Exponential B-Series: The Stiff Case

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.