Restoration of the contact surface in FORCE-type centred schemes II: Non-conservative one- and two-layer two-dimensional shallow water equations

Recently, a non-conservative well-balanced FORCE-type scheme has been proposed for solving multidimensional non-conservative equations such as the shallow water equations [4], the two-fluid flow model of Pitman and Le and the two- and three-dimensional Baer–Nunziato equations for compressible multiphase flows [17]. In the present paper the original scheme is first rewritten in a suitable form that allows easy manipulation of numerical fluxes. We then propose a modified scheme that provides a better resolution of contact waves for both one-layer and two-layer shallow water models. The improvement is particularly evident when an additional equation is solved for a passive solute. In this case, the original scheme does not satisfy the C-property for a uniformly distribute tracer, whereas the modified scheme provides a solution that is exact up to machine precision. Moreover, the modified scheme better resolves contact discontinuities, with an accuracy close to the one provided by a fully-upwind non-conservative ROE-type scheme accounting for the complete wave structure. When higher orders are achieved, the difference in accuracy between the various schemes is less pronounced. Moreover, the capability of the two-dimensional scheme to capture steady states is analyzed for both straight and meandering non-flat channels with non-zero friction, and for both one-layer and two-layer equations. Whereas the original first order FORCE scheme is able to correctly reproduce the longitudinal profile in the case of a straight channel even on a coarse mesh, when the channel is strongly meandering a modification of the scheme is necessary, otherwise a very refined mesh and a very high order of accuracy are needed, with a notable increase of computational time.