Discrete Asymptotic Equations for Long Wave Propagation

Abstract

In this paper, we discuss a new systematic method to obtain discrete asymptotic numerical models for incompressible free-surface flows. The method consists of first discretizing the Euler equations in the horizontal direction, keeping both the vertical and time derivatives continuous, and then performing an asymptotic analysis on the resulting system. The asymptotics involve the ratios wave amplitude over depth, denoted by $\varepsilon$, and depth over wavelength, denoted by $\sigma$. For simplicity, in this paper we only consider the weakly nonlinear scaling in which both $\sigma^4$ and $\varepsilon\sigma^2$ are very small and of the same order. We investigate the properties of the fully discrete Boussinesq model obtained by neglecting terms proportional to these quantities. Our study reveals that if the interaction between terms arising from the discretization and from the PDE is properly accounted for, the resulting discrete system has improved linear dispersion and shoaling approximations w.r.t. the d...