NUMERICAL SOLUTION OF NON-CONSERVATIVE LINEAR TRANSPORT PROBLEMS

Yıl 2018, Cilt: 8 Sayı: 1.1, 167 - 177, 01.09.2018

A. Korkmaz H. K. Akmaz

Öz

In this study, trigonometric cubic B-spline dierential quadrature method is developed for a linear transport problems constructed on the advection-diusion equation. The weighting coecients used in the derivative approximations are determined by using the proposed algorithm. Following the space discretization of the advectiondi usion equation, the resultant ODE system is integrated in time by using Rosenbrock implicit method of order four. The accuracy and validity of the proposed method are indicated by solving some initial boundary value problems IBVPs representing fade out of an initial positive pulse. The error between the analytical and the numerical solutions is measured by using the discrete maximum norm.

Anahtar Kelimeler

Advection-diusion equation, trigonometric cubic B-spline, dierential quadrature method, transport.

Kaynakça

• [1] Isenberg, J., & Gutfinger, C. (1973). Heat transfer to a draining film. International Journal of Heat and Mass Transfer, 16(2), 505-512.
• [2] Parlange, J. Y. (1980). Water transport in soils. Annual Review of Fluid Mechanics, 12(1), 77-102.
• [3] Chatwin P. C., & Allen, C. M. (1985). Mathematical models of dispersion in rivers and estuaries. Annual Review of Fluid Mechanics, 17(1), 119-149.
• [4] Taylor, G. (1954, May). The dispersion of matter in turbulent flow through a pipe. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 223, 1155, 446-468. The Royal Society.
• [5] Chatwin, P. C. (1980). Presentation of longitudinal dispersion data. J. Hydraul. Div. Proc. ASCE 106:71-83.
• [6] Huang, H., & Huang, S. (2011). A high-order implicit difference method for the one-dimensional convection diffusion equation. Journal of Mathematics Research, 3(3), 135-139.
• [7] Dehghan, M. (2005). On the numerical solution of the one-dimensional convection-diffusion equation. Mathematical Problems in Engineering, 2005(1), 61-74.
• [8] Dehghan, M. (2004). Weighted finite difference techniques for the one-dimensional advection-diffusion equation. Applied Mathematics and Computation, 147(2), 307-319.
• [9] Karahan, H. (2006). Implicit finite difference techniques for the advection-diffusion equation using spreadsheets. Advances in Engineering Software, 37(9), 601-608.
• [10] Karahan, H. (2007). A third-order upwind scheme for the advection-diffusion equation using spreadsheets. Advances in Engineering Software, 38(10), 688-697.
• [11] Karahan, H. (2007). Unconditional stable explicit finite difference technique for the advection-diffusion equation using spreadsheets. Advances in Engineering Software, 38(2), 80-86.
• [12] Sharma, D., Jiwari, R., & Kumar, S. (2011). Galerkin-finite element method for the numerical solution of advection-diffusion equation. International Journal of Pure and Applied Mathematics, 70(3), 389- 399.
• [13] Thongmoon, M., & McKibbin, R. (2006). A comparison of some numerical methods for the advectiondiffusion equation. Res. Lett. Inf. Math. Sci., 2006, 10, 49-62.
• [14] Dhawan, S., Bhowmik, S. K., & Kumar, S. (2015). Galerkin-least square B-spline approach toward advection-diffusion equation. Applied Mathematics and Computation, 261, 128-140.
• [15] Gurarslan, G., Karahan, H., Alkaya, D., Sari, M., & Yasar, M. (2013). Numerical solution of advectiondiffusion equation using a sixth-order compact finite difference method. Mathematical Problems in Engineering, 2013, 672936, 1-7.
• [16] M. Sari,G.Gurarslan, and A. Zeytinoglu, High-order finite difference schemes for solving the advectiondiffusion equation. Mathematical and Computational Applications, 15, 3, 449-460, 2010.
• [17] Dag, I., Irk, D., & Tombul, M. (2006). Least-squares finite element method for the advection-diffusion equation. Applied Mathematics and Computation, 173(1), 554-565.
• [18] Irk, D., Dag, I., & Tombul, M. (2015). Extended cubic B-spline solution of the advection-diffusion equation. KSCE Journal of Civil Engineering, 19(4), 929-934.
• [19] Kaya, B. (2010). Solution of the advection-diffusion equation using the differential quadrature method. KSCE Journal of civil engineering, 14(1), 69-75.
