An improved ADI-DQM based on Bernstein polynomial for solving two-dimensional convection-diffusion equations

Transcription

1 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 An improved ADI-DQM based on Bernstein polnomial for solving two-dimensional convection-diffusion equations A.S.J. Al- Saif 1 and Firas Amer Al-Saadawi 1 Department Mathematics, Education College for pure science, Basrah Universit, Basrah, Iraq Directorate of Education in Basrah, Basrah, Iraq sattaralsaif@ahoo.com, firasamer519@ahoo.com Abstract In this article, we presented an improved formulations based on Bernstein polnomial in calculate the weighting coefficients of DQM and alternating direction implicit-differential quadrature method (ADI- DQM) that is presented b (Al-Saif and Al-Kanani (01-013)), for solving convection-diffusion equations with appropriate initial and boundar conditions. Using the eact same proof for stabilit analsis as in (Al-Saif and Al-Kanani ), the new scheme has reasonable stabilit. The improved ADI-DQM is then tested b numerical eamples. Results show that the convergence of the new scheme is faster and the solutions are much more accurate than those obtained in literature. Kewords: Differential quadrature method, Convection-diffusion, Bernstein polnomial, ADI, Accurac. 1- Introduction The convection-diffusion equations are widel used in various fields such as petroleum reservoir simulation, subsurface contaminant remediation, heat conduction, shock waves, acoustic waves, gas dnamics, elasticit [,13,1]. The studies conducted for solving the convection-diffusion equations in the last half centur are still in an active area of research to develop some better numerical methods to approimate its solution. Etensive researches have been carried out to handle different tpes of convection-diffusion equation b different numerical methods [,,9,11,1,19,]. The best oldest known approimation techniques are the finite difference (FD) and finite element (FE) methods. A relativel new numerical technique is the differential quadrature method (DQM). Despite being a domain discretization method, the differential quadrature method gives accurate results using less discretization points than all the above mentioned methods (FD&FE). DQM depends on the idea of integral quadrature and approimate a spatial partial derivatives as a linear weighted sum of all functional values of the solution at all mesh points [1]. This method was proposed b Bellman and Casti [5] in One of important kes to DQM lies in the determination of weighting coefficients for the discretization of a spatial derivative of an order, where it pla the important role in the accurac of numerical solutions. Initiall, Bellman et al.[](197), suggested two methods to determine the weighting coefficients of the first order derivative. The first method solves an algebraic equation sstem. The second use a simple algebraic formulation, but with the coordinates of grid points chosen as the roots of the Legendre polnomial. Quan and Chang [0] (199a) and Shu and Richards [5] (199), derived a recursive formula to obtain these coefficients directl and irrespective of the number and positions of the sampling points. In their approach, the used the Lagrange polnomials as the trial functions and found a simple recurrence formula for the weighting coefficients, and used b Meral [1] (013) and Jiwari [13] (013). Bert et al.[7](1993) and Striz et al.[7] (1995) developed the differential quadrature method, which uses harmonic functions instead of polnomial as test function in the quadrature method to handle periodic problems efficientl, and also circumvented the limitation for the number of grid point in the conventional DQM based on polnomial test function. Their stud shows that the proper test functions are essential for the computational efficienc and reliabilit of the DQM. Shu et al.[3] (001) presented a numerical stud of natural convection in an eccentric annulus between a square outer clinder and a circular inner clinder using DQM, b using Fourier series epansion as the trial functions to compute weighting coefficients. Krowiak[15] (00) studied the methods that based on the differential quadrature in vibration analsis of plates, and using the spline functions as the trial functions to compute weighting coefficients. Korkmaz et al.[1] (011) used the quartic B-spline differential quadrature method, and applied it on the one-dimensional Burger s equation, b using the quartic B-spline functions as the trial functions to compute weighting coefficients.

