Variable steps for reversible integration methods

Abstract

Conventional variable-step implementation of symplectic or reversible integration methods destroy the symplectic or reversible structure of the system. We show that to preserve the symplectic structure of a method the step size has to be kept almost constant. For reversible methods variable steps are possible but the step size has to be equal for “reflected” steps. We demonstrate possible ways to construct reversible variable step size methods. Numerical experiments show that for the Kepler problem the new methods perform better than conventional variable step size methods or symplectic constant step size methods. In particular they exhibit linear growth of the global error (as symplectic methods with constant step size).