| Abstract: |
| We will discuss the rigorous justification of the numerical spatial discretization by means of Fourier spectral methods of equations modelling the dynamical evolution of surface gravity waves. The Fourier spectral method is especially indicated for systems such as the Boussinesq and Whitham-Boussinesq systems since the nonlinear contributions are quadratic and dispersive contributions take the form of Fourier multipliers. The spatial discretization amounts to considering the modified PDE with low-pass filters for wave numbers $\left|k \right| \leq N$ applied to the initial data and nonlinear terms. Earlier convergence results for the discretization of Boussinesq systems have the shortcoming that they lack uniformity in the non-dispersive limit. Overcoming this requires a good understanding of the underlying (non-dispersive) quasilinear system, which will be the focus of the talk.
Using energy estimates, we investigate the stability and convergence of the approximate solution as $N\to \infty$. The results depend on the regularity of the initial data and the structure of the system. We consider both sharp- and smooth low-pass filters and argue that - at least theoretically - smooth low-pass filters are operable on a larger class of systems. |
|