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

Invited Talk

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Exponential B-series: The stiff case

V.T. Luan and A. Ostermann

Abstract
For the purpose of deriving the order conditions of exponential Runge-Kutta and exponential Rosenbrock methods, we extend the well-known concept of B-series to exponential integrators. As we are mainly interested in the stiff case, Taylor series expansions have their limitations. Our approach is based on the variation-of-constants formula which allows us to treat semilinear problems, where the stiffness comes from the linear part. By truncating the arising exponential B-series to a certain order (which requires regularity of the considered exact solution) we are able to identify the sought-after order conditions. In particular, we show how the stiff order conditions of arbitrary order can be obtained in a simple way from a set of recursively defined trees.

Bibliography
[1] V.T. Luan and A. Ostermann, Exponential Rosenbrock methods of order five-construction, analysis and numerical comparisons, J. Comput. Appl. Math. 255, 417-431 (2014).
[2] V.T. Luan and A. Ostermann, Stiff order conditions for exponential Runge-Kutta methods of order five, to appear in: Proceedings of the Fifth International Conference on High Performance Scientific Computing, 2012, Hanoi, Vietnam.
[3] V.T. Luan and A. Ostermann, Explicit exponential Runge-Kutta methods of high order for parabolic problems, J. Comput. Appl. Math., DOI 10.1016/j.cam.2013.07.027.
[4] V.T. Luan and A. Ostermann, Exponential B-series: The stiff case (submitted to SIAM. J. Numer. Anal.).

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