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 boundaryvalue problems and intialboundaryvalue problems for ordinary and partial differential equations. It explains methods converting boundaryvalue problems to initialvalue problems and finitedifference methods for 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, 01MAA24, 01LA1, 01LAA2, 01NM held at the FNSPE CTU in Prague).
 Syllabus of lectures:

I.Numerical solution of ordinary differential equations  boundaryvalue problems
1.Shooting method
2Method of transformation of a boundaryvalue problem
3.Method of finite differences
4.Solution of nonlinear equations
II.Numerical solution of partial differential equations of the elliptic type
1.Finitedifference method for linear secondorder 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 onedimensional problems
2.Method of finite differences for higherdimensional problems
3.Method of lines
IV.Numerical solution of hyperbolic conservation laws
1.Formulation and properties of hyperbolic conservation laws
2.Simplest finitedifference 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 boundaryvalue problem.
5.Relation of difference approximations and of the finitevolume method
 Study Objective:

Knowledge:
Numerical methods based on transformation of a boundaryvalue problem to an initialvalue problem, finitedifference 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:
 Timetable for winter semester 2022/2023:
 Timetable is not available yet
 Timetable for summer semester 2022/2023:
 Timetable is not available yet
 The course is a part of the following study plans:

 BS Matematické inženýrství  Matematické modelování (compulsory course of the specialization)
 BS Matematická informatika (compulsory course of the specialization)