Review for the First Exam

The first exam covers the lectures listed below. Your score will represent 25% of your final grade. No calculators a permitted for this exam. If you need one, you are doing something wrong.


Date	Topics covered:
1/15	Introduction to the Class
	Use of Numerical Methods
	Initial Mathematical Model
	Transformation of Model Equations
	Steps for Use of Finite Difference
1/17	Classification of PDEs
1/22	Formation of Finite Volume Equations
1/24, 1/29	Verification and Validation of Codes and Results
1/31	Formation of Finite Difference Equations
2/5	Finite Difference Equation Examples
2/7	Predictor Corrector Method
2/12, 2/14	Iterative Solution of Linear Equations, Basics
2/19	Domain Decomposition and Multigrid Iteration
2/21	Krylov Subspace Methods
	Some considerations for direct matrix solution
2/26	Introduction to Time Dependent Conduction
2/28	More Time Dependent Conduction, (also check this link to Slides)

Topics covered are:

Steps in problem solution (1, 2, 3, 4)
Classification of PDEs (second order, systems of first order)
Well-posed problems
The Finite Volume Method
Verification and Validation
The Finite Difference Method (undetermined coefficients 1, 2, 3
Predictor Corrector Methods
Iterative Methods

Jacobi
Gauss-Seidel
SOR
Multigrid ( see 1, 2, and 3 for more detailed explanations if the basic description is confusing)
Preconditioning
Krylov Subspace (understand the general concepts, don't memorize equations)

Time Dependent Methods

Explicit (Forward Euler)
Implicit (Backward Euler)
Crank-Nicholson
ADI

Error Analysis

Substitution of Taylor series expansions
Consistency (or Compatibility) with differential Equations

Stability

What is Instability?
Gerschgorin's Theorem, its meaning and use

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