Modified Shallow Water Equations for significantly varying seabeds

In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.

💡 Research Summary

The paper addresses a fundamental limitation of the classical Nonlinear Shallow Water Equations (NSWE, also known as the Saint‑Venant system) when the seabed undergoes large spatial or temporal variations. Traditional NSWE are derived under the assumption of a gently varying bottom and neglect non‑hydrostatic pressure contributions, which makes them unsuitable for scenarios such as rapidly uplifting or subsiding sea floors during earthquakes or landslides.
To overcome this, the authors start from a variational principle and introduce a shallow‑water ansatz that respects irrotational flow while imposing constraints on the free‑surface elevation and the depth‑averaged velocity. Crucially, no small‑parameter expansion (e.g., ε = wave‑amplitude/depth) is employed; instead, the constraints directly generate additional terms that represent non‑hydrostatic pressure effects. The resulting system is a non‑dispersive, non‑hydrostatic extension of the Saint‑Venant equations.
Mathematically, the new equations retain a first‑order conservative form, and their Jacobian possesses real eigenvalues, confirming that the system remains hyperbolic. This property is essential because it guarantees that well‑established hyperbolic solvers—Godunov‑type finite‑volume methods, HLLC Riemann solvers, and TVD Runge‑Kutta time integrators—can be applied without modification. The authors therefore propose a straightforward finite‑volume discretisation that mirrors existing NSWE codes, adding only the extra source terms that arise from the non‑hydrostatic contribution and the moving‑bottom geometry.
A series of numerical experiments validates the model. In one‑dimensional tests with steep bottom slopes, the classical NSWE produce unphysical spikes in water depth and velocity, whereas the modified system damps these spikes and yields results that agree with analytical or high‑resolution reference solutions. Time‑dependent bottom motion (simulating a seismic uplift) is also captured accurately: the model reproduces the expected increase in wave speed over an uplifting region and the corresponding decrease over a subsiding region. Two‑dimensional simulations over realistic bathymetry confirm that the scheme remains stable and accurately reproduces wave refraction, diffraction, and amplitude modulation caused by complex bottom features. Finally, a tsunami‑generation scenario demonstrates the practical relevance of the approach: the added non‑hydrostatic terms significantly improve predictions of arrival time and wave height on coastal gauges, highlighting potential benefits for early‑warning systems.
In conclusion, the study provides a rigorously derived, hyperbolic, non‑dispersive extension of the NSWE that incorporates significant seabed variations without sacrificing the computational efficiency of existing finite‑volume solvers. The work opens the door to more reliable coastal‑hazard modelling, especially for events where rapid bottom deformation plays a dominant role, and suggests future extensions to three‑dimensional flows and coupled sediment‑transport processes.