  In this study, a numerical model preserving a class of nontrivial
steady-state solutions is proposed to predict waves propagation and waves
run-up on coastal zones. The numerical model is based on the Saint-Venant
system with source terms due to variable bottom topography and bed friction
effects. The resulting nonlinear system is solved using a Godunov-type finite
volume method on unstructured triangular grids. A special piecewise linear
reconstruction of the solution is implemented with a correction technique to
ensure the accuracy of the method and the positivity of the computed water
depth. Efficient semi-implicit techniques for the friction terms and a
well-balanced formulation for the bottom topography are used to exactly
preserve stationary steady-state s solutions. Moreover, we prove that the
numerical scheme preserves a class of nontrivial steady-state solutions. To
validate the proposed numerical model against experiments, we first demonstrate
its ability to preserve nontrivial steady-state solutions and then we model
several laboratory experiments for the prediction of waves run-up on sloping
beaches. The numerical simulations are in good agreement with laboratory
experiments which confirms the robustness and accuracy of the proposed
numerical model in predicting waves propagation on coastal areas.
