A mixed Lagrangian–Eulerian method for non-linear free surface flows using multigrid on hierarchical Cartesian grids

This article describes a two-dimensional finite difference method for the numerical simulation of fully non-linear irrotational water waves. The computational domain is discretised in an Eulerian fashion using hierarchical Cartesian meshes, whilst the free surface location is explicitly tracked using a Lagrangian approach. The accuracy of the method is strongly dependent on the quality of the approximation to the free surface velocities and a novel method to compute these is proposed. A multigrid strategy is implemented to take advantage of the hierarchical nature of the grids, incorporating an efficient technique to generate the coarser grids. The method is verified through simulations of quasi-linear low-amplitude waves and through a comparative study using an asymmetric sloshing wave. The non-linear behaviour of waves of moderate amplitude is also simulated.