2 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 The motivations to introduce the current work are: Firstl, Attempt of all the above researchers to develop the DQM b using different test functions in computing weighted coefficients of DQM motivating us to research about polnomial has good properties and suitable with application of DQM. Therefore, we noticed that Bernstein polnomials are incredibl useful mathematical tools as the are simpl defined. The can be calculated quickl on computer sstems and represent a tremendous variet of functions. The can be differentiated and integrated easil, and can be pieced together to form spline curves that can approimate an function to an accurac desired. One of important properties to Bernstein polnomials is surel convergence. Secondl, Most recentl, Al-Saif and Al- Kanani [3,] (01-013) proposed a new improvement for DQM that is resulting from applied ADI into DQM for convection-diffusion problems, and the results of ADI-DQM with Lagrange polnomial and Fourier series epansion as the test functions to computing the weighted coefficients show the efficienc of the proposed method to handle the problems under consideration. Moreover, we wanted to be etending the application of our new suggestion in [] to improve ADI-DQM. Depending on these reasons and according to our humble knowledge that the Bernstein polnomials are not et used to calculate weighting coefficients, in this article, we suggest Bernstein polnomials as test functions to compute the weighting coefficients of the spatial derivatives, in order to introduce a new development to the differential quadrature method that is called alternating direction implicit formulation of the Bernstein differential quadrature method (). Using the for solving convection-diffusion problems ecellent numerical results are obtained. ADI-DQM is then tested b numerical eamples. Results show that the convergence of the new scheme is faster and the solutions are much more accurate than those in literature [3,,11,1] and has reasonable stabilit. - Bernstein Differential quadrature method (BDQM) The differential quadrature is a numerical technique used to solve the initial and boundar value problems. This method was proposed b Bellman and Casti[5] in (1971). The DQM is based on the idea that the partial derivative of a field variable at the discrete points in the computational domain is approimated b a weighted linear sum of the values of the field variable along the line that passes though that point, which is parallel with coordinate direction of the derivative as following[]: where are the discrete points in the variable, is the order derivative of the function, are the function values at these points, and are the weighting coefficients for the order derivative of the function with respect to and is the number of the grid points. There are two ke points in the successful application of the DQM: how the weighting coefficients are determined and how the grid points are selected [17]. Man researchers have obtained weighting coefficients implicitl or eplicitl using various test functions [,3,13,1,1,3]. Here, we use the eact same manner in (Quan and Chang [0] (199a) and Shu and Richards[5] (199)) to determine the weighting coefficients, but with emploing the Bernstein polnomials as the test function. A Bernstein polnomial, named after Sergei Natanovich Bernstein, is a polnomial in the Bernstein form, that is a linear combination of Bernstein basis polnomials. The Bernstein basis polnomials of -degree are defined on the interval [ ] b Singh et al.[]: The general form of Bernstein polnomials of -degree that used to solve differential equation [,1] are defined on the interval [ ] as: were binomial coefficients are given b : There are -degree Bernstein polnomials. For mathematical convenience, we usuall set,, if or. These polnomials are quite eas to write down the coefficients that can be obtained from Pascal s triangle. It can easil be shown that each of the Bernstein 11

3 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 polnomials is positive and also the sum of all the Bernstein polnomials is unit for all real [ ], i.e [ ] The Bernstein polnomials can be written to an interval as following[]: where is arbitrar function, for Similar to Lagrange differential quadrature method LDQM to determined weighting coefficients, we can derive the eplicit formulationto compute the weighting coefficients b using Bernstein polnomial as a test functions, which are listed below: { where is length interval [ ] and The weighted coefficients of the second order derivative b using Bernstein polnomial as a test functions can be obtained as [ ] [ ][ ] [ ] The same technique can be used to obtained the weighting coefficients. 3- Alternating direction implicit technique - BDQM To illustrate the application of the technique of ADI to the formula of DQM, we consider the following partial differential equation in two dimensions as; where are the differential operators with respect to respectivel. The alternating direction implicit technique was introduced in the mid-50s b Peaceman and Rachford for solving equations, which result from finite difference discretization of partial differential equations (PDEs). From iterative method s perspective, ADI method can be considered as special relaation method, where a big sstem is simplified into a number of smaller sstems such that each of them can be solved efficientl and the solution of the whole sstem is got from the solutions of the sub-sstems in an iterative wa. Using alternating direction implicit method to approimate equation, we get the sstems of algebraic equations in the form [3,]: where and are the BDQM quantities that including the weighting coefficients for the differential operators and, respectivel. Equation is used to compute function values at all interval mesh points along rows and known as horizontal traverse or sweep. While, Equation is used to compute function values at all interval mesh points along columns and known as vertical traverse or sweep. 1

