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Center for Nonlinear Studies |
We introduce a fully second order self-consistent Implicit/Explicit (IMEX) method for solving multi-physics and fluid problems that exhibit multiple time scales. These problems generally consist of stiff and non-stiff terms. Our algorithm is a combination of an explicit block for the non-stiff and an implicit block for the stiff part. The explicit block is always solved inside the implicit block as part of the non-linear function evaluation making use of the Jacobian-Free Newton Krylov (JFNK) method. In this way, there is a continuous interaction between the two algorithm blocks in that the improved solutions (in terms of time accuracy) at each non-linear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of non-linear residuals. This continuous interaction results in an implicitly balanced algorithm in that all the non-linearities due to coupling of different time terms are converged. In other words, we obtain a self-consistent IMEX method that eliminates the order reduction in time accuracy that a classic IMEX method can suffer from. We note that a classic IMEX method splits the operators such that the implicit and explicit blocks are executed independent of each other leading non-converged non-linearities therefore time inaccuracies. We present computational results coming from variety of applications to validate the numerical order of our scheme. We also provide a mathematical analysis that examine/compare the time behaviour of our self-consistent IMEX method versus the classic IMEX method.
Host: Mikhail Shashkov