Wavelet-based adaptive grids as applied to hydrodynamics

An adaptive, wavelet-based, multiscale finite-volume scheme is developed and employed to investigate applications in the simulation of water waves. Firstly, two one-dimensional, strictly hyperbolic cases are investigated: shallow water and Euler equations. These are followed by two investigations using a finite-volume formulation of Madsen and Sørensen's Boussinesq equations. Converged results were obtained in all cases, which demonstrate that the adaptive grid scheme is significantly faster than that on a uniform grid. In some cases, one-seventh of the number of cells is required to obtain the same accuracy as that of the uniform grid. Issues of stability are discussed in the context of the particular problems modelled here with the Boussinesq equations, related to discretization of the high-order spatial derivatives on a non-uniform grid.