Exploring a conservative staggered scheme for Boussinesq-type equations: Insights into numerical diffusion, dispersion, and wave-breaking

Accurate and efficient modeling of coastal wave transformation, particularly under wave-breaking conditions, remains a major challenge for Boussinesq-type models. To address this, we introduce and validate a conservative staggered-grid scheme to discretize a set of weakly nonlinear Boussinesq-type equations. The presented approach revisits the staggered finite-difference strategy by ensuring a momentum-conserving solution designed to enhance numerical stability and improve shock-capturing properties. The scheme’s performance is assessed through a series of numerical tests involving monochromatic and spectral linear wave propagation. These tests demonstrate that the conservative staggered scheme is much less sensitive to grid resolution, resulting in approximately one order of magnitude lower numerical diffusion in contrast to the well-established HLLC scheme, while maintaining comparable dispersive accuracy despite using a lower-order spatial reconstruction. Additionally, the scheme introduces a slight negative phase error that compensates for the positive dispersion error inherent in the underlying equations, resulting in improved overall phase accuracy relative to the HLLC scheme. Beyond linear wave propagation, the numerical approach is validated against standard benchmark tests with solitary and spectral breaking waves. In these highly non-linear cases, coupling the conservative staggered scheme with a turbulent kinetic energy (TKE)-based eddy viscosity model yields localized and physically consistent dissipation while preserving the dispersive characteristics of the solution. Compared to conventional hybrid breaking approaches, the TKE-based closure provides enhanced stability, reduced grid sensitivity, and a more accurate representation of energy dissipation during wave breaking. These results underscore the potential of the conservative staggered scheme as an efficient and robust framework for computing complex coastal and nearshore wave processes.