Advanced Numerical Methods

Lecturer:

Michal Beneš (gar.)

Tutor:

Michal Beneš (gar.)

Supervisor:

Department of Mathematics

Synopsis:

The course is devoted to advanced numerical solution of boundary-value problems and intial-boundary-value problems for ordinary and partial differential equations. It explains the shooting method, advanced finite-difference methods and finite-volume method for nonlinear 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, 01MAB2-4, 01LA1, 01LAB2, 12NME1 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 finite differences for non-linear equations

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

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

2.Convergence and the error estimate

3.Finite volume method

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

1.Method of finite differences for nonlinear evolution problems

2.Method of lines

3. Finite volume method

IV.Numerical solution of hyperbolic conservation laws

1.Formulation and properties of hyperbolic conservation laws

2.Simplest finite-difference methods

3. Finite volume method

Syllabus of tutorials:

Study Objective:

Knowledge:

Numerical methods for nonlinear boundary-value problems, finite-difference method for ODE's and PDE's, finite-volume method.

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, ... Methods, Springer-Verlag, New York 1994

[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

Recommended references:

[5] 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 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans: