Multidimensional upwind schemes for the shallow water equations

A multidimensional discretisation of the shallow water equations governing unsteady free-surface flow is proposed. The method, based on a residual distribution discretisation, relies on a characteristic eigenvector decomposition of each cell residual, and the use of appropriate distribution schemes. For uncoupled equations, multidimensional convection schemes on compact stencils are used, while for coupled equations, either system distribution schemes such as the Lax–Wendroff scheme or scalar schemes may be used. For steady subcritical flows, the equations can be partially diagonalised into a purely convective equation of hyperbolic nature, and a set of coupled equations of elliptic nature. The multidimensional discretisation, which is second-order-accurate at steady state, is shown to be superior to the standard Lax–Wendroff discretisation. For steady supercritical flows, the equations can be fully diagonalised into a set of convective equations corresponding to the steady state characteristics. Discontinuities such as hydraulic jumps, are captured in a sharp and non-oscillatory way. For unsteady flows, the characteristic equations remain coupled. An appropriate treatment of the coupling terms allows the discretisation of these equations at the scalar level. Although presently only first-order-accurate in space and time, the classical dam-break problem demonstrates the validity of the approach. © 1998 John Wiley & Sons, Ltd.