Numerical simulation of three dimensional advection-diffusion equations by using modified cubic B-spline differential quadrature method

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1 Asia Pacific Jounal of Engineeing Science and Technolog 2 (2) (2016) Asia Pacific Jounal of Engineeing Science and Technolog jounal homepage: Full length aticle umeical simulation of thee dimensional advection-diffusion equations b using modified cubic B-spline diffeential quadatue method Mohammad Tamsi 1 Vineet K. Sivastava 2* P.D. Misha 2 1 Depatment of Mathematics & Statistics DDU Goakhpu Univesit Goakhpu India 2 Flight Dnamics Goup ISRO Telemet Tacking and Command etwok Bangaloe India (Received 29 Decembe 2015; accepted 20 June 2016; published online 5 Jul 2016) ABSTRACT In this pape modified cubic B-spline diffeential quadatue method is poposed to solve the thee dimensional advection-diffusion equations subjected to appopiate initial and bounda conditions. This method is basicall diffeential quadatue method whee the weighting coefficients ae computed b taking the modified cubic B-splines functions as basis functions. This method tansfoms the advection-diffusion equations into a sstem of fist-ode odina diffeential equations (ODEs). SSP-RK54 scheme is applied to solve the esulting sstem of fist ode ODEs. The poposed MCB-DQM is unconditionall stable fo advection-diffusion equations. The accuac and efficienc of MCB-DQM is illustated b two test poblems. It is evident that the solutions obtained b the poposed method ae in good ageement with the eact solutions. Kewods: 3D Advection-diffusion equation; MCB-DQM; Thomas algoithm; SSP-RK54 scheme 1. Intoduction Patial diffeential equations (PDEs) ae the fundamentals of vaious mathematical models of phsical chemical and biological phenomena. PDEs have also identified thei applications in the othe fields like: economics financial foecasting etc. In ode to investigate the pedictions of these mathematical models it is essential to know the behavio of these equations. In geneal the eact solution fo eve PDE is unavailable o is not an eas task to find the eact solutions of these equations and so the stud of the numeic solutions of the PDEs is a ve elevant aea of eseach. In this pape we stud the numeic solutions of one of the elevant mathematical model fo thee space dimensional advection-diffusion equation. This model equation have found its boad ange of applications in banches of science and engineeing e.g. fluid motion heat tansfe astophsics oceanogaph meteoolog semiconductos hdaulics pollutant and * addess:vineetsiiitm@gmail.com 2016 Autho(s)

2 72 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) sediment tanspot and chemical engineeing. Especiall in fluid dnamics and heat tansfe the advection-diffusion poblems pla impotant ole in computational fluid dnamics to simulate flow poblems. These applications demonstate that the development of some new schemes with accuate stable and efficient numeical solutions of advection-diffusion equations is of vital impotance. A mathematical model fo advection-diffusion equation in thee space dimension is given b: u u u u u u u t T t whee associated with initial condition: (1) u 0 (2) and the bounda conditions u 0 t f t u 1 t f t u 0 t g t u 1 t g t t 0. u 0 t h0 t u 1 t h1 t (3) whee f l g l h l ( 01 l ) ae given sufficientl smooth known functions epesent heat voticit etc. the constant coefficients velocities while constants u t ma 0 denote the convective 0 denote diffusion coefficients in the diection of and espectivel. Eq. (1) ma be seen in computational hdaulics and fluid dnamics to model convection-diffusion of quantities such as mass heat eneg voticit [24]. In the ecent eas one/two dimensional convection-diffusion equations have been solved b using seveal kind of numeical schemes: finite-diffeence scheme and highe-ode compact finite diffeence schemes (see [ ] and efeence theein) finite element method [9] Lattice Boltmann method [5] etc. oe-tan s nine-point high-ode compact implicit scheme is thid-ode accuate in space and second-ode accuate in time and has a lage one of stabilit [19]. Spot and Cae [28] pesented an etension of highe ode compact diffeence techniques fo stead-state [21] to the time-dependent poblems with accuac of ode Oh 4 t ( 2 ) but conditionall stable. Dehghan and Mohebbi [8] developed the fouth-ode 4 4 compact finite diffeence unconditionall stable scheme with thei ode of accuac O h t. A famil of unconditionall stable finite diffeence schemes pesented in [3] with accuac of ode Oh 3 t 2. The schemes pesented in [14] ae based on high-ode compact scheme and weighted time discetiation have accuac of ode 4 O h t ( 2 ). The high-ode altenating

