SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Nonlinear stability of generalized additive and partitioned implicit multirate Runge-Kutta schemes

M. Guenther and A. Sandu

Multirate schemes are an efficient tool to exploit different time scales in ordinary differential, differential-algebraic and partial differential equations by asigning appropriate step sizes to different components of the solution or the right-hand side. In this presentation we will generalize the class of additive and partitioned Runge-Kutta schemes [4] to derive multirate schemes that are characterized by excellent nonlinear stability properties. The derivation of order conditions will be based on a combination of N-trees and P series. The class of generalized Runge-Kutta, Rosenbrock-Wanner and W methods [3, 2, 1] can be embedded into this new approach. The properties of this new class of multirate schemes will be validated by various numerical results.

[1] A. Bartel and M. Günther, A multirate W-method for electrical networks in state-space formulation, J. Comput. Appl. Math., 147 (2002), pp. 411-425.
[2] M. Günther, A. Kvaerno and P. Rentrop, Multirate partitioned Runge-Kutta methods, BIT, 41 (2001), 504-515.
[3] A. Kvaerno and P. Rentrop, Low order Runge-Kutta methods in electric circuit simulation, Preprint No. 99/1, IWRMM, Universität Karlsruhe (TH), 1999.
[4] A. L. Araujo, A. Murua and J. M. Sanz-Serna, Symplectic methods based on decompositions, SIAM J. Numer. Anal., 34 (1007), 1926-1947.

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