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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