3 73 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) diection implicit (ADI) (unconditionall stable) scheme with accuac of ode Oh 4 t 2 [15]. A high-ode unconditionall stable eponential scheme fo unstead 1D convectiondiffusion equation due to Tian and Yua [32] have the accuac of ode Oh 4 t 2 in. A ational high-ode compact altenating diection implicit (ADI) method is developed fo solving 2D unstead convection-diffusion poblems is developed in [30] is unconditionall stable and having the accuac of ode Oh 4 t 2. A unconditionall stable fouth-ode compact finite diffeence appoimation fo discetiing spatial deivatives and the cubic C1 - spline collocation O h t [17]. An unconditionall stable semi- 4 4 method in time have thei accuac of ode discete based on pade appoimation scheme with accuac of ode Oh 4 t 4 is poposed in [12]. The modified cubic B-spline diffeential quadatue method was fist poposed in [1] to find the numeical solution of one dimensional Buges equation. Late on it was etended to solved 2D PDEs see [27] and aticle theein. The MCB-DQM is basicall diffeential quadatue method [2] whee the modified cubic-b-spline basis functions ae used to detemine the weighting coefficients. Thee ae vaious test functions: spline functions Lagange intepolation polnomials and sinc function (see [ ] and the efeences theein) ae alead used to compute the weighting coefficients. Thee space dimension advection-diffusion have been studied numeicall b ve few eseaches [7 20]. In this pape we ae inteested in finding out the numeic solutions of 3D advection-diffusion equation with the help of MCB- DQM. The MCB-DQM is applied to convet the sstem of PDEs into a sstem of fist ode ODEs in time which is solved b using the SSP-RK54 scheme. It is demonstated that the poposed MCB-DQM is unconditionall stable. Two test poblems have been studied to illustate the accuac and utilit of the method b obtaining the oot mean squaes eo maimum absolute eos and global pecentage eo. The numeical esults ae found to be in good ageement with the eact solutions. The est of the pape is oganied as follows. In Section 2 the desciption of the modified cubic B-spline diffeential quadatue method is given. In Sections 3 pocedue fo implementation of method is descibed. The stabilit analsis of the method is given in Section 4. Two numeic test poblems ae given to establish the applicabilit and accuac of the poposed method in Section 5. Section 6 concludes ou stud. 2. Modified cubic B-spline diffeential quadatue method in thee dimension In this section desciption of MCB-DQM [ ] fo the numeical solution of thee dimensional advection-diffusion equations is discussed. It is assumed that the computational domain is discetied unifoml b taking and gid points along and diections such as: a b1 a b2 and a 3 1 2

4 74 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) b3 with spatial step sie h 1 h j1 j and h k 1 k espectivel. The solution of i i u t on the gid i j k at time t is uijk ui j k t whee i j12... and k The th ode patial deivative of (keeping j and k p1 as fied) is appoimated as u t with espect to at the gid point i j k u ijk a u ip pjk i (4) Similal the th ode patial deivative of espect to (keeping i and k appoimated as u u ijk ijk b u jp ipk j p1 c u kp ijp k p1 whee uijk ui j k t etc. and u t at the gid point i j k with as fied) and with espect to (keeping i and a ip and jp kp j as fied) ae (5) b c 12 ae the weighting coefficients of the th-ode spatial patial deivatives with espect to and. The cubic B-spline basis functions at the knots ae defined as m 3 m2 m2 m m2 4 m 1 m1 m 3 3 m 4 m m m 3 1 h 0 othewise m2 m1 m2 (6) whee... is selected in such a wa that it foms a basis ove the computational : The values of cubic B-splines m and domain associated deivatives at the nodal points ae given b Lemma 1. Similal the cubic B-splines can be defined in the diections of and. Lemma 1. The numeical values of m and its associated deivatives at nodal point ae evaluated as