4 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, Numerical eamples and discussion In this section, we appl on two test problems to demonstrate the efficienc of the ADI- BDQM. These eamples are chosen such that their eact solutions are known. Problem 1 (Dehghan and Mohebbi[1]) Consider the unstead state linear two-dimensional convection-diffusion equation: [ ] [ ] [ ] where is a transported variable, and are arbitrar constants which show the speed of convection and the diffusion coefficients and are positive constants. The initial condition of Equation (11) has the following form: where and the boundar conditions are given b: } The eact solution is given as: Equation (11) can be approimated b using, equations (9 and ), such that and, where the and are the weighted coefficients of the first and the second order derivatives with respect to and respectivel. In this problem, we take, and, and use equall spaced grid points. Tables 1 and are shows the errors obtained from solving problem 1 b using LDQM, and at and for different values of. Fig. 1 clarifies a comparison between eact solution and numerical solutions for and respectivel. The results confirm that the is more accurate and less CPU time than the LDQM and. Table 1 Errors obtained for problem 1 with h Ma of CPU Ma of CPU Ma of CPU LDQM E E E E E E E E E E E-.9537E- 1.3 Table Errors obtained for problem 1 with h Ma of CPU Ma of CPU Ma of CPU LDQM E E E E E E E E E E E E

5 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 We choose the arbitrar function and is the arbitrar constant. In this problem, we take and for Table. and and for Table 3. respectivel at the number of grid points and Notice that in the net eample, we choose the same above arbitrar function with different values of. Eact = =1.0 u(,,t) Eact = =0.1.0 u(,,t) = =1.0 u(,,t) = =0.1.0 u(,,t) = =1.0 u(,,t) = =0.1.0 u(,,t) Fig. 1 Eact and approimate solutions of the problem 1 with, and Problem (Al-Saif and Al-kanani[]) Consider the nonlinear two-dimensional convection-diffusion equations that are called Burger's Equations: 1

6 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 where and are the velocit components to be determined, is constant and is the Renolds number, [ ] [ ] [ ] In this problem, we take, and the initial conditions of Equations and have the following form: The eact solutions are given b: The boundar conditions can be achieved easil from (1) b using Equations (15-1) can be approimated b using (( Equations(9 and )), such that and, where the and are the weighted coefficients of the first and the second order derivatives with respect to and, respectivel. For the above problem, we found numerical solutions for and and use equall spaced grid points. Tables 3 and are shows the errors obtained from solving problem b LDQM, and at and and and [ ] for different values of Fig. clarifies a comparison between eact solution and numerical solutions of the problem. The results show that the has a high accurac, good convergence and less CPU time compared with the LDQM and. Table 3 Errors obtained and for problem with of Ma of LDQM CPU Ma of CPU Ma of CPU h E E E E E E E E E E E E Table Errors obtained and for problem with of Ma of LDQM CPU Ma of CPU Ma of CPU h E E E E E E E E E E E E In this problem, we take and respectivel at the number of grid and 5- Comparison with the other methods We compare the numerical results of for problems 1 and with the results of other numerical methods such as, High-order compact boundar value method (HOCBVM)[1]and Radial basis function based meshless method(rbfbmm)[11]. The error measurements resulted from the is more accurate than the methods, HOCBVM[1], RBFBMM[11] and. Moreover, the number of grid points b using and ADI- LDQM are less than the other methods. - Stabilit analsis of BDQM The stabilit of numerical schemes is closel related to numerical error. A solution is said to be unstable if errors appear at some stage in the calculations (for eample, from erroneous initial conditions or local truncation or round-off errors) are propagated without bound throughout subsequent calculations. Thus a method is stable if small changes in the initial data produce correspondingl small changes in the final results, that is, the difference between the theoretical and numerical solutions remains bounded at a given time, as time and space steps tend to zero or time step remains fied at ever level and [1]. So stabilit, means that the numerical solution must be 15

