Numerical modelling of surface wave packet evolution on varying topography

Abstract

An initial value problem of the one-dimensional nonlinear Schr\"odinger (NLS) equation with constant dispersive and nonlinear coefficients can be solved using a compact finite difference scheme (Xie, Li, & Yi, 2009). A similar scheme is implemented in the signalling problem of the one-dimensional NLS equation with constant coefficient where it describes the propagation of surface gravity wave packet over a flat bottom. Various examples are illustrated. A similar compact finite difference scheme is modified and implemented further in the signalling problem of the one-dimensional NLS equation with variable dispersive and nonlinear coefficients which models the surface wave packet propagation over the slowly varying bottom. Several topographies are considered and illustrated. It is observed that the wave envelope signal propagates opposed to topography height. The possibility of shaping topography near shores area to minimise the effect of high amplitude solitary waves are also examined.