A ghost-cell immersed boundary method for the shallow water equations on Cartesian grids: mass conservation and wetting and drying over variable topography

Abstract

A ghost-cell immersed boundary method is proposed for the two-dimensional shallow water equations with a variable bed (topography), solved with an explicit Godunov-type finite volume scheme (unsplit second-order MUSCL, HLLC flux, hydrostatic reconstruction) on Cartesian grids. Ghost-cell methods leak mass through the wall. To find a robust and conservative treatment, eight constructions of the ghost values and a cut-cell form of the continuity equation were applied to seven test problems, which include wetting and drying on sloping beds and a laboratory dam break. In the resulting scheme, the ghost values are interpolated bilinearly at the image point; solid and dry nodes are excluded, and the weights of the remaining nodes are renormalized. Mass is conserved by solving the continuity equation in cut-cell form, while the momentum equations are left unchanged. The proposed scheme is about as accurate as the ghost-cell method without this modification and preserves a lake at rest whose shoreline crosses an immersed wall. Because only the continuity equation is modified, it avoids the treatment of the momentum equations on cut cells (wall pressure, cell merging) that a full cut-cell method requires.