7 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 close to the eact solution, meaning that whenever was the error a little the deviation in derivatives, however, this error ma accumulate at each time step and affects to the stabilit of the solution. Eact u(,,t) v(,,t) Eact 0.3 u(,,t) v(,,t) v(,,t) v(,,t) Fig. Eact and approimate solutions of the problem with, t=1 and t=001 Table 5. Comparison of the numerical results of the problem 1 for different methods at Method Number grid points Ma of u, Ma of u,.9537e-.039e E E-05 HOCBVM[ ] 1.1E E-05 RBFBMM[ ].97E-0.5E-0 Table. Comparison of the numerical results of the problem for different methods at. Theorem[] Method Number grid points Ma of u Ma of v 9.005E E E E-0 BDQM E E-05 LDQM E E-05 The sstem of ODE with a constant coefficient matri is, (1) Stable if the roots of the characteristic polnomial are purel imaginar. () Asmptoticall stable if the roots have negative real parts. 1

8 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 (3) Unstable if a root has positive real part. From application of BDQM to the an convection-diffusion equation in this work, we obtained the set ordinar differential equations: [ ] { } { } { } where { } is a vector of unknown functional values at all the interior points given b { } [ ] and { } is a known vector which is made up of the non-homogeneous part and the boundar conditions given b { } [ ] and [ ] is the coefficient matri containing the weighting coefficients, the dimension of the matri [ ] is b. For the multi-dimensional case, the matri [ ] contains man zero elements, which are irregularl distributed in the matri. The stabilit analsis of the Equation (19) is based on the eigenvalue distribution of the BDQM discretization matri [ ]. If [ ]has eigenvalues and corresponding eigenvector, being the size of the matri[ ], the similarit transformation reduces the sstem (19) of the form[,]. { } [ ]{ } { } Here the diagonal matri [D] is formed from the eigenvalues and from a nonsingular matri [P] containing the eigenvectors as columns [ ] [ ] [ ][ ] Pre-multipling b the matri [ ] on the both sides Equation (0) and setting { } [ ] [ ] { } [ ] [ ] Since [ ] is a diagonal matri, Equation (0) is an uncoupled set of ordinar differential equations. Considering the equation of (0) If is time-independent, then the solution of Equation () can be written as For this case, using Equations () and (3), the solution { } can be obtained as { } [ ][ ] [ ] Clearl, the stable solution of { } when requires for all where denotes the real part of. This is the stabilit condition for the sstem (19). In this section, we can applied the stabilit condition (7) on the problems that mentioned in the previous section b using. 17

9 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 Problem 1. From the application of to the Equation (11) and using, and, Equation (11) can be rewritten as: ( ) where { } { } From Equation (), we can obtain a sstem of algebraic equations (19). This sstem has the solution (), and this solution { } is stable as eigenvalues of the matri [ ] for respectivel, are: and the real parts of the and This means that the stabilit condition (7) is hold. Problem. From the application of to the Equation (15) and using, and Equation (15) can be rewritten as: ( ) where { } { } From Equation (9), we can obtain a sstem of algebraic equations (19). This sstem has the solution (), and this solution { } is stable as eigenvalues of the matri [ ]are: and the real parts of the This means that the stabilit condition (7) is hold. When using the equation (), we will find the same eigenvalues mentioned above of the matri [A]. Finall, the numerical results of the above problems confirm that the newl developed method ADI- BDQM is stable for the grid points. In this work, with the help of smbolic computation software Maple 13, the eigenvalues are computed. 7- Conclusions In this work, we emploed a new technique to solve convection-diffusion equations successfull. The weighting coefficients for spatial derivatives are computing b use Bernstein polnomials as test functions. The numerical results show that the new method has higher accurac, good convergence and reasonable stabilit as well as a less computation workload b using few grid points. Moreover, the results show that the application of our new suggestion in [] to improve ADI- DQM is successful and can be applied on more general problems. 1

