Augmented versions of the HLL and HLLC Riemann solvers including source terms in one and two dimensions for shallow flow applications

Shallow water flows are found in a variety of engineering problems always dominated by the presence of bed friction and irregular bathymetry. These source terms determine completely the possible evolution of the flooded area in time. It is well known that appropriate numerical schemes for this type of flows must be well-balanced. Well-balanced numerical schemes are based on the preservation of cases of quiescent equilibrium over variable bed elevation. Commonly they are formulated as an adaptation of numerical solvers defined for cases without source terms. This procedure is insufficient when applied to real situations. Then, it is possible to argue that appropriate numerical schemes cannot arise directly from those derived from the simplest homogeneous case without source terms. New solutions are presented in this work by defining weak solutions that include the presence of source terms. To do that, the solvers presented in this work extend the number of waves in the well known HLL and HLLC solvers involving a stationary jump in the solution. This is done without modifying the original solution vector of conserved quantities. The resulting approximate Riemann solvers include variable bed level surface and friction. Solvers are systematically assessed via a series of test problems with exact solutions for one and two dimensions, including steady and unsteady flow configurations, variation of the flooded area in time and comparisons with experimental data. The obtained results point out that the new method is able to predict faithfully the overall behavior of the solution and of any type of waves.