perturbeddifferential equation on uniform mes give unsatisfactory result as ϵ 0 [4]. Since for most application problems, finding te analytical solutionof singularly perturbed one dimensional convection diffusion problems is difficult even impossible, so we are applying te efficient numerical tecnique, te nonstandard finite difference sceme to singularly perturbed one dimensional convection diffusion problem for numerical simulations. Kadalbajooand Vikasgupta [5] presented a survey on numerical metods for solving singularlyperturbed problems. Spline approximation metod for solving self-adjoint singular perturbationproblems on non-uniform grids ave been investigated by Kadalbajoo andk.c. Patidar [6]. Reddy and Cakravarty [7] constructed an exponentially fitted finitedifference metod for solving singularly perturbed two-point boundary value problems. Ravikant [8] as given numerical treatment of singular boundary valueproblems.cawla and Katti [9] employed finite difference metod for a class of singulartwopoint BVPs. A class of BVPs as been solved by Rama Candra Rao [10] usingnumerical integration.parcakalyani [11] as employed numerical integration metod to solve perturbation problems, by reducing it to a differential equation of first order wit a small deviating argument.ravikantand Reddy [12] dealt wit cubic spine for a class of singulartwo-point boundary valueproblems.adomian et al. [13] solved a generalizationof Airy s equation by decomposition metod. For te numerical solution of singularlyperturbed DOI: / Page

2 two-point boundary value problems a numerical algoritm based on optimalmonitor function for mes selection as been developed by Capper and Cas [14]. Rasidinia.et.al.[15] ave developed quintic non polynomial spine functions to obtainapproximate solutions of BVPs wit singular perturbation. Linand Ceng [16] consideredspline scaling functions and wavelets for singularly perturbed problems arising inbiology and discussed teir convergence. A conventional approac for te solution of fift order boundary value problems using sixt degree spline functions as been given by ParcaKalyani et al. [17]. In tis study we applied te non-standard finite difference sceme by applying Mickens Rules on singularlyperturbed one dimensional convection diffusion problems. Te governing equation of te problem is given by Lu = u + a x u = r x, 0 < x < 1 (1) u 0 = α, u 1 = β, a x > a o > 0 Wereϵis a small parameter 0 < << 1, is used to measure te relative amount of diffusionto convection.we also solved one dimensional convection diffusion problems wit standard finite difference scemes and compared te solutions wit exact solution.we simulated te solution of te standard and nonstandard finite difference and teexact solution wit te same window.te errors obtained from te standard and non-standard scemes are plotted on te same windowand sown tat te non-standard finite difference sceme is more powerfultan te standard finite difference sceme in solving te one dimensional convectiondiffusion problems. II. Approximation Of Convection Diffusion Problem Wit Non - Standard Finite Difference Sceme In tis section we apply te non-standard modeling rulesof Mickens to find te solution of te one dimensional convection diffusion equation by constructing te appropriate denominator function. Consider te one dimensional convection diffusion equation (1) i.e. u + a x u = r x assume a(x) = 1, ten te equation (1) becomes u + u = r x (2) Te discretization is as follows u i+1 2u i + u i 1 + a u i+1 u i = r(x 2 i ) (3) As per te rules of Mickens te denominator of te igest derivative ( 2 ) of te discretized equation (3) must be replaced by te functionϕ (), were ϕ() = exp 1 1 Equation (3) becomes, 0<() <1. For a = 1 we ave u i+1 2u i + u i 1 Simplifying, we get exp u i+1 2u i + u i 1 Φ() 1 + u i+1 u i + a u i+1 u i = r(x i ) (4) u i+1 2u i + u i 1 exp a u i+1 u i = r(x i ) (6) u i+1 2u i + u i 1 + exp Arranging te coefficients of te same indices, we get u i exp u i + u i 1 = 0(8) exp 1 u i+1 u i = r x Now we find te roots of equation (8) by considering omogeneous case u i = r i exp = r(x i ) (5) 1 (7) exp r i exp r i + r i 1 = 0 (9) DOI: / Page

