Course Details

Numerical methods II

Academic Year 2023/24

DA63 course is part of 12 study plans

​ D-P-C-SI (N) / PST Winter Semester 2nd year

​ D-P-C-SI (N) / FMI Winter Semester 2nd year

​ D-P-C-SI (N) / KDS Winter Semester 2nd year

​ D-P-C-SI (N) / MGS Winter Semester 2nd year

​ D-P-C-SI (N) / VHS Winter Semester 2nd year

​ D-K-C-SI (N) / VHS Winter Semester 2nd year

​ D-K-C-SI (N) / MGS Winter Semester 2nd year

​ D-K-C-SI (N) / PST Winter Semester 2nd year

​ D-K-C-SI (N) / FMI Winter Semester 2nd year

​ D-K-C-SI (N) / KDS Winter Semester 2nd year

​ D-K-C-GK / GAK Winter Semester 2nd year

​ D-K-E-SI (N) / PST Winter Semester 2nd year

Numerical methods for the initial-value problem for one ordinary differential equation (ODE) of order one and for systems of ODE of order one, absolute stability, variational formulation of boundary-value problems for ODE and partial DE of order two, discretization of elliptic differential problems by the finite difference and the finite element methods, numerical methods for the non-stationary parabolic and hyperbolic differential problems, numerical solution of a nonlinear differential boundary-value problem.

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Objective

Getting acquainted with the basics of the theory of numerical solution of ordinary differential equations and systems of such equations and second-order partial differential equations. Learning how to use numeric methods to solve such equations.

Syllabus

1. Formulation of the initial-value problem in ordinary differential equations of degree 1, basic properties, existence and uniqueness of solutions.
2. Basic numerical methods for the initial-value problems and their absolute stability.
3. Introduction to the variational calculus, basic spaces of integrable functions.
4. Classical and variational formulations of elliptic problems for ordinary differential equations of degree 2, basic physical meanings.
5. Standard finite difference method for elliptic problems for ordinary differential equations (ODE) of degree 2 and its stable modifications.
6. Approximation of boundary-value problems for second order ODE by the finite element method.
7. Classical and variational formulation of elliptic problems for fourth-order ODE and approximation by the finite element method.
8. Classical and variational formulation of elliptic problems for second-order partial differential equations.
9. Finite element method for elliptic problems in second-order partial differential equations.
10. Finite volume method.
11. Discretization of non-stationary problems for second-order differential equations by the method of straight-lines.
12. Mathematical models of flow. Nonlinear problems and problems with dominating convection.
13. Numerical methods for the models of flow.

Prerequisites

Differential and integral calculus of one- and two-functions, interpolation and approximation of functions, numeric differentiation and intgration, numerical linear algebra.

Language of instruction

Czech

Credits

10 credits

Semester

winter

Forms and criteria of assessment

examination

Specification of controlled instruction, the form of instruction, and the form of compensation of the absences

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/272115

Lecture

13 weeks, 3 hours/week, elective

Syllabus

1. Formulation of the initial-value problem in ordinary differential equations of degree 1, basic properties, existence and uniqueness of solutions. 2. Basic numerical methods for the initial-value problems and their absolute stability. 3. Introduction to the variational calculus, basic spaces of integrable functions. 4. Classical and variational formulations of elliptic problems for ordinary differential equations of degree 2, basic physical meanings. 5. Standard finite difference method for elliptic problems for ordinary differential equations (ODE) of degree 2 and its stable modifications. 6. Approximation of boundary-value problems for second order ODE by the finite element method. 7. Classical and variational formulation of elliptic problems for fourth-order ODE and approximation by the finite element method. 8. Classical and variational formulation of elliptic problems for second-order partial differential equations. 9. Finite element method for elliptic problems in second-order partial differential equations. 10. Finite volume method. 11. Discretization of non-stationary problems for second-order differential equations by the method of straight-lines. 12. Mathematical models of flow. Nonlinear problems and problems with dominating convection. 13. Numerical methods for the models of flow.