One Dimensional Convection: Interpolation Models for CFD Gerald Recktenwald * January 24, 2012 Abstract The finite-volume method for numerical solutions of the one-dimensional convection-diffusion equation is described and demonstrated with Mat- lab. This well-known problem has an exact solution, which is used to compare the behavior and accuracy of the central difference and upwind difference schemes. Matlab codes for both schemes are developed and numerical solutions are presented on sequences of finer meshes. As the mesh size is reduced the dependency of the truncation error on mesh size for both schemes is verified. The existence and cause of oscillatory solu- tions for the central difference scheme are explained. The superior perfor- mance of the central difference method under suitable mesh refinement is demonstrated. 1 Introduction Finite volume methods are widely used in computational fluid dynamics (CFD) codes. The elementary finite volume method uses a cell-centered mesh and finite-difference approximations of first order derivatives. This paper shows how the finite volume method is applied to a model of convective transport: the one-dimensional convection-diffusion equation. There are two primary goals of this paper. The first is to expose the finite volume method. Readers interested in additional details, including application to the Navier-Stokes equations, should consult the classic text by Patankar [4]. Ferziger and Peri´ c [2] give a more up-to-date discussion of finite volume meth- ods, but without the low level details presented in this paper. Versteeg and Malalasekera [5] provide a detailed discussion of the topics described in this paper, although their presentation does not deal with the effect of non-uniform meshes. Abbott and Basco [1] provide a basic analysis of convection modeling for the transient version of the one-dimensional convection-diffusion equation. Wesseling [7] gives a mathematically rigorous treatment of the finite volume method, including a discussion of different approximations to the convective terms. The second goal of this paper is to introduce and compare the central dif- ference scheme and the upwind scheme for modeling the convective term in * Mechanical and Materials Engineering Department, Portland State University, Portland, OR, 97201, gerry@me.pdx.edu