Geometric numerical integration by means of exponentially-fitted methods

• Published in: Applied numerical mathematics Vol. 57; no. 4; pp. 415 - 435
• Main Authors: Van Daele, M., Vanden Berghe, G.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.04.2007; Elsevier
• Subjects: Exact sciences and technology; Exponential fitting; Geometric integration; Global analysis, analysis on manifolds; Hamiltonian systems; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Ordinary differential equations; Oscillating differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Hamiltonian systems; Exponential fitting; Oscillating differential equations; Geometric integration; Numerical integration; Oscillatory solution; Fitting; Differential equation; Numerical method; Numerical approximation; Numerical analysis; Applied mathematics; Hamiltonian system
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2006.06.001

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 Störmer/Verlet method. Gautschi's exponentially fitted Störmer/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.