Semi-implicit strategies for the Serre-Green-Naghdi equations in hyperbolic form. Is hyperbolic relaxation really a good idea?

The Serre-Green-Naghdi (SGN) equations provide a valuable framework for modelling fully nonlinear and weakly dispersive shallow-water flows. However, their elliptic formulation can considerably increase the computational cost compared to the Saint-Venant equations. To overcome this difficulty, hyperbolic models (hSGN) have been proposed that replace the elliptic operators with first-order hyperbolic formulations augmented by relaxation terms, which recover the original elliptic formulation in the stiff limit. Yet, as the relaxation parameter \lambda increases, explicit schemes face restrictive stability constraints that may offset these advantages. To mitigate this limitation, we introduce a semi-implicit (SI) integration strategy for the hSGN system, where the stiff acoustic terms are treated implicitly within an IMEX Runge-Kutta framework, while the advective components remain explicit. The proposed approach mitigates the CFL stability restriction and maintains dispersive accuracy at a moderate computational cost. Numerical results confirm that the combination of hyperbolization and semi-implicit time integration provides an efficient and accurate alternative to both classical SGN and fully explicit hSGN solvers.