Higher Order Computations of the Convection-Diffusion Equation With Curved Boundaries

In computational heat transfer and fluid mechanics, high order accuracy methods are desirable in order to reduce computational effort or to obtain more accurate solutions for a given mesh coarseness. On structured grids, the finite difference method is especially easy for deriving and implementing higher order schemes. In spite of this advantage, for complex geometries high order schemes have not been attractive due to the restriction of the structured grid in dealing with curved boundaries. Therefore, for complex geometries most computational methods are based on finite element or finite volume methods with unstructured or boundary-fitted mesh at the expense of difficult and complicated implementation. For this reason, few computations for complex geometries have attempted more than near-second-order accuracy. In our paper, we demonstrate a high order scheme to deal with curved boundaries of complex geometries in Cartesian coordinate system using the finite difference method, taking advantages of the ease and simplicity of structured grid. The method is based on an extension of the full second order methods presented previously by Jung et al. [2000] and Lee and Chen [2002]. The temperature distributions and maximum errors in a cylindrical solid and an annulus where the velocity distribution is given were calculated with a third order accurate scheme, and compared with exact solutions. Theoretical derivations and numerical experiments show that true third order accuracy have been attained in advection-diffusion problems with curved boundaries. The results reinforce the assertion that the same concepts can be extended to any order accuracy so far as such accuracy is deemed desirable for the problem of interest.