3 exp r exp r + 1 = 0(10) 1+exp ( ) ± (1+exp ( r 1,2 = ))2 4exp ( ) (11) 2exp ( ) 1 r 1 = 1 andr 2 = exp ( ) Tis indicates tat for all values of and ϵ, r 2 is always positive so tat it is stable and we also observed tat for all values of ϵtere will not be any oscillations. III. Approximation of convection diffusion problem wit standard finite difference scemes In tis section, we present and analyze central-difference and back ward-difference approximations for convection diffusion problem. We simulate some numerical results for different values of small parameter ϵ and discuss te beavior of te numerical solution. 3.1Approximationofte Convection Term by Central Difference Sceme We study one dimensional convection diffusion problem (1 and 2) wit central difference metod. i.e., u + a x u = r x We approximate te diffusion term wit second order central difference operator and convective term by central-difference operator as described below Rearranging te coefficients of like terms gives u i+1 2u i + u i 2 + a u i+1 u i 2 = r(x i )(12) 2 + a u i u 2 i + a u 2 2 i 1 = r(x i )(13) +a 2 u i u i + a 2 u i 1 = r(x i ) (14) Let a 1 = 2 + a 2, b 1 = 2, c 1 = 2 a 2 (15) Now let us see te solution of equation (14), by considering omogeneous case u i = r i a 1 r i+1 2b 1 r i + c 1 r i 1 = 0 (16) Te caracteristic roots of equation (17) can be obtained as r 1,2 = 2b 2 1± 4b 1 4a1 c 1 r 2a 1,2 = b 2 1± b 1 a1 c 1 1 a 1 From equation (15) we ave r 2 = c 1 a 1 = 2ϵ a 2 2 2ϵ + a 2 2 b 1 2 a 1 c 1 = ( a 1 + c 1 2 = 2ϵ a 2ϵ + a Let α = a ten we ave r 2 2 = 2 2α 2+2α ) 2 a 1 c 1 = a a 1 c 1 + c 1 2 4a 1 c 1 4 r 1,2 = b 1 ± a 1 c 1 a 1 r 1 = 1 andr 2 = c 1 = 2ϵ 2ϵa 2ϵ 2ϵ+ 2ϵa 2ϵ = 1+ α 1 α (From (15)) a 1 r 2 2b 1 r + 1 = 0(17) = ( a 1 c 1 2 Tis result sows tat if α< 1 te approximate solution to be consistent but if α>1 te numerical solution oscillates tis is because wen we take α>1,r 2 will be negative. 3.2 Approximation of te convective term by back ward difference sceme We study one dimensional convection diffusion problem (1 and 2) wit backward difference metod. i.e. DOI: / Page a 1 ) 2

4 u + a x u = r x In tis case te diffusive term is discretized wit second order central difference wereas te convective term of te equation discretized using first order back ward difference. u i+1 2u i + u i 2 + a u i u i = r(x i ) 2 u i a u i + 2 a u i 1 = r(x i )(18) Let a 2 = 2, b 2 = a, c 2 = 2 a (19) Consider te omogeneous case of equation (18)u i = r i, ten we ave te following equation. a 2 r i+1 + b 2 r i + c 2 r i 1 = 0 (20) a 2 r 2 + b 2 r + 1 = 0(21) te caracteristic roots of tis equation are b 2 ± b 2 2 4a 2 c 2 r 1,2 = 2a 2 From equation (19) we ave b a 2 c 2 = (a 2 c 2 ) 2 + 4a 2 c 2 = a a 2 c 2 + c 2 2 = ( a 2+ c 2 ) 2 ten we aver 1,2 = b 2±(a 2 + c 2 ) 2a 2 So te roots of te omogeneous case of te equation are r 1 = 1 andr 2 = c 2 + a + 2α = = = 1 + α a 2 From tis we ave tat if α>1 or α< 1, r 2 will always ave positive results and did not observeoscillations. Terefore te back ward approximation of te convective term of tegiven convection diffusion equation is more stable tan te central difference approximation ofte convective term of te one dimensional convection diffusion problem. IV. Numerical illustrations In tis section we consider two examples of one dimensional singularly perturbed convection diffusion problems. Teir numerical solution and absolute errors are given for different values of small parameter ϵ. Te approximate solution obtained by non-standard finite difference, standard finite difference, exact solutions and absolute errors at te grid points are summarized in tabular form. Te approximate solution and exact solution ave been sown grapically. Furter te comparison of numerical solutions obtained by SFDM, NSFD wit exact solutionand also absolute errors at different step lengtsas been sown grapically. 4.1Numerical solution of convection diffusion problem wit non-standard finite difference sceme Example 1. Consider te singular perturbed convection diffusion problem u + au = 1 on 0,1, u 0 = 0, u 1 = 0 (22) Te exact solution of (22) is exp 1 x exp ( 1 u x = x ) 1 exp ( 1 (23) ) Approximating te derivatives wit finite differences u i+1 2u i + u i 1 exp a u i+1 u i = 1 (24) u i+1 2u i + u i 1 exp 1 + a u i+1 u i = 1(25) DOI: / Page

