Numerical Mathematics 2

Course guarantor:

Michal Beneš

Lecturer:

Michal Beneš

Tutor:

Michal Beneš, 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:

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

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

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

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

[5] J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, 2013

Recommended references:

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

[7] 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 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans: