Discretization of the Convection Term

Abstract

So far, the discretization of the general steady diffusion equation has been formulated on orthogonal, non-orthogonal, structured, and unstructured grids. Another important term, the convection term represented by the divergence operator, is the focus of this chapter. Initially this term is discretized using a symmetrical linear profile similar to the one adopted for the discretization of the diffusion term. The shortcomings of this profile are delineated and a remedy is suggested through the use of an upwind profile. Even though it leads to physically plausible predictions, the upwind profile is shown to be highly diffusive generating results that are first order accurate. To increase accuracy, higher order profiles that are upwind biased are introduced. While reducing the discretization error, higher order profiles are shown to give rise to another type of error known as the dispersion error. Methods dealing with this error will be dealt with in the next chapter. Moreover, the flow field, which represents the driving catalyst of the convection term, is assumed to be known. The computation of the flow field will be the subject of later chapters.