Numerical Analysis (I)

Scientific Computing, Spring 2001.

A graduate course in Mathematics at the Department of Mathematics, National Tsing Hua University.

Instructor: Professor Wei-Cheng Wang,

Class meeting: General III Building, room 723. Official schedule: T567. Currently T: 1:10-2:25 PM, R: 10:10-11:25 AM

Grading: 4-5 sets of homework assignments, mostly programming.

Textbook and References: Textbook: Lecture notes distributed in class.

Course description: This course is a preparation for my next research topic: Numerical Simulation in Astrophysics. The relevant equation there is the compressible MHD equation coupled with Poisson equation. Many fields of computational PDE's are involved here. I will try to introduce them independently so that this will essentially be a course for numerical methods in PDE. Two similar courses are offered by NCTS and held in Taiwan University. I intend to make this course introductory and emphasize on numerical implementations. Part of this course (around 3 weeks on shock capturing schemes) will be held at Taiwan University on Monday mornings. Students are required to do independent programming using modern languages (any one of matlab, c, c++, fortran ...) Some knowledges on numerical programming is preferred. A preliminary undergraduate course of numerical analysis (or equivalent) is required. Undergraduate students are welcome.

Topics (temporary):

Numerical integration of ODE: Euler methods, discretization error: Runge Kutta methods: Multi-step methods: convergence and stability analysis: Stiff problems and implicit methods:
Numerical methods for PDE: General topics: Finite Difference Finite Element Spectral methods. Special Topics: introduction to hyperbolic conservation laws: Shock Capturing schemes for hyperbolic conservation laws:
Fast Fourier Transform and Fast Poisson Solver: Fast Fourier Transform. Direct and iterative methods for Laplace equation. FFT as Fast Poisson solver on rectangular domain with uniform mesh. Variants of FFT, with application to different boundary conditions. Spherical and cylindrical coordinate: Cyclic reduction. Curvilinear coordinate system and/or non-structured grids. Multi-grid methods. conjugate gradient methods. Preconditioning. Domain decomposition methods.

Other sources of useful information:

Netlib is an excellent net resource. It contains tons of reliable numerical software that can be downloaded and links to most technical information on numerical analysis and scientific computing on the internet. The ftp mirror site in Taiwan is at the National Center for High Performance Computing.

Matlab Information

Matlab is an excellent environment for experimentation in scientific computing. It is not the most efficient environment for large-scale simulations, although with some effort, Matlab can be used in conjunction with C and Fortran routines. Matlab is a product of The Mathworks. For a tutorial, see A Free Matlab Online Tutorial or Another Tutorial or Matlab Primer
You can start from our class note on matlab before reading these materials.

Source code for the Fortran compiler, DOS version.

A short tutorial for C programming language.

Announcement:

Course materials for numerical integration of ODE

Supplementary course materials for interpolation, pay attention to the behavior of the interpolating polynomial near both ends of intervals. The behavior outside the interval -- the extrapolation-- can be even worse.

Thanks to Professor I-Liang Chern of National Taiwan University. for providing the lecture notes on Theory and Numerics of Conservation laws. Updated Chap 1.

Homework assignments:

Postscript viewer: To view the Postscript files on your PC screen, you may need to install the GSview from here. If you are running MS Windows, Professor William Ferng at Chiao-Tung University has setup an installation guide here.

PDF viewer: To view the PDF files on your PC screen, you may need to install the Acrobat Reader from Adobe. (Other softwares, such as GSview, can be possible alternatives.) Acrobat Reader can be downloaded from the Adobe site