Course detail

Numerical methods II

FAST-DA63Acad. year: 2018/2019

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.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

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

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Language of instruction

Czech

Work placements

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.

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.

Classification of course in study plans

  • Programme D-K-C-SI (N) Doctoral

    branch PST , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch PST , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-K-E-SI (N) Doctoral

    branch PST , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch PST , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-K-E-SI (N) Doctoral

    branch MGS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch MGS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch KDS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-K-C-SI (N) Doctoral

    branch KDS , 2. year of study, winter semester, 10 credits, compulsory-optional
    branch MGS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch MGS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-K-E-SI (N) Doctoral

    branch FMI , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch FMI , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-K-C-SI (N) Doctoral

    branch FMI , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch FMI , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch KDS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-K-E-SI (N) Doctoral

    branch KDS , 2. year of study, winter semester, 10 credits, compulsory-optional
    branch VHS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-K-C-SI (N) Doctoral

    branch VHS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-E-SI (N) Doctoral

    branch VHS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-C-SI (N) Doctoral

    branch VHS , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-P-C-GK Doctoral

    branch GAK , 2. year of study, winter semester, 10 credits, compulsory-optional

  • Programme D-K-C-GK Doctoral

    branch GAK , 2. year of study, winter semester, 10 credits, compulsory-optional

Type of course unit

 

Lecture

39 hours, optionally

Teacher / Lecturer

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.