Alexandria Engineering Journal (2017) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations H.S. Shukla a, Mohammad Tamsir a,b,* a b
Department of Mathematics & Statistics, DDU Gorakhpur University, Gorakhpur 273009, India Department of Mathematics, Graphic Era University, Dehradun 248001, India
Received 25 January 2017; revised 19 April 2017; accepted 23 April 2017
KEYWORDS 2D and 3D convectiondiffusion equations; Expo-MCB-DQM; SSP-RK54 scheme
Abstract We present a method viz. ‘‘exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM)” to approximate the numerical solution of 2D and 3D convectiondiffusion equations (CDEs). This method is used to approximate the special derivatives whereas time derivative is approximated by using ‘‘an optimal five stages and fourth order Runge-Kutta (SSP-RK54)” scheme. Four examples are chosen to test out the efficiency and accuracy of the method. It is found that the present method turns out more accurate results compared to the results given by other methods. There is wide scope of the present method to be used for numerical solution of several PDEs governing physical, chemical and biological problems. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
with I.C.’s
1. Introduction The convection-diffusion equation (CDE) takes as key part to model the numerous physical phenomena. These equations have various physical applications. We consider the following CDEs. (a) In 2D domain
uðx; y; 0Þ ¼ uðx; yÞ 8ðx; yÞ 2 R and B.C.’s uða1 ; y; tÞ ¼ n1 ða1 ; y; tÞ; uðx; b1 ; tÞ ¼ n3 ðx; b1 ; tÞ;
uða2 ; y; tÞ ¼ n2 ða2 ; y; tÞ; uðx; b2 ; tÞ ¼ n4 ðx; b2 ; tÞ;
ð1:2Þ
t P 0; t P 0; ð1:3Þ
(b) In 3D domain @u @ u @ u @u @u ax 2 ay 2 þ bx þ by ¼ 0; ðx;y; tÞ 2 R ð0;T @t @x @y @x @y ð1:1Þ 2
2
@u @2u @2u @2u @u @u @u ax 2 ay 2 az 2 þ bx þ by þ bz @t @x @y @z @x @y @z ¼ 0;
* Corresponding author at: Department of Mathematics, Graphic Era University, Dehradun 248001, India. E-mail address:
[email protected] (M. Tamsir). Peer review under responsibility of Faculty of Engineering, Alexandria University.
ðx; y; z; tÞ 2 X ð0; T
ð1:4Þ
With I.C.’s uðx; y; z; 0Þ ¼ /ðx; y; zÞ 8ðx; y; zÞ 2 X
ð1:5Þ
and B.C.’s
http://dx.doi.org/10.1016/j.aej.2017.04.011 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: H.S. Shukla, M. Tamsir, An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.04.011
2
H.S. Shukla, M. Tamsir
8 > < uða1 ;y;z;tÞ ¼ w1 ða1 ;y;z;tÞ; uða2 ;y;z;tÞ ¼ w2 ða2 ;y;z;tÞ; t P 0; uðx;b1 ;z;tÞ ¼ w3 ðx;b1 ;z;tÞ; uðx;b2 ;z;tÞ ¼ n4 ðx;b2 ;z;tÞ; t P 0; > : uðx;y;c1 ;tÞ ¼ w5 ðx;y;c1 ;tÞ; uðx;y;c2 ;tÞ ¼ w6 ðx;y;c2 ;tÞ; t P 0; ð1:6Þ
where R ¼ ½a1 ; a2 ½b1 ; b2 and X ¼ ½a1 ; a2 ½b1 ; b2 ½c1 ; c2 are finite rectangular and cubical domains respectively. The boundaries of R and X are @R and @X respectively. The functions u, n1 ; n2 ; n3 ; n4 , /, w1 ; w2 ; w3 ; w4 ; w5 ; w6 are known functions and uðx; y; z; tÞ correspond to heat, vorticity, etc. The constants bx ; by ; bz and ax > 0; ay > 0; az > 0 denote convection and diffusion coefficients. In recent years, various numerical methods have been presented for the CDEs. Cecchi and Pirozzi [2] studied CDE via family of fully implicit finite difference (FD) methods with accuracy of order Oðh3 ; t2 Þ. Cao et al. [3] proposed Oðh4 Þ compact FD method in which Pade´ approximations are used for the consequential linear system. Chinchapatnam [4] used symmetric and unsymmetric meshless collocation techniques. The Oðh4 Þ compact FD unconditionally stable method is proposed by Dehghan and Mohebbi [7] with their order of accuracy Oðh4 ; t4 Þ, while Dhawan et al. [8] made a comprehensive study of advection-diffusion equation by using B-spline functions. They used linear as well as quadratic B-spline functions to understand the basic characteristic and advantages of the method. Ding and Zhang [9] proposed an unconditionally stable, semi-discrete method based on Pade´ approximation with their order of accuracy Oðh4 ; t5 Þ, while Kalita et al. [10] presented a weighted time discretization on high order compact method. An absolute stable high-order ADI method with accuracy of order Oðh4 ; t2 Þ is proposed by Karaa and Zhang [11]. A nine point high order compact implicit method is proposed by Noye and Tan [14]. They set up a new method with accuracy Oðh3 ; t2 Þ. Tian [15] developed a rational compact ADI method having the accuracy Oðh4 ; t2 Þ, while an exponential compact ADI method is presented by Tian and Ge [16]. An ADI method based on 4th order Pade´ approximation is developed by You [17] with an accuracy of order Oðh4 ; t2 Þ. Due to the capability of handling local phenomena and the smoothness, the B-spline basis functions are easy to implement. The differential quadrature method based on various B-spline functions has been applied to solve the convectiondiffusion and related equations. Korkmaz and Dag [26,28] proposed cubic B-spline differential quadrature methods to solve the advection-diffusion equation and the BoundaryForced regularized long wave equation respectively. Bashan et al. [27,33] proposed quintic B-spline differential quadrature method to find the numerical solution of the modified Burgers’ and Korteweg-de Vries-Burgers’ equations respectively while Thoudam [24] used aforesaid method to solve coupled KleinGordon-Zakharov equations and Mittal and Dahiya [29] used for Fisher-Kolmogorov equation. Korkmaz and Dag [32] introduced differential quadrature methods based on Bspline functions of degree four and five to solve advectiondiffusion equation. Korkmaz and Akmaz [30] used extended B-spline differential quadrature method to solve nonlinear viscous Burgers’ equation. Arora and Joshi [25] used B-spline and trigonometric B-spline by differential quadrature method to solve hyperbolic telegraph equation. Recently, Mohammadi [5] and Dag and Ersoy [6] proposed an exponential cubic Bspline collocation method while Korkmaz and Akmaz [23]
proposed an exponential cubic B-spline differential quadrature method. Tamsir et al. [13] and Shukla et al. [34] proposed an exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation and 3D nonlinear wave equations. In this study, we present an exponential modified cubic B-spline differential quadrature method to solve 2D and 3D CDEs. 2. Description of the method This section is related to the description of the method. We assume that M, N and L grid points: a1 ¼ x1 < x2 ; . . . ; < xM ¼ a2 , b1 ¼ y1 < y2 ; . . . ; < yN ¼ b2 and c1 ¼ z1 < z2 ; . . . ; < zL ¼ c2 are distributed uniformly with the spatial step size Dx ¼ ða2 a1 Þ==ðM 1Þ, Dy ¼ ðb2 b1 Þ==ðN 1Þ and Dz ¼ ðc2 c1 Þ==ðL 1Þ along x, y and z axes respectively. For two dimensional domain, the spatial derivatives of uðx; y; tÞ are approximated as follows: M @ r uij X ðrÞ ¼ aip upj ; r @x p¼1
