Course detail

Numerical Methods II

FSI-0NUAcad. year: 2011/2012

The course serves as a numerically-based counterpart to the analytical methods introduced in the concurrent course Mathematics III. The course deals with the following topics: Numerical methods for use with Taylor and Fourier series. Numerical solution of initial value problems for ODEs. Solution of linear 2nd order two-point boundary value problem by the difference method and the finite element method. Solution of two-dimensional Poisson equation by the finite difference method and the finite element method. The method of lines for heat flow along a rod and for oscillations of a string.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Libor Čermák, CSc.

Department

Institute of Mathematics (ÚM)

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. They will recognise that numerical means may be helpfull in practical computations with series. They will also deepen their programming knowledge and skills.

Prerequisites

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

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Continuous active work at seminars.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The course objective is to make students acquainted with numerical methods for differential equations. Some attention will also be devoted to numerical computations with series. Students will broaden and deepen their knowledge of MATLAB in both programming techniques and using built-in numerical routines. The course corresponds in subject matter the concurrent course Mathematics III so that students can use acquired knowledge for working out tasks assigned them in this course.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Eriksson, K. Estep, D., Hansbo, P., Johnson,C.: Computational Differential Equations, Cambridge University Press, Cambridge, 1996.
Moler, C.B.: Numerical Computing with MATLAB, SIAM, Philadelphia, 2004. Dostupný také z WWW: http://www.mathworks.com/moler.
Shampine, L.F.: Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.
Vitásek, E.: Numerické metody, SNTL, Praha, 1987.

Recommended reading

Čermák, J., Ženíšek, A.: Matematika III, CERM, Brno, 2001.
Čermák, L.: Numerické metody II - diferenciální rovnice, CERM, Brno, 2010.

Classification of course in study plans

  • Programme B2341-3 Bachelor's

    branch B-STI , 2 year of study, winter semester, elective (voluntary)

Type of course unit

 

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

RNDr. Rudolf Hlavička, CSc.
doc. RNDr. Libor Čermák, CSc.
Ing. Pavla Sehnalová, Ph.D.

Syllabus

1. MATLAB programming style, symbolic computations in MATLAB.
2. Orthogonal polynomials and numerical integration.
3. Taylor series and their applications, computing derivatives.
4. Fourier series and their applications, computing integrals using high accuracy Gaussian quadrature.
5. Initial value problem for 1st order ODEs: Euler's method, the Taylor series method.
6. The eigenvalue problem, the power method, computing roots of polynomials.
7. Systems of 1st order linear ODEs, using eigenvalues for solving systems with constant coefficients.
8. Runge-Kutta methods, automatic step-size control.
9. Adams methods, the predictor-corrector algorithm.
10. Boundary value problem for 2nd order ODEs: the difference method.
11. Boundary value problem for 2nd order ODEs: the finite element method.
12. Two-dimensional Poisson's equation: the difference method and the finite element method.
13. Heat flow in a rod, oscillations of a string: the method of lines.