Computational models for weakly dispersive nonlinear water waves

Abstract

Numerical methods for the two- and three-dimensional Boussinesq equations governing weakly nonlinear and dispersive water waves are presented and investigated. Convenient handling of grids adapted to the geometry or bottom topography is enabled by finite element discretization in space. Staggered finite difference schemes are used for the temporal discretization, resulting in only two linear systems to be solved during each time step. Efficient iterative solution of linear systems is discussed. By introducing correction terms in the equations, a fourth-order, two-level temporal scheme can be obtained. Combined with (bi-) quadratic finite elements, the truncation errors of this scheme can be made of the same order as the neglected perturbation terms in the analytical model, provided that the element size is of the same order as the characteristic depth. We present analysis of the proposed schemes in terms of numerical dispersion relations. Verification of the schemes and their implementations is performed for standing waves in a closed basin with constant depth. More challenging applications cover plane incoming waves on a curved beach and earthquake induced waves over a shallow seamount. In the latter example we demonstrate a significantly increased computational efficiency when using higher-order schemes and bathymetry-adapted finite element grids.