Course detail

Numerical Methods II

FSI-SN2Acad. year: 2005/2006

The course represents the second part of an introduction to basic numerical methods and presents further procedures for solution of selected numerical problems frequently used in technical practice. Emphasis is placed on understanding why numerical methods work. Exercises are carried out on computers and are supported by programming environment MATLAB.
Main topics: Least-squares method. Numerical differentiation and integration. Initial value problems for ordinary differential equations. Boundary value problems for ordinary differential problems. Partial differential equations of elliptic, parabolic and hyperbolic type.

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

Guarantor

doc. RNDr. Libor Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

Students will be made familiar with the extended collection of numerical methods, namely with the least-squares method, with methods for numerical computing of derivatives and integrals, with the numerical solution of initial and boundary value problems for ordinary differential equations and with the methods for the solution of elliptic, parabolic and hyperbolic partial differential equations.

Prerequisites

Differential and integral calculus for functions of one and more variables. Ordinary differential equations. Numerical methods for solving linear and nonlinear equations. Interpolation.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITION: Active participation in practicals. Elaboration of a semester assignment, where the students prove their knowledge acquired. FORM OF EXAMINATIONS: The exam is oral. Students will be provided with a list of questions at least two weeks before the exam period. ASSESSMENT is completely at the examiner's discretion. If we measure the exam success in percentage points, then the classification grades are: 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.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The aim of the course is to familiarize students with essential methods applied for solving numerical problems, and provide them with an ability to solve such problems individually on computers. Students ought to realize that only the knowledge of substantial features of particular numerical methods enables them to choose a suitable method and an appropriate software product.

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

Attendance at lectures is recommended, attendance at seminars is required. Lessons are planned according to the week schedules. Absence from lessons may be compensated by the agreement with the teacher supervising the seminars.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

M.T. Heath: Scientific Computing. An Introductory Survey. Second edition. McGraw-Hill, New York, 2002.
L.F. Shampine: Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.
C. F. Van Loan, G. H. Golub: Matrix Computations, 3th ed., the Johns Hopkins University Press, Baltimore, 1996.

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme B3901-3 Bachelor's

    branch B3904-00 , 2. year of study, summer semester, elective (voluntary)
    branch B3910-00 , 3. year of study, summer semester, compulsory

  • Programme M2301-5 Master's

    branch M3910-00 , 3. year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

doc. RNDr. Libor Čermák, CSc.

Syllabus

1. Least-squares method. Discrete and continuous variant of the least-squares method. The least-squares solution to an overdetermined linear system of equations.
2. Numerical differentiation. Basic formulas, Richardson extrapolation.
3. Numerical integration. Newton-Cotes formulas, Romberg's method.
4. Numerical integration. Gaussian formulas, adaptive integration, integration in 2D.
5. Initial value problems for ODE1. Basic notions (truncation errors, stability,...)
6. Initial value problems for ODE1. Runge-Kutta methods, step control adjustment.
7. Initial value problems for ODE1. Adams methods, predictor-corrector technique.
8. Initial value problems for ODE1. Backward differentiation formulas. Stiff problems.
9. Boundary value problems for ODE2. Difference method, finite volume method.
10. Boundary value problems for ODE2. Finite element method.
11. Elliptic PDEs. Difference method.
12. Parabolic PDEs. Method of lines, stability of an initial value problem for the system of ODE1, suitable time discretization methods.
13. Hyperbolic PDEs. Method of lines, stability of an initial value problem for the system of ODE2, suitable time discretization methods.

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer

RNDr. Rudolf Hlavička, CSc.

Syllabus

Students create elementary programs in MATLAB related to each subject-matter delivered at lectures and verify how the methods work.