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Stability and dynamics of numerical methods for nonlinear ordinary differential equations. (English) Zbl 0686.65054
Numerical methods for y ' =f(y) are considered as dynamical systems, whose fixed points are investigated. It is shown that for linear multistep methods all fixed points of the corresponding dynamical system are equilibrium points of the differential equation. For Runge-Kutta methods (as well as for predictor-corrector methods) the underlying dynamical system may possess extra fixed points which do not correspond to zeros of f(y). A characterization of all 2-stage Runge-Kutta methods, which do not allow such spurious fixed points, is given.
Reviewer: E.Hairer

MSC:
65L20Stability and convergence of numerical methods for ODE
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general