Course detail

# Numerical Methods II

FSI-3NUAcad. year: 2020/2021

The course is devoted to numerical methods for differential equations. The course deals with the following topics: numerical methods for initial value problems of ordinary differential equations. Numerical methods for solving boundary value problems in ordinary differential equations. Numerical methods for solving partial differential equations of elliptic, parabolic and hyperbolic type. The course is based on the problem-solving environment MATLAB.

Learning outcomes of the course unit

Students will understand that numerical methods are effective tools and often the only way to solve differential equations. They will learn principles of particular methods and get knowledge how to choose suitable method for a specific problem. They will manage to use high-quality numerical and graphical Matlab tools for solving problems and visualizing results.

Prerequisites

Numerical linear algebra, approximation of functions, numerical differentiation and integration, differential and integral calculus, basic MATLAB programming.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Čermák, L.: Numerické metody pro řešení diferenciálních rovnic, učební text FSI VUT Brno, [on-line], available from: http://mathonline.fme.vutbr.cz/Numericke-metody-II/sc-1246-sr-1-a-263/default.aspx.
Shampine, L.F.: Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.
Hlavička, R.: Numerické metody pro řešení diferenciálních rovnic. Průvodce softwarem a počítačová cvičení v prostředí MATLABu. [on-line], Available from: http://mathonline.fme.vutbr.cz/Numericke-metody-II/sc-1246-sr-1-a-263/default.aspx.
Shampine, L.F., Gladwell, S., Thompson, S.: Solving ODEs with MATLAB, Cambridge University Press, Cambridge, 2003.
LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, Philadelphia, 2007.
Fish, J., Belytschko, T.: A First Course in Finite Elements, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, 2007.
Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics. Pearson Prentice Hall, Harlow, 2007.
Moler, C.B.: Numerical Computing with MATLAB, Siam, Philadelphia, 2004.

Planned learning activities and teaching methods

The course is taught through exercises which are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Active participation in practisals. Students have to work out semester assignment solved by means of MATLAB (OCTAVE) and to pass successfully check test.
COURSE CLASSIFICATION: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The course objective is to make students acquainted with numerical methods for both ordinary and partial differential equations. Students will also broaden and deepen their knowledge of Matlab in programming techniques and they will manage to use Matlab functions for numerical solution of differential equations.

Specification of controlled education, way of implementation and compensation for absences

Attendance at seminars is checked. Lessons are planned according to the week schedules. Absence may be replaced by the agreement with the teacher.

Classification of course in study plans

• Programme B3S-P Bachelor's

branch B-STI , 2. year of study, summer semester, 1 credit, compulsory

#### Type of course unit

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

1. Numerical solution of initial value problems for ODE. Explicit and implicit Euler method. Accuracy and stability.
2. Explicit Runge-Kutta methods, step size control, matlab functions ode23 and ode45.
3. Adams methods, predictor corrector technique, variable-step-variable-order approach, matlab function ode113.
4. Stiff initial value problems, backward differentiation methods, matlab functions ode23t, ode15s.
5. Solving selected initial value problems in MATLAB.
6. Boundary value problem for ODE, shooting method, matlab function bvp4c.
7. Boundary value problem for ODE, difference method, finite volume method.
8. Boundary value problem for ODE, finite element method.
9. Elliptic PDE, difference method, finite element method.
10. Elliptic PDE, finite element method - continuation.
11. Parabolic PDE, methods of lines, matlab function pdepe.
12. Second order hyperbolic PDE, methods of lines.
13. First order hyperbolic PDE, method of characteristics.

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