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Abstract. Recently a lot of effort has been placed in the construction and implementation of a class of methods called exponential integrators. These methods are preferable when one has to deal with stiff and highly oscillatory semilinear problems, which often arise after spatial discretization of Partial Differential Equations (PDEs). The main idea behind the methods is to use the exponential and some closely related functions inside the numerical scheme. In this note we show that the integrating factor methods, introduced by Lawson in 1967, are also examples of exponential integrators with very special structure for the related exponential functions. In order to prove this relation, we use the approach based on bi-coloured rooted trees and B-series. We also show under what conditions every bi-coloured rooted tree can be express as a linear combination of standard non-coloured rooted trees.

Lie group methods for nonautonomous semi-discretized in space, partial differential equations are considered. The choice of frozen vector field and its corresponding algebra action on the manifold for such problems is discussed. A new exponential integrator for semilinear problems, based on commutator free Lie group methods with algebra action arising from the solutions of differential equations with nonautonomous frozen vector fields is derived. The proposed new scheme is then compared with some existing methods in several numerical experiments.

Abstract.A new effective method and its two modifications for solving Hermitian pentadiagonal block circulant systems of linear equations are proposed. New algorithms based on the proposed method are constructed. Our algorithms are then compared with some classical techniques as far as implementation time is concerned, number of operations and storage. Numerical experiments corroborating the effectiveness of the proposed algorithms are also reported.