Date / Lecture Number / Topics covered:
1/14 / 1 / Introduction to the Class
Use of Numerical Methods
Initial Mathematical Model
1/16 / 2 / Classification of PDEs
Transformation of Model Equations
Steps for Use of Finite Difference
1/21 / 3 / Formation of Finite Volume Equations
1/23 / 4 / Formation of Finite Difference Equations
1/28 / 5 / Finite Difference Equation Examples
1/30 / 6 / Predictor Corrector Method
2/4 / 7 / Iterative Solution of Linear Equations, Basics
2/6 / 8
2/11 / 9 / Domain Decomposition and Multigrid Iteration
2/13 / 10 / Krylov Subspace Methods
2/18 / 11 / Some considerations for direct matrix solution
12 / Accuracy of Boundary Conditions
2/20, 2/25, 2/27 / 13, 14, 15 / Time dependent Conduction
3/4 / 16 / Review for Exam
3/6 / 17 / More Stability Analysis
3/18, 3/20 / 18, 19 / The Advection-Diffusion Equation
3/25 and 3/27 / 20, 21 / The SIMPLE Method for CFD analysis (Dr. Kunz)
4/1 / 22 / QUICKEST and effects of Source Terms
4/3 / 23 / Quantifying Numerical Diffusion
Wiggles
4/8 / 24 / Full Flow Equation Set
4/10 / 25 / Use of Numerically Generated Jacobians
4/15 / 26 / A sample testbed program
4/17 / 27 / Non-orthogonal Grids
4/22 / 28 / Final Project Discussions
4/24, 4/29 / 29, 30 / Application to Multi-phase and/or reacting flow
5/1 / 31 / Semester Summary

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 verygood 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. You’ve 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 haven’t 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 we’re 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, Basics
Domain Decomposition and Multigrid Iteration
Krylov Subspace Methods
Some considerations for direct matrix solution
Accuracy of Boundary Conditions
Time dependent Conduction
Topics after the First Exam
More Stability Analysis (von Neumann)
The Advection-Diffusion Equation (higher order methods)
The SIMPLE Method for CFD analysis
QUICKEST and effects of Source Terms
Quantifying Numerical Diffusion
Wiggles
Full Flow Equation Set , solution of systems of non-linear equations
Use of Numerically Generated Jacobians
A sample testbed program
Non-orthogonal Grids (finite volume and finite difference)
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