AN APPLICATION OF CUBIC B-SPLINE FINITE ELEMENT METHOD FOR THE BURGERS EQUATION

Transcription

1 Aksan, E..: An Applıcatıon of Cubıc B-Splıne Fınıte Eleent Method for... THERMAL SCIECE: Year 8, Vol., Suppl., pp. S95-S S95 A APPLICATIO OF CBIC B-SPLIE FIITE ELEMET METHOD FOR THE BRGERS EQATIO by Eine esligul AKSA * Departent of Matheatics, Art and Science Faculty, Inonu niversity, Malatya, Turkey Original scientific paper Introduction It is difficult to achieve eact solution of non-linear PDE, directly. Soeties, it is possible to convert non-linear PDE into equivalent linear PDE by applying a convenient transforation. Hence, Burgers equation replaces with heat equation by eans of the Hope-Cole transforation. In this study, Burgers equation was converted to a set of non-linear ODE by keeping non-linear structure of Burgers equation. In this case, solutions for each of the non-linear ODE were obtained by the help of the cubic B-spline finite eleent ethod. Model probles were considered to verify the efficiency of this ethod. Agreeent of the solutions was shown with graphics and tables. Key words: Burgers` equation, cubic B-spline functions, finite eleent ethod It ay be regarded as a siple process to acquire the approiate solution by using the ethod of discretization in tie which provides us to with a chance of reducing an initial boundary value proble to a syste of boundary value probles which can be solved eactly or approiately []. In this study, approiate solution is obtained by applying well-known finite eleent ethod to each equation of the syste. Let us consider the Burgers equation as the odel proble. It is -D non-linear parabolic PDE: + = ε t a b, t > () where is the velocity, t the tie, the co-ordinate, and ε the kineatics viscosity, ε >. Burgers equation is transfored to heat equation through Hopf-Cole transforation, therefore eact solution can be obtained. Eact solution of the equation acquired was perfored by Cole []. Burgers equation is hyperbolic and parabolic for ε = and ε >, respectively. Thus, it is hard to obtain a solution for sall values of ε. For this reason, any researchers have used different ethods to solve the Burgers equation, such as finite eleent ethod, finite difference ethod, variational ethods, and Adoian s decoposition ethod [3-8]. In this study, each of the non-linear ODE obtained by using the ethod of discretization in tie fro Burgers equation was solved by the eans of cubic B-spline finite eleent ethod. For different viscosity values at different tie steps, the nuerical results were co- Author e-ail: nesligul.aksan@inonu.edu.tr

2 S96 Aksan, E..: An Applıcatıon of Cubıc B-Splıne Fınıte Eleent Method for... THERMAL SCIECE: Year 8, Vol., Suppl., pp. S95-S pared with the eact solutions. It is obvious that approiate solution converges to the eact solution. Moreover, it was recognized that the structure of the proble for sall viscosity values were preserved. Stateent of the probles Let us consider the Burgers eq. () with the hoogenous boundary conditions: ( ) ( ), t =,, t =, t > () and the net initial conditions: The odel proble. The initial condition is: ( ), = sin π in < < (3) The eact solution of this proble was given first by Cole []: n= n π ε t an e nsin nπ n= ( t, ) = πε n π ε t a + a e cos nπ where a and a n (n =,,...) are Fourier coefficients which are given: n (4) a cos π = ep d πε cos π an = ep cos nπd, n πε The odel proble. This proble was constructed by taking the initial condition (5) instead of the initial condition () in proble of eq. (): (,) 4 ( ) = in < < (5) In addition, eact solution of the odel proble is also given by eq. (4), where Fourier coefficients a and a n : The ethod of nuerical solution ( 3 ) a = ep d 3ε ( 3 ) an = ep cos nπd, n 3ε Model probles by eans of the ethod of discretization in tie was converted to the following probles for the successively value of =,,..., p, the functions z () which are the solutions of the probles: ε z ( ) + z( ) z ( ) + z( ) z ( ) = z () =, z () = (7) (6)

