1. PDEs and Examples of Phenomena Modeled
Ordinary differential equation: equation containing derivatives of a function of one variable
Partial differential equation: equation containing derivatives of a function of two or more variables
Models:
Air flow over an aircraft wing
Blood circulation in human body
Water circulation in an ocean
Bridge deformations as its carries traffic
Evolution of a thunderstorm
Oscillations of a skyscraper hit by earthquake
Strength of a toy
Financial Markets

2. Model of Sea Surface Temperature in Atlantic Ocean
Courtesy MICOM group
at the Rosenstiel School
of Marine and Atmospheric
Science, University of Miami

3. Solving PDEs Finite element method
Finite difference method (our focus)
Converts PDE into matrix equation
Linear system over discrete basis elements
Result is usually a sparse matrix
Matrix-based algorithms represent matrices explicitly
Matrix-free algorithms represent matrix values implicitly (our focus)

4. FINITE DIFFERENCE
In numerical analysis, two different approaches are commonly used: The finite difference and the finite element methods. In heat transfer problems, the finite difference method is used more often and will be discussed here. The finite difference method involves: Ø Establish nodal networks Ø Derive finite difference approximations for the governing equation at both interior and exterior nodal points Ø Develop a system of simultaneous algebraic nodal equations Ø Solve the system of equations using numerical schemes

5. Finite Difference Methods: Outline
Solving ordinary and partial differential equations
Finite difference methods (FDM) vs Finite Element Methods (FEM)
Vibrating string problem
Steady state heat distribution problem

6. Class of Linear Second-order PDEs
Linear second-order PDEs are of the form
where A - H are functions of x and y only
Elliptic PDEs: B2 - AC < 0
(steady state heat equations)
Parabolic PDEs: B2 - AC = 0
(heat transfer equations)
Hyperbolic PDEs: B2 - AC > 0
(wave equations)

7. Difference Quotients

8. Formulas for 1st, 2d Derivatives

9. Vibrating String Problem
Vibrating string modeled by a hyperbolic PDE

10. The Nodal Networks

11. Finite Difference Approximation

12. Finite Difference Approximation cont.

13. Finite Difference Approximation cont.

14. A System of Algebraic Equations

15. Matrix Form

16. Numerical Solutions

17. Iteration

18. Example

19. Example (cont.)

20. Example (cont.)

21. Summary of nodal finite-difference relations for various configurations:
Case 1 Interior Node

22. Case 2 Node at an internal corner with convection

23. Case 3 Node at a plane surface with convection

24. Case 4 Node at an external corner with convection

25. Case 5 Node at a plane surface with uniform heat flux