Summary of the Semester


1/11 Introduction to the Class

Use of Numerical Methods

Initial Mathematical Model
1/13 Classification of PDEs

Transformation of Model Equations

Steps for Use of Finite Difference
1/18 Formation of Finite Volume Equations 
1/20, 1/25
Verification and Validation of Codes and Results
1/27 Formation of Finite Difference Equations
2/1 Finite Difference Equation Examples
2/3 Predictor Corrector Method
2/8-2/10 Iterative Solution of Linear Equations, Basics
2/15 Domain Decomposition and Multigrid Iteration
2/17 Krylov Subspace Methods
Some considerations for direct matrix solution
2/24 Introduction to Time Dependent Conduction
(Explicit Method and Gerschgorin Analysis)
3/1 Review for Exam
More Time Dependent Conduction
3/15 More Stability Analysis
3/17, 3/22 The Advection-Diffusion Equation

The SIMPLE Method for CFD analysis
3/24 QUICKEST and effects of Source Terms
3/29 Quantifying Numerical Diffusion 
Full Flow Equation Set
Use of Numerically Generated Jacobians
A sample testbed program
Non-orthogonal Grids
4/12, 4/14
Use of Overset Grids in Computational Fluid Dynamics
4/19 Final Project Discussions 
4/21, 4/26
Application to Multi-phase and/or reacting flow
4/28 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.

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



Topics for the Second Exam

 Time dependent Conduction
(Implicit, Crank-Nicholson, ADI)

More Stability Analysis (von Neumann, Hirtís method)

The Advection-Diffusion Equation

QUICKEST and effects of Source Terms

Full Flow Equation Set

Use of Numerically Generated Jacobians

A sample testbed program

Overset Grids (general concepts from Angel reading)

Quantifying Numerical Diffusion


Non-orthogonal Grids

Application to Multi-phase and/or reacting flow






Other Lectures / Home

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