Syllabus
1) Basic concepts: Examples of evolution processes, systems of ordinary differential equation, initial problem, trajectory, vector field, phase portrait, stationary solution.
2) One-step methods: Examples of one-step methods. Analysis of convergence of a general one-step method. Adaptive choice of length of the time step. Runge-Kutta methods, Butcher's array.
3) Multi-step methods: Idea of numerical integration (Adams-Bashforth, Adams-Moulton, Nyström, Milne-Simpson), predictor-corrector methods. General linear multi-step methods.
4) Dynamical systems: Asymptotics (orbit, limit set), A-stability, Lyapunov theorem. Discrete dynamical systems.
5) A-stability: A-stability region for Runge-Kutta methods. A-stability region for linear multi-step methods. "Stiff" problems, A-stable methods.
Annotation
One-step and multi-step methods: algorithms, analysis, convergence. Discrete and continuous dynamical systems.