A consistent quasi–second-order staggered scheme for the two-dimensional shallow water equations

Abstract

Abstract A quasi–second-order scheme is developed to obtain approximate solutions of the two-dimensional shallow water equations (SWEs) with bathymetry. The scheme is based on a staggered finite volume space discretization: the scalar unknowns are located in the discretization cells while the vector unknowns are located on the edges of the mesh. A monotonic upwind-central scheme for conservation laws (MUSCL)-like interpolation for the discrete convection operators in the water height and momentum balance equations is performed in order to improve the accuracy of the scheme. The time discretization is performed either by a first-order segregated forward Euler scheme or by the second-order Heun scheme. Both schemes are shown to preserve the water height positivity under a Courants Friedrichs Lewy (CFL) condition and an important state equilibrium known as the lake at rest. Using some recent Lax–Wendroff type results for staggered grids, these schemes are shown to be LW-consistent with the weak formulation of the continuous equations, in the sense that if a sequence of approximate solutions is bounded and strongly converges to a limit, then this limit is a weak solution of the SWEs; besides, the forward Euler scheme is shown to be LW-consistent with a weak entropy inequality. Numerical results confirm the efficiency and accuracy of the schemes.