Semi-implicit strategies for the Serre-Green-Naghdi equations in hyperbolic form. Is hyperbolic relaxation really a good idea?
Emanuele Macca, Walter Boscheri, Mario Ricchiuto
TL;DR
The paper tackles the high computational cost of SGN equations due to elliptic inversions by employing hyperbolic relaxation (hSGN) and introduces a semi-implicit (SI) time integration within an IMEX Runge–Kutta framework to damp stiff acoustic modes. By splitting the dynamics into explicit convective and implicit stiff components, the method achieves stability with a material-based CFL that is largely independent of the dispersive parameter $\lambda$, while preserving dispersive accuracy. Extensive numerical tests—including solitary waves, Favre waves, Gaussian perturbations, and real‐world-like bathymetry—demonstrate that SI-hSGN is competitive with, and often faster than, explicit hyperbolic or standard SGN solvers, especially for large $\lambda$. The results provide practical guidance for tuning $\lambda$ and IMEX schemes and suggest that sem-implicit hyperbolic relaxation offers a viable, efficient alternative for dispersive shallow-water simulations, with potential extensions to multidimensional problems and higher-order discretizations.
Abstract
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 λincreases, 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.