Course detail

Numerical Methods II

FSI-9NM2Acad. year: 2024/2025

The course deals with the numerical solution of differential equations. First initial-value problems are studied (Runge-Kutta methods, linear multistep methods (especially Adams methods and backward differentiation methods), solution of stiff problems). Next solution methods for boundary value problems are introduced (the finite difference method, the control volume method and the finite element methods). The principles of those methods are explained for 1D second order boudary value problem. Main emphasis is placed on the finite element method in two dimensions. The following model problems are studied: elliptic (stationary heat transfer), parabolic (nonstationary heat transfer) and hyperbolic (membrane vibration including eigenproblems).

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

doc. Ing. Petr Tomášek, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Linear algebra, vector calculus, differential and integral calculus, basics of programming.

Rules for evaluation and completion of the course

The exam has an oral part only. The student has to answer one question from the range "numerical solution of initial value problems" and one or two questions from the range "numerical solution of partial diffetential problems" (from which always one concerns the finite element method in 2D). Emphasis is put on understanding the fundamentals of methods, formulae is not necessary to know by heart, it is howewer necessary to understand them and by means of those formulas to explain, how methods work.
Attendance at lectures is facultative, but highly recommended.

Aims

The aim of the course is to teach students the basic principles of modern computational methods used for the solution of problems described by differential equations. Based on this knowledge they ought to be able to choose suitable software product (exceptionally to write their own program) and then succesfully apply it.
Many engineering problems make for the solution of differential equation, both ordinary and partial. Skills obtained in this course equip students with the necessary minimum knowledge of basic numerical technics used in today's software packages intended for the solution of differential equations.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

K.J. Bathe: Finite Elemets Procedures. Prentice-Hall, Upper Saddle River, NJ, 1996.
O.C. Zienkiewicz, R.L. Taylor: The Finite Element Method. Volumes I,II,III. Butterworth-Heinemann, Oxford, 2000.
V. Kolář, J. Kratochvíl, F. Leitner, A. Ženíšek: Výpočet plošných a prostorových konstrukcí metodou konečných prvků. SNTL, Praha, 1979.

Recommended reading

L. Čermák: Algoritmy metody konečných prvků. Skripta FSI VUT v Brně, PC-DIR Real, Brno, 2000. http://mathonline.fme.vutbr.cz/Numericke-metody-III/sc-1151-sr-1-a-142/default.aspx
L. Čermák: Numerické metody II. Skripta FSI VUT v Brně, CERM, Brno, 2004.

Classification of course in study plans

  • Programme D-ENE-P Doctoral 1 year of study, summer semester, recommended course
  • Programme D-ENE-K Doctoral 1 year of study, summer semester, recommended course

Type of course unit

 

Lecture

20 hod., optionally

Teacher / Lecturer

doc. Ing. Petr Tomášek, Ph.D.

Syllabus

The course has 10 two-hours lessons.
1. The Runge-Kutta methods: basic notions (truncation errors, stability,...), formulas of the order 1 and 2.
2. Further Runge-Kutta formulas (of order 3 to 5), step control adjustment.
3. Adams methods, predictor-corector technique.
4. Backward differentiation formulas. Stiff problems.
5. The difference method, the control volume method and the finite element method in 1D.
6. The stationary 2D problem: classical and variational formulation, linear triangular element.
7. Stiffness matrix, load vector.
8. Assembly of global system of equations. Minimization formulation.
9. Nonstationary 2D problems: heat flow, membrane vibration, eigenvalues.
10. Izoparametric elements.