MAT7202 Numerical Solutions of PDEs

This course will discuss numerical techniques for solving partial differential equations. The emphasis will be a blend of theory and numerical application, however, rather than simply present one scheme after another, the goal is to dig deeper and understand why some schemes work and others don't.
We will begin with the parabolic PDE and end with numerical solution of elliptic PDE and finite element method.

Course Objectives
This course aims at developing applications of numerical techniques in partial differential equations. It will provide the base for research in partial differential equations.

• Students should learn the principles for designing numerical schemes for PDEs, in particular, finite difference schemes.
• Students should learn to make a connection between the mathematical equations or properties and the corresponding physical meanings.
• Students should be able to analyze the consistency, stability and convergence of a numerical scheme.
• Students should know, for each type of PDEs (hyperbolic, parabolic and elliptic), what kind of numerical methods are best suited for and the reasons behind these choices.
• Students should be able to use a programming language or math software (Matlab, Maple or Mathematica) to implement and test the numerical schemes. 

Learning Outcomes
The student will be able to:

• Solve different applied problems in engineering and science.
• Use this course in Marine engineering, fluid dynamics, hydrodynamics and fluid flow 
• Demonstrate the principles for designing numerical schemes for PDEs, in particular, finite difference schemes. 
• Establish the connection between the mathematical equations or properties and the corresponding physical meanings. 
• Analyze the consistency, stability and convergence of a numerical scheme. 
• Apply the numerical techniques to each type of PDEs (hyperbolic, parabolic and elliptic), what kind of numerical methods are best suited for and the reasons behind these choices. 
• Apply a programming language or math software (Matlab, Maple or Mathematica) to implement and test the numerical schemes.