Analysis of stability, energy conservation, and convergence of coupled advection-diffusion models for ocean-atmosphere interactions

The main focus of this work is to study ocean-atmosphere coupling, modeled using coupled advection-diffusion equations with two coupling conditions: the Dirichlet-Neumann (DN) and the heat flux condition, defined across two non-overlapping domains. For numerical approximation, the finite-volume method (FVM) and finite-difference method (FDM) are applied. The convergence analysis of the coupled problem is conducted using the Generalized Minimal Residual (GMRES) method. We conclude that central difference schemes ensure conservation under heat flux coupling, while one-sided differences introduce errors leading to energy non-conservation. While for the DN-coupling one sided maintain the conservation. The stability analysis, based on Fourier analysis and normal mode techniques from Godunov-Ryabenkii (GR) stability theory, reveals stricter stability constraints for explicit schemes compared to implicit ones. The GMRES method is used to achieve numerical convergence. The results demonstrate how variations in the Péclet number (Pe) influence the behavior of the solution, transitioning from diffusion-driven smoothness to advection-driven sharpness, while maintaining physical consistency across the domains. The proposed algorithms undergo rigorous numerical validation, with results illustrated through detailed graphs and numerical tables that demonstrate strong agreement between theoretical predictions and computational outcomes.