Maps of implicit, linearized θ-methods for the logistic differential equation

摘要

Implicit, linearized θ-methods for the logistic differential equation which provide explicit expressions for the value of the dependent variable at one time level as function of that at the previous time and the values of the time step and implicitness parameter, are developed. The fixed points of the maps of implicit,linearized θ-methods and their linear stability are determined analytically, together with their basins of attraction and singularity sets. It is shown that the basin of attraction of second-order accurate, implicit, linear methods technique is the right-most fixed point, whereas that of first-order accurate ones depends on the time step. The monotonic or oscillatory convergence to the fixed points is also determined analytically and numerically.