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

TL;DR

This paper compares the stability and accuracy of velocity Verlet and Liouville-operator-derived algorithms in integrating non-Hamiltonian systems, highlighting their strengths and limitations through exact Jacobian analysis.

Contribution

It provides a detailed stability analysis of LOD and velocity Verlet schemes for non-Hamiltonian systems, revealing conditions under which each method succeeds or fails.

Findings

LOD schemes are stable for dissipative systems with conserved shadow Hamiltonians.

Velocity Verlet fails for dissipative systems but succeeds for conservative ones.

Jacobian analysis explains the stability differences between schemes.

Abstract

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, (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 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.