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

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

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

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

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

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

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

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

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

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

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

12. UNC Chapel Hill M. C. Lin Problems

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

14. UNC Chapel Hill M. C. Lin

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

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

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

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

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

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

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

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