Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique

In the context of numerical techniques for solving unsteady free surface problems, finite element and finite volume approximations are widely used. A class of upwind methods which attempts to model the equations in a genuinely multidimensional manner has been recently introduced as an alternative. Multidimensional upwind schemes (MUS) were developed initially for the approximation of steady-state solutions of the two-dimensional Euler equations on unstructured grids, although they can be applicable to any system of hyperbolic conservation laws, such as the shallow water equations. The formal analogy between the two systems of equations is useful for simple cases. However, in practical applications of interest in hydraulics, complex geometries and bottom slope variation can lead to important numerical errors produced by an inadequate source term discretization. This problem has been analyzed and, in this work, the necessity of a multidimensional upwind discretization of the source terms is justified. The basis of the numerical method is stated and the particular adaptation to unsteady shallow water flows over irregular geometry is described. As test cases, laboratory experimental data are used together with academic tests for validation.