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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.).