Linearized methods for ordinary differential equations

摘要

The conservation properties, singularities and nonlinear dynamics of both time-linearized methods which provide piecewise analytical solutions and keep the independent variable continuous, and implicit, linearized θ-techniques which are based on discretization and linearization, are analyzed in this paper. It is shown that both methods are implicit and provide explicit maps, but they do not preserve the energy in conservative systems. It is also shown that time-linearized methods preserve both the fixed points and the linear stability of the original ordinary differential equation, whereas linearized θ-techniques do preserve the fixed points and the linear stability of attractors, but the stability of the repellers depends on the time step and the implicitness parameter. The results clearly indicate that the linearized θ-techniques which more faithfully reproduce the nonlinear dynamics of the original ordinary differential equation are second-order accurate in time.