Uppsala universitet
Information Technology

Scientific Computing

Jan Nordström

Analysis of Numerical Methods

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Examination

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Uppsala University
Department of Information Technology
Division of Scientific Computing
Analysis of Numerical Methods
2007-10-22

Analysis of Numerical Methods, Fall 2007

(This information is available on http://www.it.uu.se/edu/course/homepage/anm/ht07/)


The course will start Monday 5 November. The first lecture will be given at 10.15 in room 2345 at Polacksbacken, cf. schedule. You can find the schedule via http://www.teknat.uu.se/student/schema/index.php or http://www.it.uu.se/edu/schedule. The written examination will take place on Tuesday 11 December 2007. It is important that you register for the exam at least 14 days before the exam. Passing the exam and completing the 6 compulsory assignments gives 5 study points = 7.5 ECTS points.

Teacher
Jan Nordström
Room 2441, tel 018 - 471 2766
Jan.Nordstrom@it.uu.se




Why should I take this course ?
Partial differential equations (PDE's) are used to describe most of the existing physical phenomena. The most general and practical way to produce solutions to the PDE's is to use numerical methods and computers. It is essential that the solutions produced by the numerical scheme are accurate and reliable. This course will give you advanced knowledge about various ways to analyse such situations. The course is particulary useful if you intend to work with computer simulations in science and engineering.

Aims of the course (in Swedish)
After finishing the course, the student should
  • explain basic concepts in the numerical solution of partial differential equations like consistency, convergence, stability, efficiency
  • analyze consistency, convergence, stability, efficiency of finite difference methods for partial differential equations
  • analyze stability and efficiency of finite difference methods for linear hyperbolic and parabolic problems with periodic boundary conditions by means of the Fourier method
  • analyze stability of finite difference methods for simple initial-boundary value problems by means of the energy method and the normal mode analysis, i.e. GKS analysis
  • analyze properties of nonlinear partial differential equations and finite difference methods like hyperbolicity, Rankine-Hugoniot conditions, conservativity, total variation diminishing
  • analyze iterative methods for elliptic problems like two-grid multigrid method for model problems
  • program finite difference methods for simple one-dimensional hyperbolic, parabolic and elliptic problems
  • choose and implement suitable numerical methods for solving scientific and engineering problems described by partial differential equations
  • identify deficiencies and limitations of the considered method with regard to efficiency, accuracy and stability


Course contents (in Swedish)
Fundamental properties for numerical methods to solve partial differential equations: consistency, convergence, stability, efficiency. Fourier method to analyze stability and efficiency of finite difference methods for time-dependent partial differential equations. Energy method and normal mode analysis, i.e. GKS analysis, to analyze stability of finite difference methods for simple initial-boundary value problems. Special properties of non-linear partial differential equations and finite difference methods: hyperbolicity, Rankine-Hugoniot conditions, conservativity, total variation diminishing. Multigrid methods for elliptic partial differential equations.

Application
Examples for the application of numerical methods in computational fluid dynamics, aeroacoustics and electromagnetics are the research topics of the TDB research group Waves and fluids.