Analysis of the self-consistent IMEX method for tightly coupled non-linear systems

We present a mathematical analysis for our self-consistent Implicit/Explicit (IMEX) method that we have introduced in Kadioglu and Knoll (2010, 2013), Kadioglu et al. (2009, 2010) and Kadioglu (2017). The self-consistent IMEX algorithm is designed to produce second order time convergent solutions to multi-physics and multiple time scale fluid dynamics problems. The algorithm is a combination of an explicit block that solves the non-stiff part and an implicit block that solves the stiff part of the problem. The explicit block is always solved inside the implicit block as part of the nonlinear function evaluation making use of the Jacobian-Free Newton Krylov (JFNK) method (Knoll and Keyes, 2004; Nourgaliev et al., 2010; Saad, 2003). In this way, there is a continuous interaction between the implicit and explicit blocks meaning that the improved solutions (in terms of the time accuracy) at each nonlinear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of nonlinear residuals. This continuous interaction between the two algorithm blocks results in an implicitly balanced algorithm in that all the nonlinearities due to coupling of different time terms are converged. In other words, we obtain a self-consistent IMEX method that eliminates the possible order reduction in time accuracy for certain types of problems that a classical IMEX method can suffer from. We note that the classic IMEX methods split the operators such a way that the implicit and explicit blocks are executed independent of each other, and this may lead to non-converged nonlinearities therefore time inaccuracies for certain models. We also note that the well-known Strang-Splitting operator split technique (Strang, 1968 ) can suffer from the above mentioned time order reduction for certain applications, even though the method itself formally a second order numerical procedure. In this study, we provide a mathematical analysis (modified equation analysis) that examines and compares the time convergence behavior of our self-consistent IMEX method versus the classic IMEX method. We provide computational results to verify our analysis and analytical findings. We also computationally compare our IMEX procedure to the Strang-Splitting method.