ð2:1Þ
N @ r uij X ðrÞ ¼ bjp uip ; @yr p¼1
ð2:2Þ
where uij ¼ uðxi ; yj ; tÞ; i ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; N, For three dimensional domain, the spatial derivatives of uðx; y; z; tÞ are approximated as follows: M @ r uijk X ðrÞ ¼ aip upjk ; r @x p¼1
ð2:3Þ
N @ r uijk X ðrÞ ¼ bjp uipk ; @yr p¼1
ð2:4Þ
N @ r uijk X ðrÞ ¼ ckp uijp ; r @z p¼1
ð2:5Þ
where uijk ¼ uðxi ; yj ; zk ; tÞ; i ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; N and k ¼ 1; 2; . . . ; L. The terms ðrÞ ðrÞ ðrÞ aij , bij and cij , r ¼ 1; 2 denote the weighting coefficients of the rth order derivatives w. r. t. x, y and z axes respectively. The exponential cubic B-spline basis functions are defined as [5,6,13,23,34]: 8 1 > > > b2 ðxi2 xÞ c ðsinhðcðxi2 xÞÞÞ ; > > > > > < a1 þ b1 ðxi xÞ þ c1 expðcðxi xÞÞ þ d1 expðcðxi xÞÞ; Ei ðxÞ ¼ a1 þ b1 ðx xi Þ þ c1 expðcðx xi ÞÞ þ d1 expðcðx xi ÞÞ; > > > > > b2 ðx xiþ2 Þ 1c ðsinhðpðx xiþ2 ÞÞÞ ; > > > : 0;
x 2 ½xi2 ; xi1 Þ x 2 ½xi1 ; xi Þ x 2 ½xi ; xiþ1 Þ x 2 ½xiþ1 ; xiþ2 Þ otherwise ð2:6Þ
where chr1 c r1 ðr1 1Þ þ r22 ; ; b1 ¼ 2 ðchr1 r2 Þð1 r1 Þ chr1 r2 1 expðchÞð1 r1 Þ þ r2 ðexpðchÞ 1Þ c1 ¼ 4 ðchr1 r2 Þð1 r1 Þ a1 ¼
Please cite this article in press as: H.S. Shukla, M. Tamsir, An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.04.011
An exponential cubic B-spline algorithm
3
Values of Ei ðxÞ, E0i ðxÞ and E00i ðxÞ.
Table 1
xi2
xi1
xi
xiþ1
xiþ2
Ei ðxÞ
0
r2 ch 2ðchr1 r2 Þ
1
r2 ch 2ðchr1 r2 Þ
0
E0i ðxÞ
0
cðr1 1Þ 2ðchr1 r2 Þ
0
cðr1 1Þ 2ðchr 1 r2 Þ
0
E00i ðxÞ
0
c2 r2 2ðchr1 r2 Þ
2 chrc 1rr 2
c2 r2 2ðchr1 r2 Þ
0
2
1 expðchÞðr1 1Þ þ r2 ðexpðchÞ 1Þ ; 4 ðchr1 r2 Þð1 r1 Þ c ; r1 ¼ coshðchÞ; r2 ¼ sinhðchÞ: b2 ¼ 2ðchr1 r2 Þ
2
d1 ¼
In Eq. (2.6), p is free parameter which is used to get special forms of exponential cubic B-spline functions. Here fE0 ; E1 ; . . . ; EN ; ENþ1 g forms basis in ½a1 ; a2 . The values Ei ðxÞ, E0i ðxÞ and E00i ðxÞ are depicted in Table 1. The exponential cubic B-spline basis functions are modified as [20–22,31]: 9 /1 ðxÞ ¼ E1 ðxÞ þ 2E0 ðxÞ > > > > > > /2 ðxÞ ¼ E2 ðxÞ E0 ðxÞ > > = /m ðxÞ ¼ Em ðxÞ for m ¼ 3; . . . ; M 2 ð2:7Þ > > > > > /M1 ðxÞ ¼ EM1 ðxÞ EMþ1 ðxÞ > > > ; /M ðxÞ ¼ EM ðxÞ þ 2EMþ1 ðxÞ where f/1 ; /2 ; . . . ; /M g forms basis in a1 6 x 6 a2 .
By substituting r ¼ 1 and the values of /m ðxÞ in Eq. (2.1), we find M X ð1Þ aip /m ðxp Þ; for i; m ¼ 1; 2; . . . ; M:
ð2:8Þ
p¼1
By Eq. (2.6) and Table 1, the system of Eq. (2.8) is replaced into !
!
A a ð1Þ ½i ¼ R½i;
ð2:9Þ
h iT ð1Þ ð1Þ ð1Þ are vectors at xi and the where a ½i ¼ ai1 ; ai2 ; . . . ; aiM !ð1Þ
matrix A is tri-diagonal which is 2 chr1 ch 6 6 6 6 6 6 6 6 A¼6 6 6 6 6 6 6 4
3
chr1 r2
r2 ch 2ðchr1 r2 Þ
0
1
r2 ch 2ðchr1 r2 Þ
r2 ch 2ðchr1 r2 Þ
1 ..
r2 ch 2ðchr1 r2 Þ
.
..
.
r2 ch 2ðchr1 r2 Þ
..
.
