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
Mode of study
Guarantor
Department
Entry knowledge
Rules for evaluation and completion of the course
Attendance at lectures is facultative, but highly recommended.
Aims
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
Prerequisites and corequisites
Basic literature
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: Numerické metody II. Skripta FSI VUT v Brně, CERM, Brno, 2004.
Classification of course in study plans
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
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.