5 75 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) if i j 0 3 h if i j 1 12 h if i j 0 2 ij i j ij h i j ij h i j 1 if 1; 3 if 1; 6 if 1 0 else 0 else 0 else whee etc. ij i j We modif cubic B-spline basis functions in such a wa that the esulting mati sstem of the equations emains diagonall dominant. The modified cubic B-splines basis functions ae given as: m m fo m whee foms a basis in the computational egion : Computation of the weighting coefficients Since -ais is fied in Eq. (4) consequentl b substituting the values of m( ) m in Eq. (4) we get a m (8) (1) m i ip m p p1 with the help of Eq. (7) and Lemma1 Eq. (8) eplaced into a sstem of linea equations: 1 Aa i R i fo i T i1 i2 i is a vecto coesponding to i whee a i a a... a vecto 1 i 2 i 1 i i (7) (9) known as weighting coefficient Ri and the coefficient mati A is given b: T

6 76 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) A ote that the sstem of linea equations (9) is a tidiagonal sstem. Also we obseve that the coefficient mati A being the diagonall dominant is invetible. Theefoe the sstem of linea equations (9) is solved fo each i using the Thomas algoithm. This gives the weighting coefficients 1 a fo the fist ode spatial patial deivatives. ow the weighting coefficients ip a 2 1 i p M fo the second ode spatial patial deivatives ae detemined b using the ip fomula poposed b Shu [33] which is defined below: a ip aip aip aii fo i p and i ; i p aii aip fo i p p1 pi (10) Similal we obtain the weighting coefficients 1 b and jp 1 c. kp 3. Implementation of MCB-DQM fo 3D advection-diffusion equations On substituting the appoimate values of the spatial deivatives obtained b using MCB-DQM. Eq. (1)-(2) is changed into the following fom du ijk a u b u c u a u b u c u dt 0 uijk t 0 u ijk. ip pjk jp ipk kp ijp ip pjk jp ipk kp ijp p1 p1 p1 p1 p1 p1 At the bounda the solution can be ewitten fom Eq. (3) as (11) u1 jk f0 j k t u 1 jk f j k t ui1k g0 i k t ui 1 and 0. k g i k t i j k t t uij1 h0 i j t uij h1 i j t (12)

7 77 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) ow on substituting the bounda values Eq. (11) can be witten in mati fom: du BU F dt 0 U 0 U whee a) T U u is a solution vecto: U u u... u u u... u... u and initial vecto b) F Fijk is the vecto of ode U epesents the containing the bounda values i.e. (2) (2) (2) (2) (2) (2) i i 1 1 j j k k (1) (1) (1) (1) (1) (1) a u 1 1 a u b u 1 1 b u c u 1 1 c u. i i j j k k F a u a u b u b u c u c u ijk jk jk i k i k ij ij jk jk i k i k ij ij (13) c) B A B C A B C be a squae mati of ode whee and 1 2 the weighting coefficients A B C ae squae block diagonal matices of ( ) ( ) ( ) a b and c espectivel as given below ij ij ij A ( ) ( ) ( ) a I 22 a I 23 a I 2 1 ( ) ( ) ( ) a I 32 a I 33 a I 3 1 ; ( ) ( ) ( ) a I 12 a I 13 a I 1 1 (14) B ( ) ( ) ( ) b I 22 b I 23 b I M 2 1 O O (2) ( ) ( ) ( ) a M 21 O b I 32 b I 33 b I 3 1 M ; O ( ) ( ) ( ) O M b I b I b I (15)

