1. Disclaimer
The following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright  2001 by David Baraff at Pixar).
UNC Chapel Hill
M. C. Lin

2. Determining Step Size Explicit Integration Implicit Methods
Too big, unstable!
Too small, too slow
Adaptive, maybe
Ultimately the constants decide!
Implicit Methods
Taking large steps when possible
UNC Chapel Hill
M. C. Lin

3. An Example
UNC Chapel Hill
M. C. Lin

4. Speed Limitation of Euler’s Method
UNC Chapel Hill
M. C. Lin

5. Stiff Equations
UNC Chapel Hill
M. C. Lin

6. A Stiff Energy Landscape
UNC Chapel Hill
M. C. Lin

7. Example: Particle-on-line
UNC Chapel Hill
M. C. Lin

8. Example: Particle-on-line
UNC Chapel Hill
M. C. Lin

9. Example: Particle-on-line
UNC Chapel Hill
M. C. Lin

10. Example: Particle-on-line
UNC Chapel Hill
M. C. Lin

11. Explicit Integration
UNC Chapel Hill
M. C. Lin

12. Problems
UNC Chapel Hill
M. C. Lin

13. Explicit vs. Implicit Euler Method
UNC Chapel Hill
M. C. Lin

14. UNC Chapel Hill
M. C. Lin

15. One Step: Implicit vs. Explicit
UNC Chapel Hill
M. C. Lin

16. Large Systems
UNC Chapel Hill
M. C. Lin

17. Implicit Integration
UNC Chapel Hill
M. C. Lin

18. Implicit Integration
UNC Chapel Hill
M. C. Lin

19. Implicit Integration
UNC Chapel Hill
M. C. Lin

20. Linearized Implicit Integration
UNC Chapel Hill
M. C. Lin

21. Single-Step Implicit Euler Method
UNC Chapel Hill
M. C. Lin

22. Solving Large Systems Matrix structure reflects force-coupling:
(i , j)th entry exists iff fi depends on Xj
Conjugate gradient a good first choice
UNC Chapel Hill
M. C. Lin