10 Mathematical Theor and Modeling ISSN -50 (Paper) ISSN 5-05(Online) Vol 3, No.1, 013 References [1] A. Ali, Mesh free collocation method for numerical solution of initial-boundar- value problems using radial basis functions, Ph.D. thesis, Ghulam Ishaq Khan Institute of Engineering Sciences and Technolog, Pakistan, (009). [] A. S. J. Al-Saif and Firas A. Al-Saadawi, Bernstein differential quadrature method for solving the unstead state convection-diffusion equation, Indian J. of Appl. Reasearch, 3(9)(013),0-. [3] A. S. J. Al-Saif and M. Al-kanani, Solution of nonlinear initial-value problems b the alternating direction implicit formulation on differential quadrature method, Int. J. of Pure and Appl. Research in Engin. and Technolog, 1(9)(013), [] A. S. J. Al-Saif and M. J. Al-kanani, Alternating direction implicit formulation of the differential quadrature method for solving Burger equations, Inter. J. Modern Math. Sc. 30(1)(01), [5] R. Bellman and J. Casti, Differential quadrature and long-term integration, J. Math. Anal. Appl., 3(1971), [] R. Bellman, B.G. Kashef and J. Casti, Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phs, (197), 0-5. [7] C. W. Bert, X. Wang and A. G. Striz, Differential quadrature for static and free vibrational analsis of anisotropic plates, Int. J. Solids Structures, 30(1993), [] M. I. Bhatti and P. Bracken, Solution of differential equations in a Bernstein polnomial basis, J. Comput. Appl. Math., 05(007), 7-0. [9] J. Chan and J. A. Evans, A minimum-residual finite element method for the convection-diffusion equation, Institute for Comput. Engineering and Sciences ICES, 13(13)(013), [] W. Chen, Differential quadrature method and its applications in engineering, Ph. D. thesis, Shanghai JiaoTong Universit, China, (199). [11] P. P. Chinchapatnam, Radial basis function based meshless methods for fluid flow problems, Ph.D. thesis, Southampton Universit, Facult of Engineering Science and Mathematics, UK, (00). [1] M. Dehghan and A. Mohebbi, High-order compact boundar value method for the solution of unstead convection-diffusion problems, Math. Comput. Simul.,79(00), [13] R. Jiwari, R. Mittal and K. K. Sharma, A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burger s equation, Appl. Math. Comput., 19(013), [1] A. Korkmaz,A. M. Akso and I. Dag, Quartic B-spline differential quadrature method, International J. of Nonlinear Sc., 11(011), [15] A. Krowiak, methods based on the differential quadrature in vibration analsis of plates, J. Theor. Appl. Mech., (1)(00), [1] G. Meral, Differential quadrature solution of heat-and mass-transfer equations, Appl. Math. Modelling, 37(013), [17] M. Mozaffari, A. A. Atai and N. Mostafa, Large deformation and mechanics of fleible isotropic membrane ballooning in three dimensions b differential quadrature method, J. Mech. Sc. Tech., 3(009), [1] Y. Ordokhani and S. D. Far, Application of the Bernstein polnomials for solving the nonlinear Fredholm Integro-differential equations, J. Appl. Math. And Bioinformatics,1(011), [19] F. U. Prieto, J.B. Muñaz and L.G. Corvinos, Application of the generalized finite difference method to solve the advection diffusion equation, J. of Comp. and App. Math., 35(011), [0] J. Quan and C. Chang, New insights in solving distributed sstem equations b the quadrature methods - I, Comput. Chem. Engng, 13(199a), [1] M. Ramezani, M. Shahrezaee, H. Kharazi and L. H. Kashan, Numerical solutions of differential algebraic equation b differential quadrature method, J. Basic. Appl. Sc. Res, (11)(01), [] D. A. Sanchez, Ordinar Differential Equations and Stabilit Theor: An Introduction, W.H. Freeman and Compan, USA, (19). [3] C. Shu, H. Xue and Y. D. Zhu, Numerical stud of natural convection in an eccentric annulus between a square outer clinder and a circular inner clinder using DQ method, Int. J. Heat and Mass Transfer, (001), [] C. Shu, Differential Quadrature and Its Application in Engineering, springer-verlag, London, (000). [5] C. Shu and B. Richards, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, Int. J. Numer. Meth. In Fluids,15(199), [] A. K. Singh, V. K. Singh and O. P. Singh, The Bernstein operational matri of integration, Appl. Math. Sc., 3(9)(009), 7-3. [7] A. G. Striz, X. Wang and C. W. Bert, Harmonic differential method and applications to structural components, Acta Mechanica, 111(1995), 5-9. [] H. Zhu, H. Shu and M. Ding, Numerical solutions of two-dimensional Burger s equations b discrete Adomian decomposition method, Comput. Math. Appl., 0(0),