An hp/spectral element model for efficient long-time integration of Boussinesq-type equations

We present an hp/spectral element method for modelling one-dimensional nonlinear dispersive water waves, described by a set of enhanced Boussinesq-type equations. The model uses nodal basis functions of arbitrary order in space and the third-order Adams-Bashforth scheme to advance in time. Numerical computations are used to show that the hp/spectral element model exhibits exponential convergence. The model is compared to two numerical methods frequently used for solving Boussinesq-type equations; a finite element model using linear basis functions and a finite difference model using a five-point stencil for estimating the first-order derivatives. Using numerical examples, we show that the hp/spectral element model gives great savings in computational time, compared to the other models, if: (i) highly accurate results are requested, or, more importantly, (ii) results of "engineering accuracy" are called for in combination with long-time integration.