Collaborative Research: Structure Preserving Numerical Methods for Hyperbolic Balance Laws with Applications to Shallow Water and Atmospheric Models

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This project will significantly contribute toward development of computational methods for shallow water and related models and will provide considerably more powerful tools for studying a variety of water waves and atmospheric phenomena. Special attention will be paid to applications arising in oceanography, atmospheric sciences, hydraulic, coastal, civil engineering, in which rapid changes in the bottom topography, Coriolis forces, friction, multiscale regimes, and uncertain phenomena factor heavily. The studied problems will include shallow water flows in multi-connected river channel systems, tsunami wave propagation and low Froude regime shallow water models, dynamics models of tropical cyclones and clouds with uncertain data.The newly developed tools may have a great potential in designing coastal protection systems and investigating the effects of sediment transport on shelf drilling platforms as well as contributing to a better prediction of tropical cyclones trajectories and tsunami wave propagation and on-shore arrival.The project focuses on development of new structure preserving numerical methods for hyperbolic balance laws with applications to shallow water equations and related models. 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 applications, especially in cases of high space dimensions, require development and implementation of special numerical methods that are not only consistent with the governing system of PDEs, but also preserve certain structural and asymptotic properties of the underlying problem at the discrete level. The development of new numerical techniques will be based on high-order shock-capturing finite-volume schemes, asymptotic preserving, adaptive moving mesh and stochastic Galerkin methods utilizing major advantages of each one of these methods in the context of studied problems. Besides providing examples that corroborate the numerical approach, the foregoing applications are of a substantial independent value for a broad class of problems arising in today's science.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.