A New Finite Element Scheme for the Boussinesq Equations

Abstract

In this paper we consider the Boussinesq equations to simulate the motion of water waves with a moderate curvature of the free surface. The mathematical model describing the wave dynamics is introduced together with a short description of its derivation, posing emphasis on the related assumptions. The discrete representation of the Boussinesq equations is faced with numerical difficulties of two kinds: the nonsymmetric character of the (nonlinear) advection–propagation operator and the presence of third-order differential terms accounting for dispersion phenomena. In this paper it is shown how it is possible to use a finite element Taylor–Galerkin method to discretize the equations, ensuring high order accuracy both in time and space and obtaining a numerical solution free of spurious oscillations.