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Course, academic year 2023/2024
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Numerical Solution of ODE - NMNV539
Title: Numerické řešení ODR
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Václav Kučera, Ph.D.
Class: M Mgr. NVM
M Mgr. NVM > Povinně volitelné
Classification: Mathematics > Differential Equations, Potential Theory, Numerical Analysis
Incompatibility : NNUM010
Interchangeability : NNUM010
Is interchangeable with: NNUM010
Annotation -
Last update: T_KNM (29.04.2015)
One-step and multi-step methods: algorithms, analysis, convergence. Discrete and continuous dynamical systems.
Course completion requirements -
Last update: Scott Congreve, Ph.D. (29.10.2019)

The condition for the credit is an active participation at exercices. The nature of the condition excludes the repetition of this check.

Literature -
Last update: Scott Congreve, Ph.D. (29.10.2019)

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

Requirements to the exam -
Last update: Scott Congreve, Ph.D. (29.10.2019)

The exam is composed from the two parts: written and oral. The written part comes first and its

passing is the necessary condition to pass the whole exam. The final mark corresponds to the level

of knowledge in both parts of the exam.

The written part contains examples related to those in the course syllabus.

The oral part corresponds to the syllabus and the lectures.

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.

Entry requirements -
Last update: prof. RNDr. Vladimír Janovský, DrSc. (15.05.2018)

Bc in mathematics

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