Advanced Numerical Methods
- Garant předmětu:
- Department of Mathematics
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.
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
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:
Numerical methods for nonlinear boundary-value problems, finite-difference method for ODE's and PDE's, finite-volume method.
Application of given methods in particular examples in physics and engineering including computer implementation and error assessment.
- Study materials:
 A.A. Samarskij, Theory of Difference Schemes, CRC Press, New York, 2001
 J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, 2013
 R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007
 R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002
 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.
- Further information:
- No time-table has been prepared for this course
- 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)