An efficient second order ImEx scheme for the shallow water model in low Froude regime

Abstract

This paper presents the development and analysis of a second order numerical method tailored for shallow water flows in regimes characterized by low Froude numbers. The focus is on modeling oceanic and coastal dynamics across different scales, with particular attention on the variation of the Froude number from 1 near the shoreline to significantly lower values offshore. Classical hyperbolic schemes, such as Riemann solvers, become inefficient in these deep water conditions. To address this challenge, a hybrid numerical approach is proposed where part of the system is treated implicitly, resulting in an ImEx (Implicit–Explicit) scheme that allows long time simulation using a CFL condition that is independent of the Froude number. To minimize the computational cost associated with solving linear systems, a fully segregated approach is used. In this method, the water height and hybrid mass fluxes are handled implicitly, while velocities are treated explicitly, thus avoiding large linear system resolutions. While various Runge–Kutta schemes are available for a second-order time integration, we chose here a Crank–Nicolson scheme to reduce the number of linear systems required. Spatial discretization is performed using a second-order MUSCL reconstruction. The novel scheme is demonstrated to be Asymptotic Preserving (AP), ensuring that a consistent discretization of the limit model, known as the “lake equations” is obtained as the Froude number approaches zero. Through a series of one- and two-dimensional test cases, the method is shown to achieve second-order accuracy for different Froude numbers. Additionally, the computational efficiency of the proposed method is compared with that of a fully explicit scheme, demonstrating significant time savings with the ImEx approach, particularly in scenarios governed by low Froude numbers.