
Corrected Refactorized Midpoint Method for Coupled Problems
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We consider Magnetohydrodynamic (MHD) flows - a coupled system stemming from the Navier-Stokes and Maxwell's equations. A stable, yet only first order accurate implicit-explicit decoupling method for MHD was presented in [1]. Since then, efforts have been made to create a second order accurate decoupling method, that would also be stable when used with variable time steps. The Refactorized Midpoint (RM) approach, [2], was modified in [3] and applied to the MHD system, producing a second order accurate and stable solution. This was achieved by a fixed point iteration, requiring from 3 to 9 iterations per time step, depending on the size of the time step. The extra computational effort can prove prohibitively expensive, even for the decoupled MHD system. Herein we propose a novel RM-based approach, with the mid-step correction of the decoupling method's solution. The method is naturally parallelizable, the resulting solution is stable and more accurate than the solution of the decoupling method - and the computational cost is approximately two iterations per time step, regardless of the size of the time step. Additionally, the correction step makes the new method compatible with the recently proposed techniques for resolving turbulent MHD flows, [4]. We provide the results of numerical tests to compare the methods.