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Course, academic year 2023/2024
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Numerical Solution of Differential Equations - NNUM010
Title: Numerické řešení diferenciálních rovnic
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: winter
E-Credits: 6
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: cancelled
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. RNDr. Vladimír Janovský, DrSc.
Classification: Mathematics > Numerical Analysis
Interchangeability : NMNV539
Is incompatible with: NNUM044, NNUM045, NMNV539
Is interchangeable with: NMNV539
Annotation -
Last update: T_KNM (27.04.2006)
One-Step and Multistep Methods: algorithms, convergence analysis. Dynamical Systems (continuous and discrete time).
Aim of the course -
Last update: T_KNM (17.05.2008)

Dynamical systems: theory and numerical approximation.

Literature - Czech
Last update: T_KNM (17.05.2008)

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

Teaching methods -
Last update: T_KNM (17.05.2008)

The course consists of lectures in a lecture hall and exercises in a computer laboratory.

Requirements to the exam -
Last update: T_KNM (17.05.2008)

Examination according to the syllabus.

Syllabus -
Last update: T_KNM (17.05.2008)

ODE: Mathematical model of evolution (examples: population dynamics, chemical oscilations, etc.)

Revising ODE theory: The Existence and Uniqueness Theorems, geometric interpretation of a solution: vector field, phase flow, phase portrait. Taylor expansion of the flow.

One-Step Metods: elementary examples, convegence analysis, adaptive step-size, Runge-Kutta Methods, Implicit methods (Gauss, Radau, linearly Implicit RK, etc.)

Multistep Metods: The methods based on numerical integration and differentiation, Linear Multistep Methods (discretisation error, D-stability, covergence analysis).

Dynamical systems: A long time evolution (orbit, limit set), steady state, A-stability, linearization, Lyapunov Theorem. Discrete time dynamical systems.

A-stability of a method: Domain of A-stability for Runge-Kutta Methods and for linear Multistep Methods. "Stiff" problems. A-stable methods.

Entry requirements -
Last update: T_KNM (17.05.2008)

There are no special entry requirements.

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