In this study, high-order multi-implicit spectral deferred correction (MISDC) methods are presented for solving A-D-R equations. MISDC methods are similar to operating-splitting methods both types of methods give rise to multiple implicit equations, for which different time steps may be used. In theory, however, the temporal order of accuracy of MISDC methods can be arbitrarily high because both the integration and splitting errors are eliminated during the deferred correction process. MISDC methods, which are a generalization of semi-implicit spectral deferred correction (SISDC) methods introduced by Minion (which are in turn modifications of the explicit and implicit spectral deferred correction methods previously developed by Dutt et al.), use a low-order numerical method to compute a high-order approximation. This is achieved by using the low-order numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation by one. The accuracy and stability of MISDC methods for A-D-R equations and its efficiency relative to SISDC methods are investigated in this study.

The above figure shows results of an efficiency comparison between MISDC methods and semi-implicit additive Runge-Kutta (ARK) methods developed by Kennedy et al., using a simple, one-dimensional model of flamelets, with a large reaction coefficient, as a test problem. These results indicate that for problems with stiff reactions, MISDC methods can be constructed to be competitive with ARK methods.

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