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Course, academic year 2017/2018
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Numerical Solution of ODE - NMNV539
Czech title: Numerické řešení ODR
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2013
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: prof. RNDr. Vladimír Janovský, DrSc.
Class: M Mgr. NVM
M Mgr. NVM > Povinně volitelné
Classification: Mathematics > Differential Equations, Potential Theory, Numerical Analysis
Incompatibility : NNUM010
Interchangeability : NNUM010
Annotation -
Last update: T_KNM (29.04.2015)

One-step and multi-step methods: algorithms, analysis, convergence. Discrete and continuous dynamical systems.
Literature - Czech
Last update: T_KNM (15.09.2013)

Deuflhart P., Bornemann F.: Scientific Computing with Ordinary Differential Equations, Springer Verlag, 2002

Hairer E., Norset S.P., Wanner G.: Solving Ordinary Differential Equations I (Nonstiff Problems), Second Revised Edition, Springer Verlag, 1993

Hairer E., Wanner G.: Solving Ordinary Differential Equations II (Stiff and Differential-Algebraic Problems), Springer Verlag, 1991

Syllabus -
Last update: T_KNM (29.04.2015)

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.

 
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