ESAM 446-1

Numerical Methods for Partial Differential Equations

Finite Difference Methods

/~riecke

Overview

  1. Introduction
    1. Numerical Solution of ODE's: Order of Scheme. Variable Time Step. Duffing Oscillator
    2. Fourier Transformation
  2. Classification and Some Properties of PDE
    1. Well-Posedness. Elliptic vs. Hyperbolic
    2. Dispersion and Dissipation
      Demo
  3. Difference Approximations of Derivatives
    1. Central Differences - 2nd order
    2. Error & Points/Wavelength
    3. Higher-Order Difference Approximations
    4. Pictorial Representation of Central Differencing
    5. One-sided Differencing
    6. Differencing of 2nd Derivatives
  4. Temporal Errors: Continuous Time Approximation
    1. Central Differences. Numerical Dispersion
    2. One-sided Differences. Numerical Dissipation & Stability
  5. Difference Schemes in Space and Time
    1. Forward Euler
    2. Neumann Analysis
    3. Backward Euler
    4. Lax-Friedrich
    5. Leap-Frog
    6. Interpretation of Stability Condition for Lambda
  6. General Formulation of Difference Schemes
  7. More Schemes
    1. Lax-Wendroff
    2. MacCormack
    3. Runge-Kutta
    4. Crank-Nicholson
    5. Compact Difference Schemes
    6. Phase Errors from Neumann Analysis
    7. Artificial Dissipation
      Demo
    8. Some Criteria for Choice of Scheme
      1. Simple Scaling Arguments
      2. Explicit vs. Implicit
  8. Initial-Boundary Value Problems: PDE-Theory
    1. One-Way Wave Equation
    2. Two-Way Wave Equation
    3. General System of Wave Equations
    4. Example: 1-dimensional Euler Equations
  9. IBV Problems: Finite Differences
    1. Physical Boundary Conditions
    2. Numerical Boundary Conditions
    3. Boundary-Condition Schemes
    4. Coupled Equations
  10. Stability of Boundary Treatment
    1. Examples of Stable and Unstable Boundary Treatments
  11. Two-dimensional Problems
    1. Scalar Equations
    2. Coupled Equations
    3. Operator Splitting
    4. Approximate Factorization and ADI
    5. Anisotropy
    6. Boundary Conditions
  12. Parabolic Equations
    1. Nonlinear Diffusion Equations


    Text.There is no text book for this course. However, there is a nice manuscript by Prof. D. Chopp, who has taught the course previously, that covers most of the material.
    MATLAB.There are a number of tutorials and more detailed manuals of MATLAB on the web. A good starting page is the tutorial from UMD which also points to more detailed and extensive web resources. Please note that the current version of Matlab is 5 and it differs in some relevant respects from the older versions 3 and 4.
    Matlab sample code:
    Duffing oscillator: duffing.m euler.m f.m stepeuler.m
    Wave equation: wave.m

    Kuramoto-Sivashinsky equation



    Assignments:
    There will be homework assignments that cover theoretical and practical (including coding) aspects. There will also be some larger programming projects.

    HW 1 (postscript) HW 1 (html) Solution Sample for VSRK5
    HW 2 (postscript) HW 2 (html)
    Project 1 (postscript) Project 1(html)
    Project 2 (postcript)

    If your browser does not show the html-file appropriately you may tune it by following these instructions

    Please let me know if you have difficulty printing out the postscript files.
    Office hours: Mo Tu Th 5-6 in M458

    Blackboard Web Page


    There will be no class on Wednesday November 22 and Monday November 27. To make up the time the first 10 lectures will be 15 minutes longer.
    H. Riecke, ESAM