Course detail
Numerical Methods II
FSI-SN2 Acad. year: 2024/2025 Summer semester
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 a semester assignement.
Language of instruction
Czech
Number of ECTS credits
4
Supervisor
Department
Entry knowledge
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.
Rules for evaluation and completion of the course
COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in practicals. Elaboration of tasks, where the students prove their knowledge acquired. At least half of all possible 30 points in a credit test using also own programs.
FORM OF EXAMINATIONS: The exam is of test (max. 75 pts.) and oral part (max 25 pts.). As a result of the exam students will obtain 0--100 points.
FINAL COURSE CLASSIFICATION is based on the exam point 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.
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.
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. The development of individual semester assignements constitutes an important experience enabling to verify how the subject matter was managed.
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 acquainted knowledge by elaborating of several assignements.
The study programmes with the given course
Programme B-MAI-P: Mathematical Engineering, Bachelor's, compulsory
Programme B-MET-P: Mechatronics, Bachelor's, compulsory
Programme C-AKR-P: , Lifelong learning
specialization CLS: , elective
Type of course unit
Lecture
26 hours, optionally
Syllabus
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.
Computer-assisted exercise
26 hours, compulsory
Syllabus
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.