1 r2 ch 2ðchr1 r2 Þ
2
cðr1 1Þ chr1 r2
0 0 .. . 0 0
3
2
6 7 6 7 6 7 6 7 6 7 6 7 6 7 ! 7; R½2 ¼ 6 6 7 6 7 6 7 6 7 6 7 6 7 4 5
r2 ch 2ðchr1 r2 Þ
1
0
r2 ch 2ðchr1 r2 Þ
chr1 ch chr1 r2
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
! The coefficient vectors R½i ¼ /01 ðxi Þ; /02 ðxi Þ; . . . ; /0M1 ðxi Þ; T /0M ðxi Þ
cðr1 1Þ 2ðchr 1 r2 Þ
0 cðr1 1Þ 2ðchr1 r2 Þ
0 .. . 0
3 7 7 7 7 7 7 ! 7 7; . . . ; R½N 1 7 7 7 7 7 7 5
0
3
0 2 3 0 6 7 6 7 0 6 7 6 7 0 6 7 6 7 .. 6 7 6 7 6 7 .. . 6 7 6 7 . 6 7 6 7 6 7 ! 6 7 0 6 7 6 7 0 ½M ¼ ¼6 ; R 7 6 7 6 cðr1 1Þ 7 6 7 6 2ðchr1 r2 Þ 7 6 7 0 6 7 6 7 6 7 6 cðr1 1Þ 7 6 7 6 7 6 7 4 chr1 r2 5 6 7 0 4 5 cðr1 1Þ cðr1 1Þ 2ðchr1 r2 Þ
2.1. Evaluation of the weighting coefficients
/0m ðxi Þ ¼
6 6 6 6 6 6 ! 6 R½1 ¼ 6 6 6 6 6 6 6 4
1 1Þ cðr chcr2
chr1 r2
The system of linear Eq. (2.9) is solved for each i by using the ‘‘Thomas algorithm”. This provides the weighting coeffið1Þ ð2Þ cients aij . The weighting coefficients aij ; 1 6 i; j 6 N, are determined by using the Shu’s recurrence relation [1] which is defined as follows: 8 ð1Þ aij > ð2Þ ð1Þ ð1Þ > ¼ 2ða a Þ; for i – j a > ij ij ii xi xj < ð2:10Þ
M X ð2Þ ð2Þ > > ¼ aij ; for i ¼ j; a > ii : j¼1;j–i
ð1Þ
ð1Þ
The weighting coefficients bij and cij w. r. t. y and z are obtained by using same the approach while the weighting coefð2Þ ð2Þ ficients bij and cij are obtained by using Shu’s recurrence relation. 3. Implementation procedure of the method On discretizing the spatial derivatives of convection-diffusion Eq. (1.1) by using Expo-MCB-DQM, Eq. (1.1) is reduced into the form: M N M N X X X X duij ð2Þ ð2Þ ð1Þ ð1Þ ¼ ax aip upj þ ay bjp uip bx aip upj by bjp uip ; dt p¼1 p¼1 p¼1 p¼1
ð3:1Þ which can be written as
Please cite this article in press as: H.S. Shukla, M. Tamsir, An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.04.011
4
H.S. Shukla, M. Tamsir
duij ¼ L1 ðuij Þ: dt
ð3:2Þ
On discretizing the spatial derivatives of convectiondiffusion Eq. (1.4), we get Eq. (1.4) into the following form:
(iii) B ¼ ax A2x þ ay B2y þ az C 2z bx A1x by B1y bz C 1z is a square matrix of order ðM 2ÞðN 2ÞðL 2Þ where Arx ; Bry andC rz ðr ¼ 1; 2Þ are square block diagonal matrices ðrÞ
M N L X X X duijk ð2Þ ð2Þ ð2Þ ¼ ax aip upjk þ ay bkp uipk þ az cjp uijp dt p¼1 p¼1 p¼1
2
M N L X X X ð1Þ ð1Þ ð1Þ bx aip upjk by bjp uipk by ckp uijp p¼1
p¼1
ð3:3Þ
p¼1
which can be written as duijk ¼ L2 ðuijk Þ dt
ð3:4Þ
where L1 and L2 are linear operators. Eqs. (3.2) and (3.4) together with initial and boundary conditions are solved by using ‘‘SSP-RK54” scheme [12]. The ‘‘SSP-RK54” is strongly stable method and requires low storage space. 4. Stability analysis In this section, we analyze the stability analysis for three dimensional convection-diffusion equation. After discretization via Expo-MCB-DQM, Eq. (3.3) is reduced into a set of ordinary differential equations in time as follows: dU ¼ BU þ F; dt
2
My 6 ð2Þ 6 a21 6 Bry ¼ 6 . 6 . 4 .
