On the Stability of Loose and Strong Partitioned Algorithms for Thermal Coupling of Domains Using Higher Order Implicit Time Integration Schemes

Thermal interaction of fluids and solids, or conjugate heat transfer (CHT), is encountered in many engineering applications. Since time-accurate computations of such coupled problems can be computationally expensive, we consider loosely-coupled and stronglycoupled solution algorithms in which higher order multi-stage Runge-Kutta schemes are employed for time integration. The higher order time integration schemes have the potential to improve the computational efficiency at arriving at a certain accuracy relative to the traditionally used 1 and 2 order implicit schemes. The spatial coupling between the subdomains is realized using Dirichlet-Neumann interface conditions and the coupled domains are solved in a sequential manner at each stage (Block Gauss-Seidel). In this paper, the stability of two partitioned algorithms is analyzed by considering a one dimensional model problem. The model problem consists of two thermally coupled domains where the governing equation within each subdomain is unsteady linear heat conduction. In the loosely-coupled approach, a family of multi-stage IMEX schemes is used for time integration. By observing similarities between the second stage of the IMEX schemes and the θ scheme with θ = 0.5 (Crank-Nicolson), the stability of the partitioned algorithm in which the Crank-Nicolson scheme is used for time integration is first analyzed by applying the stability theory of Godunov-Ryabenkii. The stability of the IMEX schemes is next investigated by numerically solving the model problem and comparing the results to the conclusions of the stability analysis for the Crank-Nicolson scheme. Due to partly explicit nature of the IMEX schemes, the loosely-coupled algorithm becomes unstable for sufficiently large Fourier numbers (similar to the Crank-Nicolson scheme). When the ratio of the thermal effusivities of the coupled domains is much smaller than unity, time step restriction due to stability is sufficiently weak that computations can be performed with reasonably large Fourier numbers. Furthermore, the results show better stability properties of the IMEX schemes compared to the Crank-Nicolson scheme. In the strongly-coupled approach, the stability and rate of convergence of performing (Gauss-Seidel) subiterations at each stage of the higher order implicit ESDIRK time integration schemes are analyzed. From the stability analysis, an expression for the rate of convergence of the iterations (σ) is obtained. For cases where σ ≪ 1, subiterations will convergence rapidly. However, when σ ≈ 1, the convergence rate of the iterations is slow. The results obtained by solving the model problem numerically are in line with the performed analytical stability analysis.