517 IGPM517.pdf        July 2021
TITLE A unified surface-gradient and hydrostatic reconstruction scheme for the shallow water equations
AUTHORS Guoxian Chen, Sebastian Noelle
ABSTRACT We propose a new second-order accurate hydrostatic reconstruction scheme for the Saint-Venant system. Such a scheme needs to overcome several difficulties: besides the well-known issues of positivity and well-balancing there is also the difficulty of unphysical reflections from bottom reconstructions which create artificial steps. We address all of these problems at once by changing the logic of the reconstruction of the bottom, the water depth and the water surface level. Notably, our bottom reconstruction is continuous across cell interfaces and remains unchanged during the computation, except if the original topography has a jump, or if a wet-dry front passes through a cell. Only in these exceptional cases we apply the new discontinuous bottom approximation and compute the residual via the subcell hydrostatic reconstruction method. The scheme gives excellent results in one and two space dimensions. To highlight the novel reconstruction of bottom and water surface, we call the scheme bottom-surface-gradient method (BSGM).
KEYWORDS Saint-Venant system, well-balanced property, positivity preserving property, subcell hydrostatic reconstruction, bottom-surface-gradient method, maximum-minimum property