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Explicit, time reversible, adaptive step size control

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Ernst Hairer and Gustaf Söderlind

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Abstract. Adaptive step size control is difficult to combine with geometric numerical integration. As classical step size control is based on ``past'' information only, time symmetry is destroyed and with it also the qualitative properties of the method. In this paper we develop completely explicit, reversible and symmetry preserving, adaptive step size selection algorithms for geometric numerical integrators such as the St\"ormer--Verlet method. A new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory. For integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and, in particular, the total energy for Hamiltonian systems. It has the same excellent long term behaviour as if constant steps were used. With variable steps, however, both accuracy and efficiency are greatly improved.

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Key Words. Adaptive integration, geometric integration, time reversible and symmetric methods, St\"ormer--Verlet method, Hamiltonian systems, explicit and reversible step size control, backward error analysis, reversible perturbation theory.

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