We discuss our initial approach that employs a block Gauss-Seidel iteration procedure. In general, this strategy is attractive because of its simplicity for solving problems that may be represented by linking, via shared boundary conditions, existing codes using a modular design. Its success hinges upon a novel coupling scheme which employs two parameters to define a specific exchange of temperature and heat flux information between the two models. The iteration behavior depends strongly on the specific choices of these coupling parameters.
Improved robustness and rate of convergence could be achieved using a full Newton method; however, one would need a complete Jacobian matrix for the coupled problem. In general, this is not the case. Because of the loose coupling of the individual models, we do not have knowledge of the sensitivities of one model's unknowns to those of the other. Instead, we discuss an extremely promising approach that implements an approximate, Block-Newton (ABN) method that uses only the solvers themselves, without the need to access the Jacobian matrices. This new algorithm is based on several important ideas. First, the Newton iteration is recast using the Schur compliment. Next, the iteration can be posed so that subproblems involving the unknown cross-Jacobian matrices, when solved using Krylov-subspace iterative solvers, appear only in matrix-vector products that can be approximated by simple finite-difference formulae. In this sense, the formulation is a manifestation of more general Jacobian-free Newton--Krylov methods. Results from this formulation are presented and compared to the prior strategy.