Serre–Green–Naghdi Equations with Optimized Dispersion Properties Through a Modified Auxiliary Elliptic Equation

The ability to capture frequency dispersion over a wide range is of fundamental importance for nearshore wave models that are designed for practical use in coastal engineering. Considering the Serre–Green–Naghdi equations, which are only weakly dispersive, we present and outline a strategy that can adapt the level of dispersion through a simple and easily implementable modification of the equations. The momentum equations considered as starting point contain an elliptic equation, which determines the non-hydrostatic pressure. The proposed modified system is designed in a way to facilitate the optimization of its dispersion properties through a finite difference discretization of the linearized equations. Due to the fact that the improvement of the dispersion properties is performed through modification of the elliptic part of the governing equations, the introduced term for improving the dispersion properties is of order \(\sigma ^4\) ${\sigma }^4$ according to Airy wave theory, where \(\sigma \) $\sigma$ is the shallowness parameter. This differs from classical methods for dispersion improvement and is of advantage for both numerical and theoretical reasons. Although the presented justification is based on a linear dispersion analysis assuming zero bathymetry variation, numerical comparison with experimental data shows that the modified system provides encouraging results even over a spatially varying bottom.