Last updated at 10:59 PM on 28 Jan 2008

This page provides learning objectives and other supplementary iformation to support lectures in ME 448 during Winter 2008. Lectures are presented in reverse chronological information, i.e. the most recent lecture is listed first.

6. Star CCM+ Demo

Lecture on 28 January 2008

Reading: On-line Star CCM+ Documentation

Learning Objectives

  1. To be able to follow the Star CCM+ Tutorial for the Pipe Flow Problem
  2. To be able to set up and solve a flow problem with the Star CCM+ interface when the mesh is given

Tutorial Files

Download the pipe flow tutorial (updated and slightly expanded) that we followed in the MCAE lab.

Download the mesh file for the pipe flow problem. This mesh is necessary for the tutorial.

5. Finite-Volume Solution to the 1D Convection-Diffusion Equation

Lecture on 23 January 2008

Reading: Handout

Learning Objectives

  1. To be able to follow the Star CCM+ Tutorial for the Pipe Flow Problem
  2. To be able to set up and solve a flow problem with the Star CCM+ interface when the mesh is given
  3. To recognize the finite-volume stencil using compass point notation.
  4. To be able to relate compass point notation to conventional grid indices, P -> (i,j), E -> (i+1,j), etc.
  5. To be able to derive the discrete form of the diffusion term for the one-dimensional convection-diffusion equation.
  6. To be able to read and understand the derivation of the discrete form of the convection term for the one-dimensional convection-diffusion equation using Central difference and upwind schemes
  7. To be able to relate the "wiggles" in the solution to the one-dimensional convection-diffusion equation to the mesh Peclet number. What is the limit for the avoidance of wiggles?
  8. To be able to explain the trend in truncation error as mesh size is reduced for a fixed PeL.
  9. To be able to describe the effect of using non-uniform meshes in the solution to the convection-diffusion equation.

4. Crank-Nicolson Solution to the Heat Equation

Lecture on 16 January 2008

Reading: Chapter 14

3. BTCS and Crank-Nicolson Schemes to the Heat Equation

Lecture on 14 January 2008

Reading: Chapter 14, pp. 547 -- 556

Learning Objectives

  1. To be able to derive the computational formulas for the BTCS and Crank-Nicolson schemes for the heat equation.
  2. To be able to identify the computational molecules for the BTCS and Crank-Nicolson schemes.
  3. To be able to identify the stability limit for the BTCS and Crank-Nicolson schemes applied to the heat equation.
  4. To be able to interpret the results of measuring the truncation error for the BTCS and Crank-Nicolson schemes scheme. To be able to explain the trends in plots of truncation error versus mesh spacing.
  5. To be able to describe the important implementation differences between the explicit FTCS scheme and the implicit BTCS and Crank-Nicolson schemes
  6. To be able to explain the meaning of the terms "explicit" and "implicit" when describing FTCS, BTCS, and Crank-Nicolson schemes.
  7. To be able to rank FTCS, BTCS, and Crank-Nicolson schemes according to their truncation errors.

2. FTCS Solution to the Heat Equation

Lecture on 10 January 2008

Reading: Chapter 14, pp. 532 -- 547

Learning Objectives

  1. Recognize formulas for forward and backward difference approximations to first derivatives.
  2. Recognize formulas for the central difference approximations to second derivatives.
  3. To be able to define the term ``truncation error'' and identify its origin in finite-difference formulas.
  4. To be able to derive the computational formulas for the FTCS scheme for the heat equation.
  5. To be able to identify the computational molecules for the FTCS scheme.
  6. To be able to define stability of a finite-difference scheme and to correctly identify the stability limit for the FTCS scheme applied to the heat equation.
  7. To be able to interpret the results of measuring the truncation error for the FTCS scheme. To be able to explain the trends in plots of truncation error versus mesh spacing.

MATLAB Codes

Learning objectives for Finite Difference Solution to the 1D Heat Equation (PDF)

Handouts

I demonstrated a MATLAB implementation of the FTCS method for the one-dimensional heat equation. A zip archive of the entire suite of finite-difference codes for the one-dimensional heat equation can be downloaded directly, or by visiting the web page for toy codes.

1. Course Introduction, Classification of PDEs, Introduction to the Heat Equation

Lecture on 7 January 2008

Reading: PDE Classification Notes, pp. 527 -- 531 in "Chapter 14"

Learning Objectives

  1. Understand the purpose, scope, and primary objectives of the course
  2. Know about the course project
  3. Be able to describe the qualitative differences between parabolic, hyperbolic, and elliptic PDEs
  4. Be able to meaningfully discuss the components of the analytical solution to the 1D Heat Equation with Dirichlet boundary conditions
  5. Be able to give a physical interpretation to the analytical solution to the 1D Heat Equation with Dirichlet boundary conditions

Handouts

  1. Syllabus
  2. Calendar
  3. Problem Set 1

I also handed out excerpts from Chapter 14, "Numerical Solution of Partial Differential Equations". That will not be made available as a PDF. If you need a copy, ask me in person.