Multi-algorithmic numerical strategies for the solution of shallow water models

This dissertation addresses a class of finite element methods for numerically solving the shallow water equations. The coupling of continuous and discontinuous Galerkin finite element methods is presented as a multi-algorithmic strategy for capturing the complex solution behavior of the shallow water hydrodynamic model.; We present the mathematical framework of the partial differential equations modeling such regimes, including a model advection-diffusion transport problem as well as the two-dimensional shallow water equations themselves. These systems typically display a complicated solution structure within widely varying flow regimes, including advection-dominated regions with sharp fronts. Various existing numerical algorithms perform quite differently within such domains in terms of stability, accuracy and localized mass conservation properties.; To address these difficulties, we develop new multi-algorithmic strategies for capturing the complex behavior of solutions in an accurate and efficient manner. This technique can, for the first time, successfully handle the well-known difficulties of the numerical system in one complete model. Our work addresses the multi-discliplinary aspect of the Computational and Applied Mathematics program in the following manner:; Area A. This research involves the numerical solution of advection-diffusion systems using multi-algorithmic strategies. Our approach involves formulating the partial differential equations on subdomains, and determining appropriate transmission conditions between subdomains. Then, a numerical method which “honors” the PDEs and the transmission conditions is formulated. Tools from functional analysis and approximation theory are used to determine the stability, accuracy, and convergence properties of the method.; Area B. Novel coupled discontinuous and continuous Galerkin finite elements methods are formulated for the numerical solution of transport and shallow water systems. Stability and a priori error estimates are derived and subsequently verified computationally.; Area C. The strategies are applied to transport and shallow water equations. The resulting numerical models are compared to each other and to existing models of flow. We conduct parametric studies to identify solution behavior.