2
(ii) F ¼ F ijk is the vector of containing the boundary values which is given by ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð1Þ
ð1Þ
Fijk ¼ ax ðai1 u1jk þ aiM uMjk Þ þ ay ðbj1 ui1k þ bjN uiNk Þ þaz ðck1 uij1 þ ckNz uijL Þ bx ðai1 u1jk þ aiNx uMjk Þ ð1Þ
ð1Þ
ð1Þ
ð1Þ
by ðbj1 ui1k þ bjN uiNk Þ bz ðck1 uij1 þ ckL uijL Þ:
a23 Ix
ðrÞ
...
a2ðM1Þ Ix
a33 Ix
ðrÞ
...
a3ðM1Þ Ix
.. .
..
.. .
aðM1Þ3 Ix
...
ðrÞ
Oy
...
My .. . Oy
... .. . ...
Oy 2 ðrÞ b22 Iz 6 6 bðrÞ I 6 32 z My ¼ 6 6 .. 6 . 4 ðrÞ bðN1Þ2 Iz
Crz
U ¼ ðu222 ; u223 ; . . . ; u22ðL1Þ ; u322 ; u323 ; . . . ; u32ðL1Þ ; . . . ; uðM1ÞðN1ÞðL1Þ Þ;
ðrÞ
a22 Ix 6 6 aðrÞ I 6 32 x Arx ¼ 6 6 .. 6 . 4 ðrÞ aðM1Þ2 Ix
ð4:1Þ
where (i) U ¼ ½uT is a solution vector:
ðrÞ
ðrÞ
of the weighting coefficients aij ; bij andcij respectively as given below
Oy
...
b33 Iz
ðrÞ
...
.. .
..
Mz ...
... .. .
Oz ...
Oz
Oz
...
Mz
7 7 7 7; 7 5
...
c33
ðrÞ
...
.. .
..
ðrÞ
7 ðrÞ b3ðN1Þ Iz 7 7 7; 7 .. 7 . 5 ðrÞ bðN1ÞðN1Þ Iz
3
ðrÞ c23
cðL1Þ3
3
ðrÞ
b2ðN1Þ Iz
. ...
ðrÞ
bðN1Þ3 Iz
6 ð2Þ 6 a21 6 ¼6 . 6 .. 4
6 6 ðrÞ 6 c32 6 Mz ¼ 6 6 .. 6 . 4 ðrÞ cðL1Þ2
ðrÞ
aðM1ÞðM1Þ Ix
3
ðrÞ
Oz
ðrÞ c22
.
b23 Iz
...
2
7 7 7 7; 7 7 5
ðrÞ
7 Oy 7 7 ; .. 7 7 . 5 My
Oz
Mz
3
ðrÞ
.
...
ðrÞ
c2ðL1Þ
3
7 7 ðrÞ c3ðL1Þ 7 7 7 .. 7 7 . 5
;
ðrÞ
cðL1ÞðL1Þ
ðL2Þ
Oz are null matrices of order where Oy , ðN 2ÞðL 2ÞandðL 2Þ; respectively. Ix and Iy are identity matrices of order ðN 2ÞðL 2ÞandðL 2Þ;, respectively. The stability of the method depends on the Eigen values of the coefficient matrix B. To show that the present method is stable, it is sufficient to show that the Reðki Þ 6 0 where ki ði ¼ 1; 2; . . . ; nÞ represent the Eigen values of matrix B. Fig. 1 shows the Eigen values of the matrix B with ax ¼ ay ¼ az ¼ 0:01 and bx ¼ by ¼ bz ¼ 0:8 for different values of h whereh ¼ Dx ¼ Dy ¼ Dz. It is evident that all real
Table 2 Various error norms obtained for Example 1 with h ¼ 0:025; Dt ¼ 0:0125.
Fig. 1 Eigen values of matrix B for different values of h in domain [0, 1].