3 Aksan, E..: An Applıcatıon of Cubıc B-Splıne Fınıte Eleent Method for... THERMAL SCIECE: Year 8, Vol., Suppl., pp. S95-S S97 where z () = (, ) []. The boundary value probles of eqs. (6) and (7) can be solved either eactly or nuerically. Since obtaining of eact solutions is difficult as increases, it is preferable to solve the proble nuerically. The finite eleent ethod gives systeatics eans of generating nuerical solutions to a proble forulating of the odel probles. Therefore, the cubic B-spline finite eleent ethod to solve each of the probles is applied. The weak for of eq. (6) over interval [, ] is given: { ε } v( ) z ( ) + z ( ) z ( ) + z ( ) z ( ) d=, =,,..., p (8) where v() is weighted function. After a siple arrangeent, we get: ε v ( z ) ( )d + vz ( ) ( z ) ( )d + vz ( ) ( )d= = { ε } + t vz ( ) ( ) vz ( ) ( d ) where =,,..., p. In the finite eleent ethod, the interval [, ] is divided into finite eleents of equal length h by the nodes i such that = < <... < =. Set of splines {ϕ, ϕ,..., ϕ + } for a basis for the functions defined on [, ]. Cubic B-splines ϕ () with the required properties are defined: 3 ( ), [, ] 3 3 h + 3 h ( ) + 3 h( ) 3( ), [, ] 3 3 φ = 3 ( 3 h + h + ) + 3 h( + ) 3( + ), [, ] h ( + ), [ +, + ], otherwise where h= +, =,,..., + [9]. The values of ϕ () and its first and second derivatives ϕ () and ϕ () at the nodes are given by the tab.. Hence, an approiate solution ( ) of the function z () is given: z nφn n= z ( ) = c ( ) () where c n are coefficients to be deterined. sing eq. (), the nodal values z and z, at the nodes can be given: z, = z( ) = c + 4c+ c+ 3 () z, = z ( ) = ( c + c+ ) h Because c ( ) should satisfy the boundary conditions (), we get c = (4 c + c ) and c + = ( c + 4 c ). So we have: Table. The B-spline values at the nodes + + ϕ () 4 ϕ () 3/h 3/h ϕ () 6/h /h 6/h (9)

4 S98 Aksan, E..: An Applıcatıon of Cubıc B-Splıne Fınıte Eleent Method for... THERMAL SCIECE: Year 8, Vol., Suppl., pp. S95-S where nψ n n= z ( ) = c ( ) () ψ ( ) = [ 4 φ ( ) + φ ( )], ψ( ) = [ φ ( ) + φ( )], ψ( ) = φ( ), ( =,3,..., ) ψ = [ φ ( ) φ+ ( )], ψ = [ φ( ) 4 φ + ( )] Thus the unknown c ( =,,..., ) ust be obtained. Let us construct approiate solution of eq. (9) by using Galerkin ethod. In this ethod, the weighted function v() is taken as v () = ψ (), ( =,,...,). Substituting eqs. () and () into eq. (9), we get: ε ψψ k nd c ψψψ n k n d ccn k nd cn kz d n n t ψψ + + = n t ψ = = = = (3) where =,,..., p and k =,,...,. After soe operations, eq. (3) becoes in the atri for: ε Ac + B( c ) c + Cc = D (4) T where c = ( c, c, c ), =,,..., p. Equation (4) is a (+) (+) non-linear equations syste for each. The atrices A, B, and C are (+) (+) 7-diagonal atrices and th row of these atrices have the for: A : 3, 7, 45, 4, 45, 7, 3 h ( ) B: [( 5,,, 5,,, ) c,( 8, 944,,944,8,,) c, 84 ( 9, 83, 784, 784, 83, 9, ) c, (, 3888, 3568,, 3568, 3888, ) c, (, 9, 83, 784, 784, 83, 9) c,(,, 8, 944,, 944,8) c, (,,, 5,,, 5) c ] c = ( c, c, c, c, c, c, c ), T where h C :,,9, 46,9,, 4 ( ) T where c = ( c 4, c 3, c, c, c, c+, c+ ). The D is an + colun vector. D is calculated by using coposite Sipson s rule [] for both of proble and th row of D is given: h ( : D c 4, c 3,9 c, 46 c,9 c, c+, c+ ), 4 for each proble.

5 Aksan, E..: An Applıcatıon of Cubıc B-Splıne Fınıte Eleent Method for... THERMAL SCIECE: Year 8, Vol., Suppl., pp. S95-S S99 The nuerical results and conclusion The non-linear systes, eq. (4), were solved for each -value by using ewton ethod which rapidly converges to solution. Hence, Jacobian atri in the algorith of the ewton ethod is 7-diagonal atri because of choosing the basis functions in the finite eleent ethod as cubic B-spline functions. Syste obtained for each -value has been solved with a direct ethod. In the right hand side of the first syste, D was calculated by using coposite Sipson s rule. We consider the two odel probles to show the efficiency of the given ethod. The nuerical solutions obtained by this way have been copared with the eact solutions for various ε values at different ties. The nuber of both divisions of the interval [, T] and basis eleents was increased to appreciate the nuerical solutions. It is seen that as p is increasing nuerical solution converge to the eact solution, tab.. Siilarly, as the nuber of basic functions increase the nuerical solutions converge to eact solution, too tab. 3. Table. uerical solutions and the eact solution of proble at various values of for ε =, = at t =. Δt =. uerical solutions Δt =.5 Δt =. Eact solution Table 3. uerical solutions and the eact solution of proble with different nuber of basis eleents for ε =, =. at t =. = uerical solutions = = 4 Eact solution The nuerical solutions obtained for. ε at different tie steps were given by figs. -6 and tabs Obviously, it was found that the atheatical structure of probles for the obtained nuerical solutions in very sall values of viscosity (ε <.) at different tie steps did not change, figs. 4 and 5.

