Course detail

# Modern Numerical Methods

The course deals with some numerical methods that are used to find the numerical solution of the problem that we can not or are not able to solve analytically. All methods are correctly implemented and in most cases proved. Therefore, the first we focus on the theory of errors introduced in terms of metrics and standards and their relationships. Furthermore, we discuss proceeds with Banach fixed point theorem, which is the basis of a number of numerical methods. Explanation of its action is carried out on systems of linear algebraic equations. The interpretation starts from the finite methods and iterative solution methods. Similarly, we discuss the solution of nonlinear equations, algebraic equations and their systems. We also deal with eigenvalues of the matrix and with the search for solutions to the initial and boundary value problems for ordinary differential equations and their systems and also for partial differential equations. For each numerical methods are included that guarantee convergence of the method.

Learning outcomes of the course unit

After completing the course the student will be able to:
• Work with various matrix and vector norms and make their estimates.
• Solve systems of linear algebraic equations. Decide whether it is possible to solve the system using a given method.
• Find roots of nonlinear and algebraic equations with required accuracy.
• Solve systems of equations.
• Determine the dominant eigenvalue of a matrix.
• Find all eigenvalues. To the suitability of the specified procedure for finding eigenvalues.
• Find the numerical solution of initial value problems for ordinary differential equations and their systems with required accuracy.
• Find the numerical solution of partial differential equations. Work with boundary and internal points system.
• Explain the nature of the finite element method and know how to use it to solve problems on a computer.
• Select the appropriate method for a given type of task and estimate the rate of convergence of certain methods.
• Determine accuracy estimates for certain methods.

Prerequisites

We require knowledge at the level of bachelor's degree, i.e. that students must be able to work with matrices and vectors, handle the calculation of determinants, calculate the product of a matrix and inverse matrix, know the graphs of elementary functions and methods of construction, differentiate and integrate of basic functions, solve basic types of ordinary differential equations of the first order.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Jin J.: The finite element method in electromagnetics. JOHN-WILEY & SONS, INC. New York 1993.
P.P. Sivester, R.L. Ferrari: Finite Elements for Electrical Engineers, Second Edition, Cambridge University Press, 1990,ISBN 0-471-90677-8
J.D.Lambert: Computational Methods in Ordinary Differential Equations, John Wiley & sons, Londyn, 1973,ISBN 0-471-51194-3
Steven C. Chapra, Raymond P. Canale: Numerical Methods for Engineers, Fifth edition, McGraw-Hill 2006,ISBN 007-124429-8

Planned learning activities and teaching methods

Teaching methods depend on types of classes. They are described in Article 7 of the Study and Examination Regulations of Brno University of Technology.

Assesment methods and criteria linked to learning outcomes

Students may be awarded
Up to 40 points for computer exercises for a written test (10 points) and 30 points for individual homework (max. 15 points for the program and the maximum 15 points for presentation and protocol).
Up to 60 points for the written final exam. The test contains both theoretical and numerical tasks that are used to verify the orientation in the problems of numerical methods and their application. This includes tasks such as "adjust to the shape of convergence", without interpolating the end.

Language of instruction

English

Work placements

Not applicable.

Course curriculum

1. The principle of numerical methods, classification and propagation of errors in the numerical process, increasing the accuracy of the calculation, Banach fixed point theorem.
2. Solving systems of linear equations: an overview of finite and iterative solution methods.
3. Review of methods for solving nonlinear equations.
4. Algebraic equations and their properties, estimates of the root position, the method of determining roots of algebraic equations.
5. Solving systems of nonlinear equations. Newton and iterative methods for systems of equations.
6. Eigenvalues. Identification of the dominant eigenvalue.
7. The solution of ordinary differential equations of the first order. Basic concepts, the initial problem, one-step and multi-step methods of solution, Taylor series method.
8. Ordinary differential equations of higher order. Systems of ordinary differential equations of first order and their solutions.
9. Boundary value problems for ordinary differential equation and its solution by finite differences and finite volumes.
10. Finite element methods for ordinary differential equations.
11. Partial Differential Equations. Basic concepts, solutions of partial differential equations of the first order.
12. Classification of partial differential equations of second order. The solution of partial differential equations of second order using method of finite differences.
13. The solution of partial differential equations of second order using the finite elements method.

Aims

The aim is to extend and intesify knowledge from the previous courses, namely in connexion with practical applications of the methods for solving the ordinary a partial differential equations. For this purpose two chapters summarizing the methods for solving linear and nonlinear equations precede.

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

Computer exercises are compulsory. Properly excused absence can be replaced by individual homework, which focuses on the issues discussed during the missed exercise.
Specifications of the controlled activities and ways of implementation are provided in annual public notice.
Date of the written test is announced in agreement with the students at least one week in advance. The new term for properly excused students is usually during the credit week.

Classification of course in study plans

• Programme AUDIO-PU Master's

branch PU-AUD , 1. year of study, summer semester, 5 credits, optional interdisciplinary

• Programme EEKR-MN Master's

branch MN-BEI , 1. year of study, summer semester, 5 credits, theoretical subject
branch MN-TIT , 1. year of study, summer semester, 5 credits, theoretical subject
branch MN-KAM , 1. year of study, summer semester, 5 credits, theoretical subject
branch MN-EST , 1. year of study, summer semester, 5 credits, theoretical subject
branch MN-MEL , 1. year of study, summer semester, 5 credits, theoretical subject
branch MN-SVE , 1. year of study, summer semester, 5 credits, theoretical subject
branch MN-EEN , 1. year of study, summer semester, 5 credits, theoretical subject

• Programme AUDIO-PU Master's

branch PU-AUD , 2. year of study, summer semester, 5 credits, optional interdisciplinary

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Computer exercise

13 hours, compulsory

Teacher / Lecturer

eLearning

eLearning: opened course