Upwind-biased FORCE schemes with applications to free-surface shallow flows

In this paper we develop numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple-space dimensions on structured meshes. The proposed method is an extension of the multidimensional FORCE flux developed by Toro et al. (2009) [14]. Here we introduce upwind bias by modifying the shape of the staggered mesh of the original FORCE method. The upwind bias is evaluated using an estimate of the largest eigenvalue, which in any case is needed for selecting a time step. The resulting basic flux is first-order accurate and monotone. For the linear advection equation, the proposed UFORCE method reproduces exactly the upwind Godunov method. Extension to non-linear systems has been done empirically via the two-dimensional inviscid shallow water equations. Second order of accuracy in space and time on structured meshes is obtained in the framework of finite volume methods. The proposed method improves the accuracy of the solution for small Courant numbers and intermediate waves associated with linearly degenerate fields (contact discontinuities, shear waves and material interfaces). It achieves comparable accuracy to that of upwind methods with approximate Riemann solvers, though retaining the simplicity and efficiency of centred methods. The performance of the schemes is assessed on a suite of test problems for the two-dimensional shallow water equations.