5 u i+1 2u i + u i 1 + a exp Arranging te coefficients of te same indices gives 1 u i+1 u i = exp 1 (26) (1 + a exp (27) 1))u i+1 + ( 2 a(exp 1))u i + u i 1 = [exp 1] In tis article, in all experiments of MATLAB coding te value of ais considered as 1. Te comparison of numerical solution of te discretized equation (27) obtained by non-standard finite difference metod for several values of ϵ wit exact solution as been sown grapically. Example 2: Te exact solution is u x = 2x + x 2 + u i+1 2u i + u i 1 exp 1 1 u i+1 2u i + u i 1 exp 1 ((2+1)(exp 1 + u i+1 u i + u i+1 u i (1 exp 1 exp x 1 ) u + u = 2xon 0,1, u 0 = 0, u 1 = 0 (28) ) (29) = 2x i (30) = 2x i (31) Arranging te coefficients of te same indices gives (exp 1))u i+1 + ( 1 exp ( ))u i + u i 1 = 2. i.. [exp 1] (32) DOI: / Page

6 Te comparison of numerical solution of te discretized equation (32) obtained by non-standard finite difference metod for several values of ϵ wit exact solution as been sown grapically. DOI: / Page

7 4.2 NumericalSolution of Convection Diffusion Problem wit Standard Finite Difference Metod In tis section we found te numerical solution of convection diffusion problem using central and backward difference scemes. We ave cosen te same problem for te sake of comparison. i.e, u + au = Te Solution wit Central Difference Sceme u i+1 2u i + u i 2 + a u i+1 u i 2 = 1 +a 2 u i u i + a 2 u i 1 = 1 (33) Now let us consider te numerical solution of equation (33) for different values of ϵandcompare wit te exact solution as follows. DOI: / Page

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9 4.2.2.Solution Wit Back Ward Difference Sceme u i+1 2u i + u i 2 + a u i u i = 1 2 u i a u i + 2 a u i 1 = 1 DOI: / Page

10 From te figures (8-15), we observed tat te back ward discretization of te convective term of te one dimensional convection diffusion problem is more stable tan te central difference approximation of te convective term of te one dimensional convection diffusion problem. V. Comparative Study of Non-Standard and Standard Finite Difference Metods. In tis section te performance of standard and non-standard finite difference scemes are compared. Te performance of te sceme was evaluated by comparing te result wit exact solution. As discussed earlier te back ward discretization of te convective term of te one dimensional convection diffusion problem is more stable tan te central difference approximation of te convective term of te one dimensional convection diffusion problem. So te performance of te non-standard finite difference sceme is compared wit back ward difference approximation. Te comparison of numerical solution obtained by non-standard finite difference metod for several values of ϵ and te solution obtained by back ward difference approximation, wit exact solutions is given in tabular form and as been sown grapically. We also plotted te grap of exact solution for different values of ϵ. DOI: / Page

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13 We ave also sowntat te comparison of te errors of standard and non-standard finite difference scemes of te numerical solution of example 1 for several values of ϵ and different step lengts grapically. DOI: / Page

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15 Te comparison of numerical solution obtained by non-standard finite difference metod for several values of ϵ and te solution obtained by back ward difference approximation, of example 2 wit exact solutions as been sown grapically. Te following figures sows te comparison of absolute errors of example 2 for different step lengts and ϵ. DOI: / Page

