Adaptive Non-Hydrostatic Model for Moving Bottom-Generated Waves

The Shallow Water Equations (SWE) is widely used to simulate the ocean waves, particularly tsunami waves, given its simplicity and robustness for wide range of wave dynamics. This model is limited to the hydrostatic pressure assumption. However, in some scenarios such as landslide tsunamis and slow earthquake-generated waves, the non-hydrostatic pressure plays a crucial role. In that case, two approaches are mainly used: Boussinesq-type equations and non-hydrostatic SWE extensions. The SWE extensions can be achieved by splitting the pressure terms into hydrostatic and non-hydrostatic pressure while deriving a depth-averaged form.

We extend the work by Jeschke et al. (2017), where the quadratic pressure relation was used instead of the linear one showing equivalence to Boussinesq-type equations. This model was improved for moving-bottom generated waves and manipulated such that it can be solved by a projection method without the simplifications in the mentioned publication [Firdaus and Behrens (2024)]. Furthermore, this method also allows us to make the non-hydrostatic correction adaptively on a particular area, where the dispersion might play a significant role. In this work, we investigate such an adaptive model in simulating moving bottom-generated waves. We compare both global and local correction simulations with measured data along with their computational time. It can be shown that we can achieve a similar accuracy with lower computational effort.

References:

• Jeschke, A., Pedersen, G. K., Vater, S., and Behrens, J. (2017) Depth-averaged non-hydrostatic extension for shallow water equations with quadratic vertical pressure profile: equivalence to Boussinesq-type equations. Int. J. Numer. Meth. Fluids, 84: 569–583. doi: 10.1002/fld.4361. 
• Firdaus, K., Behrens, J. (2024) Non-Hydrostatic Model for Simulating Moving Bottom-Generated Waves: A Shallow Water Extension with Quadratic Vertical Pressure Profile. *arXiv*. https://arxiv.org/abs/2410.23707.