A simple LBE wave runup model

The suitability of a Lattice Bhatnagar-Gross-Krook (LBGK) modelling approach is tested to examine the behaviour of free surface waves in shallow water. The free surface model is fully non-linear. Mathematically, this is taken into account through the collision integral. The present LBGK model discretises the non-linear shallow water equation in rotational flow and approximates the collision between particles using a single time relaxation. It is further assumed that the waves do not overturn. The test case includes a long-wave run-up study on a sloping beach. Two methods have been evaluated to treat the wet-dry interface: a thin film in dry areas of the beach slope a linear extrapolation scheme. The standard LBGK solutions are second-order accurate in space and first-order accurate in time. The uniform grid solutions compared well with other findings reported in the literature when using a shoreline algorithm.