TY - JOUR

T1 - Stability of velocity-Verlet- and Liouville-operator-derived algorithms to integrate non-Hamiltonian systems

AU - Watanabe, Hiroshi

N1 - Funding Information:
We would like to thank N. Kawashima and H. Noguchi for helpful discussions. This work was supported by JSPS KAKENHI Grant No. 15K05201 and by MEXT as “Exploratory Challenge on Post-K computer” (Challenge of Basic Science-Exploring Extremes through Multi-Physics and Multi-Scale Simulations).
Publisher Copyright:
© 2018 Author(s).

PY - 2018/10/21

Y1 - 2018/10/21

N2 - 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.

AB - 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.

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U2 - 10.1063/1.5030034

DO - 10.1063/1.5030034

M3 - Article

C2 - 30342459

AN - SCOPUS:85054988135

VL - 149

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 15

M1 - 154101

ER -