The solution of fluid dynamics problems has recently witnessed a remarkable increase in the formulation of Boltzmann-based approaches, mainly triggered by the numerous advantages stemming from the concrete possibility of employing a low number of allowed velocities, which is the basis of the Lattice-Boltzmann (LB) methods. The field of Shallow Water (SW) modeling also took advantage of these techniques, but there are still open problems related to the practical impossibility to simulate transcritical flows while retaining the intrinsic simplicity of the LB approaches. This problem is even more crucial if one considers that transcritical flows always develop whenever a transition over dry bed occurs. Since such vertically integrated models are currently the mostly employed ones for simulating technically interesting flows, and since these flows often require the flooding of an initially dry bed, the application of LB methods seems to be facing its limits. The Gas Kinetic Method (GKM) overcomes this issue, integrating the Boltzmann equations in continuous velocity space. In this work, we formulate and extensively test a GKM-based model for solving the SW equations, which is able to manage the propagation over uneven dry bed. The benchmarking, carried out against analytical, experimental and previously proposed numerical reference solutions, shows promising results of the proposed approach.
Prestininzi P, La Rocca M, Montessori A, & Sciortino G (2014). A gas-kinetic model for 2D transcritical shallow water flows propagating over dry bed. COMPUTERS & MATHEMATICS WITH APPLICATIONS, 68(4), 439-453 [10.1016/j.camwa.2014.06.022].