Stability, convergence, and energy preservation robust methods for fully implicit and fully explicit coupling schemes

Abstract

This paper presents an analysis of the Godunov–Ryabenkii stability, Generalized Mini-mal Residual(GMRES) convergence, and energy-preserving properties of partitioned and monolithic approaches (fully implicit and fully explicit schemes) for solving coupled parabolic problems. Specifically, we consider a bi-domain parabolic diffusion problem with two types of coupling conditions at the interface: Dirichlet–Neumann and heat-flux coupling. Our findings shows that the Dirichlet–Neumann coupling is unconditionally stable for both approaches. In contrast, the heat-flux coupling requires additional conditions to ensure the stability of the coupled problem. For numerical approximations, finite volume and finite difference schemes are used. The results show that energy preservation is achieved with one-sided differences in the finite volume method, while the finite difference method achieves conservation when central difference approximations are used for both the coupling and boundary conditions in the heat-flux coupling case. Additionally, Dirichlet–Neumann coupling maintains stability and energy preservation in both methods using the one-sided approach without requiring extra conditions. However, for heat-flux coupling, an additional restriction is necessary to ensure stability. The challenge for the convergence of coupled interface problems arise due to strong domain interactions and sensitive interface conditions, like Dirichlet–Neumann or heat-flux coupling. The poor system conditioning and discretization choices can slow the rate of convergence. For this purpose we used the GMRES method. This work provides a comprehensive framework for addressing coupled parabolic diffusion problems using robust, stable, and energy-preserving numerical methods.