Studies on Nodal Integral Methods for the Convection-Diffusion Equation

The computational efficiencies of two nodal integral methods for the numerical solution of linear convection-diffusion equations are studied. Although the first, which leads to a second-order spatial truncation error, has been reported earlier, it is reviewed in order to lead logically to the development here of the second, which has a third-order error. This third-order nodal integral method is developed by introducing an upwind approximation for the linear terms in the "pseudo-sources" that appear in the transverse-averaged equations introduced in the formulation of nodal integral methods. This upwind approximation obviates the need to develop and solve additional equations for the transverse-averaged first moments of the unknown, as would have to be done in a more straightforwardly developed higher-order nodal integral method. The computational efficiencies of the second-order nodal method and the third-order nodal method - of which there are two versions: one, a full third-order method and the other, which uses simpler second-order equations near the boundaries - are compared with those of both a very traditional method and a recently developed state-of-the-art method. Based on the comparisons reported here for a challenging recirculating flow benchmark problem it appears that, among the five methods studied, the second-order nodal integral method has the highest computational efficiency (the lowest CPU computing times for the same accuracy requirements) in the practical 1% error regime.