Geometric numerical integration by means of exponentially-fitted methods

The subject of geometrical numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to an improved long-time behavior. Exponential fitting deals in the case of numerical methods for differential equations with tuned methods, which are developed for situations where the solution is oscillatory. In this paper both concepts are combined for the well-known Stormer/Verlet method. Gautschi's exponentially fitted Stormer/Verlet method is discussed and its various interpretations are given. Attention is paid to geometric properties such as reversibility, symplecticity, volume interpretation and conservation of first integrals. Also the extension to Hamiltonian systems on manifolds is described. Finally the problem of choosing the optimal frequency for such exponentially fitted methods is discussed. (c) 2006 IMACS. Published by Elsevier B.V. All rights reserved.