Forward-Backward Alternating Parallel Shooting Method for Multi-layer Boundary Value Problems

Yıl 2020, Cilt: 4 Sayı: 4, 432 - 442, 30.12.2020

Mohamed Ali Hajji

https://doi.org/10.31197/atnaa.753561

Cited By: 1

Öz

Multi-layer boundary value problems have received a great deal of attention in the past few years. This is due to the fact that they model many engineering applications. Examples of applications include fluid flow though multi-layer porous media such as ground water and oil reservoirs.
In this work, we present a new method for solving multi-layer boundary value problems.
The method is based on an efficient adaption of the classical shooting method, where a boundary value problem is solved by means of solving a sequence of initial value problems. We propose, an alternating forward-backward shooting strategy that reduces computational cost. Illustration of the method is presented on application to fluid flow through multi-layer porous media. The examples presented suggested that the method is reliable and accurate.

Anahtar Kelimeler

Boundary value problem, Shooting method, Fluid flow, Porous media

Kaynakça

• [1] R.L. Burden, and J.D. Faires, Numerical Analysis, 7th ed., Brooks/Cole (2001).
• [2] R. Holsapple, R. Venkataraman, and D. Doman, A modified simple shooting method for solving two-point boundary value problems, Proc. of IEEE Aerospace Conference 6 (2003) 2783-2790.
• [3] J.R. Munchen, R.B. Munchen, and J.P. Innsbrusk, A multiple shooting method for the numerical treatment of stellar structure and evolution, Astron. Nachr . 315 (1994) 205-234.
• [4] D.D. Morrison, J.D. Relay, and J.F. Zancanaro, Multiple shooting method for two-point boundary value problems, Communication of the ACM 5(12) (1962) 613-514.
• [5] G. Gheri, and P. Marzulli, Parallel shooting with error estimate for increasing the accuracy, J. Computational and Applied Mathematics 115 (2000) 213-227 .
• [6] H.H.H. Homeier, A modified Newton method for root finding with cubic convergence, J. Comput. Appl. Math. 157 (2003) 227-230.
• [7] H.H.H. Homeier, A modified Newton method with cubic convergence: the multivariate case, J. Comput. Appl. Math. 169 (2004) 161-169.
• [8] M. Frontini, E. Sormani, Modified Newton's method with third-order convergence and multiple roots, J. Comput. Appl. Math. 156 (2003) 345 - 354.
• [9] M. Frontini, E. Sormani, Some variant of Newton's method with third-order convergence, Appl. Math. Comput. 140 (2003) 419-426.
• [10] F. Allan and M.A. Hajji, (2008), Mathematical Manipulation of the Interface Region of Multilayer Flow, International Journal of Porous Media 12 (2009) 461-475.
• [11] F. Allan and M. Hamdan, Fluid Mechanics of The Interface Region Between Two porous Layers, Applied Mathematics Computation 128(1) (2002) 37-43.
• [12] R.A. Ford and M. Hamdan, Coupled Parallel Flow through Composite Porous Layers, Applied Mathematics Computation 97 (1998) 261-271.
• [13] F. Allan, M.A. Hajji, and M. Anwar, The characteristics of fluid flow through multilayer porous media, J. Applied Mechanics 76 (2009).
• [14] K. Vafai, S.J. Kim, Fluid mechanics of the interface region between a porous medium and a fluid layer: an exact solution, Int. J. Heat Fluid Flow 11 (1990) 254-256.
• [15] Z. Chen, G. Huan, Y. Ma, Computational methods for multiphase flows in porous media, Soc. Indust. Appl. Math. (2006).
• [16] M.A. Hajji, Multi-point special boundary value problems and applications to fluid flow through porous media, Proceedings of the 2009 IAENG International Conference on Scientific Computing, Hong Kong (2009) 18-20.
• [17] M.A. Hajji, A numerical scheme for multi-point special boundary-value problems and application to fluid flow through porous layers, Applied Mathematics and Computation 217 (2011) 5632-5642.
• [18] A. Ramírez, A. Romero, F. Chavez, F. Carrillo, and S. Lopez, Mathematical simulation of oil reservoir properties, Chaos, Solitons & Fractals, 38 (3) (2008) 778-788.