6 S Aksan, E..: An Applıcatıon of Cubıc B-Splıne Fınıte Eleent Method for... THERMAL SCIECE: Year 8, Vol., Suppl., pp. S95-S Table 4. uerical solutions and the eact solution of proble at various values of ε and =, =. at different ties ε =. ε =. t uerical uerical Eact solution solution solution Eact solution Table 5. Coparison of the nuerical solutions of proble with the both result fro [6] and the eact solutions for ε = at different ties uerical solutions t = 8, Δt =. [6] =, Δt =. Present Eact solution The ost fundaental difference of this study fro the literature is that although larger and less nuber of base eleent are selected ore good results have been obtained. This iplies that the ethod used is very econoics. The ethod of solution presented provides high accuracy. Finally, it can be used to solve Burgers -like equations.

7 Aksan, E..: An Applıcatıon of Cubıc B-Splıne Fınıte Eleent Method for... THERMAL SCIECE: Year 8, Vol., Suppl., pp. S95-S S Table 6. Coparison of the nuerical solutions of proble with the both result fro [6] and the eact solutions for ε =. at different ties uerical solutions t = 8, Δt =. [6] =, Δt =. Present Eact solution t =. t =.5 t =. t =. t =.3 t =.4 t =. t = 3. Figure. Solutions of proble at different ties for ε =, =, and t =. Figure. Solutions of proble at different ties for ε =., =, and t =. t =. t =. t =.5 t =. t =.5 t =. t =.5 t =.4 t =.6 t =.8 t =. Figure 3. Solutions of proble at different ties for ε =., =, and t =. Figure 4. Solutions of proble at different ties for ε =., =, and t =.

8 S Aksan, E..: An Applıcatıon of Cubıc B-Splıne Fınıte Eleent Method for... THERMAL SCIECE: Year 8, Vol., Suppl., pp. S95-S t =.5 t =. t =.5 t =. t =.5 t =. t =.5 t =. t =.5 t =.5 Figure 5. Solutions of proble at different ties for ε =., =, and t =. Figure 6. Solutions of proble at different ties for ε =, =, and t =. References [] Rektorys, K, The Method of Discretization in Tie and Partial Differential Equations, D. Reidel Publishing Copany, Dordrecht, The etherland, 98 [] Cole, J. D., On a Quasi-Linear Parabolic Equation Occurring in Aerodynaics, Quart. Apply. Math. 9 (95), 3, pp [3] Ozis, T., et al., The Sei-Approiate Approach for Solving Burgers Equation with High Reynolds uber, Appl. Math. Coput., 63 (5),, pp [4] Aksan, E.., A uerical Solution of Burgers Equation by Finite Eleent Method Constructed on the Method of Discretization in Tie, Appl. Math. Coput, 7 (5),, pp [5] Abbasbandy, S., et al., A uerical Solution of Burgers Equation by Modified Adoian Method, Appl. Math. Coput, 63 (5), 3, pp [6] Ozis, T., et al., uerical Solution of Burgers Equation by Quadratic B-Spline Finite Eleent, Appl. Math. Coput., 65 (5),, pp [7] Ozis, T., et al., A Direct Variational Methods Applied to Burgers Equation, J. Coput. Appl. Math., 7 (996),, pp [8] Aksan, E.., et al., A uerical Solution of Burgers Equation, Appl. Math. Coput., 56 (3),, pp [9] Prenter, P. M., Splines and Variational Methods, John Wiley and Sons, ew York, SA, 975 [] Burden, R. L., et al., uerical Analysis, Brooks/Cole Pub. Co., Pacific Grove, Cal., SA, 5 Paper subitted: June 3, 7 Paper revised: oveber 5, 7 Paper accepted: oveber 8, 7 8 Society of Theral Engineers of Serbia Published by the Vinča Institute of uclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-C-D 4. ters and conditions