We construct a symplectic integrator for non-separable Hamiltonian systems combining an extended phase space approach of Pihajoki and the symmetric projection method. The resulting method is semiexplicit in the sense that the main time evolution step is explicit whereas the symmetric projection step is implicit. The symmetric projection binds potentially diverging copies of solutions, thereby remedying the main drawback of the extended phase space approach. Moreover, our semiexplicit method is symplectic in the original phase space. This is in contrast to existing extended phase space integrators, which are symplectic only in the extended phase space. We demonstrate that our method exhibits an excellent long-time preservation of invariants, and also that it tends to be as fast as and can be faster than Tao's explicit modified extended phase space integrator particularly for small enough time steps and with higher-order implementations and for higher-dimensional problems.

翻译：我们结合Pihajoki扩展相空间方法和对称投影法，构造了一种适用于不可分离哈密顿系统的辛积分器。所得方法具有半显式特性：主时间演化步为显式，而对称投影步为隐式。对称投影步骤能够约束可能发散的解副本，从而弥补了扩展相空间方法的主要缺陷。此外，该半显式方法在原始相空间中具有辛性，这与现有仅能在扩展相空间中保持辛性的扩展相空间积分器形成对比。我们证明该方法能够长期完美保持不变量，且特别在时间步长足够小、采用高阶实现及高维问题场景下，其计算速度可与Tao显式修正扩展相空间积分器相当甚至更快。