Space-time Least-squares Finite-element Method for Shallow-water Equations

In this paper, a space-time least-squares finite-element method for the 2D nonlinear shallow-water equations (SWE) is developed. The method can easily handle complex geometry, bed slope (source term), and radiation boundary condition without any special treatment. Other advantages of the method include: high order approximations in space and time can easily be employed, no upwind scheme is needed, as well as the resulting system equations is symmetric and positivedefinite, therefore, can be solved efficiently with the pre-conditioned conjugate gradient method. The model is applied to several benchmark tests, including standing wave in a flat closed basin, propagation of sinusoidal wave in a flat channel with open boundary, flow past an elliptic hump, and wavecylinder interactions. Simulation of standing wave in a closed flat basin, with water depth ranging from shallow water to deep water, shows that prediction of SWE model is accurate for shallow waters but inaccurate for deep waters due to the hydrostatic pressure and non-dispersive assumption. Computation of propagation of sinusoidal wave in a flat channel shows open boundary condition is treated satisfactorily. Simulation of flow past an elliptical hump shows good conservation property, and complicate and detailed fine wave structures which were not observed using the low order approximations in previous study. Computed result of wave-cylinder interactions compare well with other numerical results.