Method
Average |error|
Maximum |error|
Noye and Tan [14] Kalita et al. [10] Dehghan and Mohebbi [7] Present
1.43E05 1.59E05 9.483E06 1.2296E06
4.84E04 4.48E04 2.4693E04 2.6733E6
Please cite this article in press as: H.S. Shukla, M. Tamsir, An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.04.011
An exponential cubic B-spline algorithm
5
Table 3 Error norms obtained for Example 1 with h ¼ 0:025; Dt ¼ 0:00625. Method
Average |error|
Maximum |error|
Noye and Tan [14] Kalita et al. [10] Dehghan and Mohebbi [7] P-R ADI [16] Karra and Zhang ADI [11] Tian and Ge ADI [16] Present
1.971E05 1.597E05 9.4931E06 9.218E06 2.500E04 9.663E06 6.4819E07
6.509E04 4.447E04 2.4766E04 2.664E04 2.500E04 2.664E04 1.6090E6
Fig. 2
Fig. 3
parts of each Eigen value are negative. Thus the proposed method is unconditionally stable. 5. Result and discussions In this section we choose four examples to test the efficiency and accuracy of the present method. Example 1. Consider the CDE (1.1) with exact solution [7,10,11,14,16]:
Contour plots (a) numerical and (b) exact solutions with h ¼ 0:025; Dt ¼ 0:0025 and ax ¼ ay ¼ 0:01 at t ¼ 1:25 Example 1.
Surface plot of numerical (left) and exact (right) solution with h ¼ 0:025; Dt ¼ 0:0125 and ax ¼ ay ¼ 0:01 at t ¼ 1:25 Example 1.
Fig. 4
Surface plots of solutions with h ¼ 0:02; Dt ¼ 0:001 and ax ¼ ay ¼ 0:1 at t ¼ 1 for Example 2.
Please cite this article in press as: H.S. Shukla, M. Tamsir, An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.04.011
6
H.S. Shukla, M. Tamsir
Fig. 5
Surface plots of solutions with h ¼ 0:02; Dt ¼ 0:001 and ax ¼ ay ¼ 0:05 at t ¼ 1 for Example 2.
Table 4 The RMS error norm for Example 3 with Dt ¼ 0:01 at different t. t
h ¼ 0:1
h ¼ 0:05
h ¼ 0:025
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.66211E04 1.21649E04 9.15324E05 7.44655E05 5.01330E05 3.35858E05 1.91012E05 6.01720E06 1.208670E06 2.497980E07
2.54808E06 2.08096E06 2.62743E06 6.53605E06 5.56751E06 5.49092E06 4.22667E06 1.30143E06 2.08647E07 2.05296E08
2.25582E07 2.85001E07 6.60018E07 1.76211E06 1.50829E06 1.48677E06 1.13366E06 3.37305E07 5.19397E08 4.91607E09
(
2
1 ðx 0:2 þ th3 Þ ðy 0:2 þ th4 Þ uðx; y; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi exp h1 ð1 þ 4tÞ h2 ð1 þ 4tÞ 1 þ 4t
2
)
I.C.’s and B.C.’s are taken out from Eq. (5.1). Numerical solution for Example 1 is obtained for parameters: h ¼ 0:025; Dt ¼ 0:0125 and 0.00625, ax ¼ ay ¼ 0:01, bx ¼ by ¼ 0:8 in domain ½1; 2 at t ¼ 1:25 and shown in Tables 2 and 3. It is found that present method generated more accurate results in comparison with results given by high accurate
methods presented in [7,10,11,14,16]. Figs. 2 and 3 show the contour and surface plots of the solutions with h ¼ 0:025; Dt ¼ 0:0025 and ax ¼ ay ¼ 0:01; at t ¼ 1:25 in domains [1, 2] and [0, 2] respectively. Example 2. Consider the CDE (1.1) in domain [0, 1] with the exact solution [4]: uðx; y; tÞ ¼ a expðbtÞðexpðcx xÞ þ expðcy yÞÞ;
FTCS [18] h ¼ 0:008
FTBSCS [18]
Lax-Wendroff [18]
Dt ¼ 0:0008 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,
ð5:2Þ
Comparison of the proposed method with three different methods presented in [18] for Example 3.
ðx; y; zÞ
(0.1, (0.2, (0.3, (0.4, (0.5, (0.6, (0.7, (0.8, (0.9,
RMS error norm versus time t for Example 3.