• [20] Kaya, B. & Gharehbaghi, A. (2014). Implicit Solutions of Advection Diffusion Equation by Various Numerical Methods. Australian Journal of Basic and Applied Sciences, 8(1), 381-391.
• [21] Korkmaz, A., & Dag, I. (2012). Cubic B-spline differential quadrature methods for the advectiondiffusion equation. International Journal of Numerical Methods for Heat & Fluid Flow, 22(8), 1021- 1036.
• [22] Korkmaz, A., & Dag, I. (2016). Quartic and quintic B-spline methods for advection-diffusion equation. Applied Mathematics and Computation, 274, 208-219.
• [23] Korkmaz, A., & Akmaz, H. K. (2015). Numerical Simulations for Transport of Conservative Pollutants. Selcuk Journal of Applied Mathematics, 16(1).
• [24] Nazir, T., Abbas, M., Ismail, A. I. M., Majid, A. A., & Rashid, A. (2016). The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach. Applied Mathematical Modelling, 40(7), 4586-4611.
• [25] Bellman, R., Kashef, B. G., & Casti, J. (1972). Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. Journal of computational physics, 10(1), 40-52.
• [26] Jiwari, R., & Yuan, J. (2014). A computational modeling of two dimensional reaction-diffusion Brusselator system arising in chemical processes. Journal of Mathematical Chemistry, 52(6), 1535-1551.
• [27] Jiwari, R., Pandit, S., & Mittal, R. C. (2012). Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method. Computer Physics Communications, 183(3), 600-616.
• [28] Mittal, R. C., & Jiwari, R. (2011). Numerical study of two-dimensional reaction-diffusion Brusselator system by differential quadrature method. International Journal for Computational Methods in Engineering Science and Mechanics, 12(1), 14-25.
• [29] Jiwari, R., Pandit, S., & Mittal, R. C. (2012). A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions. Applied Mathematics and Computation, 218(13), 7279-7294.
• [30] Korkmaz, A., & Dag, I. (2011). Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation. Journal of the Franklin Institute, 348(10), 2863-2875.
• [31] Jiwari, R. (2015). Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions. Computer Physics Communications, 193, 55-65.
• [32] Bashan, A., Karakoc, S. B. G., & Geyikli, T. (2015). Approximation of the KdVB equation by the quintic B-spline differential quadrature method. Kuwait Journal of Science, 42(2).
• [33] Karakoc, S. B. G., Bashan, A., & Geyikli, T. (2014). Two different methods for numerical solution of the Modified Burgers’ equation. The Scientific World Journal, 2014.
• [34] Bashan, A., Karakoc, S. B. G., & Geyikli, T. (2015). B-spline Differential Quadrature Method for the Modified Burgers’ Equation. Cankaya University Journal of Science and Engineering, 12, 1, 1-13.
• [35] Karakoc, S. B. G., Geyikli, T., & Bashan, A. (2013). A numerical solution of the modified regularized long wave (MRLW) equation using quartic B-splines. TWMS Journal of Applied and Engineering Mathematics, 3(2), 231-244.
• [36] Korkmaz, A., Aksoy, A. M., & Dag, I. (2011). Quartic B-spline differential quadrature method. Int. J. Nonlinear Sci, 11(4), 403-411.
• [37] Hamid, N. N., Majid, A. A., & Ismail, A. I. M. (2010). Cubic trigonometric B-spline applied to linear two-point boundary value problems of order two. World Academic of Science, Engineering and Technology, 47, 478-803.
• [38] Ersoy, O., & Dag, I. (2016). The exponential cubic B-spline collocation method for the KuramotoSivashinsky equation. Filomat, 30(3), 853-861.
• [39] Dag, I., & Ersoy, O. (2016). The exponential cubic B-spline algorithm for Fisher equation. Chaos, Solitons & Fractals, 86, 101-106.
• [40] Abbas, M., Majid, A. A., Ismail, A. I. M., & Rashid, A. (2014). Numerical method using cubic trigonometric B-spline technique for nonclassical diffusion problems. Abstract and applied analysis, 2014, 849682, 1-11.
• [41] Hairer, E., Wanner, G., Solving Ordinary Differential Equations II, Springer, 2002.
• [42] Noye B.J., Tan H.H. (1988). A third-order semi-implicit finite difference method for solving the onedimensional convection-diffusion equation, International Journal for Numerical Methods in Engineering, 26,1615-1629.
• [43] Mittal, R. C., & Jain, R. K. (2012). Redefined cubic B-splines collocation method for solving convection-diffusion equations. Applied Mathematical Modelling, 36(11), 5555-5573.