Two -dimensional analysis of spatial discretizations of the shallow water equations
The goal of the work presented here is to investigate the behavior of a variety of numerical methods for solving the shallow water equations, to establish the accuracy and overall quality of the solutions obtained by these numerical methods, and to make recommendations about further development and use. This study is motivated by limitations of one of the existing methods and by a lack of rigorous analysis and testing to characterize the alternate methods that are being offered as replacements to the current approach.
For the past twenty years, efforts to solve the shallow water equations in the finite element community have been dominated by the “wave equation” approach. This approach has been very successful in overcoming the most significant drawback of early finite element solutions of the shallow water equations, the generation of artificial oscillations that obscured the true solution. However, recent advances in computer hardware have made possible the use of much finer computational grids and the use of more complicated domains that have revealed some inherent deficiencies in wave equation solutions. Most significant is that the solutions generated by this class of methods do not always conserve mass locally. Numerous papers exist in the literature that document the substantial effort made to improve the wave equation models, but problems with the method persist despite the ever increasing complexities of successive improvements. It is clear that a new approach is needed to supplement or replace the wave equation based algorithms.
This study will investigate some promising spatial discretization schemes for the two dimensional shallow water equations, and establish their fundamental characteristics. Two dimensional dispersion analysis, truncation error analysis, and numerical experiments will be presented. The dispersion surfaces and their folding characteristics are carefully examined and numerical experiments are used to demonstrate that two dimensional dispersion analysis can precisely predict noisiness of the solutions for various algorithms and grid configurations while one dimensional analysis may not. The goal of this study is to identify possible numerical schemes to succeed the wave equation approach for solving the shallow water equations.