Linearized methods for ordinary differential equations
The linearized \(\Theta\)-techniques and time-linearized methods for the solution of ordinary differential equations are considered. 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 fixed points and the linear stability of the original ordinary differential equation, whereas linearized \(\Theta\)-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 indicate that the linearized \(\Theta\)-techniques which more faithfully reproduce the nonlinear dynamics of the original ordinary differential equation are second-order accurate in time.
scientific article; zbMATH DE number 1355409