In this paper, we consider exponential integrators for semilinear Poisson systems. Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs are presented. These geometric exponential integrators exhibit better long time stability properties as compared to non-geometric integrators, and are computationally more efficient than traditional symplectic integrators and energy-preserving methods based on the discrete gradient method.

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<p style='text-indent:20px;'>A variational integrator of arbitrarily high-order on the special orthogonal group <inline-formula><tex-math id="M2">\begin{document}$ SO(n) $\end{document}</tex-math></inline-formula> is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie–Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on <inline-formula><tex-math id="M3">\begin{document}$ SO(3) $\end{document}</tex-math></inline-formula> which demonstrate that they are comparable to variational Runge–Kutta–Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.</p>