Course detail

Numerical methods 2

FAST-DAB035Acad. year: 2022/2023

Mathematical approaches to the analysis of engineering problems, namely ordinary and partial differential equations, directed to numerical calculations - deeper knowledge than from the course DA01.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

prof. Ing. Jiří Vala, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

At the level of the course DA61.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

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.

Work placements

Not applicable.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

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

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

DALÍK J., PŘIBYL O., VALA J.: Numerické metody 2 (pro doktorandy). FAST VUT v Brně 2010. (CS)

Type of course unit

Lecture

39 hod., optionally

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

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.

VUT

Faculties

University Institutes

Parts

Numerical methods 2

Type of course unit