Optimal Hermite Collocation Applied to a One-dimensional Convection-diffusion Equation Using a Hybrid Optimization Algorithm

The Hermite collocation method of discretization can be used to determine highly accurate solutions to the steady state one-dimensional 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 search for 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 is close to a locally optimal feasible mesh refinement. Following the genetic algorithm, a hill-climbing approach is used to determine a truly local optimal mesh refinement that is feasible. In most cases the mesh refinements determined with this hybrid method are equally optimal or a significant improvement over previous mesh refinements determined through direct search methods.