A Least-Squares Finite-Element Method for Shallow-Water Equations

A wave-structure interaction model based on the least-squares finite-element formulation of the depth-averaged, nonlinear, non-conservative 2D shallow-water equations is developed. Advantages of the model include: (1) a single approximating space can be used for all variables, and its choice of approximating space is not subject to the Ladyzhenskaya-Babuska-Brezzi (LBB) condition; (2) upwind scheme is no needed; (3) sources terms, such as the bottom slope, surface stresses and bed frictions, can be treated easily without any special treatment; and (4) the resulting system of equations is symmetric and positive-definite (SPD) which can be solved efficiently with the preconditioned conjugate gradient method. The model was verified with flow past a bump, shoaling and dam-breaking where flow exhibits sharp gradient changes. The model was then applied to flow past a vertical circular cylinder. Computed results are compared with experiment data and other numerical results. Important flow characteristics, such as reflection, diffraction, run-up around the cylinder and vortex shedding behind the cylinder are investigated.