We investigate the difference between the velocity Verlet and the Liouville-operator-derived (LOD) algorithms by studying two non-Hamiltonian systems, one dissipative and the other conservative, for which the Jacobian of the transformation can be determined exactly. For the two systems, we demonstrate that (1) the velocity Verlet scheme fails to integrate the former system while the first- and second-order LOD schemes succeed and (2) some first-order LOD fails to integrate the latter system while the velocity Verlet and the other first- and second-order schemes succeed. We have shown that the LOD schemes are stable for the former system by determining the explicit forms of the shadow Hamiltonians which are exactly conserved by the schemes. We have shown that the Jacobian of the velocity Verlet scheme for the former system and that of the first-order LOD scheme for the latter system are always smaller than the exact values, and therefore, the schemes are unstable. The decomposition-order dependence of LOD schemes is also considered.

中文翻译：

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基于速度Verlet和Liouville算子的算法可集成非哈密顿系统的稳定性

通过研究两个非哈密顿系统，一个耗散的系统和另一个保守的系统，我们可以准确地确定变换的雅可比性，从而研究速度Verlet算法和Liouville-operator-derived（LOD）算法之间的差异。对于这两个系统，我们证明（1）速度Verlet方案无法集成前者系统，而一阶和二阶LOD方案成功，并且（2）某些一阶LOD不能集成后者的系统，而速度Verlet和其他一阶和二阶方案成功了。我们通过确定阴影哈密顿量的显式形式来确定LOD方案对于前一系统是稳定的，该显式形式由该方案精确守恒。我们已经证明，对于前一个系统，速度Verlet方案的雅可比行列式和对于后一个系统的一阶LOD方案的雅可比行列总是小于精确值，因此，这些方案是不稳定的。还考虑了LOD方案的分解顺序依赖性。