Rigorous open boundary condition for Boussinesq-type models, applicable to complex wave fields

We present a novel approach to handle open boundary conditions for a Boussinesq-type wave model coupled with the nonlinear shallow water equations. Traditional methods for managing open boundaries — such as sponge layers and source functions — are computationally intensive and require ad hoc calibration. To address this, we reformulate the Boussinesq equations as a system of conservation laws with nonlocal flux and a rapidly decaying source term. This reformulation is adapted to generate waves at the boundary of the numerical domain, from surface elevation data in situations where both incoming and outgoing waves are present. The proposed numerical scheme employs a MacCormack prediction–correction strategy combined with finite volume and finite difference methods, preserving key physical properties and ensuring stability. Comparison with laboratory experiments demonstrates that our approach avoids boundary reflection issues. In particular, it is able to accurately reproduce infragravity waves associated with a random wave field propagating over a sloping beach. This work opens important perspectives for improving phase-resolving coastal wave models, with the aim of forecasting complex random wave conditions in natural environments. • Wave generation and absorption is hard to manage in Boussinesq-type models. • Sponge layers and source function methods are computationally costly and need tuning. • We propose a rigorous, efficient boundary treatment with no calibration required. • Comparisons with experiments show accurate handling of incoming and outgoing waves.