Objectives and significance:

Many application problems leads to partial differential equations in which analytic solutions are rarely available or too complicated. Finite difference methods are often easy to use but powerful to obtain an approximate solution of the PDE. It is strongly believed that the knowledge of Finite Difference Methods for PDEs is a MUST for mathematicians, scientists, and engineers,  who are interested in solving their problems approximately. It is a required course in many universities.

 

This course is designed for students in applied mathematics, engineering, and the sciences to learn the basic theories and algorithms of finite difference methods for differential equations including elliptic, parabolic and hyperbolic PDE's. While theoretical foundations will be described, emphasis will also be placed on algorithm design and implementation. We will also explore available software packages in this field.

Calculus I-III, some background  in linear algebra, numerical analysis, and partial differential equations; Some programming experience (Matlab, recommended, Fortran, C, C++, ...)

Grading Policy:

There will be homework assignments about every couple of weeks, including both analytic work and computer projects.You can choose 4 out of 6.

• HW & Project: 70%
• Final project or a take home exam: 30%
  
• Class participation extra credit: 5%, questions and activities.
  

Materials:

Introduction (ODE/PDE and classification, analytic approaches versus numerical approximation, finite difference versus finite element method)

A model problem (two point boundary value problem) and the finite difference method. Finite difference methods basics, stability and consistency, etc.

Finite difference method for general one dimensional elliptic boundary value problems with different boundary conditions

Finite difference methods for two dimensional elliptic PDEs (may include multi-grid and fast Poisson solvers).

Finite difference methods for one and two dimensional parabolic PDEs, e.g. the heat equation; von Neumann stability analysis and Fourier transforms, ADI method.

Finite difference methods for one and two dimensional hyperbolic PDEs, e.g. the wave equation, numerical methods for conservation laws.

Advanced topics, irregular domain, the level set method etc. if time permits.

Text :

• Instructor's Book Draft: forthcoming by Cambridge,  Numerical Solutions of Partial Differential Equations– An Introduction to Finite Difference and Finite Element Methods
  
• Finite Difference Methods for Ordinary and Partial Differential Equations, R. J. LeVeque, SIAM 2007
• Instructor's Notes will be updated constantly.

• Finite Difference Schemes and Partial Differential Equations, J. C. Strilwerda, Chapman & Hall, 1989.
• Numerical Solution of Partial Differential Equations,  K. W. Morton and D. F. Mayers, Cambridge press, 1995.

Calendar:

                            August              September        
                      Su Mo Tu We Th Fr Sa  Su Mo Tu We Th Fr Sa  
                                                   1  2  3  4  5  
                                             6  7  8  9 10 11 12  
                         10 11 12 13 14 15  13 14 15 16 17 18 19  
                      16 17 18 19 20 21 22  20 21 22 23 24 25 26  
                      23 24 25 26 27 28 29  27 28 29 30           
                      30 31                                       

      October               November                      
Su Mo Tu We Th Fr Sa  Su Mo Tu We Th Fr Sa    
             1  D  3   1  2  3  4  5  6  7      
 4  5  6  7  8  9 10   8  9 10 11 12 13 14     
11 12 13 14 15 16 17  15 16  L 18  F 20 21    
18 19 20 21 22 23 24  22 23 24 25 26 27 28    
25 26 27 28 29 30 31            

H: Holiday;   D: drop date;  V: Vacation (No class);  L: Last day of instruction.