Numerical Simulation of Non-linear Water Waves over Variable Bathymetry

We present a numerical method for the computation of surface gravity waves in the presence of variable bathymetry. The evolution is described by an alternative form of the usual Hamiltonian system where the substrate boundary value problem for the potential gives its place to an infinite system of second order partial differential equations on horizontal fields with appropriate boundary conditions and a linear algebraic constraint. We employ the finite difference method to solve this time-independent problem and the classic Runge-Kutta method for the time-integration of the evolution equations. Special care is taken for the consistent wave generation and absorption at the lateral boundaries of a reasonably sized computational domain. Simulations of non-linear waves in the presence of undulated bottoms (Bragg reflection) are presented and validated against experimental measurements.