A mass, energy, vorticity, and potential enstrophy conserving lateral boundary scheme for the shallow water equations using piecewise linear boundary approximations

A numerical scheme for treating fluid–land boundaries in inviscid shallow water flows is derived that approximates boundary profiles with piecewise linear segments (shaved cells) while conserving the domain-summed mass, energy, vorticity, and potential enstrophy. The new scheme is a generalization of a previous scheme that also conserves these quantities but uses stairsteps to approximate boundary profiles. Numerical simulations are carried out demonstrating the conservation properties and accuracy of the piecewise linear boundary scheme (the PLS) for inviscid flows and comparing its performance with that of the stairstep scheme (the STS). It is found that while both schemes conserve all four domain-summed quantities, the PLS generates depth and velocity fields that are one-half to one order more accurate than those generated by the STS, and it generates vorticity and potential vorticity fields that are at least as accurate as those generated by the STS and often more accurate. The higher accuracy of the PLS is due to its ability to generate smoother flow fields near boundaries of arbitrary shape.