Stabilized quasi-monotone semi-Lagrangian high-order finite element method for a nonhydrostatic ocean model

Abstract

We present a high-order finite element method combined with a stabilized quasi-monotone semi-Lagrangian scheme to compute numerical solutions for a nonhydrostatic ocean model. This model is governed by the full incompressible Navier–Stokes equations, coupled with convection–diffusion equations for salinity and temperature, and an equation of state for seawater. Ocean models based on semi-Lagrangian schemes can lead to spurious oscillations near sharp gradients in the solution, often resulting in numerical instabilities and unphysical values such as negative salinity. A novel feature of the proposed method is the use of mesh adaptation strategies to identify elements affected by spurious oscillations, applying the so-called quasi-monotone scheme. The approach is tested on classical two-dimensional convection–diffusion problems, which share governing equations with those used for salinity and temperature conservation, and is validated against published results for stratified oceanic flows. The numerical results demonstrate the effectiveness of the proposed method in achieving high-order accuracy in smooth regions while reducing spurious oscillations near sharp gradients, resulting in stable and accurate numerical solutions for convection–diffusion problems and stratified ocean flows.