Optimal collocation applied to a one-dimensional convection-diffusion equation using a hybrid optimization algorithm

Abstract

The method of collocation can be used to determine highly accurate solutions to the one-dimensional steady-state convection-diffusion equation (which can be used to model the transport of contaminants dissolved in groundwater). This accuracy is dependent upon sufficient refinement of the finite element mesh as well as applying upstream weighting to the convective term through the determination of collocation locations which meet specified constraints. Due to an increase in computational intensity of the application of the method of collocation associated with increases in the mesh refinement, minimal mesh refinement is sought. A hybrid method that utilizes a genetic algorithm and a hill-climbing approach is used to determine the optimal mesh refinement for a number of models differentiated by their velocity fields. The genetic algorithm is used to determine a mesh refinement that results in feasible collocation locations that is close to optimal. Following the genetic algorithm, a hill-climbing approach is used to determine a local optimal mesh refinement that is feasible. In most cases the optimal mesh refinements determined with this hybrid method are equal to or better than previous mesh refinements determined through direct search methods.