Three-dimensional shallow water system: A relaxation approach

We study a three-dimensional shallow water system, which is obtained from the three-dimensional Navier-Stokes equations after Reynolds averaging and under the simplifying hydrostatic pressure assumption. Since the three-dimensional shallow water system is generically not hyperbolic, it cannot be numerically solved using hyperbolic shock capturing schemes. At the same time, existing simple finite-difference and finite-volume methods may fail in simulations of unsteady flows with sharp gradients, such as dam-break and flood flows. To overcome this limitation, we propose a novel numerical method, which is based on a relaxation approach utilized to “hyperbolize” the three-dimensional shallow water system. The extended relaxation system is hyperbolic and we develop a second-order semi-discrete central-upwind scheme for it. The proposed numerical method can preserve “lake at rest” steady states and positivity of water depth over irregular bottom topography. The accuracy, stability and robustness of the developed numerical method is verified on five numerical experiments.