| Research was conducted in order to develop more efficient solution techniques for the ShallowWater Equations (SWE) for naturally occurring free-surface flows in natural and engineered channels. Methods relating to numerical solution of the two-dimensional equations utilizing graphical processing units (GPU) as the main computational device and combined one-and two-dimensional schemes are presented and tested. Different numerical methods were investigated for inclusion to the model. General requirements for the proposed schemes included the ability to be solved using a finite-difference conservative solution algorithm on a fixed rectangular grid and the ability to both withstand and provide reasonable approximation of shocks and bores within the solution domain. Two such schemes were investigated that met initial criteria: A graphical processing unit (GPU) implementation of the well established MacCormack method, and a selectively under-relaxed implicit method. Both methods included the addition of a TVD (total variation diminishing) term to help maintain stability around high gradient flow areas.;The implicit method incorporates an algorithm for selectively under-relaxing the iterative process to maintain stability in the presence of shock interfaces. The value of the Courant number and the frequency at which the TVD term was incorporated were constantly updated during the computation to achieve optimal speed of execution while maintaining stability. The method was tested against published results from experiments and from computations employing alternative algorithms and the results obtained demonstrate both the economy and accuracy of the proposed algorithm.;The MacCormack-based scheme was chosen for both optimizing procedure attempts. Methodology was tested that allowed for one and two-dimensional TVD-MacCormack equation coupling, reducing grid-size dependency for the solution domain, while permitting simultaneous calculation of both one and two dimensional domains, and the explicit, finite-difference formulation of the solution methodology was well suited for inclusion into simultaneous GPU calculation. Cell alignment and cell-neighbor management is shifted from matrix to array form, which allows for a new framework, optimally constructed for inclusion of the dimensionally coupled solution scheme. The code contains adaptive time-stepping, based on maximum local Courant number, and special wetting/drying schemes to maximize stability while maintaining accuracy. The method was tested against published results, showing it's effectiveness in minimizing computational resources while comparing well with experimentally derived results. The coupled code is tempered for insertion into a parallel computing array.;Ultimately, while dimensional coupling provided a slight optimization in terms of computational efficiency, the dimensional interface methodology and limited domain types the solution technique was constructed for restrict it to a specific-use tool. The extension of the MacCormack method to GPU processing ultimately proved more useful, showing speed increases of 4-40 times depending on the domains geomorphological characteristics. |
