Discrete dynamics of implicit time integration schemes for a dissipative system

A dissipative one degree of freedom dynamical system is investigated. Three time integration schemes are applied to the time continuous system leading to time discrete systems. Their energy dissipation and phase space contraction behaviour is analysed and differs from the properties of the differential equation. One of the schemes guarantees energy dissipation, whereas another is characterized by phase space contraction; the third scheme does not accurately discretize both properties. It was found that the scheme with unconditional phase space contraction and the standard scheme possess spurious steady states, higher order fixed points, as a result of tangled stable and unstable manifolds of a saddle. However, the consistently energy dissipative scheme exhibits no spurious steady states. It was concluded that the property of correct energy dissipation has a beneficial influence on the quality of the numerical solution. This result is analogous to that obtained for Hamiltonian mechanical systems for which algorithms with energy-conservation are often considered preferable to symplectic schemes.