Course detail

Numerical methods II

FAST-DA63Acad. year: 2024/2025

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.

Language of instruction

Czech

Number of ECTS credits

10

Mode of study

Not applicable.

Guarantor

prof. Ing. Jiří Vala, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Entry knowledge

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

Rules for evaluation and completion of the course

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

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.

Study aids

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)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme D-K-C-SI (N) Doctoral

    branch VHS , 2 year of study, winter semester, compulsory-optional
    branch MGS , 2 year of study, winter semester, compulsory-optional
    branch PST , 2 year of study, winter semester, compulsory-optional
    branch FMI , 2 year of study, winter semester, compulsory-optional
    branch KDS , 2 year of study, winter semester, compulsory-optional

  • Programme D-K-C-GK Doctoral

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

  • Programme D-K-E-CE (N) Doctoral

    branch FMI , 2 year of study, winter semester, compulsory-optional
    branch KDS , 2 year of study, winter semester, compulsory-optional
    branch MGS , 2 year of study, winter semester, compulsory-optional
    branch VHS , 2 year of study, winter semester, compulsory-optional
    branch PST , 2 year of study, winter semester, compulsory-optional

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.