8 78 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) C ( ) ( ) ( ) M O O c c c (2) ( ) ( ) ( ) a M 21 O c c c M ( ) ( ) ( ) O O M c c c (16) whee O O ae null matices of ode I ae identit matices of ode 2 2 and and 2 espectivel. I and espectivel. Finall SSP-RK54 scheme is emploed to solve Eq. (11) with the inial and bounda conditions. 4. Stabilit analsis The stabilit of the MCB-DQM fo the numeical solution of advection-diffusion equation (1) is diectl depends upon the stabilit of the sstem of odina diffeential equation (13). If the sstem of odina diffeential equation (13) is unstable then the numeic scheme fo tempoal discetiation ma not be geneate conveged solution. The stabilit of (13) is depends on the eigen values of the coefficient mati B because of the eact solutions can be obtained diectl found b using the eigen values. Thus to show the stabilit of the eact solution U of (1) it is sufficient to show that the eal pat non-positive i.e. Re 0 of eve eigenvalue i of the mati B is i Re 0 fo all i n. Moeove if the eigenvalues ae comple then i thee eist some toleance that the eal pats of eigenvalues ma be a small positive numbes. Fig. 1 shows the eigenvalues of the mati B with 0.01 and 0.8 fo diffeent gid sies. It is evident that all eigenvalues ae negative. Thus the poposed method is unconditionall stable. Fig. 1. Eigenvalues of mati B fo diffeent gid sies.

9 79 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) umeical esults Conside the following eamples to confim the accuac of the scheme. Eample 1. In this eample the numeical solution of 3D advection-diffusion equation (1) with 0.01 and 0.8 ove the egion [0 1] [0 1] [0 1] with the initial condition: u0 ep be etacted fom the eact solution [7] and bounda conditions can t t t 0.5 u ep t t t t Table 1. The oot mean squaes eos in E. 1 with the paametes t h 0.1 h e e e e e e e e e e e e e e e e e e e e-08 t 0.01 at diffeent time levels. Fig. 2. Absolute eos ( a ) fo h 0.1 ( b ) fo h 0.05 at t 1.

10 80 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) The paamete values ae t 0.01 h 0.1. The oot mean squaes (RMS) eos ae computed fo diffeent time levels t 1 and epoted in Table 1. It shows that MCB-DQM solutions ae ageed with the eact solutions. The absolute eos fo h h h and t 1 ae depicted in Fig. 2 while the behavio ae depicted in Fig. 3. Fig. 3. Appoimate solution ( a ) fo h 0.1 ( b ) fo h 0.05 at t 1. Eample 2. In this eample the numeical solution of 3D advection-diffusion equation (1) with and 1ove the egion [0 1] [0 1] [0 1] with the initial condition: u solution [20] 0 ep and bounda conditions can be etacted fom the eact u t ep 3 1. Fig. 4. The global % eo fo at t 1 fo t and diffeent values of h.

11 81 Tamsi et al. Asia Pacific J. Eng. Sci. Tech. 2 (2) (2016) The eos ae evaluated fo diffeent values of t h and. The global % eo fo at t 1 is depicted in Fig. 4 fo t and diffeent values of h. The global % is depicted fo 0.5 at t 1 with h 0.1 fo diffeent values of t in Fig. 5. It is evident fom Fig. 4 that the global % eo is deceases with inceasing and deceasing space step sie. Futhe fom Fig. 5 it is obseved that the global % eo is deceases with deceasing time step sie. It is also obseved that ou esults ae compaable with the esults in [20]. Fig. 5. The global % eo fo 0.5 at t 1 with h 0.1 fo t Conclusions A numeical teatment fo 3D advection-diffusion equation is done b means of MCB-DQM. Fist the modified cubic B-spline basis functions ae used in diffeential quadatue to evaluate the weighting coefficients. Thus the equations ae conveted into a sstem of fist-ode ODEs which we solve b SSP-RK54 scheme. The efficienc and accuac of the method is illustated with the help of two test poblems. It is shown that the obtained esults ae in good ageement with the eisting esults and appoach towads the eact solutions. Refeences [1] G. Aoa B. K. Singh umeical solution of Buges equation with modified cubic B-spline diffeential quadatue method. Appl. Math. Comput. 224 (2013) [2] R. Bellman B. G. Kashef J. Casti Diffeential quadatue: a technique fo the apid solution of nonlinea diffeential equations. J. Comput. Ph. 10 (1972) [3] M. M. Cecchi M. A. Pioi High ode finite diffeence numeical methods fo time-dependent convecion-dominated poblems Appl. ume. Math. 55 (2005)

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