Abstract
Implicit–explicit (IMEX) time stepping methods can efficiently solve differential equations with both stiff and nonstiff components. IMEX Runge–Kutta methods and IMEX linear multistep methods have been studied in the literature. In this paper we study new implicit–explicit methods of general linear type. We develop an order conditions theory for high stage order partitioned general linear methods (GLMs) that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms. Next, we propose a family of IMEX schemes based on diagonally-implicit multi-stage integration methods and construct practical schemes of order up to three. Numerical results confirm the theoretical findings.
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Acknowledgments
This work has been supported in part by NSF through awards NSF OCI-8670904397, NSF CCF-0916493, NSF DMS-0915047, NSF CMMI-1130667, NSF CCF-1218454 AFOSR FA9550-12-1-0293-DEF, FOSR 12-2640-06, DoD G&C 23035, and by the Computational Science Laboratory at Virginia Tech. Sébastien Blaise is a Postdoctoral Researcher with the Belgian Fund for Research (FNRS).
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This paper is dedicated to Prof. J.C. Butcher’s 80-th birthday.
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Zhang, H., Sandu, A. & Blaise, S. Partitioned and Implicit–Explicit General Linear Methods for Ordinary Differential Equations. J Sci Comput 61, 119–144 (2014). https://doi.org/10.1007/s10915-014-9819-z
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DOI: https://doi.org/10.1007/s10915-014-9819-z