16 VI. Conclusion Non-standard and standard finite difference scemes are applied to find te numerical solution of example 1 and 2at differentstep lengts for different values of small parameter ϵ. Numerical solutions are summarized in te tables and te comparison as been sown in figures.from te figures 12, 13, 14 and 15, we observed tat te back ward discretization of te convectiveterm of te one dimensional convection diffusion problem is more stable tan tecentralapproximation of te convective termof te one dimensional convection diffusionproblem. Terefore we compared te backward sceme wit NSFD. From te figures 1-7, we observed tat even if te small parameter ϵgets smallerand smaller, te nonstandard finite difference sceme performed well and tere is no oscillations observed so tat itis stable on te given domain.it is also observed fromte tables,eventoug te standard finite difference metodyield good result wen te small parameter ϵ large enoug, te non-standard finitedifference sceme perform better tan te standard finite difference metod. Te graps(figure 23 and 27) of te errors sows tat te error of te standard finite differencesceme increases as te value of te small parameter ϵdecreases and te error plots sows tat instability of te numerical sceme for different values of n. Te error plots (figures 20-27) ofnonstandard finite sceme sows tat te error decreases as te value of n increasestis sows tat te sceme is dynamically consistent and it is stable for all values of ϵ. From all te tables and graps we conclude tat te non-standard finite difference sceme is more powerfultan te standard finite difference metod. References [1]. R. E. Mickens, Difference equation models of differential equations aving zero local truncation errors, in: I. W. Knowles and R. T. Lewis (Editors), Differential Equations, Nort-Holland, Amsterdam, 1984, [2]. R. E.Mickens, Exact solutions to difference equation models of Burgers equation, Numerical Metods for Partial Differential Equations 2, (1986). [3]. R.E. Mickens, Exact solution to a finite-difference model of a nonlinear reaction Advection equation: implications for numerical analysis, Numerical Metods for Partial Differential Equations, (1989). [4]. Deepti Sakti. Numerics of Singularly perturbed Differential Equations. Dept. of Matematics, NIT- Rourkela, May [5]. Kadalbajoo, M. K. and Vikas Gupta, A brief survey on numerical metods for solving singularly perturbed problems, Applied Matematics and Computation, vol.217, pp (2010). [6]. Kadalbajoo, M.K. and Patidar, K.C., Spline approximation metod for solving self- singular perturbation problems on non-uniform grids, J. Comput. Anal. Appl., 5, pp (2003). [7]. Reddy, Y.N. and PramodCakravarty, P., An exponentially fitted finite difference metod for singular perturbation problems, Appl. Mat. Comput. vol. 154, (2004). [8]. Ravikant, A.S.V., Numerical Treatment of Singular Boundary Value problems, P.D. Tesis, National Institute of Tecnology, Warangal, India (2002). [9]. Cawla,M.M and Katti,P.A Finite difference Metod for a class of two point boundary value problems, IMA.J.Number.Anal, pp (1984). [10]. Rama Candra Rao, P S., Solution of a Class of Boundary Value Problems using Numerical Integration, Indian Journal of Matematics and Matematical Sciences.Vol.2.No.2, pp (2006). [11]. Parcakalyani et al Numerical solution of singular perturbation problems via deviating argument troug te numerical metods, Researc Journal of Matematical and statistical Sciences, Vol. 2(9), 9-19, September (2014). [12]. Ravikant, A.S.V. and Reddy, Y.N., Cubic spline for a class of singular two point boundary value problems, Appl. Mat. Comput. (170), pp (2005). [13]. Adomian, G., Elrod, M. and Rac, R., A new approac to boundary value equations and application to a generalization of Airy s equation, J. Mat. Anal. Appl., (140), pp (1989). [14]. Capper, S. and Cas, J., On te development of effective algoritms for te numerical solution of singularly perturbed two-point boundary value problems, Int. J. Comput. Sc. Mat. 1, pp (2007). [15]. Rasidinia, J., Moannadi, R. and Moatamedosariati, S.H., Quintic spline metod for te solution of singularly perturbed boundary value problems, Int. J. Comput. Metods Eng. Mec., 11, pp (2010). [16]. Lin, B., Li, K. and Ceng, Z.., B-spline solution of singularly perturbed boundary value problem arising in biology, Caos SolitonsFract. 42, pp (2009). [17]. Parcakalyani et al A conventional approac for te solution of fift order boundary value problems using sixt degree spline functions ttp: //dx.doi.org/ /am 44082, DOI: / Page