Numerical Methods II
FSI-SN2Acad. year: 2020/2021
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: Eigenvalue problems. Initial value problems for ordinary differential equations. Boundary value problems for ordinary differential problems. Partial differential equations of elliptic, parabolic and hyperbolic type. The students will demonstrate the acquinted knowledge by elaborating at least two semester assignements.
Learning outcomes of the course unit
Students will be made familiar with the extended collection of numerical methods, namely with methods for approximation of eigenvalues and eigenvectors, with the numerical solution of initial and boundary value problems for ordinary differential equations and with methods for the solution of elliptic, parabolic and hyperbolic partial differential equations. Students will demonstrate the acquinted knowledge by elaborating of several semester assignements.
Differential and integral calculus for functions of one and more variables. Fundamentals of linear algebra. Ordinary differential equations. Numerical methods for solving linear and nonlinear equations. Interpolation. Programming in MATLAB.
Recommended optional programme components
Recommended or required reading
M.T. Heath: Scientific Computing. An Introductory Survey. Second edition. McGraw-Hill, New York, 2002.
L. Čermák: Numerické metody pro řešení diferenciálních rovnic, [online], available from: http://mathonline.fme.vutbr.cz/Numericke-metody-II/sc-1246-sr-1-a-263/default.aspx.
L.F. Shampine: Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.
L. Čermák: Vybrané statě z numerických metod. [on-line], available from: http://mathonline.fme.vutbr.cz/Numericke-metody-II/sc-1227-sr-1-a-238/default.aspx.
E. Vitásek: Základy teorie numerických metod pro řešení diferenciálních rovnic. Academia, Praha, 1994.
C. F. Van Loan, G. H. Golub: Matrix Computations, 3th ed., the Johns Hopkins University Press, Baltimore, 1996.
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes
COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in practicals. Elaboration of a semester assignments, where the students prove their knowledge acquired. Students, who gain course-unit credits, will also obtain 0--30 points, which will be included in the final course classification.
FORM OF EXAMINATIONS: The exam is oral. As a result of the exam students will obtain 0--70 points.
FINAL ASSESSMENT: The final point course classicifation is the sum of points obtained from both the practisals (0--30) and the exam (0--70).
FINAL 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.
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.
Language of instruction
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. The development of individual semester assignements constitutes an important experience enabling to verify how the subject matter was managed.
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.
Type of course unit
26 hours, optionally
Teacher / Lecturer
1. Eigenvalue problems: basic knowledge.
2. Eigenvalue problems: power method, QR method.
3. Eigenvalue problems: Arnoldi method, Jacobi method, bisection method, computing the singular value decomposition.
4. Initial value problems for ODE1: basic notions (truncation error, stability,...)
5. Initial value problems for ODE1: Runge-Kutta methods, step control adjustment.
6. Initial value problems for ODE1: Adams methods, predictor-corrector technique.
7. Initial value problems for ODE1: backward differentiation formulas, stiff problems.
8. Boundary value problems for ODE2: shooting method, difference method, finite volume method.
9. Boundary value problems for ODE2: finite element method.
10. Elliptic PDEs: difference method, finite volume method.
11. Elliptic PDEs: finite element method.
12. Parabolic and hyperbolic PDEs: method of lines, stability, time discretization methods.
13. First order hyperbolic equation: method of lines, stability, method of characteristics.
26 hours, compulsory
Teacher / Lecturer
Students create elementary programs in MATLAB related to each subject-matter delivered at lectures and verify how the methods work. Furthermore students individually elaborate semester assignemets.