- Introduction
- Numerical Solution of ODE's: Order of Scheme. Variable Time Step. Duffing Oscillator
- Fourier Transformation

- Classification and Some Properties of PDE
- Well-Posedness. Elliptic vs. Hyperbolic
- Dispersion and Dissipation

Demo

- Difference Approximations of Derivatives
- Central Differences - 2nd order
- Error & Points/Wavelength
- Higher-Order Difference Approximations
- Pictorial Representation of Central Differencing
- One-sided Differencing
- Differencing of 2nd Derivatives

- Temporal Errors: Continuous Time Approximation
- Central Differences. Numerical Dispersion
- One-sided Differences. Numerical Dissipation & Stability

- Difference Schemes in Space and Time
- Forward Euler
- Neumann Analysis
- Backward Euler
- Lax-Friedrich
- Leap-Frog
- Interpretation of Stability Condition for Lambda

- General Formulation of Difference Schemes
- More Schemes
- Lax-Wendroff
- MacCormack
- Runge-Kutta
- Crank-Nicholson
- Compact Difference Schemes
- Phase Errors from Neumann Analysis
- Artificial Dissipation

Demo - Some Criteria for Choice of Scheme
- Simple Scaling Arguments
- Explicit vs. Implicit

- Initial-Boundary Value Problems: PDE-Theory
- One-Way Wave Equation
- Two-Way Wave Equation
- General System of Wave Equations
- Example: 1-dimensional Euler Equations

- IBV Problems: Finite Differences
- Physical Boundary Conditions
- Numerical Boundary Conditions
- Boundary-Condition Schemes
- Coupled Equations

- Stability of Boundary Treatment
- Examples of Stable and Unstable Boundary Treatments

- Two-dimensional Problems
- Scalar Equations
- Coupled Equations
- Operator Splitting
- Approximate Factorization and ADI
- Anisotropy
- Boundary Conditions

- Parabolic Equations
- 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

**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

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*

**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.

**Matlab sample code:**

Duffing oscillator: duffing.m euler.m f.m stepeuler.m

Forward Euler for wave equation:

**Assignments:**

There will be homework assignments that cover theoretical and practical (including coding) aspects. Some will involve extensive coding and running of the codes.

The assignments will be handed in and graded. Late homework will not be accepted.

Except for analytical calculations the text should be typed. The clarity and neatness of the presentation will be part of the grade (is the text ordered logically and comprehensible? Is it convincing? Are the figures labeled properly and described adequately? ) The code will be handed in as well. The clarity of the code and its documentation will also enter the grade (is the code structured, using a good level of modularity? Is the purpose of the various sections stated so somebody else can understand what the code is doing?)

HW 1 Results

Office hours: We 3-4, Th 4-5 in M458

There will be no class on Monday October 25. To make up the time the first 5 lectures will be 15 minutes longer.