الفهرس Only 14 pages are availabe for public view
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Abstract
An important question concerning computational solutions is what guarantee can be given that the computational solution will be close to the exact solution of the partial differential equation(s), and under what circumstances the computational solution will coincide with the exact one. The second part of this question can be answered (superficially) by requiring that the approximate (computational) solution should converge to the exact one as the grid spacings shrink to zero. However, convergence is very difficult to establish directly so that an indirect route. The indirect route requires that the system of algebraic equations formed by the discretization process should be consistent with the governing partial differential equation(s). Consistency implies that the discretization process can be reversed, through a Taylor series expansion, to recover the governing equation(s).
The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth.
The aim of this work is to investigate the numerical solution of convection-diffusion-reaction equation. The solutions of the one and two dimensional problems are discussed in two cases: Firstly, we use
the implicit scheme, namely alternating-direction-impliciy (ADI) scheme which originally proposed by Peaceman and Rachford . The solution is based on the well known discretization scheme, which is applied at any stage of (ADI) scheme steps. We approximate the time derivative by using a forward time-stepping method. After that, we use the finite difference method to approximate the resulting equation in one dimension. Secondly, we have been used throughout this study, viz, Chebyshev polynomial method. Our problem is discussed using the above two schemes, and we discuss the best of them. Finally, we introduce numerical solutions for different examples.
A formulation of (ADI) method is given by extending Peaceman and Rachford scheme to three dimensions. The scheme becomes conditionally stable. The Von Neumann stability analysis is performed. Numerical results for solving the heat diffusion equation have been obtained for different specified boundary conditions problems to obtain a simple explicit stability.