 CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2022/2023

# Numerical Mathematics 2

Code Completion Credits Range
01NUM2 Z,ZK 3 2+1
Lecturer:
Michal Beneš (guarantor)
Tutor:
Michal Beneš (guarantor), Tomáš Oberhuber
Supervisor:
Department of Mathematics
Synopsis:

The course is devoted to numerical solution of boundary-value problems and intial-boundary-value problems for ordinary and partial differential equations. It explains methods converting boundary-value problems to initial-value problems and finite-difference methods for elliptic, parabolic and first-order hyperbolic partial differential equations.

Requirements:

Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, 01NM held at the FNSPE CTU in Prague).

Syllabus of lectures:

I.Numerical solution of ordinary differential equations - boundary-value problems

1.Shooting method

2Method of transformation of a boundary-value problem

3.Method of finite differences

4.Solution of non-linear equations

II.Numerical solution of partial differential equations of the elliptic type

1.Finite-difference method for linear second-order equations

2.Convergence and the error estimate

3.Method of lines

III.Numerical solution of partial differential equations of the parabolic type

1.Method of finite differences for one-dimensional problems

2.Method of finite differences for higher-dimensional problems

3.Method of lines

IV.Numerical solution of hyperbolic conservation laws

1.Formulation and properties of hyperbolic conservation laws

2.Simplest finite-difference methods

Syllabus of tutorials:

1.Taylor expansion in the context of difference formulas with particular properties

2.Normalized conversion method

3.Nonlinear difference schemes.

4.Definition of the weak solution of an elliptic boundary-value problem.

5.Relation of difference approximations and of the finite-volume method

Study Objective:

Knowledge:

Numerical methods based on transformation of a boundary-value problem to an initial-value problem, finite-difference method for ODE's and PDE's.

Skills:

Application of given methods in particular examples in physics and engineering including computer implementation and error assessment.

Study materials:

Key references:

 A.A. Samarskij, Theory of Difference Schemes, CRC Press, New York, 2001

 I. Babuška, M. Práger, E. Vitásek, Numerical Processes in Differential Equations, Wiley, London 1966

 R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007

 R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002

 J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science &amp; Business Media, 2013

Recommended references:

 R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007

 E. Godlewski a P.-A. Raviart, Numerical approximation of hyperbolic systems of conversation laws, New York, Springer 1996

Media and tools:

Computer training room with Windows/Linux and programming languages C, Pascal, Fortran.

Note:
Time-table for winter semester 2022/2023:
Time-table is not available yet
Time-table for summer semester 2022/2023:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2023-01-28
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