A Central Finite Volume Surface Gradient Method for the Two-Layer Shallow Water Equations with Rigid Lid

Abstract

We develop here a new formulation of the two-layer shallow water equations with rigid-lid on variable-bottom topography. This formulation is presented as a conservative system where the function describing the geometry of the waterbed or bottom topography is embedded within the divergence of the flux term. The resulting system is then solved using an adaptation of the unstaggered central scheme (UCS) we previously developed. To satisfy the lake-at-rest physical constraint, we blend the surface gradient method (SGM) with our UCS scheme. The resulting UCS-SGM scheme is then validated and successfully implemented, representing an extension of the method previously presented for shallow water equations (one-layer flow) to address the complexities of two-layer rigid-lid flows while ensuring well-balanced characteristics. The performed numerical tests confirm the potential of the proposed method to handle two-layer flow problems on either smooth or non-smooth bottom topographies and also to handle steady- and unsteady-state flow problems.