Course detail
Numerical Methods II
FSI-SN2 Acad. year: 2027/2028 Summer semester
Numerical Methods II builds upon the Numerical Methods I course and introduces students to fundamental procedures for solving advanced numerical problems frequently encountered in engineering practice. Students verify their comprehension of algorithms by solving computer-based tasks, utilizing commercial software as well as writing their own code. Covered topics: Eigenvalue and eigenvector calculations, initial and boundary value problems for ODEs, and elliptic, parabolic, and hyperbolic PDEs.
Language of instruction
Czech
Number of ECTS credits
4
Supervisor
Department
Entry knowledge
Students are expected to possess foundational knowledge of differential and integral calculus of functions of one and several variables, basic linear algebra, analytical and numerical methods for solving ordinary and non-linear equations, basics of interpolation, and programming/algorithmization skills.
Rules for evaluation and completion of the course
Requirements for Awarding Course Credit: Compulsory and recorded attendance in computer labs. Submission of assigned individual term projects and computer tasks. Scoring at least 50% of total possible points (min. 15 out of 30 points) combined across the theoretical credit test and the computer practical test.
Exam and grading: The exam consists of a written part (max. 75 points) and an oral part (max. 25 points). Total exam score ranges from 0 to 100 points.
Grading scale: 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.
Lecture attendance is recommended; lab attendance is compulsory. Replacement of missed classes is determined by the lab instructor.
Aims
The aim of the Numerical Methods II course is to introduce students to basic techniques for solving selected advanced numerical problems and to equip them with the ability to solve these problems independently using computers. Students will understand the essential properties of individual numerical methods, enabling them to make an effective choice of suitable methods and corresponding software products. Independent work on assigned projects forms an essential part of the course.
Graduates will acquire knowledge of eigenvalue and eigenvector approximations, solving initial and boundary value problems for ordinary differential equations, and solving elliptic, parabolic, and hyperbolic partial differential equations. Students will verify and deepen their knowledge through term projects.
The study programmes with the given course
Programme B-MAI-P: Mathematical Engineering, Bachelor's, compulsory
Programme B-MET-P: Mechatronics, Bachelor's, compulsory
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-size control.
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
Computer labs take place in PC classrooms. For each lecture topic, students create MATLAB scripts to verify method performance. Independent work on term projects and taking credit tests form an integral part of the lab sessions.