Collaborative Research: Numerical Methods for Partial Differential Equations Arising in Shallow Water Modeling

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This research project will contribute significantly toward development of computational methods for shallow water and related models. Special attention will be paid to applications arising in oceanography, in atmospheric sciences, and in hydraulic, coastal, civil, and enhanced oil recovery engineering, in which rapid changes in the bottom topography, Coriolis forces, friction, nonconservative terms, and uncertain phenomena have to be taken into account. The problems under study include rainwater drainage and flooding in urban areas, shallow water models of turbidity currents, multilayer flows, and shallow water models with uncertain data. The new tools under development promise to have great potential in designing coastal protection systems and investigating the effects of sediment transport on shelf drilling platforms as well as contributing toward the development of flood mitigation systems and planning of new urban areas. The project is aimed at developing accurate, efficient, and robust numerical methods for shallow water equations and related models, with particular reference to problems that admit nonsmooth (discontinuous) solutions and involve complicated nonlinear waves, moving interfaces, and uncertain data. Shallow water models are systems of time-dependent partial differential equations (PDEs) that are derived using physical properties such as conservation of mass and momentum, and hydrostatic or barotropic approximations. Naturally these models, especially in the cases of high space dimensions, require development and implementation of special numerical techniques such as numerical balancing between the terms that are balanced in the original system of PDEs (development of well-balanced schemes), ensuring positivity of all fluid layers (this is absolutely necessary for both accurate description of dry and near dry states and enforcement of nonlinear stability), operator splitting methods, interface tracking approaches, and others that will be in the focus of the research project. The development of new techniques will be based on high-order shock-capturing finite-volume schemes, accurate and efficient ODE solvers, and stochastic Galerkin methods, utilizing major advantages of each one of these methods in the context of the problems under study.