Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications ______________________________________________________________________________________________________ 16 Analysis of Convection-Diffusion Problems at Various Peclet Numbers Using Finite Volume and Finite Difference Schemes Anand Shukla Department of Mathematics Motilal Nehru National Institute of Technology, Allahabad, 211 004, U.P., India E-mail: anandshukla86@gmail.com Surabhi Tiwari (Corresponding author) Department of Mathematics Motilal Nehru National Institute of Technology, Allahabad, 211 004, U.P., India E-mail: surabhi@mnnit.ac.in; au.surabhi@gmail.com P. Singh Department of Mathematics Motilal Nehru National Institute of Technology, Allahabad, 211 004, U.P., India E-mail: psingh11@rediffmail.com Abstract Convection-diffusion problems arise frequently in many areas of applied sciences and engineering. In this paper, we solve a convection-diffusion problem by central differencing scheme, upwinding differencing scheme (which are special cases of finite volume scheme) and finite difference scheme at various Peclet numbers. It is observed that when central differencing scheme is applied, the solution changes rapidly at high Peclet number because when velocity is large, the flow term becomes large, and the convection term dominates. Similarly, when velocity is low, the diffusion term dominates and the solution diverges, i.e., mathematically the system does not satisfy the criteria of consistency. On applying upwinding differencing scheme, we conclude that the criteria of consistency is satisfied because in this scheme the flow direction is also considered. To support our study, a test example is taken and comparison of the numerical solutions with the analytical solutions is done. Keywords: Finite volume method, Partial differential equation 1. Introduction Mathematical models of physical [2, 7], chemical, biological and environmental phenomena are governed by various forms of differential equations. The partial differential equations describing the transport phenomena in fluid dynamics are difficult to solve, particularly, due to the convection terms. Such equations represent the hyperbolic conservation law for which their solutions always contain discontinuity and high gradient. Thus accurate numerical solutions are very difficult to obtain. Special treatment must be applied to suppress spurious oscillations of the computed solutions for both the convection and convection-dominated problems. In the present scenario, better ways to approximate the convection term are still needed, and thus development of accurate numerical modeling for the convection-diffusion equations remains a challenging task in computational fluid dynamics. In recent years (see [7]), with the rapid development of energy resources and environmental science, it is very important to study the numerical computation of underground fluid flow and the history of its changes under heat. In actual numerical simulation, the nonlinear three-dimensional convection-dominated diffusion problems need to be considered. When velocity is higher, that is, flow term is larger, a simple convection-diffusion problem is converted to convection-dominated diffusion problem because Peclet number is greater than two. A Peclet number is a dimensionless number relevant in the study of transport phenomena in fluid flows. It is defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient. In the context of the transport of heat, Peclet number is equivalent to the product of Reynolds number and Prandtl number. In