Summary of the Semester


 

Date

Lecture Number

Topics covered:

1/13

1

Introduction to the Class

 

 

Use of Numerical Methods

 

 

Initial Mathematical Model

1/15

2

Classification of PDEs 

 

 

Transformation of Model Equations

 

 

Steps for Use of Finite Difference

1/20

3

Formation of Finite Volume Equations 

1/22

4

Formation of Finite Difference Equations

1/27

5

Finite Difference Equation Examples

1/29

6

Predictor Corrector Method

2/4

7

Iterative Solution of Linear Equations,

2/5

8

 

2/10

9

Domain Decomposition and Multigrid

2/12

10

Krylov Subspace Methods

2/17

11

 Donsiderations for direct matrix solution

 

12

Accuracy of Boundary Conditions

2/19, 2/24, 2/26

13, 14, 15

Time dependent Conduction

3/2

16

Review for Exam

3/4

17

More Stability Analysis

3/16, 3/18

18, 19

The Advection-Diffusion Equation

 

 

The SIMPLE Method for CFD analysis

3/23

20

QUICKEST and effects of Source Terms

3/25

21

Full Flow Equation Set

3/30

22

Use of Numerically Generated Jacobians

4/1

23

A sample testbed program

4/6 and 4/8

24,25

Verification and Validation

4/13

26

Quantifying Numerical Diffusion  Wiggles

4/15

27

Non-orthogonal Grids

4/20

28

Final Project Discussions 

4/22, 4/27

29, 30

Application to Multi-phase and/or reacting flow

4/29

31

Semester Summary


 

 


 

The Big Picture

 

What have we been doing?  Computer simulation is a method of performing experiments without costly hardware.  To be in this business, you need to become a very good experimentalist.  Always remember that your experiments are performed in virtual worlds where the physical laws may be close to those of the real world, but never really match.  Youíve got to use your full training as an engineer to detect the differences in physical laws and understand their impact.  Never trust the results of a computer simulation until you (or someone you know and trust) have thoroughly tested the relationship between the virtual and real worlds. 

 

If you havenít learned the skills already, spend time learning how to construct controlled experiments.  One major advantage of computer simulations is that you have far more opportunities for highly controlled experiments than you do in the real world.  During this process you will be applying basic scientific method:

1.    Make careful observations of a system;

2.    Make a hypothesis to explain those observations;

3.    Design a test (or tests) for the hypothesis;

4.    Perform the test;

5.    Either confirm your hypothesis, or revise it (loop back to 2).

When designing a test, limit the changes you introduce into the system.  In computer simulation, there is almost never an excuse for introducing more than one change at a time.

 

While weíre talking about science, I want to introduce a broader definition.  One of our basic characteristics as human beings is that we see what we want to see.  This is not wholly a weakness.  Science is a discipline that we have built over millennia, to help us see what is really there.  When properly used Computer Simulation is a tool that can help us see what is really there.  However, be cautious of your fundamental nature.  Do not except the results of a computer simulation (or any other observation) because they are what you want to see.  Use all of your experience and training to be certain that the results adequately reflect reality.



 


Topics covered through the First Exam

Use of Numerical Methods

Initial Mathematical Model

Classification of PDEs 

Transformation of Model Equations

Steps for Use of Finite Difference

Formation of Finite Volume Equations 

Formation of Finite Difference Equations

Finite Difference Equation Examples

Predictor Corrector Method

Iterative Solution of Linear Equations,

Domain Decomposition and Multigrid

Krylov Subspace Methods

 Donsiderations for direct matrix solution

Accuracy of Boundary Conditions

Time dependent Conduction

 



 

Topics after the First Exam

 

More Stability Analysis (von Neumann, Hirtís method)

The Advection-Diffusion Equation

The SIMPLE Method for CFD analysis

QUICKEST and effects of Source Terms

Full Flow Equation Set

Use of Numerically Generated Jacobians

A sample testbed program

Verification and Validation

Quantifying Numerical Diffusion  Wiggles

Non-orthogonal Grids

Application to Multi-phase and/or reacting flow

 

 

 

 



Summary of steps in problem solution

 

1.   Determine appropriate mathematical model

2.   Classification of partial differential equation

3.   Transformation of mathematical model

4.   Select grid pattern

5.   Formation of finite difference equations

6.   Solution algorithm

7.   Perform auxiliary calculations

 


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