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A Meta-algorithm to Symmetrize Any One-step Time-integration Scheme

In this paper we present a meta-algorithm which can symmetrize any one-step timeintegration scheme. The meta-algorithm creates time symmetry, even for those algorithms which are intrinsically non-symmetric. We also present the way to use an adaptive stepsize control with this meta-algorithm. It is shown that the adaptive stepsize control can be implemented naturally and easily. We demonstrate how it works by applying it to the classical Runge-Kutta method. The result shows that our idea works well.