1. Overview Class #2 (Jan 16) Brief introduction to time-stepping ODEs
Application: Particle systems
Numerical stiffness issues

2. No class next Tuesday Instead: http://www.cs.cmu.edu/~djames/pbmis
assignment
readings

3. What is stiffness? From [AscherPetzold97]
“Loosely speaking, the initial value problem is referred to as being stiff if the absolute stability requirement dictates a much smaller time step than is needed to satisfy approximation requirements alone.”
Definition 3.1 [p48]: An IVP is stiff in some interval [0,b] if the step size needed to maintain stability of the forward Euler method is much smaller than the step size required to represent the solution accurately.

7. What is stiffness? Stiffness depends on your problem and…
Accuracy criterion
Length of the interval of integration, [0,b]
Region of absolute stability of the method
Related to local Jacobian eigenvalues {j}
Stiff on [0,b] if
“Stiff decay”

8. Some higher-order explicit methods

9. Some higher-order implicit methods

10. Complications
Smoothness assumption in Taylor expansion