:
ð5:1Þ
Table 5
Fig. 6
0.1) 0.2) 0.3) 0.4) 0.5) 0.6) 0.7) 0.8) 0.9)
5.5E03 5.4E03 5.4E03 5.3E03 5.1E03 5.0E03 4.8E03 4.7E03 4.5E03
4.6E03 4.6E03 4.4E03 4.3E03 4.2E03 4.0E03 4.1E03 4.4E03 4.5E03
5.4E04 5.5E04 5.3E04 5.2E04 5.1E04 4.9E04 5.0E04 4.8E04 5.1E04
Present method h ¼ 0:1 t ¼ 1 Dt ¼ 0:01
Dt ¼ 0:001
5.1E07 2.2E07 2.6E06 1.9E06 5.7E05 1.5E05 5.5E05 7.7E06 2.1E05
7.9E07 2.2E07 3.0E06 2.6E06 6.9E05 1.7E05 6.6E05 8.8E06 2.2E05
Please cite this article in press as: H.S. Shukla, M. Tamsir, An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.04.011
An exponential cubic B-spline algorithm
7
Table 6 RMS error norm for Example 4 with free parameter p ¼ 0:01 and Dt ¼ 0:001 at t ¼ 1. h
a ¼ 0:05
a ¼ 0:1
a ¼ 0:5
0.2 0.125 0.1 0.0625
5.92E05 2.61E05 1.75E05 7.29E06
6.51E05 2.88E05 1.92E05 8.02E06
1.20E04 5.30E05 3.54E05 1.48E05
uðx; y; zÞ ¼
(
1 ð1 þ 4tÞ
3 2
exp
ðx 0:8t 0:5Þ2 0:01ð1 þ 4tÞ
) ðy 0:8t 0:5Þ2 ðz 0:8t 0:5Þ2 : 0:01ð1 þ 4tÞ 0:01ð1 þ 4tÞ
ð5:3Þ
The I.C.’s and B.C.’s are taken out from Eq. (5.3). The numerical solution of Example 3 is obtained with parameters: ax ¼ ay ¼ az ¼ 0:01, bx ¼ by ¼ bz ¼ 0:8, Dt ¼ 0:01, p ¼ 0:00001. The root-mean-square (RMS) error norms are obtained and shown in Tables 4, 5 and Fig. 6. It is found that proposed method produces high accurate results and errors decrease on increasing grid sizes. The proposed method results are compared with three different methods viz. FTCS, FTBSCS, Lax-Wendroff presented in [18]. The results are more accurate than those presented in [18] even for small grid size and time intervals. Example 4. Consider the CDE (1.4) with ax ¼ ay ¼ az ¼ a and bx ¼ by ¼ bz ¼ 1 over the region [0, 1] [0, 1] [0, 1]. The exact solution is taken as [19]: uðx; y; z; tÞ ¼ expf3ð1 aÞx þ y þ zg:
ð5:4Þ
The I.C.’s and B.C.’s are taken out from Eq. (5.4). The numerical solution of Example 4 is obtained with parameters: a ¼ 0:05; 0:1; 0:5 for different grid sizes and presented in Table 6 in terms of RMS error norm. It is marked that error norm decreases on increasing grid size. Fig. 7 shows the solution profile with p ¼ 0:01, h ¼ 0:05, a ¼ 0:5, Dt ¼ 0:01 at t ¼ 1. 6. Conclusion
Fig. 7 (a) Numerical and (b) exact solution profile with h ¼ 0:05, a ¼ 0:5, Dt ¼ 0:001 at t ¼ 1.
where cx ¼
bx
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2x þ 4bax 2ax
> 0; cy ¼
by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2y þ 4bay 2ay
> 0:
I.C.’s and B.C.’s are taken out from Eq. (5.2). The solutions of Example 2 for a ¼ 1, b ¼ 0:1, bx ¼ by ¼ 1, ax ¼ ay ¼ 0:1, t ¼ 1 are shown in Fig. 4 and those for a ¼ 1, b ¼ 0:1, bx ¼ by ¼ 1, ax ¼ ay ¼ 0:05, t ¼ 1 are shown in Fig. 5. It is understandable from Figs. 4 and 5 that solutions have a sharp discontinuity near the boundary which is not able to capture by meshless collocation techniques presented by Chinchapatnam et al. [4] for the analytical solution even for t ¼ 0:1. Example 3. Consider the CDE (1.4) over the region [0, 1] [0,1] [0,1] with the exact solution [18]:
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