Numerical Mathematics 2
Code | Completion | Credits | Range |
---|---|---|---|
01NMA2 | Z,ZK | 3 | 2P+1C |
- Course guarantor:
- Michal Beneš, Tomáš Oberhuber
- Lecturer:
- Michal Beneš, Tomáš Oberhuber
- 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:
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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:
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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:
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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: