Advanced Numerical Methods
Code  Completion  Credits  Range 

01PNM  KZ  2  2+0 
 Lecturer:
 Michal Beneš (guarantor)
 Tutor:
 Michal Beneš (guarantor)
 Supervisor:
 Department of Mathematics
 Synopsis:

The course is devoted to advanced numerical solution of boundaryvalue problems and intialboundaryvalue problems for ordinary and partial differential equations. It explains the shooting method, advanced finitedifference methods and finitevolume method for nonlinear elliptic, parabolic and firstorder hyperbolic partial differential equations.
 Requirements:

Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAB24, 01LA1, 01LAB2, 12NME1 held at the FNSPE CTU in Prague).
 Syllabus of lectures:

I.Numerical solution of ordinary differential equations  boundaryvalue problems
1.Shooting method
2Method of finite differences for nonlinear equations
II.Numerical solution of partial differential equations of the elliptic type
1.Finitedifference method for nonlinear secondorder 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 finitedifference methods
3. Finite volume method
 Syllabus of tutorials:
 Study Objective:

Knowledge:
Numerical methods for nonlinear boundaryvalue problems, finitedifference method for ODE's and PDE's, finitevolume 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] J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, 2013
[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:
 Timetable for winter semester 2020/2021:
 Timetable is not available yet
 Timetable for summer semester 2020/2021:
 Timetable is not available yet
 The course is a part of the following study plans:

 Informatická fyzika (compulsory course of the specialization)
 Aplikace softwarového inženýrství (compulsory course of the specialization)