Scientific Computing (I)

Numerical Methods for PDE, Fall 2004.

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

Instructor: Professor Wei-Cheng Wang,

Class meeting: 綜三 734. Meeting schedule: W567.

First meeting: Wednesday 9/22, 1:10 PM. (Not 9/15)

Grading: 4-5 sets of homework assignments, including paper work and basic programming.

Course description: The fluid equation (compressible and incompressible, Euler and Navier-Stokes) is a complicated system of PDE of mixed type. The efficient computation of the fluid interacting with the physical boundary is the central issue of computational fluid dynamics in past decades. The purpose of this course (Numerical methods for PDE + CFD) is to ultimate lead you to the state-of-the art modern development of effective computation of the fluid equation (with emphasize in incompressible Navier-Stokes). In the first semester, we will introduce numerical methods for ODE and PDE of elliptic, parabolic and hyperbolic type. These are on one hand, the building block for CFD and on the other hand, self-contained as an introductory course in numerical analysis/scientific computation. The course will emphasize on the analysis aspect with some programming requirement. Students are required to do independent homework assignments on paper works and programming using modern languages (any one of matlab, c, c++, fortran ...). Some knowledges on numerical programming. and a preliminary undergraduate course of numerical analysis (or equivalent) is required. Undergraduate students are welcome.

Textbook and References: For the first semester (NMPDE), most materials can be found in standard materials such as John C. Strikwerda: Finite Difference Schemes and Partial Differential Equations; or R. Leveque: Finite Difference Method for Differential Equations. (available on line at http://www.amath.washington.edu/~rjl/booksnotes.html) Some recent research results will be distributed in class as lecture notes. For the second semester (CFD), typed lecture notes will be distributed in class.

Topics and Contents:

Numerical Integration of ODE: Explicit and implicit methods, basic facts about stability and convergence; applications. Various notions of stability. Local Truncation Error and Global error. The Duhamel's Principle. Two point boundary value problems.
Introduction to elliptic, parabolic and hyperbolic PDE.
Direct and Iterative Methods for Elliptic Equations: Finite difference methods on regular domain; Fourier Transform and cyclic reduction. Elliptic interface problems. Differential operator in curvilinear coordinate systems. Finite element methods; Jacobi and Gauss-Siedel Iteration, Successive Over Relaxation and Variants. Preconditioned Conjugate Gradient Method and Variants. Maximum Principle and Energy Estimate.
Explicit, Implicit and Semi-Explicit methods for Parabolic Equations: The von-Neumann Analysis. Explicit time step constraints for Parabolic equations. Implicit methods: Backward Euler and Crank-Nicholson. Semi-Implicit method.
Linear and quasi-linear Hyperbolic PDE's: Linear transport equation. Introduction to Hyperbolic Conservation Laws. Lax Equivalence Theorem. Yee's scheme and Maxwell's Equation. Transport Diffusion Equation and Operator Splitting.

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.

A short tutorial for C programming language.

Homework assignments:

Postscript viewer: To view the Postscript files on your PC screen, you may need to install the GSview from 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

Assignment 1: Numerical Integration of ODE. .

Assignment 2: Direct and Iterative Solvers for Elliptic PDE .

Assignment 3: .

Assignment 4: .